**Extracellular Matrix**: A meshwork-like substance found within the extracellular space and in association with the basement membrane of the cell surface. It promotes cellular proliferation and provides a supporting structure to which cells or cell lysates in culture dishes adhere.

**Extracellular Matrix Proteins**: Macromolecular organic compounds that contain carbon, hydrogen, oxygen, nitrogen, and usually, sulfur. These macromolecules (proteins) form an intricate meshwork in which cells are embedded to construct tissues. Variations in the relative types of macromolecules and their organization determine the type of extracellular matrix, each adapted to the functional requirements of the tissue. The two main classes of macromolecules that form the extracellular matrix are: glycosaminoglycans, usually linked to proteins (proteoglycans), and fibrous proteins (e.g., COLLAGEN; ELASTIN; FIBRONECTINS; and LAMININ).

**Matrix Metalloproteinase 2**: A secreted endopeptidase homologous with INTERSTITIAL COLLAGENASE, but which possesses an additional fibronectin-like domain.

**Matrix Metalloproteinases**: A family of zinc-dependent metalloendopeptidases that is involved in the degradation of EXTRACELLULAR MATRIX components.

**Nuclear Matrix**: The residual framework structure of the CELL NUCLEUS that maintains many of the overall architectural features of the cell nucleus including the nuclear lamina with NUCLEAR PORE complex structures, residual CELL NUCLEOLI and an extensive fibrogranular structure in the nuclear interior. (Advan. Enzyme Regul. 2002; 42:39-52)

**Bone Matrix**: Extracellular substance of bone tissue consisting of COLLAGEN fibers, ground substance, and inorganic crystalline minerals and salts.

**Matrix Metalloproteinase 1**: A member of the metalloproteinase family of enzymes that is principally responsible for cleaving FIBRILLAR COLLAGEN. It can degrade interstitial collagens, types I, II and III.

**Matrix Metalloproteinase Inhibitors**: Compounds that inhibit the enzyme activity or activation of MATRIX METALLOPROTEINASES.

**Matrix Metalloproteinase 3**: An extracellular endopeptidase of vertebrate tissues similar to MATRIX METALLOPROTEINASE 1. It digests PROTEOGLYCAN; FIBRONECTIN; COLLAGEN types III, IV, V, and IX, and activates procollagenase. (Enzyme Nomenclature, 1992)

**Matrix Metalloproteinase 14**: A transmembrane domain-containing matrix metalloproteinase. It is synthesized as an inactive zymogen that is activated by the action of PROPROTEIN CONVERTASES such as FURIN. Matrix metalloproteinase 14 plays a direct role in the cleavage of proteins in the pericellular environment. In addition, it can function indirectly by enzymatically activating the proprotein form of MATRIX METALLOPROTEINASE 15.

**Matrix Metalloproteinase 7**: The smallest member of the MATRIX METALLOPROTEINASES. It plays a role in tumor progression.

**Matrix Metalloproteinase 13**: A secreted matrix metalloproteinase that plays a physiological role in the degradation of extracellular matrix found in skeletal tissues. It is synthesized as an inactive precursor that is activated by the proteolytic cleavage of its N-terminal propeptide.

**Collagen**: A polypeptide substance comprising about one third of the total protein in mammalian organisms. It is the main constituent of SKIN; CONNECTIVE TISSUE; and the organic substance of bones (BONE AND BONES) and teeth (TOOTH).

**Fibronectins**: Glycoproteins found on the surfaces of cells, particularly in fibrillar structures. The proteins are lost or reduced when these cells undergo viral or chemical transformation. They are highly susceptible to proteolysis and are substrates for activated blood coagulation factor VIII. The forms present in plasma are called cold-insoluble globulins.

**Matrix Metalloproteinases, Membrane-Associated**: Matrix metalloproteinases that are associated with the CELL MEMBRANE, either through transmembrane domains or GLYCOSYLPHOSPHATIDYLINOSITOL ANCHORS. Membrane-type matrix metalloproteinases may act within the pericellular environment to influence the process of CELL MIGRATION.

