**Calculi**: An abnormal concretion occurring mostly in the urinary and biliary tracts, usually composed of mineral salts. Also called stones.

**Urinary Calculi**: Low-density crystals or stones in any part of the URINARY TRACT. Their chemical compositions often include CALCIUM OXALATE, magnesium ammonium phosphate (struvite), CYSTINE, or URIC ACID.

**Ureteral Calculi**: Stones in the URETER that are formed in the KIDNEY. They are rarely more than 5 mm in diameter for larger renal stones cannot enter ureters. They are often lodged at the ureteral narrowing and can cause excruciating renal colic.

**Kidney Calculi**: Stones in the KIDNEY, usually formed in the urine-collecting area of the kidney (KIDNEY PELVIS). Their sizes vary and most contains CALCIUM OXALATE.

**Urinary Bladder Calculi**: Stones in the URINARY BLADDER; also known as vesical calculi, bladder stones, or cystoliths.

**Dental Calculus**: Abnormal concretion or calcified deposit that forms around the teeth or dental prostheses.

**Lithotripsy**: The destruction of a calculus of the kidney, ureter, bladder, or gallbladder by physical forces, including crushing with a lithotriptor through a catheter. Focused percutaneous ultrasound and focused hydraulic shock waves may be used without surgery. Lithotripsy does not include the dissolving of stones by acids or litholysis. Lithotripsy by laser is LITHOTRIPSY, LASER.

**Salivary Duct Calculi**: Presence of small calculi in the terminal salivary ducts (salivary sand), or stones (larger calculi) found in the larger ducts.

**Ureteroscopy**: Endoscopic examination, therapy or surgery of the ureter.

**Nephrostomy, Percutaneous**: The insertion of a catheter through the skin and body wall into the kidney pelvis, mainly to provide urine drainage where the ureter is not functional. It is used also to remove or dissolve renal calculi and to diagnose ureteral obstruction.

**Salivary Gland Calculi**: Calculi occurring in a salivary gland. Most salivary gland calculi occur in the submandibular gland, but can also occur in the parotid gland and in the sublingual and minor salivary glands.

**Lithotripsy, Laser**: Fragmentation of CALCULI, notably urinary or biliary, by LASER.

**Prostatic Diseases**: Pathological processes involving the PROSTATE or its component tissues.

**Urolithiasis**: Formation of stones in any part of the URINARY TRACT, usually in the KIDNEY; URINARY BLADDER; or the URETER.

**Calcium Oxalate**: The calcium salt of oxalic acid, occurring in the urine as crystals and in certain calculi.

**Urography**: Radiography of any part of the urinary tract.

**Salivary Calculi**: The presence of calculi in a salivary duct or gland.

**Gallstones**: Solid crystalline precipitates in the BILIARY TRACT, usually formed in the GALLBLADDER, resulting in the condition of CHOLELITHIASIS. Gallstones, derived from the BILE, consist mainly of calcium, cholesterol, or bilirubin.

**Cholelithiasis**: Presence or formation of GALLSTONES in the BILIARY TRACT, usually in the gallbladder (CHOLECYSTOLITHIASIS) or the common bile duct (CHOLEDOCHOLITHIASIS).

**Reference Books, Medical**: Books in the field of medicine intended primarily for consultation.

**Urologic Surgical Procedures**: Surgery performed on the urinary tract or its parts in the male or female. For surgery of the male genitalia, UROLOGIC SURGICAL PROCEDURES, MALE is available.

**Magnesium Compounds**: Inorganic compounds that contain magnesium as an integral part of the molecule.

**Cystinuria**: An inherited disorder due to defective reabsorption of CYSTINE and other BASIC AMINO ACIDS by the PROXIMAL RENAL TUBULES. This form of aminoaciduria is characterized by the abnormally high urinary levels of cystine; LYSINE; ARGININE; and ORNITHINE. Mutations involve the amino acid transport protein gene SLC3A1.

**Calcium Phosphates**: Calcium salts of phosphoric acid. These compounds are frequently used as calcium supplements.

**Colic**: A clinical syndrome with intermittent abdominal pain characterized by sudden onset and cessation that is commonly seen in infants. It is usually associated with obstruction of the INTESTINES; of the CYSTIC DUCT; or of the URINARY TRACT.

**Kidney Pelvis**: The flattened, funnel-shaped expansion connecting the URETER to the KIDNEY CALICES.

**Ethylene Glycol**: A colorless, odorless, viscous dihydroxy alcohol. It has a sweet taste, but is poisonous if ingested. Ethylene glycol is the most important glycol commercially available and is manufactured on a large scale in the United States. It is used as an antifreeze and coolant, in hydraulic fluids, and in the manufacture of low-freezing dynamites and resins.

**Postcholecystectomy Syndrome**: Abdominal symptoms after removal of the GALLBLADDER. The common postoperative symptoms are often the same as those present before the operation, such as COLIC, bloating, NAUSEA, and VOMITING. There is pain on palpation of the right upper quadrant and sometimes JAUNDICE. The term is often used, inaccurately, to describe such postoperative symptoms not due to gallbladder removal.

**Urology**: A surgical specialty concerned with the study, diagnosis, and treatment of diseases of the urinary tract in both sexes, and the genital tract in the male. Common urological problems include urinary obstruction, URINARY INCONTINENCE, infections, and UROGENITAL NEOPLASMS.

**Urologic Diseases**: Pathological processes of the URINARY TRACT in both males and females.

**Diverticulum**: A pouch or sac developed from a tubular or saccular organ, such as the GASTROINTESTINAL TRACT.

**Hydronephrosis**: Abnormal enlargement or swelling of a KIDNEY due to dilation of the KIDNEY CALICES and the KIDNEY PELVIS. It is often associated with obstruction of the URETER or chronic kidney diseases that prevents normal drainage of urine into the URINARY BLADDER.

**Holmium**: Holmium. An element of the rare earth family of metals. It has the atomic symbol Ho, atomic number 67, and atomic weight 164.93.

**Apatites**: A group of phosphate minerals that includes ten mineral species and has the general formula X5(YO4)3Z, where X is usually calcium or lead, Y is phosphorus or arsenic, and Z is chlorine, fluorine, or OH-. (McGraw-Hill Dictionary of Scientific and Technical Terms, 4th ed)

**Lithiasis**: A condition characterized by the formation of CALCULI and concretions in the hollow organs or ducts of the body. They occur most often in the gallbladder, kidney, and lower urinary tract.

**Cystic Duct**: The duct that is connected to the GALLBLADDER and allows the emptying of bile into the COMMON BILE DUCT.

**Pyelonephritis, Xanthogranulomatous**: A chronic inflammatory condition of the KIDNEY resulting in diffuse renal destruction, a grossly enlarged and nonfunctioning kidney associated with NEPHROLITHIASIS and KIDNEY STONES.

**Testicular Diseases**: Pathological processes of the TESTIS.

**Cholangiography**: An imaging test of the BILIARY TRACT in which a contrast dye (RADIOPAQUE MEDIA) is injected into the BILE DUCT and x-ray pictures are taken.

**Bile Duct Diseases**: Diseases in any part of the ductal system of the BILIARY TRACT from the smallest BILE CANALICULI to the largest COMMON BILE DUCT.

**Submandibular Gland Diseases**

**Oxalates**: Derivatives of OXALIC ACID. Included under this heading are a broad variety of acid forms, salts, esters, and amides that are derived from the ethanedioic acid structure.

**Oxalic Acid**: A strong dicarboxylic acid occurring in many plants and vegetables. It is produced in the body by metabolism of glyoxylic acid or ascorbic acid. It is not metabolized but excreted in the urine. It is used as an analytical reagent and general reducing agent.

**Hypercalciuria**: Excretion of abnormally high level of CALCIUM in the URINE, greater than 4 mg/kg/day.

**Triazines**: Heterocyclic rings containing three nitrogen atoms, commonly in 1,2,4 or 1,3,5 or 2,4,6 formats. Some are used as HERBICIDES.

**Kidney Calices**: Recesses of the kidney pelvis which divides into two wide, cup-shaped major renal calices, with each major calix subdivided into 7 to 14 minor calices. Urine empties into a minor calix from collecting tubules, then passes through the major calix, renal pelvis, and ureter to enter the urinary bladder. (From Moore, Clinically Oriented Anatomy, 3d ed, p211)

**Urine**: Liquid by-product of excretion produced in the kidneys, temporarily stored in the bladder until discharge through the URETHRA.

**Dental Prophylaxis**: Treatment for the prevention of periodontal diseases or other dental diseases by the cleaning of the teeth in the dental office using the procedures of DENTAL SCALING and DENTAL POLISHING. The treatment may include plaque detection, removal of supra- and subgingival plaque and calculus, application of caries-preventing agents, checking of restorations and prostheses and correcting overhanging margins and proximal contours of restorations, and checking for signs of food impaction.

**Urachus**: An embryonic structure originating from the ALLANTOIS. It is a canal connecting the fetal URINARY BLADDER and the UMBILICUS. It is normally converted into a fibrous cord postnatally. When the canal fails to be filled and remains open (patent urachus), urine leaks through the umbilicus.

**Periodontal Index**: A numerical rating scale for classifying the periodontal status of a person or population with a single figure which takes into consideration prevalence as well as severity of the condition. It is based upon probe measurement of periodontal pockets and on gingival tissue status.

**Uric Acid**: An oxidation product, via XANTHINE OXIDASE, of oxypurines such as XANTHINE and HYPOXANTHINE. It is the final oxidation product of purine catabolism in humans and primates, whereas in most other mammals URATE OXIDASE further oxidizes it to ALLANTOIN.

**Gingivitis**: Inflammation of gum tissue (GINGIVA) without loss of connective tissue.

**Ureteroscopes**: Endoscopes for examining the interior of the ureter.

**Persia**: An ancient civilization, known as early as 2000 B.C. The Persian Empire was founded by Cyrus the Great (550-529 B.C.) and for 200 years, from 550 to 331 B.C., the Persians ruled the ancient world from India to Egypt. The territory west of India was called Persis by the Greeks who later called the entire empire Persia. In 331 B.C. the Persian wars against the Greeks ended disastrously under the counterattacks by Alexander the Great. The name Persia in modern times for the modern country was changed to Iran in 1935. (From Webster's New Geographical Dictionary, 1988, p546 & Asimov, Words on the Map, 1962, p176)

**Ureteral Diseases**: Pathological processes involving the URETERS.

**Dental Pellicle**: A thin protein film on the surface of DENTAL ENAMEL. It is widely believed to result from the selective adsorption of precursor proteins present in SALIVA onto tooth surfaces, and to reduce microbial adherence to the TEETH.

**Hyperoxaluria**: Excretion of an excessive amount of OXALATES in the urine.

**Oral Hygiene Index**: A combination of the debris index and the dental calculus index to determine the status of oral hygiene.

**Cystoscopy**: Endoscopic examination, therapy or surgery of the urinary bladder.

**Endoscopes**: Instruments for the visual examination of interior structures of the body. There are rigid endoscopes and flexible fiberoptic endoscopes for various types of viewing in ENDOSCOPY.

**Scrotum**: A cutaneous pouch of skin containing the testicles and spermatic cords.

**Nephrolithiasis**: Formation of stones in the KIDNEY.

**Nephrocalcinosis**: A condition characterized by calcification of the renal tissue itself. It is usually seen in distal RENAL TUBULAR ACIDOSIS with calcium deposition in the DISTAL KIDNEY TUBULES and the surrounding interstitium. Nephrocalcinosis causes RENAL INSUFFICIENCY.

**Cholangiopancreatography, Endoscopic Retrograde**: Fiberoptic endoscopy designed for duodenal observation and cannulation of VATER'S AMPULLA, in order to visualize the pancreatic and biliary duct system by retrograde injection of contrast media. Endoscopic (Vater) papillotomy (SPHINCTEROTOMY, ENDOSCOPIC) may be performed during this procedure.

**Urethral Obstruction**: Partial or complete blockage in any part of the URETHRA that can lead to difficulty or inability to empty the URINARY BLADDER. It is characterized by an enlarged, often damaged, bladder with frequent urges to void.

**Dental Plaque**: A film that attaches to teeth, often causing DENTAL CARIES and GINGIVITIS. It is composed of MUCINS, secreted from salivary glands, and microorganisms.

**Cystoscopes**: Endoscopes for visual examination of the urinary bladder.

**Nigella sativa**: A plant genus of the family RANUNCULACEAE that contains alpha-hederin, a triterpene saponin in the seeds, and is the source of black seed oil.