**Matrix Metalloproteinase 12**: A secreted matrix metalloproteinase which is highly expressed by MACROPHAGES where it may play a role in INFLAMMATION and WOUND HEALING.

**Metalloendopeptidases**: ENDOPEPTIDASES which use a metal such as ZINC in the catalytic mechanism.

**Cells, Cultured**: Cells propagated in vitro in special media conducive to their growth. Cultured cells are used to study developmental, morphologic, metabolic, physiologic, and genetic processes, among others.

**Tissue Inhibitor of Metalloproteinase-1**: A member of the family of TISSUE INHIBITOR OF METALLOPROTEINASES. It is a N-glycosylated protein, molecular weight 28 kD, produced by a vast range of cell types and found in a variety of tissues and body fluids. It has been shown to suppress metastasis and inhibit tumor invasion in vitro.

**Matrix Metalloproteinases, Secreted**: A subclass of matrix metalloproteinases that are secreted into the pericellular space.

**Viral Matrix Proteins**: Proteins associated with the inner surface of the lipid bilayer of the viral envelope. These proteins have been implicated in control of viral transcription and may possibly serve as the "glue" that binds the nucleocapsid to the appropriate membrane site during viral budding from the host cell.

**Tissue Inhibitor of Metalloproteinases**: A family of secreted protease inhibitory proteins that regulates the activity of SECRETED MATRIX METALLOENDOPEPTIDASES. They play an important role in modulating the proteolysis of EXTRACELLULAR MATRIX, most notably during tissue remodeling and inflammatory processes.

**Collagenases**: Enzymes that catalyze the degradation of collagen by acting on the peptide bonds.

**Tissue Inhibitor of Metalloproteinase-2**: A member of the family of TISSUE INHIBITOR OF METALLOPROTEINASES. It is a 21-kDa nonglycosylated protein found in tissue fluid and is secreted as a complex with progelatinase A by human fibroblast and uncomplexed from alveolar macrophages. An overexpression of TIMP-2 has been shown to inhibit invasive and metastatic activity of tumor cells and decrease tumor growth in vivo.

**Laminin**: Large, noncollagenous glycoprotein with antigenic properties. It is localized in the basement membrane lamina lucida and functions to bind epithelial cells to the basement membrane. Evidence suggests that the protein plays a role in tumor invasion.

**Proteoglycans**: Glycoproteins which have a very high polysaccharide content.

**Fibroblasts**: Connective tissue cells which secrete an extracellular matrix rich in collagen and other macromolecules.

**Gelatinases**: A class of enzymes that catalyzes the degradation of gelatin by acting on the peptide bonds. EC 3.4.24.-.

**Matrix Metalloproteinase 10**: A secreted matrix metalloproteinase that may play a role in matrix degradation during WOUND HEALING. It is expressed at high levels by KERATINOCYTES, suggesting its role in keratinocyte migration.

**Cartilage Oligomeric Matrix Protein**: Major component of chondrocyte EXTRACELLULAR MATRIX of various tissues including bone, tendon, ligament, SYNOVIUM and blood vessels. It binds MATRILIN PROTEINS and is associated with development of cartilage and bone.

**Cell Movement**: The movement of cells from one location to another. Distinguish from CYTOKINESIS which is the process of dividing the CYTOPLASM of a cell.

**Collagen Type I**: The most common form of fibrillar collagen. It is a major constituent of bone (BONE AND BONES) and SKIN and consists of a heterotrimer of two alpha1(I) and one alpha2(I) chains.

**Cartilage**: A non-vascular form of connective tissue composed of CHONDROCYTES embedded in a matrix that includes CHONDROITIN SULFATE and various types of FIBRILLAR COLLAGEN. There are three major types: HYALINE CARTILAGE; FIBROCARTILAGE; and ELASTIC CARTILAGE.

**RNA, Messenger**: RNA sequences that serve as templates for protein synthesis. Bacterial mRNAs are generally primary transcripts in that they do not require post-transcriptional processing. Eukaryotic mRNA is synthesized in the nucleus and must be exported to the cytoplasm for translation. Most eukaryotic mRNAs have a sequence of polyadenylic acid at the 3' end, referred to as the poly(A) tail. The function of this tail is not known for certain, but it may play a role in the export of mature mRNA from the nucleus as well as in helping stabilize some mRNA molecules by retarding their degradation in the cytoplasm.

**Molecular Sequence Data**: Descriptions of specific amino acid, carbohydrate, or nucleotide sequences which have appeared in the published literature and/or are deposited in and maintained by databanks such as GENBANK, European Molecular Biology Laboratory (EMBL), National Biomedical Research Foundation (NBRF), or other sequence repositories.

**Tenascin**: Hexameric extracellular matrix glycoprotein transiently expressed in many developing organs and often re-expressed in tumors. It is present in the central and peripheral nervous systems as well as in smooth muscle and tendons. (From Kreis & Vale, Guidebook to the Extracellular Matrix and Adhesion Proteins, 1993, p93)

**Matrix Metalloproteinase 11**: A secreted matrix metalloproteinase that is believed to play a role in EXTRACELLULAR MATRIX remodeling and cell fate determination during normal and pathological processes. Matrix metalloproteinase 11 was originally isolated in primary BREAST NEOPLASMS and may be involved in the process of tumorigenesis.

**Immunohistochemistry**: Histochemical localization of immunoreactive substances using labeled antibodies as reagents.

**Integrins**: A family of transmembrane glycoproteins (MEMBRANE GLYCOPROTEINS) consisting of noncovalent heterodimers. They interact with a wide variety of ligands including EXTRACELLULAR MATRIX PROTEINS; COMPLEMENT, and other cells, while their intracellular domains interact with the CYTOSKELETON. The integrins consist of at least three identified families: the cytoadhesin receptors(RECEPTORS, CYTOADHESIN), the leukocyte adhesion receptors (RECEPTORS, LEUKOCYTE ADHESION), and the VERY LATE ANTIGEN RECEPTORS. Each family contains a common beta-subunit (INTEGRIN BETA CHAINS) combined with one or more distinct alpha-subunits (INTEGRIN ALPHA CHAINS). These receptors participate in cell-matrix and cell-cell adhesion in many physiologically important processes, including embryological development; HEMOSTASIS; THROMBOSIS; WOUND HEALING; immune and nonimmune defense mechanisms; and oncogenic transformation.

**Position-Specific Scoring Matrices**: Tabular numerical representations of sequence motifs displaying their variability as likelihood values for each possible residue at each position in a sequence. Position-specific scoring matrices (PSSMs) are calculated from position frequency matrices.

**Chondrocytes**: Polymorphic cells that form cartilage.

**Calcification, Physiologic**: Process by which organic tissue becomes hardened by the physiologic deposit of calcium salts.

**Matrilin Proteins**: PROTEOGLYCANS-associated proteins that are major components of EXTRACELLULAR MATRIX of various tissues including CARTILAGE; and INTERVERTEBRAL DISC structures. They bind COLLAGEN fibers and contain protein domains that enable oligomer formation and interaction with other extracellular matrix proteins such as CARTILAGE OLIGOMERIC MATRIX PROTEIN.

**Cartilage, Articular**: A protective layer of firm, flexible cartilage over the articulating ends of bones. It provides a smooth surface for joint movement, protecting the ends of long bones from wear at points of contact.

**Amino Acid Sequence**: The order of amino acids as they occur in a polypeptide chain. This is referred to as the primary structure of proteins. It is of fundamental importance in determining PROTEIN CONFORMATION.

**Hyaluronic Acid**: A natural high-viscosity mucopolysaccharide with alternating beta (1-3) glucuronide and beta (1-4) glucosaminidic bonds. It is found in the UMBILICAL CORD, in VITREOUS BODY and in SYNOVIAL FLUID. A high urinary level is found in PROGERIA.