**Pedaliaceae**: The sesame family of the order Lamiales that are mainly herbs and shrubs growing in warm regions.

**Cholecystectomy**: Surgical removal of the GALLBLADDER.

**Dysuria**: Painful URINATION. It is often associated with infections of the lower URINARY TRACT.

**Potassium Citrate**: A powder that dissolves in water, which is administered orally, and is used as a diuretic, expectorant, systemic alkalizer, and electrolyte replenisher.

**Knowledge of Results (Psychology)**: A principle that learning is facilitated when the learner receives immediate evaluation of learning performance. The concept also hypothesizes that learning is facilitated when the learner is promptly informed whether a response is correct, and, if incorrect, of the direction of error.

**Potassium Compounds**: Inorganic compounds that contain potassium as an integral part of the molecule.

**Hematuria**: Presence of blood in the urine.

**Sialography**: Radiography of the SALIVARY GLANDS or ducts following injection of contrast medium.

**Tooth Discoloration**: Any change in the hue, color, or translucency of a tooth due to any cause. Restorative filling materials, drugs (both topical and systemic), pulpal necrosis, or hemorrhage may be responsible. (Jablonski, Dictionary of Dentistry, 1992, p253)

**Anuria**: Absence of urine formation. It is usually associated with complete bilateral ureteral (URETER) obstruction, complete lower urinary tract obstruction, or unilateral ureteral obstruction when a solitary kidney is present.

**Cystostomy**: Surgical creation of an opening (stoma) in the URINARY BLADDER for drainage.

**Pharmaceutic Aids**: Substances which are of little or no therapeutic value, but are necessary in the manufacture, compounding, storage, etc., of pharmaceutical preparations or drug dosage forms. They include SOLVENTS, diluting agents, and suspending agents, and emulsifying agents. Also, ANTIOXIDANTS; PRESERVATIVES, PHARMACEUTICAL; COLORING AGENTS; FLAVORING AGENTS; VEHICLES; EXCIPIENTS; OINTMENT BASES.

**Gingival Pocket**: An abnormal extension of a gingival sulcus not accompanied by the apical migration of the epithelial attachment.

**Ultrasonography**: The visualization of deep structures of the body by recording the reflections or echoes of ultrasonic pulses directed into the tissues. Use of ultrasound for imaging or diagnostic purposes employs frequencies ranging from 1.6 to 10 megahertz.

**Ureteral Obstruction**: Blockage in any part of the URETER causing obstruction of urine flow from the kidney to the URINARY BLADDER. The obstruction may be congenital, acquired, unilateral, bilateral, complete, partial, acute, or chronic. Depending on the degree and duration of the obstruction, clinical features vary greatly such as HYDRONEPHROSIS and obstructive nephropathy.

**Urethral Diseases**: Pathological processes involving the URETHRA.

**Cholecystectomy, Laparoscopic**: Excision of the gallbladder through an abdominal incision using a laparoscope.

**Lasers, Solid-State**: Lasers which use a solid, as opposed to a liquid or gas, as the lasing medium. Common materials used are crystals, such as YAG (YTTRIUM aluminum garnet); alexandrite; and CORUNDUM, doped with a rare earth element such as a NEODYMIUM; ERBIUM; or HOLMIUM. The output is sometimes additionally modified by addition of non-linear optical materials such as potassium titanyl phosphate crystal, which for example is used with neodymium YAG lasers to convert the output light to the visible range.

**Parotid Diseases**

**History, Medieval**: The period of history from the year 500 through 1450 of the common era.

**Urinalysis**: Examination of urine by chemical, physical, or microscopic means. Routine urinalysis usually includes performing chemical screening tests, determining specific gravity, observing any unusual color or odor, screening for bacteriuria, and examining the sediment microscopically.

**Citrates**

**Spectrophotometry, Infrared**: Spectrophotometry in the infrared region, usually for the purpose of chemical analysis through measurement of absorption spectra associated with rotational and vibrational energy levels of molecules. (McGraw-Hill Dictionary of Scientific and Technical Terms, 4th ed)

**Root Planing**: A procedure for smoothing of the roughened root surface or cementum of a tooth after subgingival curettage or scaling, as part of periodontal therapy.

**Pyelonephritis**: Inflammation of the KIDNEY involving the renal parenchyma (the NEPHRONS); KIDNEY PELVIS; and KIDNEY CALICES. It is characterized by ABDOMINAL PAIN; FEVER; NAUSEA; VOMITING; and occasionally DIARRHEA.

**Alcian Blue**: A copper-containing dye used as a gelling agent for lubricants, for staining of bacteria and for the dyeing of histiocytes and fibroblasts in vivo.

**Tomography, Spiral Computed**: Computed tomography where there is continuous X-ray exposure to the patient while being transported in a spiral or helical pattern through the beam of irradiation. This provides improved three-dimensional contrast and spatial resolution compared to conventional computed tomography, where data is obtained and computed from individual sequential exposures.

**Periodontal Diseases**: Pathological processes involving the PERIODONTIUM including the gum (GINGIVA), the alveolar bone (ALVEOLAR PROCESS), the DENTAL CEMENTUM, and the PERIODONTAL LIGAMENT.

**Iran**

**Metabolic Diseases**: Generic term for diseases caused by an abnormal metabolic process. It can be congenital due to inherited enzyme abnormality (METABOLISM, INBORN ERRORS) or acquired due to disease of an endocrine organ or failure of a metabolically important organ such as the liver. (Stedman, 26th ed)

**Treatment Outcome**: Evaluation undertaken to assess the results or consequences of management and procedures used in combating disease in order to determine the efficacy, effectiveness, safety, and practicability of these interventions in individual cases or series.

**Calcium Carbonate**: Carbonic acid calcium salt (CaCO3). An odorless, tasteless powder or crystal that occurs in nature. It is used therapeutically as a phosphate buffer in hemodialysis patients and as a calcium supplement.

**Operative Time**: The duration of a surgical procedure in hours and minutes.

**Petroselinum**: A plant genus of the family APIACEAE used for flavoring food.

**Dental Plaque Index**: An index which scores the degree of dental plaque accumulation.

**Pancreatic Diseases**: Pathological processes of the PANCREAS.

**Dental Scaling**: Removal of dental plaque and dental calculus from the surface of a tooth, from the surface of a tooth apical to the gingival margin accumulated in periodontal pockets, or from the surface coronal to the gingival margin.

**Urinary Tract Infections**: Inflammatory responses of the epithelium of the URINARY TRACT to microbial invasions. They are often bacterial infections with associated BACTERIURIA and PYURIA.

**Cholecystitis**: Inflammation of the GALLBLADDER; generally caused by impairment of BILE flow, GALLSTONES in the BILIARY TRACT, infections, or other diseases.

**Propantheline**: A muscarinic antagonist used as an antispasmodic, in rhinitis, in urinary incontinence, and in the treatment of ulcers. At high doses it has nicotinic effects resulting in neuromuscular blocking.

**Endoscopy**: Procedures of applying ENDOSCOPES for disease diagnosis and treatment. Endoscopy involves passing an optical instrument through a small incision in the skin i.e., percutaneous; or through a natural orifice and along natural body pathways such as the digestive tract; and/or through an incision in the wall of a tubular structure or organ, i.e. transluminal, to examine or perform surgery on the interior parts of the body.

**Phosphates**: Inorganic salts of phosphoric acid.

**Crystallization**: The formation of crystalline substances from solutions or melts. (McGraw-Hill Dictionary of Scientific and Technical Terms, 4th ed)

**Gallbladder**: A storage reservoir for BILE secretion. Gallbladder allows the delivery of bile acids at a high concentration and in a controlled manner, via the CYSTIC DUCT to the DUODENUM, for degradation of dietary lipid.

**Pancreatic Ducts**: Ducts that collect PANCREATIC JUICE from the PANCREAS and supply it to the DUODENUM.

**Gingival Hemorrhage**: The flowing of blood from the marginal gingival area, particularly the sulcus, seen in such conditions as GINGIVITIS, marginal PERIODONTITIS, injury, and ASCORBIC ACID DEFICIENCY.

**Sphincter of Oddi**: The sphincter of the hepatopancreatic ampulla within the duodenal papilla. The COMMON BILE DUCT and main pancreatic duct pass through this sphincter.

**Cholecystography**: Radiography of the gallbladder after ingestion of a contrast medium.

**Islam**: A monotheistic religion promulgated by the Prophet Mohammed with Allah as the deity.

**Cholangiopancreatography, Magnetic Resonance**: Non-invasive diagnostic technique for visualizing the PANCREATIC DUCTS and BILE DUCTS without the use of injected CONTRAST MEDIA or x-ray. MRI scans provide excellent sensitivity for duct dilatation, biliary stricture, and intraductal abnormalities.

**Urinary Bladder**: A musculomembranous sac along the URINARY TRACT. URINE flows from the KIDNEYS into the bladder via the ureters (URETER), and is held there until URINATION.

**Common Bile Duct**: The largest bile duct. It is formed by the junction of the CYSTIC DUCT and the COMMON HEPATIC DUCT.

**Sphincterotomy, Endoscopic**: Incision of Oddi's sphincter or Vater's ampulla performed by inserting a sphincterotome through an endoscope (DUODENOSCOPE) often following retrograde cholangiography (CHOLANGIOPANCREATOGRAPHY, ENDOSCOPIC RETROGRADE). Endoscopic treatment by sphincterotomy is the preferred method of treatment for patients with retained or recurrent bile duct stones post-cholecystectomy, and for poor-surgical-risk patients that have the gallbladder still present.

**Proteus Infections**: Infections with bacteria of the genus PROTEUS.

**Citric Acid**: A key intermediate in metabolism. It is an acid compound found in citrus fruits. The salts of citric acid (citrates) can be used as anticoagulants due to their calcium chelating ability.

**Retrospective Studies**: Studies used to test etiologic hypotheses in which inferences about an exposure to putative causal factors are derived from data relating to characteristics of persons under study or to events or experiences in their past. The essential feature is that some of the persons under study have the disease or outcome of interest and their characteristics are compared with those of unaffected persons.

**Magnesium**: A metallic element that has the atomic symbol Mg, atomic number 12, and atomic weight 24.31. It is important for the activity of many enzymes, especially those involved in OXIDATIVE PHOSPHORYLATION.

**Urinary Bladder Neck Obstruction**: Blocked urine flow through the bladder neck, the narrow internal urethral opening at the base of the URINARY BLADDER. Narrowing or strictures of the URETHRA can be congenital or acquired. It is often observed in males with enlarged PROSTATE glands.

**Retroperitoneal Space**: An area occupying the most posterior aspect of the ABDOMINAL CAVITY. It is bounded laterally by the borders of the quadratus lumborum muscles and extends from the DIAPHRAGM to the brim of the true PELVIS, where it continues as the pelvic extraperitoneal space.

**Ureter**: One of a pair of thick-walled tubes that transports urine from the KIDNEY PELVIS to the URINARY BLADDER.

**Radiography, Abdominal**: Radiographic visualization of the body between the thorax and the pelvis, i.e., within the peritoneal cavity.

**Durapatite**: The mineral component of bones and teeth; it has been used therapeutically as a prosthetic aid and in the prevention and treatment of osteoporosis.

**Urinary Retention**: Inability to empty the URINARY BLADDER with voiding (URINATION).

**Optical Fibers**: Thin strands of transparent material, usually glass, that are used for transmitting light waves over long distances.

**Powder Diffraction**: Method of using a polycrystalline powder and Rietveld refinement (LEAST SQUARES ANALYSIS) of X-RAY DIFFRACTION or NEUTRON DIFFRACTION. It circumvents the difficulties of producing single large crystals.

**Sulfadiazine**: One of the short-acting SULFONAMIDES used in combination with PYRIMETHAMINE to treat toxoplasmosis in patients with acquired immunodeficiency syndrome and in newborns with congenital infections.

**Methods**: A series of steps taken in order to conduct research.

**Calcinosis**: Pathologic deposition of calcium salts in tissues.

**Oral Hygiene**: The practice of personal hygiene of the mouth. It includes the maintenance of oral cleanliness, tissue tone, and general preservation of oral health.

**Postoperative Complications**: Pathologic processes that affect patients after a surgical procedure. They may or may not be related to the disease for which the surgery was done, and they may or may not be direct results of the surgery.

**Kidney Diseases**: Pathological processes of the KIDNEY or its component tissues.

**Gallbladder Diseases**: Diseases of the GALLBLADDER. They generally involve the impairment of BILE flow, GALLSTONES in the BILIARY TRACT, infections, neoplasms, or other diseases.

**Choledocholithiasis**: Presence or formation of GALLSTONES in the COMMON BILE DUCT.

**Kidney**: Body organ that filters blood for the secretion of URINE and that regulates ion concentrations.

**Tomography, X-Ray Computed**: Tomography using x-ray transmission and a computer algorithm to reconstruct the image.

**Paraplegia**: Severe or complete loss of motor function in the lower extremities and lower portions of the trunk. This condition is most often associated with SPINAL CORD DISEASES, although BRAIN DISEASES; PERIPHERAL NERVOUS SYSTEM DISEASES; NEUROMUSCULAR DISEASES; and MUSCULAR DISEASES may also cause bilateral leg weakness.

**Urinary Tract**: The duct which coveys URINE from the pelvis of the KIDNEY through the URETERS, BLADDER, and URETHRA.

**Oral Health**: The optimal state of the mouth and normal functioning of the organs of the mouth without evidence of disease.

**Urinary Catheterization**: Passage of a CATHETER into the URINARY BLADDER or kidney.

**Cystine**: A covalently linked dimeric nonessential amino acid formed by the oxidation of CYSTEINE. Two molecules of cysteine are joined together by a disulfide bridge to form cystine.

**Ampulla of Vater**: A dilation of the duodenal papilla that is the opening of the juncture of the COMMON BILE DUCT and the MAIN PANCREATIC DUCT, also known as the hepatopancreatic ampulla.

**Silicon**: A trace element that constitutes about 27.6% of the earth's crust in the form of SILICON DIOXIDE. It does not occur free in nature. Silicon has the atomic symbol Si, atomic number 14, and atomic weight [28.084; 28.086].

**Phosphorus**: A non-metal element that has the atomic symbol P, atomic number 15, and atomic weight 31. It is an essential element that takes part in a broad variety of biochemical reactions.

**Nephrectomy**: Excision of kidney.

**Diphosphates**: Inorganic salts of phosphoric acid that contain two phosphate groups.

**Microscopy, Electron, Scanning**: Microscopy in which the object is examined directly by an electron beam scanning the specimen point-by-point. The image is constructed by detecting the products of specimen interactions that are projected above the plane of the sample, such as backscattered electrons. Although SCANNING TRANSMISSION ELECTRON MICROSCOPY also scans the specimen point by point with the electron beam, the image is constructed by detecting the electrons, or their interaction products that are transmitted through the sample plane, so that is a form of TRANSMISSION ELECTRON MICROSCOPY.

**Periodontal Pocket**: An abnormal extension of a gingival sulcus accompanied by the apical migration of the epithelial attachment and bone resorption.

**Prospective Studies**: Observation of a population for a sufficient number of persons over a sufficient number of years to generate incidence or mortality rates subsequent to the selection of the study group.

**Calcium**: A basic element found in nearly all organized tissues. It is a member of the alkaline earth family of metals with the atomic symbol Ca, atomic number 20, and atomic weight 40. Calcium is the most abundant mineral in the body and combines with phosphorus to form calcium phosphate in the bones and teeth. It is essential for the normal functioning of nerves and muscles and plays a role in blood coagulation (as factor IV) and in many enzymatic processes.

## Dacryolith formation around an eyelash retained in the lacrimal sac. (1/113)

A dacryolith was discovered in the lacrimal sac during a dacryocystorhinostomy for chronic dacryocystitis in which there was mucocele formation. Morphological examination confirmed the presence of an eyelash at the centre of the stone and electron microscopy demonstrated the presence of fungi (Candida sp.) in a matrix which was of markedly vairable morphology. The mechanism by which a hair enters the punctum and passes along the canaliculus may be attributed to the step-like pattern of ridges on the surface of a hair. The directional nature of these ridges dictates preferential movement towards the root end of the hair and prevents movements in the opposite direction. (+info)## Long term follow up of patients with chronic pancreatitis and pancreatic stones treated with extracorporeal shock wave lithotripsy. (2/113)

BACKGROUND: There have been conflicting reports as to whether pancreatic ductal drainage achieved by endoscopy and lithotripsy improves the clinical outcome of patients with chronic pancreatitis. AIMS: To determine the clinical outcome in patients with chronic pancreatitis who received extracorporeal shock wave lithotripsy (ESWL), and were followed up for two to eight years. METHODS: Eighty patients with severe chronic pancreatitis and endoscopically unretrievable obstructive stones underwent ESWL with a piezoelectric lithotripter between 1989 and 1996. Clinical status, relief of symptoms, further endoscopic or surgical interventions, and mortality were defined. RESULTS: Forty three (54%) patients were treated successfully with ESWL. The only feature associated with treatment success was the presence of a single stone rather than multiple stones. Successfully treated patients tended to experience less pain, although this did not reach statistical significance. A slight increase in weight was noted in our patients; however, there was no notable improvement in anomalous stools and diabetes mellitus. Five patients died due to extrapancreatic reasons. No pancreatic carcinomas developed. CONCLUSIONS: ESWL associated with endoscopic drainage is a safe technique that is particularly successful in patients with a single stone. However, pancreatic drainage by endoscopy and ESWL has almost no effect on pain in chronic pancreatitis. Furthermore, endoscopic management and ESWL does not prevent or postpone the development of glandular insufficiency. (+info)## Pulmonary alveolar microlithiasis: a rare familial inheritance with report of six cases in a family. Contribution of six new cases to the number of case reports in Turkey. (3/113)

Pulmonary alveolar microlithiasis (PAM) is a lung disease characterized by deposits of calcium within the alveoli. Our aim was to emphasize the familial character and the clinical features of the disease, and to draw attention to the increasing number of Turkish patients reported in the world. We detected 6 cases of PAM. Three cases had been diagnosed 4 years earlier, and 3 new cases were detected during the screening of the family members. All patients were male and the mean age was 11.5 ranging between 5 and 29 years. Five of the patients were cousins and the other one was their uncle. Radiographic studies showed a sand-like appearance in all patients. One case showed small subpleural bullae and bronchiectatic changes in both lower lobes in recent high-resolution CT scans, while his CT performed 4 years ago showed only sand-like appearance. The cases were diagnosed with the demonstration of microliths by bronchoalveolar lavage in 5 patients and transbronchial biopsy in 1. Recently reported cases from Turkey have constituted a considerable percentage among all cases in the world. In conclusion, (1) our patients constitute one of the largest series of cases reported in one family in the world. The disease seems to have familial and racial characteristics. The Turkish race has to be further investigated for genetic transmission. (2) Contrary to female predominance in previous reports, all 6 cases were male and 5 of them were below 12 years of age. (3) The disorder may show rapid progression in some cases probably due to the severity of the genetic disturbance. (+info)## Histochemical studies of pancreatic calculi. (4/113)

Fourteen pancreatic calculi from the corresponding number of pancreatic lithiasis patients were examined mineralogically and histochemically. The following results were obtained. The main components of calculi were calcium carbonate in 13 of the 14 cases, and calcium phosphate in the remaining one. Calcium carbonate calculi were all so-called intraductal calculi, with acidic glycoprotein apparently enwrapping the component particles. Acidic glycoproteins acted to bridge calcium carbonate particles, as in the cases of gallstone and urinary stone. The calcium phosphate calculus had a histochemical feature of pathologic calcification with degenerated collagen fibrils as the matrix, suggesting the calcification of the fibrotic pancreatic parenchyma after pancreatitis. (+info)## Urinary stones following renal transplantation. (5/113)

BACKGROUND: The formation of urinary tract stones following renal transplantation is a rare complication. The clinical features of stones after transplantation differ from those of non-transplant patients. Renal colic or pain is usually absent and rarely resembles acute rejection. METHODS: We retrospectively studied 849 consecutive kidney transplant patients in The Rogosin Institute/The Weill-Cornell Medical Center, New York who were transplanted between 1980 and 1997 and had functioning grafts for more than 3 months, to determine the incidence of stone formation, composition, risk factors and patient outcome. RESULTS: At our center, urinary stones were diagnosed in 15 patients (1.8%) of 849 functioning renal grafts for 3 or more months. Of the 15 patients, 10 were males and 5 were females in their third and fourth decade. Eight patients received their transplant from living donors and 7 from cadaveric donors. The stones were first diagnosed between 3 and 109 months after transplantation (mean 17.8 months) and 5 patients had recurrent episodes. The stones were located in the bladder in 11 cases (73.3%), transplanted kidney in 3 cases and in multiple sites in one case. The size of stones varied from 3.4 mm to 40 mm (mean 12 mm). The composition of stones was a mixed form of calcium oxalate and calcium phosphate in 5 cases and 4 patients had infected stones consisting of struvite or mixed form of struvite and calcium phosphate. Factors predisposing to stone formation included tertiary hyperparathyroidism (n = 8), hypercalciuria (n = 5), recurrent urinary tract infection (n = 5), hypocitraturia (n = 4), and obstructive uropathy (n = 2). Many cases had more than one risk factor. Clinically, painless hematuria was observed in 6 patients and dysuria without bacteriuria in 5 patients. None had renal colic or severe pain at any time. There were no changes in graft function at diagnosis and after removal of stones. Five patients passed stones spontaneously and 8 patients underwent cystoscopy for stone removal. CONCLUSION: Urinary stone formation following kidney transplantation is a rare complication (1.8%). Hyperparathyroidism, hypercalciuria, recurrent urinary tract infection and hypocitraturia are the most common risk factors, but often there are multiple factors which predispose to stone formation. To detect stones and determine their location and size, ultrasonography appears to be the most useful diagnostic tool. Prompt diagnosis, the removal of stones and stone-preventive measures can prevent adverse effects on renal graft outcome. (+info)## Radiological changes in pica. (6/113)

The significance of pica and geophagia as a public health problem is well known. The objective radiographic diagnosis of geophagia depends on the abnormal opacification of the bowel as an immediate manifestitation of the condition. The chance of detectability of geophagia is highest in the colon and can be improved by using low penetration films, particularly for smaller amounts of ingested clay. Other radiologic changes frequently associated with the prolonged practice of geophagic are an atonic pattern of the colon, secondary radiographic changes due to iron-deficiency anemia and bone-age retardation. The occurrences of intestinal obstruction due to pica in the presence of preexisting bowel stricture is demonstrated. (+info)## Genetic susceptibility to fibrocalculous pancreatic diabetes in Bangladeshi subjects: a family study. (7/113)

Fibrocalculous pancreatic diabetes (FCPD) is an uncommon cause of diabetes, seen mainly in developing countries. A family-based study was carried out in 67 Bangladeshi families, consisting of a proband with FCPD and both parents, to determine whether an association exists between FCPD susceptibility and either the major histocompatiblity complex (MHC) or insulin gene (INS) loci. HLA-DQB1 typing was done using allele-specific primers, and INS was typed using the restriction enzyme HphI. Three microsatellites (TNFa, TNFc and TNFd), from within and flanking the TNF-LT locus, were used for MHC Class IV typing and a PCR-RFLP assay was used to define the -308G/A TNF promoter polymorphism. The extended transmission disequilibrium test (ETDT) was used for statistical analysis. An overall association was observed between FCPD and HLA-DQB1 (P = 0.003), that was largely due to a positive association with HLA-DQB1*0302 and a negative association with HLA-DQB1*0202. Although no association was found between FCPD and TNF-LT microsatellite markers a trend was observed for TNFc (P = 0.037, Pc = 0.15). No association was found between FCPD and INS (P = 0.26). This study confirms an association between FCPD and the MHC using a family-based study design and the stringent ETDT analysis; a novel protective association was found with HLA-DQB1*0202 in Bangladeshi FCPD subjects. The genetic susceptibility to FCPD has features both similar and dissimilar to T1DM. (+info)## Positional down beating nystagmus in 50 patients: cerebellar disorders and possible anterior semicircular canalithiasis. (8/113)