**Tissue Engineering**: Generating tissue in vitro for clinical applications, such as replacing wounded tissues or impaired organs. The use of TISSUE SCAFFOLDING enables the generation of complex multi-layered tissues and tissue structures.

**Gelatin**: A product formed from skin, white connective tissue, or bone COLLAGEN. It is used as a protein food adjuvant, plasma substitute, hemostatic, suspending agent in pharmaceutical preparations, and in the manufacturing of capsules and suppositories.

**Matrix Attachment Regions**: Regions of the CHROMATIN or DNA that bind to the NUCLEAR MATRIX. They are found in INTERGENIC DNA, especially flanking the 5' ends of genes or clusters of genes. Many of the regions that have been isolated contain a bipartite sequence motif called the MAR/SAR recognition signature sequence that binds to MATRIX ATTACHMENT REGION BINDING PROTEINS.

**Glycosaminoglycans**: Heteropolysaccharides which contain an N-acetylated hexosamine in a characteristic repeating disaccharide unit. The repeating structure of each disaccharide involves alternate 1,4- and 1,3-linkages consisting of either N-acetylglucosamine or N-acetylgalactosamine.

**Cell Line**: Established cell cultures that have the potential to propagate indefinitely.

**Basement Membrane**: A darkly stained mat-like EXTRACELLULAR MATRIX (ECM) that separates cell layers, such as EPITHELIUM from ENDOTHELIUM or a layer of CONNECTIVE TISSUE. The ECM layer that supports an overlying EPITHELIUM or ENDOTHELIUM is called basal lamina. Basement membrane (BM) can be formed by the fusion of either two adjacent basal laminae or a basal lamina with an adjacent reticular lamina of connective tissue. BM, composed mainly of TYPE IV COLLAGEN; glycoprotein LAMININ; and PROTEOGLYCAN, provides barriers as well as channels between interacting cell layers.

**Protease Inhibitors**: Compounds which inhibit or antagonize biosynthesis or actions of proteases (ENDOPEPTIDASES).

**Blotting, Western**: Identification of proteins or peptides that have been electrophoretically separated by blot transferring from the electrophoresis gel to strips of nitrocellulose paper, followed by labeling with antibody probes.

**Elastin**

**Transforming Growth Factor beta**: A factor synthesized in a wide variety of tissues. It acts synergistically with TGF-alpha in inducing phenotypic transformation and can also act as a negative autocrine growth factor. TGF-beta has a potential role in embryonal development, cellular differentiation, hormone secretion, and immune function. TGF-beta is found mostly as homodimer forms of separate gene products TGF-beta1, TGF-beta2 or TGF-beta3. Heterodimers composed of TGF-beta1 and 2 (TGF-beta1.2) or of TGF-beta2 and 3 (TGF-beta2.3) have been isolated. The TGF-beta proteins are synthesized as precursor proteins.

**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.

**Signal Transduction**: The intracellular transfer of information (biological activation/inhibition) through a signal pathway. In each signal transduction system, an activation/inhibition signal from a biologically active molecule (hormone, neurotransmitter) is mediated via the coupling of a receptor/enzyme to a second messenger system or to an ion channel. Signal transduction plays an important role in activating cellular functions, cell differentiation, and cell proliferation. Examples of signal transduction systems are the GAMMA-AMINOBUTYRIC ACID-postsynaptic receptor-calcium ion channel system, the receptor-mediated T-cell activation pathway, and the receptor-mediated activation of phospholipases. Those coupled to membrane depolarization or intracellular release of calcium include the receptor-mediated activation of cytotoxic functions in granulocytes and the synaptic potentiation of protein kinase activation. Some signal transduction pathways may be part of larger signal transduction pathways; for example, protein kinase activation is part of the platelet activation signal pathway.

**Reverse Transcriptase Polymerase Chain Reaction**: A variation of the PCR technique in which cDNA is made from RNA via reverse transcription. The resultant cDNA is then amplified using standard PCR protocols.