OBJECTIVES: To clarify the clinical significance of positional down beat nystagmus (pDBN). METHODS: A discussion of the neuro-otological findings in 50 consecutive patients with pDBN. RESULTS: In 38 patients there was evidence of CNS disease (central group) but in 12 there was not (idiopathic group). In the CNS group, presenting symptoms were gait, speech, and autonomic dysfunction whereas in the idiopathic group patients mostly reported positional vertigo. The main neurological and oculomotor signs in the CNS group were explained by cerebellar dysfunction, including 13 patients with multiple system atrophy. In patients with multiple system atrophy with a prominent extrapyramidal component, the presence of pDBN was helpful in the differential diagnosis of atypical parkinsonism. No patient with pDBN had the Arnold-Chiari malformation, a common cause of constant down beat nystagmus (DBN). In the idiopathic group, the pDBN had characteristics which suggested a peripheral labyrinthine disorder: vertigo, adaptation, and habituation. In six patients an additional torsional component was found (concurrently with the pDBN in three). Features unusual for peripheral disorder were: bilateral positive Dix-Hallpike manoeuvre in nine of 12 patients and selective provocation by the straight head-hanging manoeuvre in two. CONCLUSION: It is argued that some patients with idiopathic pDBN have benign paroxysmal positional vertigo (BPPV) with lithiasis of the anterior canal. The torsional component may be weak, because of the predominantly sagittal orientation of the anterior canal, and may not be readily seen clinically. Nystagmus provocation by bilateral Dix-Hallpike and straight head-hanging may be explained by the vertical upwards orientation of the ampullary segment of the anterior canal in the normal upright head position. Such orientation makes right-left specificity with the Dix-Hallpike manoeuvre less important than for posterior canal BPPV. This orientation requires a further downwards movement of the head, often achieved with the straight head-hanging position, to provoke migration of the canaliths. The straight head-hanging manoeuvre should be carried out in all patients with a history of positional vertigo and a negative Dix-Hallpike manoeuvre. (+info)**Calculus**[edit]. Alhazen discovered the sum formula for the fourth power, using a method that could be generally used to ... Katz, Victor J. (1995). "Ideas of

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Renal

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**Calculi**in the salivary glands are called salivary

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**Calculi**in the tonsils are called tonsillar

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**Calculi**in the veins are called venous

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**Calculi**in the urinary system are called urinary

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**calculus**(plural

**calculi**), often called a stone, is a concretion of material, usually mineral salts, that forms in an organ ...

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- I have most recently helped tutor a student in pre- calculus . (purplemath.com)

- Calculus , originally called infinitesimal calculus or "the calculus of infinitesimals ", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations . (wikipedia.org)
- It may be taught as part of geometry, Algebra 2, pre- calculus , or some combination of these classes depending on the school. (purplemath.com)
- That year everyone of my students passed the math portion of the the TAKS test.Taught algebra, geometry and pre calculus for over 10 years in public school. (purplemath.com)
- Calculus (from Latin calculus, literally 'small pebble', used for counting and calculations, like on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. (wikipedia.org)
- In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). (wikipedia.org)
- The phrase "Schubert calculus" is sometimes used to mean the enumerative geometry of linear subspaces, roughly equivalent to describing the cohomology ring of Grassmannians, and sometimes used to mean the more general enumerative geometry of nonlinear varieties. (wikipedia.org)
- Vector calculus plays an important role in differential geometry and in the study of partial differential equations. (wikipedia.org)
- Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions by integrating with respect to the Euler characteristic as a finitely-additive measure. (wikipedia.org)
- Noncommutative geometry Quantum differential calculus Time scale calculus q-analog FUNCTIONS q-ORTHOGONAL WITH RESPECT TO THEIR OWN ZEROS, LUIS DANIEL ABREU, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 04-32 F. H. Jackson (1908), "On q-functions and a certain difference operator", Trans. (wikipedia.org)
- In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra A {\displaystyle A} over a field k {\displaystyle k} means the specification of a space of differential forms over the algebra. (wikipedia.org)
- In the quantum groups approach to noncommutative geometry one starts with the algebra and a choice of first order calculus but constrained by covariance under a quantum group symmetry. (wikipedia.org)
- Spivak's five-volume textbook A Comprehensive Introduction to Differential Geometry states in its preface that Calculus on Manifolds serves as a prerequisite for a course based on this text. (wikipedia.org)

- During that time I taught math ranging from basic math and pre algebra to calculus 2. (purplemath.com)
- I have been tutoring for five years, I was also an University facilitator for the Calculus 2 and Algebra courses, and I have also been a Student Teacher for the 7th, 8th, 9th and 11th grades. (purplemath.com)
- I taught a high school student Algebra 2 (from scratch) who then passed his final exam and entered Honors Pre-Calculus with a strong math background. (purplemath.com)
- Anybody who has taken Algebra 1 and has ever ridden in a car or pushed a toy train down a track can understand the basic concepts behind calculus. (metafilter.com)
- It provides a way to transition from advanced algebra and trigonometry into calculus . (purplemath.com)
- Graphing is very important in Algebra II to get you ready for Pre calculus . (purplemath.com)
- the process calculi were originally inspired by algebra (Milner 1993). (wikipedia.org)
- In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. (wikipedia.org)
- The relational algebra and the relational calculus are essentially logically equivalent: for any algebraic expression, there is an equivalent expression in the calculus, and vice versa. (wikipedia.org)
- In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize, as discussed below. (wikipedia.org)
- The algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field. (wikipedia.org)
- The PI Calculus is an extension of the extension of the process algebra CCS, a tool that has algebraic languages that are specific to processing and formulating statements. (wikipedia.org)
- Tomography of constructible functions, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes Lecture Notes in Computer Science, 1995, Volume 948/1995, 427-435, doi:10.1007/3-540-60114-7_33 Viro, O. Some integral calculus based on Euler characteristic, Lecture Notes in Math. (wikipedia.org)
- The q-calculus, while dating in a sense back to Leonhard Euler and Carl Gustav Jacobi, is only recently beginning to see more usefulness in quantum mechanics, having an intimate connection with commutativity relations and Lie algebra. (wikipedia.org)
- In mathematics, the continuous functional calculus of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. (wikipedia.org)
- Given an operator T, the range of the continuous functional calculus h → h(T) is the (abelian) C*-algebra C(T) generated by T. The Borel functional calculus has a larger range, that is the closure of C(T) in the weak operator topology, a (still abelian) von Neumann algebra. (wikipedia.org)
- A higher order differential calculus can mean an exterior algebra, or it can mean the partial specification of one, up to some highest degree, and with products that would result in a degree beyond the highest being unspecified. (wikipedia.org)
- The above definition is minimal and gives something more general than classical differential calculus even when the algebra A {\displaystyle A} is commutative or functions on an actual space. (wikipedia.org)
- As a byproduct, this enlarged definition includes finite difference calculi and quantum differential calculi on finite sets and finite groups (finite group Lie algebra theory). (wikipedia.org)
- Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient. (wikipedia.org)
- DES - An educational tool for working with Domain Relational Calculus and other formal languages WinRDBI - An educational tool for working with Domain Relational Calculus and other formal languages Relational algebra Relational calculus Tuple relational calculus (TRC) Aldat Relational Algebra Domain algebra Michel Lacroix, Alain Pirotte: Domain-Oriented Relational Languages. (wikipedia.org)
- More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of T to the bounded operators. (wikipedia.org)
- Lacroix and Pirotte proposed domain calculus, which is closer to first-order logic and together with Codd showed that both of these calculi (as well as relational algebra) are equivalent in expressive power. (wikipedia.org)
- Calculus on Manifolds aims to present the topics of multivariable and vector calculus in the manner in which they are seen by a modern working mathematician, yet simply and selectively enough to be understood by undergraduate students whose previous coursework in mathematics comprises only one-variable calculus and introductory linear algebra. (wikipedia.org)

- Check with the mathematics department for major requirements, and calculus course registration. (calculuscourse.org)
- The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics . (wikipedia.org)
- In mathematics education , calculus denotes courses of elementary mathematical analysis , which are mainly devoted to the study of functions and limits. (wikipedia.org)
- From the age of Greek mathematics , Eudoxus ( c. 408-355 BC) used the method of exhaustion , which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes ( c. 287-212 BC) developed this idea further , inventing heuristics which resemble the methods of integral calculus. (wikipedia.org)
- Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus. (wikipedia.org)
- The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. (wikipedia.org)
- In mathematics , time-scale calculus is a unification of the theory of difference equations with that of differential equations , unifying integral and differential calculus with the calculus of finite differences , offering a formalism for studying hybrid discrete-continuous dynamical systems . (wikipedia.org)
- Building on previous texts in the Modular Mathematics series, in particular 'Vectors in Two or Three Dimensions' and 'Calculus and ODEs', this book introduces the student to the concept of vector calculus. (worldcat.org)
- I've taken higher level mathematics classes up to Multivariable calculus (Calculus III) and Engineering classes inclu. (purplemath.com)
- Editors' note: Countless students learn integral calculus-the branch of mathematics concerned with finding the length, area or volume of an object by slicing it into small pieces and adding them up. (scientificamerican.com)
- Calculus is a part of modern mathematics education. (wikipedia.org)
- A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. (wikipedia.org)
- In mathematics, tensor calculus or tensor analysis is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). (wikipedia.org)
- Tensor calculus has many real-life applications in physics and engineering, including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), and general relativity (see mathematics of general relativity). (wikipedia.org)
- In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to 'prove' them. (wikipedia.org)
- Rod calculus played a key role in the development of Chinese mathematics to its height in Song Dynasty and Yuan Dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie. (wikipedia.org)
- In mathematics, a multiplicative calculus is a system with two multiplicative operators, called a "multiplicative derivative" and a "multiplicative integral", which are inversely related in a manner analogous to the inverse relationship between the derivative and integral in the classical calculus of Newton and Leibniz. (wikipedia.org)
- The non-Newtonian calculi are alternatives to the classical calculus of Newton and Leibniz, and are used in science, engineering, and mathematics. (wikipedia.org)
- An ethical calculus is the application of mathematics to calculate issues in ethics. (wikipedia.org)
- In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields. (wikipedia.org)
- In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. (wikipedia.org)
- In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. (wikipedia.org)
- Moreover, the new method also solves some problems unsolvable by calculus, and allows many incredible generalizations yet unknown in mathematics. (wikipedia.org)
- Stochastic calculus is a branch of mathematics that operates on stochastic processes. (wikipedia.org)
- In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. (wikipedia.org)
- In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. (wikipedia.org)
- In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. (wikipedia.org)
- Alonzo Church invented the lambda calculus in the 1930s, originally to provide a new and simpler basis for mathematics. (wikipedia.org)
- In the 14th century, Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics stated components of calculus such as the Taylor series and infinite series approximations. (wikipedia.org)
- In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectra), which has particularly broad scope. (wikipedia.org)
- In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates with uniform scales) is a line in the plane. (wikipedia.org)
- In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. (wikipedia.org)
- In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs and finite element meshes. (wikipedia.org)
- All Students Take Calculus is a mnemonic in mathematics that is used to help people memorize the sign values of all the trigonometric functions in the 2-dimensional Cartesian coordinate system. (wikipedia.org)
- Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (Mathematics Monograph Series) (PDF). (wikipedia.org)

- Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz . (wikipedia.org)
- Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. (wikipedia.org)
- Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. (wikipedia.org)

- Chemistry vs Ap Calculus Ab? (yahoo.com)
- I now have Ap Calculus Ab and i was just wondering if it is as hard or harder than Chemistry. (yahoo.com)
- I use an easy approach to AP chemistry, AP physics, and AP calculus . (purplemath.com)
- How to use quantity calculus for unit conversion and keeping track of units in algebraic manipulations is explained in the handbook on Quantities, Units and Symbols in Physical Chemistry. (wikipedia.org)

- Contrasted with the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold. (wikipedia.org)
- Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. (wikipedia.org)
- Thus, the epsilon, delta techniques that some believe to be the essence of analysis can be implemented once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts on pretense of being taught infinitesimal calculus", to quote a recent study. (wikipedia.org)
- Calculus Made Easy is a book on infinitesimal calculus originally published in 1910 by Silvanus P. Thompson. (wikipedia.org)
- The infinitesimal calculus can be expressed either in the notation of fluxions or in that of differentials, or, as noted above, it was also expressed by Newton in geometrical form, as in the Principia of 1687. (wikipedia.org)

- Fundamental introduction for beginning student of absolute differential calculus and for those interested in applications of tensor calculus to mathematical physics and engineering. (google.com)
- As a physics specialist, calculus has been the "tool of the trade" most often used. (purplemath.com)
- I was a physics major and math minor in college and have a masters in physics, so I have studied and used calculus extensively! (purplemath.com)
- I'm of the opinion that basic Physics, at least the force/acceleration/velocity/displacement relationship, needs to be taught before calculus. (metafilter.com)
- Calculus is a genius, who demonstrates himself throughout the series to be an expert in many fields of science, holding three PhDs in nuclear and theoretical physics,and planetary astronomy. (wikipedia.org)
- Guided greatly by intuition and his wealth of knowledge on the physics behind his circuit studies, [Heaviside] developed the operational calculus now ascribed to his name. (wikipedia.org)
- Borel functional calculus Holomorphic functional calculus Theorem VII.1 p. 222 in Modern methods of mathematical physics, Vol. 1, Reed M., Simon B. (wikipedia.org)
- Bondi k-calculus is a method of teaching special relativity popularised by Professor Sir Hermann Bondi, and now common in university and college-level physics classes. (wikipedia.org)