**Cattle**: Domesticated bovine animals of the genus Bos, usually kept on a farm or ranch and used for the production of meat or dairy products or for heavy labor.

**Microscopy, Electron**: Microscopy using an electron beam, instead of light, to visualize the sample, thereby allowing much greater magnification. The interactions of ELECTRONS with specimens are used to provide information about the fine structure of that specimen. In TRANSMISSION ELECTRON MICROSCOPY the reactions of the electrons that are transmitted through the specimen are imaged. In SCANNING ELECTRON MICROSCOPY an electron beam falls at a non-normal angle on the specimen and the image is derived from the reactions occurring above the plane of the specimen.

**Tissue Scaffolds**: Cell growth support structures composed of BIOCOMPATIBLE MATERIALS. They are specially designed solid support matrices for cell attachment in TISSUE ENGINEERING and GUIDED TISSUE REGENERATION uses.

**Neoplasm Invasiveness**: Ability of neoplasms to infiltrate and actively destroy surrounding tissue.

**Matrix Metalloproteinase 20**: A secreted matrix metalloproteinase that is the predominant proteolytic activity in the enamel matrix. The enzyme has a high specificity for dental enamel matrix protein AMELOGENIN.

**Cell Culture Techniques**: Methods for maintaining or growing CELLS in vitro.

**Glycoproteins**: Conjugated protein-carbohydrate compounds including mucins, mucoid, and amyloid glycoproteins.

**Models, Biological**: Theoretical representations that simulate the behavior or activity of biological processes or diseases. For disease models in living animals, DISEASE MODELS, ANIMAL is available. Biological models include the use of mathematical equations, computers, and other electronic equipment.

**Time Factors**: Elements of limited time intervals, contributing to particular results or situations.

**Wound Healing**: Restoration of integrity to traumatized tissue.

**Cell Differentiation**: Progressive restriction of the developmental potential and increasing specialization of function that leads to the formation of specialized cells, tissues, and organs.

**Aggrecans**: Large HYALURONAN-containing proteoglycans found in articular cartilage (CARTILAGE, ARTICULAR). They form into aggregates that provide tissues with the capacity to resist high compressive and tensile forces.

**Protein Binding**: The process in which substances, either endogenous or exogenous, bind to proteins, peptides, enzymes, protein precursors, or allied compounds. Specific protein-binding measures are often used as assays in diagnostic assessments.

**Antigens, CD29**: Integrin beta-1 chains which are expressed as heterodimers that are noncovalently associated with specific alpha-chains of the CD49 family (CD49a-f). CD29 is expressed on resting and activated leukocytes and is a marker for all of the very late activation antigens on cells. (from: Barclay et al., The Leukocyte Antigen FactsBook, 1993, p164)

**Matrix Metalloproteinase 15**: A transmembrane domain-containing matrix metalloproteinase that plays a role in the cleavage of proteins in the pericellular environment. It is synthesized as an inactive zymogen that is activated by the action of ENDOPEPTIDASES such as MATRIX METALLOPROTEINASE 14.

**Biocompatible Materials**: Synthetic or natural materials, other than DRUGS, that are used to replace or repair any body TISSUES or bodily function.

**Dental Enamel Proteins**: The proteins that are part of the dental enamel matrix.

**Gene Expression Regulation**: Any of the processes by which nuclear, cytoplasmic, or intercellular factors influence the differential control (induction or repression) of gene action at the level of transcription or translation.

**Stress, Mechanical**: A purely physical condition which exists within any material because of strain or deformation by external forces or by non-uniform thermal expansion; expressed quantitatively in units of force per unit area.

**Base Sequence**: The sequence of PURINES and PYRIMIDINES in nucleic acids and polynucleotides. It is also called nucleotide sequence.