- Lambda calculus (also written as λ-calculus ) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution . (wikipedia.org)
- Lambda calculus consists of constructing lambda terms and performing reduction operations on them. (wikipedia.org)
- Lambda calculus is Turing complete , that is, it is a universal model of computation that can be used to simulate any Turing machine . (wikipedia.org)
- Lambda calculus may be untyped or typed . (wikipedia.org)
- In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. (wikipedia.org)
- in the simply typed lambda calculus it is, for example, a theorem that every evaluation strategy terminates for every simply typed lambda-term, whereas evaluation of untyped lambda-terms need not terminate. (wikipedia.org)
- One reason there are many different typed lambda calculi has been the desire to do more (of what the untyped calculus can do) without giving up on being able to prove strong theorems about the calculus. (wikipedia.org)
- Lambda calculus has played an important role in the development of the theory of programming languages . (wikipedia.org)
- Functional programming languages implement the lambda calculus. (wikipedia.org)
- Lambda calculus is also a current research topic in Category theory . (wikipedia.org)
- Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. (wikipedia.org)
- In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus . (wikipedia.org)
- Until the 1960s when its relation to programming languages was clarified, the lambda calculus was only a formalism. (wikipedia.org)
- Thanks to Richard Montague and other linguists' applications in the semantics of natural language, the lambda calculus has begun to enjoy a respectable place in both linguistics and computer science. (wikipedia.org)
- There is a bit of controversy over the reason for Church's use of the Greek letter lambda (λ) as the notation for function-abstraction in the lambda calculus, perhaps in part due to conflicting explanations by Church himself. (wikipedia.org)
- It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus , Ricci calculus , calculus of variations , lambda calculus , and process calculus . (wikipedia.org)
- This question is a bit off topic, but I hope it is of sufficient interest [because of lambda calculus] to this group anyway. (upenn.edu)
- My favorites would be lambda calculus, RAM's, and imperative language models (e.g., the LOOP language). (upenn.edu)
- Occasionally, a book will cover the RAM model as a side-light, but I have been unable to find a book, for example, that covers lambda calculus. (upenn.edu)
- Though of course the lambda calculus was contemporaneous - or even earlier than - Turing machines. (upenn.edu)
- It covers the basic theory of pi-calculus, typed pi-calculi, higher-order processes, the relationship between pi-calculus and lambda-calculus, and applications of pi-calculus to object-oriented design and programming. (upenn.edu)
- The term calculus (plural calculi) is also used for naming specific methods of calculation or notation as well as some theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus. (wikipedia.org)
- The first is a formalism intended to combine the higher-order facilities of lambda calculus with the pattern matching of term rewriting. (wikipedia.org)
- Lambda calculus Type theory API-Calculus Cardelli, L. (wikipedia.org)
- Juxtaposition is often used as an abbreviation for a combination of " lift {\displaystyle \operatorname {lift} } " and composition: e 1 e 2 = d e f e 1 ∘ lift ( e 2 ) {\displaystyle e_{1}e_{2}{\overset {def}{=}}e_{1}\circ \operatorname {lift} (e_{2})} The presentation here uses sequents ( Γ ⊢ e : τ {\displaystyle \Gamma \vdash e:\tau } ) rather than hypothetical judgments in order to ease comparison with the simply typed lambda calculus. (wikipedia.org)
- A simple example of a formula from Palm calculus is Little's law L = λ W {\displaystyle L=\lambda W} , which states that the time-average number of users (L) in a system is equal to the product of the rate ( λ {\displaystyle \lambda } ) at which users arrive and the Palm-average waiting time (W) that a user spends in the system. (wikipedia.org)
- Like lambda calculus, it supports a uniform treatment of function evaluation. (wikipedia.org)
- Deductive lambda calculus considers what happens when lambda terms are regarded as mathematical expressions. (wikipedia.org)
- One interpretation of the untyped lambda calculus is as a programming language where evaluation proceeds by performing reductions on an expression until it is in normal form. (wikipedia.org)
- However soon after inventing it major logic problems were identified with the definition of the lambda abstraction: The Kleene-Rosser paradox is an implementation of Richard's paradox in the lambda calculus. (wikipedia.org)
- The existence of these paradoxes meant that the lambda calculus could not be both consistent and complete as a deductive system. (wikipedia.org)
- Combinatory logic is closely related to lambda calculus, and the same paradoxes exist in each. (wikipedia.org)
- Later the lambda calculus was resurrected as a definition of a programming language. (wikipedia.org)
- Lambda calculus is the model and inspiration for the development of functional programming languages. (wikipedia.org)
- The problems are related to the definition of the lambda abstraction and the definition and use of functions as the basic type in lambda calculus. (wikipedia.org)
- This is not a criticism of pure lambda calculus, and lambda calculus as a pure system is not the primary topic here. (wikipedia.org)
- The problems arise with the interaction of lambda calculus with other mathematical systems. (wikipedia.org)
- Function application will be represented using the lambda calculus syntax. (wikipedia.org)
- Another difficulty for the interpretation of lambda calculus as a deductive system is the representation of values as lambda terms, which represent functions. (wikipedia.org)
- The untyped lambda calculus is implemented by performing reductions on a lambda term, until the term is in normal form. (wikipedia.org)
- A typed lambda calculus is a typed formalism that uses the lambda-symbol ( λ {\displaystyle \lambda } ) to denote anonymous function abstraction. (wikipedia.org)
- From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type. (wikipedia.org)
- Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages. (wikipedia.org)
- Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry-Howard isomorphism and they can be considered as the internal language of classes of categories, e.g. the simply typed lambda calculus is the language of Cartesian closed categories (CCCs). (wikipedia.org)
- Various typed lambda calculi have been studied. (wikipedia.org)
- Lambda calculi with dependent types are the base of intuitionistic type theory, the calculus of constructions and the logical framework (LF), a pure lambda calculus with dependent types. (wikipedia.org)
- Based on work by Berardi on pure type systems, Henk Barendregt proposed the Lambda cube to systematize the relations of pure typed lambda calculi (including simply typed lambda calculus, System F, LF and the calculus of constructions). (wikipedia.org)
- There exist, however, typed lambda calculi that are not strongly normalizing. (wikipedia.org)
- For example the dependently typed lambda calculus with a type of all types (Type : Type) is not normalizing due to Girard's paradox. (wikipedia.org)
- Kappa calculus-an analogue of typed lambda calculus which excludes higher-order functions Henk Barendregt, Lambda Calculi with Types, Handbook of Logic in Computer Science, Volume II, Oxford University Press, pp. 117-309. (wikipedia.org)
- In mathematical logic and computer science, the lambda-mu calculus is an extension of the lambda calculus introduced by M. Parigot. (wikipedia.org)
- According to the Curry-Howard isomorphism, lambda calculus on its own can express theorems in intuitionistic logic only, and several classical logical theorems can't be written at all. (wikipedia.org)
- We can augment the definition of a lambda expression to gain one in the context of lambda-mu calculus. (wikipedia.org)
- The three main expressions found in lambda calculus are as follows: V, a variable, where V is any identifier. (wikipedia.org)
- In addition to the traditional λ-variables, the lambda-mu calculus includes a distinct set of μ-variables. (wikipedia.org)
- The terms that are added by the lambda-mu calculus are of the form: [α]t is a named term, where α is a μ-variable and t is an unnamed term. (wikipedia.org)
- Classical pure type systems for typed generalizations of lambda calculi with control Michel Parigot. (wikipedia.org)
- Formal definitions of the Lambda calculus. (wikipedia.org)
- Lambda calculus is a programming language based on lambda abstraction and function application. (wikipedia.org)
- Frequently in uses of lambda calculus, α-equivalent terms are considered to be equivalent. (wikipedia.org)
- Only the limit λ → 0 {\displaystyle \lambda \to 0} has functions commuting with 1-forms, which is the special case of high school differential calculus. (wikipedia.org)
- The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. (wikipedia.org)
- The Curry-Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. (wikipedia.org)
- SKI combinator calculus is a combinatory logic, a computational system that may be perceived as a reduced version of untyped lambda calculus. (wikipedia.org)
- All operations in lambda calculus can be encoded via abstraction elimination into the SKI calculus as binary trees whose leaves are one of the three symbols S, K, and I (called combinators). (wikipedia.org)
- Note that from these definitions it can be shown that SKI calculus is not the minimum system that can fully perform the computations of lambda calculus, as all occurrences of I in any expression can be replaced by (SKK) or (SKS) or (SK whatever) and the resulting expression will yield the same result. (wikipedia.org)
- In lambda calculus, a branch of mathematical logic concerned with the formal study of functions, a reduction strategy is how a complex expression is reduced to a simple expression by successive reduction steps. (wikipedia.org)
- Roughly, a reduction strategy is a function that maps a lambda calculus term with reducible expressions to one particular reducible expression, the one to be reduced next. (wikipedia.org)
- Ariola and Felleisen and independently Maraist, Odersky and Wadler completed this line of work with the design of a lambda calculus that precisely relates the notion of call-by-need aka lazy functional programming to an equational system of calculation. (wikipedia.org)
- The simply typed lambda calculus ( λ → {\displaystyle \lambda ^{\to }} ), a form of type theory, is a typed interpretation of the lambda calculus with only one type constructor: → {\displaystyle \to } that builds function types. (wikipedia.org)
- It is the canonical and simplest example of a typed lambda calculus. (wikipedia.org)
- The simply typed lambda calculus was originally introduced by Alonzo Church in 1940 as an attempt to avoid paradoxical uses of the untyped lambda calculus, and it exhibits many desirable and interesting properties. (wikipedia.org)
- The term simple type is also used to refer to extensions of the simply typed lambda calculus such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). (wikipedia.org)
- The syntax of the simply typed lambda calculus is essentially that of the lambda calculus itself. (wikipedia.org)
- x\mid \lambda x.e\mid e\,e} We see that in typed lambda calculus every function (abstraction) must specify the type of its argument. (wikipedia.org)

- Process calculi that make such distinctions typically define an input operator (e.g. x ( v ) {\displaystyle x(v)} ) and an output operator (e.g. x ⟨ y ⟩ {\displaystyle x\langle y\rangle } ), both of which name an interaction point (here x {\displaystyle {\mathit {x}}} ) that is used to synchronise with a dual interaction primitive. (wikipedia.org)
- Only when this input has occurred will the process P {\displaystyle {\mathit {P}}} be activated, with the received data through x {\displaystyle {\mathit {x}}} substituted for identifier v {\displaystyle {\mathit {v}}} . The key operational reduction rule, containing the computational essence of process calculi, can be given solely in terms of parallel composition, sequentialization, input, and output. (wikipedia.org)
- The simplest judgment form is used in Hilbert-style deduction systems, where a judgment has the form B {\displaystyle B} where B {\displaystyle B} is any formula of first-order-logic (or whatever logic the deduction system applies to, e.g., propositional calculus or a higher-order logic or a modal logic). (wikipedia.org)
- This enables one to replace occurrences of B n ( x ) {\displaystyle B_{n}(x)} by L ( ( z + x ) n ) {\displaystyle L((z+x)^{n})} , that is, move the n from a subscript to a superscript (the key operation of umbral calculus). (wikipedia.org)
- The process constructs available in the calculus are the following (a precise definition is given in the following section): concurrency, written P ∣ Q {\displaystyle P\mid Q} , where P {\displaystyle P} and Q {\displaystyle Q} are two processes or threads executed concurrently. (wikipedia.org)
- creation of a new name, written ( ν x ) P {\displaystyle \left(\nu x\right)P} , which may be seen as a process allocating a new constant x within P {\displaystyle P} . The constants of π-calculus are defined by their names only and are always communication channels. (wikipedia.org)
- Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals F by replacing the derivative kernel used above by the "Malliavin derivative" denoted D t {\displaystyle D_{t}} in the above statement of the result. (wikipedia.org)
- In vector calculus, the del operator ( ∇ {\displaystyle \nabla } ) is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. (wikipedia.org)
- P {\displaystyle open\;m.P} instructs the surrounding ambient to dissolve the boundary of an ambient m {\displaystyle m} located at the same level c o p y m . {\displaystyle copy\;m.} makes any number of copies of something m {\displaystyle m} The ambient calculus provides a reduction semantics that formally defines what the results of these operations are. (wikipedia.org)
- The simplest example of application of the operational calculus is to solve: py=H(t), which gives y = p − 1 H = ∫ 0 t H ( u ) d u = t H ( t ) . {\displaystyle y=p^{-1}H=\int _{0}^{t}H(u)du=tH(t). (wikipedia.org)
- Euler calculus begins from the observation that the Euler characteristic with compact supports obeys one of the main properties of a measure: χ ( A ∪ B ) = χ ( A ) + χ ( B ) − χ ( A ∩ B ) {\displaystyle \chi (A\cup B)=\chi (A)+\chi (B)-\chi (A\cap B)} . As a result, for a suitably restricted class of functions, it is possible to define an integral with respect to this measure. (wikipedia.org)
- In the q-calculus and h-calculus, differentials of functions are defined as d q ( f ( x ) ) = f ( q x ) − f ( x ) {\displaystyle d_{q}(f(x))=f(qx)-f(x)\,} and d h ( f ( x ) ) = f ( x + h ) − f ( x ) {\displaystyle d_{h}(f(x))=f(x+h)-f(x)\,} respectively. (wikipedia.org)
- The expression [ n ] q x n − 1 {\displaystyle [n]_{q}x^{n-1}} is then the q-calculus analogue of the simple power rule for positive integral powers. (wikipedia.org)
- John Wallis refined earlier techniques of indivisibles of Cavalieri and others by exploiting an infinitesimal quantity he denoted 1 ∞ {\displaystyle {\tfrac {1}{\infty }}} in area calculations, preparing the ground for integral calculus. (wikipedia.org)
- Although D q D − q = I {\displaystyle \mathbb {D} ^{q}\mathbb {D} ^{-q}=\mathbb {I} } wherein D−q is the left inverse of Dq, the converse is not necessarily true: D − q D q ≠ I {\displaystyle \mathbb {D} ^{-q}\mathbb {D} ^{q}\neq \mathbb {I} } It is instructive to consider elementary integer-order calculus to see what's happening. (wikipedia.org)
- D t q f ( t ) = 1 Γ ( n − q ) d n d t n ∫ 0 t ( t − τ ) n − q − 1 f ( τ ) d τ + Ψ ( x ) {\displaystyle \mathbb {D} _{t}^{q}f(t)={\frac {1}{\Gamma (n-q)}}{\frac {d^{n}}{dt^{n}}}\int _{0}^{t}(t-\tau )^{n-q-1}f(\tau )\,d\tau +\Psi (x)} Working with a properly initialized differintegral is the subject of initialized fractional calculus. (wikipedia.org)
- Dedicated to the discrete logarithm in ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} where q {\displaystyle q} is a prime, index calculus lead to a family of algorithms adapted to finite fields and to some families of elliptic curves. (wikipedia.org)
- Since all axioms and rules of the calculus are schemata, derivation is closed under substitution: If Γ ⊢ A , {\displaystyle \Gamma \vdash A,} then σ ( Γ ) ⊢ σ ( A ) , {\displaystyle \sigma (\Gamma )\vdash \sigma (A),} where σ is any substitution (of formulas using only implication). (wikipedia.org)
- The implicational propositional calculus also satisfies the deduction theorem: If Γ , A ⊢ B {\displaystyle \Gamma ,A\vdash B} , then Γ ⊢ A → B . {\displaystyle \Gamma \vdash A\to B.} As explained in the deduction theorem article, this holds for any axiomatic extension of the system containing axiom schemas 1 and 2 above and modus ponens. (wikipedia.org)
- Frequently the calculus with only one base type, usually o {\displaystyle o} , is considered. (wikipedia.org)

- For example, both natural deduction and sequent calculus systems facilitate the elimination and introduction of universal and existential quantifiers so that unquantified logical expressions can be manipulated according to the much simpler rules of propositional calculus. (wikipedia.org)
- In a typical argument, quantifiers are eliminated, then propositional calculus is applied to unquantified expressions (which typically contain free variables), and then the quantifiers are reintroduced. (wikipedia.org)
- Among the later-found applications of cirquent calculus was the use of it to define a semantics for purely propositional independence-friendly logic. (wikipedia.org)
- Propositional calculus (also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic) is the branch of logic concerned with the study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives. (wikipedia.org)
- Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics. (wikipedia.org)
- It combines elements of predicate logic, propositional calculus, and multi-valued logic. (wikipedia.org)
- Propositional proof system Proof nets Cirquent calculus Calculus of structures Formal proof Anita Wasilewska. (wikipedia.org)
- Taking propositional logic as given, every formula in the monadic predicate calculus expresses something that can likewise be formulated in term logic. (wikipedia.org)
- In mathematical logic Frege's propositional calculus was the first axiomatization of propositional calculus. (wikipedia.org)
- Axioms THEN-1: A → (B → A) THEN-2: (A → (B → C)) → ((A → B) → (A → C)) THEN-3: (A → (B → C)) → (B → (A → C)) FRG-1: (A → B) → (¬B → ¬A) FRG-2: ¬¬A → A FRG-3: A → ¬¬A Inference Rule MP: P, P→Q ⊢ Q Frege's propositional calculus is equivalent to any other classical propositional calculus, such as the "standard PC" with 11 axioms. (wikipedia.org)
- In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. (wikipedia.org)
- The implicational propositional calculus is semantically complete with respect to the usual two-valued semantics of classical propositional logic. (wikipedia.org)

- the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. (wikipedia.org)
- The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. (wikipedia.org)
- The course exposes students to the fundamental concepts of differential calculus and integral calculus. (centennialcollege.ca)
- Calculus I is the study of limits, rates of change of a function, derivative of algebraic and trigonometric functions, applications of derivatives, integration and applications of the definite integral. (bucks.edu)
- One could use these "indivisibles," he said, to calculate length, area and volume-an important step on the way to modern integral calculus. (scientificamerican.com)
- His "method of indivisibles" became a forerunner of integral calculus-but not before surviving attacks from Swiss mathematician Paul Guldin, ostensibly for empirical reasons. (scientificamerican.com)
- It was on the subject of integral calculus and contained many of Euler's discoveries about differential equations. (wikipedia.org)
- It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves). (wikipedia.org)
- Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (13th dynasty, c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components. (wikipedia.org)
- The multiplicative calculi provide alternatives to the classical calculus, which has an additive derivative and an additive integral. (wikipedia.org)
- These calculi all have a derivative and/or integral that is not a linear operator. (wikipedia.org)
- In single-variable calculus, the fundamental theorem of calculus establishes a link between the derivative and the integral. (wikipedia.org)
- The link between the derivative and the integral in multivariable calculus is embodied by the integral theorems of vector calculus: Gradient theorem Stokes' theorem Divergence theorem Green's theorem. (wikipedia.org)
- As a computing machine, the ideal calculus ratiocinator would perform Leibniz's integral and differential calculus. (wikipedia.org)
- Here the term calculus should be understood in its broader sense of "a system of computation", rather than in the sense of differential calculus and integral calculus. (wikipedia.org)
- Visual calculus, invented by Mamikon Mnatsakanian (known as Mamikon), is an approach to solving a variety of integral calculus problems. (wikipedia.org)
- The Itô integral is central to the study of stochastic calculus. (wikipedia.org)
- It is one of the two traditional divisions of calculus, the other being integral calculus. (wikipedia.org)
- Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration. (wikipedia.org)
- Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning. (wikipedia.org)
- It was not until the 17th century that the method was formalized by Cavalieri as the method of Indivisibles and eventually incorporated by Newton into a general framework of integral calculus. (wikipedia.org)
- The book itself Silvanus P. Thompson, Calculus Made Easy: Being a Very-Simplest Introduction to Those Beautiful Methods of Reckoning which Are Generally Called by the Terrifying Names of the Differential Calculus and the Integral Calculus (New York: MacMillan Company, 2nd Ed., 1914). (wikipedia.org)
- The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. (wikipedia.org)
- Conversely, the second part of the theorem, sometimes called the second fundamental theorem of calculus, states that the integral of a function f over some interval can be computed by using any one, say F, of its infinitely many antiderivatives. (wikipedia.org)
- Indefinite product Multiplicative calculus Product integral M. Grossman and R. Katz, Non-Newtonian Calculus, ISBN 0-912938-01-3, Lee Press, 1972. (wikipedia.org)

- The epsilon operator and epsilon substitution method are typically applied to a first-order predicate calculus, followed by a showing of consistency. (wikipedia.org)
- Predicate calculus proofs are generally much easier to discover with this approach, and are often shorter. (wikipedia.org)
- Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus. (wikipedia.org)
- In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols. (wikipedia.org)
- Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments. (wikipedia.org)
- The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. (wikipedia.org)
- It is so weak that, unlike the full predicate calculus, it is decidable-there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains). (wikipedia.org)
- Inferences in term logic can all be represented in the monadic predicate calculus. (wikipedia.org)
- Conversely, monadic predicate calculus is not significantly more expressive than term logic. (wikipedia.org)
- The formal system described above is sometimes called the pure monadic predicate calculus, where "pure" signifies the absence of function letters. (wikipedia.org)
- It was invented by Gottlob Frege, who also invented predicate calculus, in 1879 as part of his second-order predicate calculus (although Charles Peirce was the first to use the term "second-order" and developed his own version of the predicate calculus independently of Frege). (wikipedia.org)

- Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one. (wikipedia.org)
- A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions. (wikipedia.org)
- In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds. (wikipedia.org)
- Techniques of multivariable calculus are used to study many objects of interest in the material world. (wikipedia.org)
- In particular, Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. (wikipedia.org)
- The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. (wikipedia.org)
- While Spivak's elementary treatment of modern mathematical tools is broadly successful, and this approach has made Calculus on Manifolds a standard introduction to the rigorous theory of multivariable calculus, the text is also well-known for its laconic style, lack of motivating examples, and frequent omission of non-obvious steps and arguments. (wikipedia.org)
- A brief, rigorous, and modern treatment of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates] Spivak, Michael (1999) . (wikipedia.org)

- The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. (wikipedia.org)
- Hilbert's epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consistency for the extended formal language. (wikipedia.org)
- Zhou Chaochen and Michael R. Hansen, Duration Calculus: A Formal Approach to Real-Time Systems. (wikipedia.org)
- Sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology. (wikipedia.org)
- In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. (wikipedia.org)
- The applied π-calculus due to Abadi and Fournet put these various extensions on a formal footing by extending the π-calculus with arbitrary datatypes. (wikipedia.org)
- A formal philosophy of ethical calculus is a development in the study of ethics, combining elements of natural selection, self-organizing systems, emergence, and algorithm theory. (wikipedia.org)
- Attributional calculus provides a formal language for natural induction, an inductive learning process whose results are in forms natural to people. (wikipedia.org)
- The analytic point of view understands that the calculus ratiocinator is a formal inference engine or computer program, which can be designed so as to grant primacy to calculations. (wikipedia.org)
- That part of formal logic relevant to the calculus comes under the heading of proof theory. (wikipedia.org)
- The PI Calculus provides a formal theory for modeling systems and reasoning about their behaviors. (wikipedia.org)
- Quantity calculus is the formal method for describing the mathematical relations between abstract physical quantities. (wikipedia.org)
- Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. (wikipedia.org)
- Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system. (wikipedia.org)
- In mathematical logic, category theory, and computer science, kappa calculus is a formal system for defining first-order functions. (wikipedia.org)
- The join-calculus was developed to provide a formal basis for the design of distributed programming languages, and therefore intentionally avoids communications constructs found in other process calculi, such as rendezvous communications, which are difficult to implement in a distributed setting. (wikipedia.org)

- In computer science , the process calculi (or process algebras ) are a diverse family of related approaches for formally modelling concurrent systems . (wikipedia.org)
- Process calculi provide a tool for the high-level description of interactions, communications, and synchronizations between a collection of independent agents or processes. (wikipedia.org)
- Leading examples of process calculi include CSP , CCS , ACP , and LOTOS . (wikipedia.org)
- is the key primitive distinguishing the process calculi from sequential models of computation. (wikipedia.org)
- In some process calculi (notably the π-calculus ) channels themselves can be sent in messages through (other) channels, allowing the topology of process interconnections to change. (wikipedia.org)
- Some process calculi also allow channels to be created during the execution of a computation. (wikipedia.org)
- The choice of the kind of data that can be exchanged in an interaction is one of the key features that distinguishes different process calculi. (wikipedia.org)
- COWS combines in an original way a number of ingredients borrowed from well-known process calculi, e.g. asynchronous communication, polyadic synchronization, pattern matching, protection, delimited receiving and killing activities, while resulting different from any of them. (springer.com)
- In computer science, the Actor model and process calculi are two closely related approaches to the modelling of concurrent digital computation. (wikipedia.org)
- there are numerous process calculi, developed for reasoning about a variety of different kinds of concurrent systems at various levels of detail (including calculi that incorporate time, stochastic transitions, or constructs specific to application areas such as security analysis). (wikipedia.org)
- The publications on the Actor model and on process calculi have a fair number of cross-references, acknowledgments, and reciprocal citations (see Actor model and process calculi history). (wikipedia.org)
- Some process calculi differ from the Actor model in their use of channels as opposed to direct communication. (wikipedia.org)
- However, simple synchronous channels do not suffice for process calculi such as Communicating Sequential Processes (CSP) [Hoare 1978 and because use of the guarded choice (after Dijkstra) command (called the alternative command in CSP). (wikipedia.org)
- While the variety of existing process calculi is very large (including variants that incorporate stochastic behaviour, timing information, and specializations for studying molecular interactions), there are several features that all process calculi have in common: Representing interactions between independent processes as communication (message-passing), rather than as modification of shared variables. (wikipedia.org)
- In process calculi, the sequentialisation operator is usually integrated with input or output, or both. (wikipedia.org)
- The π-calculus belongs to the family of process calculi, mathematical formalisms for describing and analyzing properties of concurrent computation. (wikipedia.org)
- The Actor model and process calculi share an interesting history and co-evolution. (wikipedia.org)
- The calculus has since been applied to study linear logic, classical logic, modal logic, and process calculi, and many benefits are claimed to follow in these investigations from the way in which deep inference is made available in the calculus. (wikipedia.org)

- Differential calculus of vector-valued functions, transformation of coordinates, change of variables in multiple integrals. (indiana.edu)
- I thought you might be interested in this item at http://www.worldcat.org/oclc/835594152 Title: Vector calculus Author: Bill Cox Publisher: Burlington, MA : Elsevier, 1998. (worldcat.org)
- This course covers functions of several variables, vector-valued functions, partial derivatives and applications, double and triple integrals, conic sections, polar coordinates, vectors and vector calculus, first order ordinary differential equations. (upenn.edu)
- Vector analysis Matrix calculus Ricci calculus Tensors in curvilinear coordinates Multilinear subspace learning Dimitrienko, Yuriy (2002). (wikipedia.org)
- Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. (wikipedia.org)
- The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions: In two dimensions, the divergence and curl theorems reduce to the Green's theorem: Linear approximations are used to replace complicated functions with linear functions that are almost the same. (wikipedia.org)
- As a first example, consider the gradient from vector calculus. (wikipedia.org)
- Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965, ISBN 978-0-8053-9021-6) by Michael Spivak is a brief (146 pp.) monograph on the theory of vector-valued functions of several real variables (f : Rn→Rm) and differentiable manifolds in Euclidean space. (wikipedia.org)

- The course is traditionally taken after precalculus and is the first calculus course offered at most schools except for the regular calculus class. (wikipedia.org)
- Some schools do this, though many others only require precalculus as a prerequisite for Calculus BC. (wikipedia.org)

- Calculi of the gallbladder and bile ducts are called gallstones and are primarily developed from bile salts and cholesterol derivatives. (wikipedia.org)
- Calculus II continues to develop the basic ideas introduced in Calculus I: limits, derivatives, and integrals. (hsc.edu)
- Nobody actually explained that we were learning calculus at the time, but by the time I was being taught derivatives, I already had a really intuitive notion of how and why this stuff all worked. (metafilter.com)
- By the end of this course, the student will have learnt the fundamental theorem of calculus, chain rule, derivatives of transcendental functions, integration, and applications of all these in the real world. (wikipedia.org)
- More specifically, the topics are Analysis of graphs (predicting and explaining behavior) Limits of functions (one and two sided) Asymptotic and unbounded behavior Continuity Derivatives Concept At a point As a function Applications Higher Order derivatives Techniques Integrals Interpretations Properties Applications Techniques Numerical approximations Fundamental theorem of calculus Antidifferentiation L'Hôpital's rule, starting in the 2016-17 school year. (wikipedia.org)

- Too many of my classmates saw calculus as complicated algebraic manipulation instead of the inevitable need to describe the relationship between the basic parameters of motion. (metafilter.com)
- The icosian calculus is a non-commutative algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. (wikipedia.org)
- In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. (wikipedia.org)
- Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation. (wikipedia.org)
- A different approach to operational calculus was developed in the 1930s by Polish mathematician Jan Mikusiński, using algebraic reasoning. (wikipedia.org)

- The situation calculus is a logic formalism designed for representing and reasoning about dynamical domains. (wikipedia.org)
- Chronological calculus is a formalism for the analysis of flows of non-autonomous dynamical systems. (wikipedia.org)
- Gentzen's sequent calculus, which is the most studied formalism of structural proof theory. (wikipedia.org)
- The fluent calculus is a formalism for expressing dynamical domains in first-order logic. (wikipedia.org)

- These two branches are related to each other by the fundamental theorem of calculus , and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit . (wikipedia.org)
- These two branches are related to each other by the fundamental theorem of calculus. (wikipedia.org)
- Since its inception, the ambient calculus has grown into a family of closely related ambient calculi The fundamental primitive of the ambient calculus is the ambient. (wikipedia.org)
- The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. (wikipedia.org)
- The fundamental theorem of calculus relates differentiation and integration, showing that these two operations are essentially inverses of one another. (wikipedia.org)
- The historical relevance of the Fundamental Theorem of Calculus is not the ability to calculate these operations, but the realization that the two seemingly distinct operations (calculation of geometric areas, and calculation of velocities) are actually closely related. (wikipedia.org)
- This is the crux of the Fundamental Theorem of Calculus. (wikipedia.org)
- To rebut this case it is sufficient to show that he: saw some of Newton's papers on the subject in or before 1675 or at least 1677 obtained the fundamental ideas of the calculus from those papers. (wikipedia.org)

- Sequent calculus is one of several extant styles of proof calculus for expressing line-by-line logical arguments. (wikipedia.org)
- Cirquent calculus is a proof calculus which manipulates graph-style constructs termed cirquents, as opposed to the traditional tree-style objects such as formulas or sequents. (wikipedia.org)
- In mathematical logic, a proof calculus or a proof system is built to prove statements. (wikipedia.org)
- The calculus of structures is a proof calculus with deep inference for studying the structural proof theory of noncommutative logic. (wikipedia.org)

- In the geometric calculus, the exponential functions are the functions having a constant derivative. (wikipedia.org)
- The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. (wikipedia.org)
- Matrix calculus refers to a number of different notations that use matrices and vectors to collect the derivative of each component of the dependent variable with respect to each component of the independent variable. (wikipedia.org)
- One may proceed further and develop, for example, equivalent notions of Taylor expansion, et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the sine function whose q-derivative is the appropriate analogue for the cosine. (wikipedia.org)
- is the derivative used in standard calculus. (wikipedia.org)
- The first derivative of a linear function, in the sense of calculus, is exactly this slope of the graph of the function. (wikipedia.org)
- A motivational example, of central interest in geometric topology, is the functor of embeddings of one manifold M into another manifold N, whose first derivative in the sense of calculus of functors is the functor of immersions. (wikipedia.org)

- The event calculus solves the frame problem in a way that is similar to the successor state axioms of the situation calculus: a fluent is true at time t if and only if it has been made true in the past and has not been made false in the meantime. (wikipedia.org)

- Infinitely many non-Newtonian calculi are not multiplicative. (wikipedia.org)
- for example, each of the infinitely many non-Newtonian calculi. (wikipedia.org)

- Newton and Leibniz invent calculus. (metafilter.com)

- Currently, umbral calculus refers to the study of Sheffer sequences, including polynomial sequences of binomial type and Appell sequences, but may encompass in its penumbra systematic correspondence techniques of the calculus of finite differences. (wikipedia.org)
- The analogy to Taylor's expansion is utilized in the calculus of finite differences. (wikipedia.org)
- The h-calculus is just the calculus of finite differences, which had been studied by George Boole and others, and has proven useful in a number of fields, among them combinatorics and fluid mechanics. (wikipedia.org)

- Relational calculus consists of two calculi, the tuple relational calculus and the domain relational calculus, that are part of the relational model for databases and provide a declarative way to specify database queries. (wikipedia.org)
- In computer science, domain relational calculus (DRC) is a calculus that was introduced by Michel Lacroix and Alain Pirotte as a declarative database query language for the relational data model. (wikipedia.org)

- Sano, K.: Sound and Complete Tree-Sequent Calculus for Inquisitive Logic. (springer.com)
- In other words, natural deduction and sequent calculus systems are particular distinct kinds of Gentzen-style systems. (wikipedia.org)
- Sequent calculus systems are more suited to theoretical analysis. (wikipedia.org)
- For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. (wikipedia.org)

- Rod calculus or rod calculation is the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were replaced by the more convenient and faster abacus. (wikipedia.org)
- The π-calculus allows channel names to be communicated along the channels themselves, and in this way it is able to describe concurrent computations whose network configuration may change during the computation. (wikipedia.org)
- Functional programs can be encoded into the π-calculus, and the encoding emphasises the dialogue nature of computation, drawing connections with game semantics. (wikipedia.org)
- Pattern calculus bases all computation on pattern matching of a very general kind. (wikipedia.org)

- It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), developed by Gregorio Ricci-Curbastro in 1887-1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. (wikipedia.org)

- While Gödel's results on incompleteness mooted Hilbert's Program to a great extent, modern researchers find the epsilon calculus to provide alternatives for approaching proofs of systemic consistency as described in the epsilon substitution method. (wikipedia.org)
- As most first-order calculi, superposition tries to show the unsatisfiability of a set of first-order clauses, i.e. it performs proofs by refutation. (wikipedia.org)

- The calculus has been applied to stochastic partial differential equations as well. (wikipedia.org)
- Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold. (wikipedia.org)
- In Java, a famous programming language used by various corporations such as IBM, TCS, and Google, the API Calculus is commonly used to solve equations and programs involving calculus. (wikipedia.org)
- An important application of stochastic calculus is in quantitative finance, in which asset prices are often assumed to follow stochastic differential equations. (wikipedia.org)
- Just as the Lindblad master equation provides a quantum generalization to the Fokker-Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classical Langevin equations. (wikipedia.org)
- In non-relativistic quantum mechanics, the Hamiltonian operator H models the total energy observable of a quantum mechanical system S. The unitary group generated by iH corresponds to the time evolution of S. We can also use the Borel functional calculus to abstractly solve some linear initial value problems such as the heat equation, or Maxwell's equations. (wikipedia.org)

- The process is the list of expressions used within the program to solve a calculus problem or equation. (wikipedia.org)
- Because its functions are not first-class objects, evaluation of kappa calculus expressions does not require closures. (wikipedia.org)
- One of the main goals of this extended calculus is to be able to describe expressions corresponding to theorems in classical logic. (wikipedia.org)

- After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum . (wikipedia.org)
- The textbook, which is licensed under Creative Commons Attribution Non-commercial Share Alike License, incorporated some of its example and exercise problems from Elementary calculus: An approach using Infinitesimals. (wikipedia.org)
- Elementary Calculus: An Infinitesimal Approach, a textbook by Jerome Keisler. (wikipedia.org)
- Unlike cirquent calculus, neither approach thus permits mixed cases where some identical subformulas are shared and some not. (wikipedia.org)
- The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation. (wikipedia.org)
- The refinement calculus is a formalized approach to stepwise refinement for program construction. (wikipedia.org)
- This approach formalized by Weierstrass came to be known as the standard calculus. (wikipedia.org)
- After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by Abraham Robinson in the 1960s. (wikipedia.org)
- This approach used technical machinery from mathematical logic to create a theory of hyperreal numbers that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus. (wikipedia.org)

- algebras are the appropriate where to and prove the continuous functional calculus. (planetmath.org)
- The following result is perhaps the for the definition of the continuous functional calculus. (planetmath.org)
- In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. (wikipedia.org)
- Thus one is interested in functional calculi more general than polynomials. (wikipedia.org)
- In particular, this implies that bounded normal operators on a Hilbert space have a continuous functional calculus. (wikipedia.org)
- The Borel functional calculus is more general than the continuous functional calculus. (wikipedia.org)
- More precisely, the Borel functional calculus allows us to apply an arbitrary Borel function to a self-adjoint operator, in a way which generalizes applying a polynomial function. (wikipedia.org)
- However, it is desirable to formulate the functional calculus in a way in which it is clear that it does not depend on the particular representation of T as a multiplication operator. (wikipedia.org)
- Any self-adjoint operator T has a unique Borel functional calculus. (wikipedia.org)
- This defines the functional calculus for bounded functions applied to possibly unbounded self-adjoint operators. (wikipedia.org)
- Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups: Theorem. (wikipedia.org)
- The existence of a mapping with the properties of a functional calculus requires proof. (wikipedia.org)
- For the case of a bounded self-adjoint operator T, the existence of a Borel functional calculus can be shown in an elementary way as follows: First pass from polynomial to continuous functional calculus by using the Stone-Weierstrass theorem. (wikipedia.org)
- Extending by continuity defines f(T) for a continuous function f on the spectrum of T. The Riesz-Markov theorem then allows us to pass from integration on continuous functions to spectral measures, and this is the Borel functional calculus. (wikipedia.org)
- In particular, T can be a square matrix with complex entries, a case which will be used to illustrate functional calculus and provide some heuristic insights for the assumptions involved in the general construction. (wikipedia.org)
- This is the polynomial functional calculus. (wikipedia.org)
- Thus a more general functional calculus is needed. (wikipedia.org)
- i.e., f(λ) must be defined for all eigenvalues λ of T. For a general bounded operator this condition translates to "f must be defined on the spectrum of T". This assumption turns out to be an enabling condition such that the functional calculus map, f → f(T), has certain desirable properties. (wikipedia.org)
- He validates this idea with two different calculi: one for call-by-name and another one for call-by-value, each for purely functional programming languages. (wikipedia.org)

- In mathematical analysis, initialization of the differintegrals is a topic in fractional calculus. (wikipedia.org)
- Multiplicative calculus and its applications", Journal of Mathematical Analysis and Applications, 2008. (wikipedia.org)

- A diagrammatic logical calculus for the syllogistic reasoning is introduced and discussed. (springer.com)

- The situation calculus represents changing scenarios as a set of first-order logic formulae. (wikipedia.org)
- The main elements of the situation calculus are the actions, fluents and the situations. (wikipedia.org)
- The situation calculus is based on a sorted domain with three sorts: actions, situations, and objects, where the objects include everything that is not an action or a situation. (wikipedia.org)
- In the situation calculus, a dynamic world is modeled as progressing through a series of situations as a result of various actions being performed within the world. (wikipedia.org)
- In the Reiter version of the situation calculus described here, a situation does not represent a state, contrarily to the literal meaning of the term and contrarily to the original definition by McCarthy and Hayes. (wikipedia.org)
- Fluent (artificial intelligence) Frame problem Situation calculus Event calculus M. Thielscher (1998). (wikipedia.org)

- There are infinitely many multiplicative non-Newtonian calculi, including the geometric calculus and the bigeometric calculus discussed below. (wikipedia.org)
- The geometric calculus is useful in image analysis, and in the study of growth/decay phenomena (e.g., in economic growth, bacterial growth, and radioactive decay). (wikipedia.org)
- Furthermore, just as the arithmetic average (of functions) is the 'natural' average in the classical calculus, the well-known geometric average is the 'natural' average in the geometric calculus. (wikipedia.org)
- The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). (wikipedia.org)

- Duration calculus (DC) is an interval logic for real-time systems. (wikipedia.org)
- Interval Temporal Logic (ITL) Temporal logic Temporal Logic of Actions (TLA) Modal logic Zhou Chaochen, C. A. R. Hoare and Anders P. Ravn, A Calculus of Durations, Information Processing Letters, 40(5):269-276, December 1991. (wikipedia.org)
- The first sequent calculi, systems LK and LJ, were introduced in 1934/1935 by Gerhard Gentzen as a tool for studying natural deduction in first-order logic (in classical and intuitionistic versions, respectively). (wikipedia.org)
- Since this early work, sequent calculi, also called Gentzen systems, and the general concepts relating to them, have been widely applied in the fields of proof theory, mathematical logic, and automated deduction. (wikipedia.org)
- The basic version of cirquent calculus in was accompanied with an "abstract resource semantics" and the claim that the latter was an adequate formalization of the resource philosophy traditionally associated with linear logic. (wikipedia.org)
- Furthermore, he argued that not only the deductive power but also the expressive power of linear logic was weak, for it, unlike cirquent calculus, failed to capture the ubiquitous phenomenon of resource sharing. (wikipedia.org)
- G.Japaridze, "The taming of recurrences in computability logic through cirquent calculus, Part I".Archive for Mathematical Logic 52 (2013), pages 173-212. (wikipedia.org)
- The 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. (wikipedia.org)
- The superposition calculus is a calculus for reasoning in equational first-order logic. (wikipedia.org)
- As of 2007, most of the (state-of-the-art) theorem provers for first-order logic are based on superposition (e.g. the E equational theorem prover), although only a few implement the pure calculus. (wikipedia.org)
- Attributional calculus is a logic and representation system defined by Ryszard S. Michalski. (wikipedia.org)
- Modern research in logic teems with rival proof calculi: Several systems have been proposed which replace the usual textual syntax with some graphical syntax. (wikipedia.org)
- Recently, many logicians interested in structural proof theory have proposed calculi with deep inference, for instance display logic, hypersequents, the calculus of structures, and bunched implication. (wikipedia.org)
- In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. (wikipedia.org)

- We are pleased to announce the book: ------------------------------------------------------- The Pi-Calculus: A Theory of Mobile Processes Davide Sangiorgi and David Walker Cambridge University Press, 592 pp, ISBN 0521781779 ------------------------------------------------------- Below is the blurb and the table of contents. (upenn.edu)
- Conophytum calculus - SANBI article Conophytum calculus - Information page Hammer,S.(2002) Dumpling and his wife: New views of the genus Conophytum EAE Creative Colour Ltd. ISBN 0-9539326-1-3. (wikipedia.org)
- Some other languages based on CCS: Calculus of broadcasting systems Language Of Temporal Ordering Specification (LOTOS) Models that have been used in the study of CCS-like systems: History monoid Actor model Robin Milner: A Calculus of Communicating Systems, Springer Verlag, ISBN 0-387-10235-3. (wikipedia.org)

- In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. (wikipedia.org)

- The word calculus (plural calculi ) is a Latin word, meaning originally "small pebble" (this meaning is kept in medicine - see Calculus (medicine) ). (wikipedia.org)
- A calculus (plural calculi ), often called a stone , is a concretion of material, usually mineral salts, that forms in an organ or duct of the body. (wikipedia.org)
- Calculus (from Latin calculus meaning 'pebble', plural calculī) in its most general sense is any method or system of calculation. (wikipedia.org)

- λμ-Calculus: An algorithmic interpretation of classical natural deduction. (wikipedia.org)

- The resemblance to pebbles and the firmness of its flesh is what got it its name ( "calculus" is Latin for "pebble" ). (wikipedia.org)

- Subsets of Duration Calculus have been studied (e.g., using discrete time rather than continuous time). (wikipedia.org)
- In fractional calculus, however, since the operator has been fractionalized and is thus continuous, an entire complementary function is needed, not just a constant or set of constants. (wikipedia.org)
- Alternatively, the continuous calculus can be obtained via the Gelfand transform, in the context of commutative Banach algebras. (wikipedia.org)

- Itô calculus, named after Kiyoshi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). (wikipedia.org)
- In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. (wikipedia.org)
- In the study of stochastic processes, Palm calculus, named after Swedish teletrafficist Conny Palm, is the study of the relationship between probabilities conditioned on a specified event and time average probabilities. (wikipedia.org)

- In 1989 the PI- Calculus was created by Robin Milner and was very successful throughout the years. (wikipedia.org)
- Milner claimed the detailed characteristics of the API Calculus to be its "Communication Ability, Capacity for Cooperation, Capacity for Reasoning and Learning, Adaptive Behavior and Trustworthiness. (wikipedia.org)
- Milner later removed some of these restrictions in his work on the Pi calculus (see section Milner, et al. (wikipedia.org)
- The calculus of communicating systems (CCS) is a process calculus introduced by Robin Milner around 1980 and the title of a book describing the calculus. (wikipedia.org)
- The pi-calculus, developed by Milner in the late 80's, provides mobility of communication links by allowing processes to communicate the names of communication channels themselves. (wikipedia.org)

- This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than Rn. (wikipedia.org)
- At more than twice the length of Calculus on Manifolds, Munkres's work presents a more careful and detailed treatment of the subject matter at a leisurely pace. (wikipedia.org)
- In fact, several of the concepts introduced in Calculus on Manifolds reappear in the first volume of this classic work in more sophisticated settings. (wikipedia.org)
- The cover of Calculus on Manifolds features snippets of a July 2, 1850 letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem (i.e., the Kelvin-Stokes' theorem). (wikipedia.org)
- Differentiable manifolds Multilinear form The formalisms of differential forms and the exterior calculus used in Calculus on Manifolds were first formulated by Élie Cartan. (wikipedia.org)

- Calculus Made Easy ignores the use of limits with its epsilon-delta definition, replacing it with a method of approximating (to arbitrary precision) directly to the correct answer in the infinitesimal spirit of Leibniz, now formally justified in modern non-standard analysis and smooth infinitesimal analysis. (wikipedia.org)

- This language uses the same operators as tuple calculus, the logical connectives ∧ (and), ∨ (or) and ¬ (not). (wikipedia.org)
- Tuple calculus is a calculus that was created and introduced by Edgar F. Codd as part of the relational model, in order to provide a declarative database-query language for data manipulation in this data model. (wikipedia.org)

- Functions that maximize or minimize functionals may be found using the Euler-Lagrange equation of the calculus of variations. (wikipedia.org)
- In fact, the π-calculus, like the λ-calculus, is so minimal that it does not contain primitives such as numbers, booleans, data structures, variables, functions, or even the usual control flow statements (such as if-then-else, while). (wikipedia.org)
- According to the College Board, Calculus BC is a full-year course in the calculus of functions of a single variable. (wikipedia.org)
- Network calculus can also be used to express traffic arrival and departure functions as well as service curves. (wikipedia.org)
- In network calculus, a flow is modelled as cumulative functions A, where A(t) represents the amount of data (number of bits for example) send by the flow in the interval [0,t). (wikipedia.org)
- The key element of the operational calculus is to consider differentiation as an operator p = d⁄dt acting on functions. (wikipedia.org)
- Here is an analogy: with the Taylor series method from calculus, you can approximate the shape of a smooth function f around a point x by using a sequence of increasingly accurate polynomial functions. (wikipedia.org)
- They define calculi for a language with variables, functions, function application and assignment statement that capture the conventional notions of parameter passing and evaluation strategies of a wide array of programming languages. (wikipedia.org)
- Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi. (wikipedia.org)

- An AP Calculus AB course is typically equivalent to one semester of college calculus. (wikipedia.org)
- The RTC Toolbox is an academic Java/MATLAB implementation of the Real-Time calculus framework, a theory quasi equivalent to network calculus. (wikipedia.org)
- Roger Fenn and Colin Rourke exhibited an equivalent construction in terms of a single move, the Fenn-Rourke move, that appears in many expositions and extensions of the Kirby calculus. (wikipedia.org)

- The second is a reflective higher-order variant of the asynchronous polyadic pi calculus. (wikipedia.org)

- It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while q stands for quantum. (wikipedia.org)
- First, a calculus defines an evaluation function that maps closed terms (representations of programs) to answers (representations of outputs). (wikipedia.org)

- Calculi in the urinary system are called urinary calculi and include kidney stones (also called renal calculi or nephroliths) and bladder stones (also called vesical calculi or cystoliths). (wikipedia.org)
- Tubeless percutaneous nephrolithotomy (PCNL) is a well-accepted procedure for uncomplicated renal calculi. (urotoday.com)
- Total tubeless PCNL was planned in 40 easily accessible patients with uncomplicated renal calculi, with single infracostal punctures, normal intraoperative events, and acceptable postoperative parameters (visual analogue pain score, parenteral analgesic requirement, bleeding, urinary soakage, hemodynamic stability), allowing an early discharge within 24 hours. (urotoday.com)
- Calculi too large to pass spontaneously range in size from 1 cm to the staghorn stones that occupy the renal pelvis and calyces. (labome.org)