**Collagen Type IV**: A non-fibrillar collagen found in the structure of BASEMENT MEMBRANE. Collagen type IV molecules assemble to form a sheet-like network which is involved in maintaining the structural integrity of basement membranes. The predominant form of the protein is comprised of two alpha1(IV) subunits and one alpha2(IV) subunit, however, at least six different alpha subunits can be incorporated into the heterotrimer.

**Fluorescent Antibody Technique**: Test for tissue antigen using either a direct method, by conjugation of antibody with fluorescent dye (FLUORESCENT ANTIBODY TECHNIQUE, DIRECT) or an indirect method, by formation of antigen-antibody complex which is then labeled with fluorescein-conjugated anti-immunoglobulin antibody (FLUORESCENT ANTIBODY TECHNIQUE, INDIRECT). The tissue is then examined by fluorescence microscopy.

**Cell Line, Tumor**: A cell line derived from cultured tumor cells.

**Hydrogels**: Water swollen, rigid, 3-dimensional network of cross-linked, hydrophilic macromolecules, 20-95% water. They are used in paints, printing inks, foodstuffs, pharmaceuticals, and cosmetics. (Grant & Hackh's Chemical Dictionary, 5th ed)

**Cell Proliferation**: All of the processes involved in increasing CELL NUMBER including CELL DIVISION.

**Gels**: Colloids with a solid continuous phase and liquid as the dispersed phase; gels may be unstable when, due to temperature or other cause, the solid phase liquefies; the resulting colloid is called a sol.

**Gene Expression**: The phenotypic manifestation of a gene or genes by the processes of GENETIC TRANSCRIPTION and GENETIC TRANSLATION.

**Fibrosis**: Any pathological condition where fibrous connective tissue invades any organ, usually as a consequence of inflammation or other injury.

**Chondroitin Sulfate Proteoglycans**: Proteoglycans consisting of proteins linked to one or more CHONDROITIN SULFATE-containing oligosaccharide chains.

**Gene Expression Regulation, Enzymologic**: Any of the processes by which nuclear, cytoplasmic, or intercellular factors influence the differential control of gene action in enzyme synthesis.

**Recombinant Proteins**: Proteins prepared by recombinant DNA technology.

**Mice, Inbred C57BL**

**Transforming Growth Factor beta1**: A subtype of transforming growth factor beta that is synthesized by a wide variety of cells. It is synthesized as a precursor molecule that is cleaved to form mature TGF-beta 1 and TGF-beta1 latency-associated peptide. The association of the cleavage products results in the formation a latent protein which must be activated to bind its receptor. Defects in the gene that encodes TGF-beta1 are the cause of CAMURATI-ENGELMANN SYNDROME.

**Up-Regulation**: A positive regulatory effect on physiological processes at the molecular, cellular, or systemic level. At the molecular level, the major regulatory sites include membrane receptors, genes (GENE EXPRESSION REGULATION), mRNAs (RNA, MESSENGER), and proteins.

**Disease Models, Animal**: Naturally occurring or experimentally induced animal diseases with pathological processes sufficiently similar to those of human diseases. They are used as study models for human diseases.

**Electrophoresis, Polyacrylamide Gel**: Electrophoresis in which a polyacrylamide gel is used as the diffusion medium.

**Matrix Attachment Region Binding Proteins**: Proteins that bind to the MATRIX ATTACHMENT REGIONS of DNA.

**Enzyme Precursors**: Physiologically inactive substances that can be converted to active enzymes.

**Skin**: The outer covering of the body that protects it from the environment. It is composed of the DERMIS and the EPIDERMIS.

**Fibrillar Collagens**: A family of structurally related collagens that form the characteristic collagen fibril bundles seen in CONNECTIVE TISSUE.

**Nuclear Matrix-Associated Proteins**: A broad category of nuclear proteins that are components of or participate in the formation of the NUCLEAR MATRIX.

**Mice, Knockout**: Strains of mice in which certain GENES of their GENOMES have been disrupted, or "knocked-out". To produce knockouts, using RECOMBINANT DNA technology, the normal DNA sequence of the gene being studied is altered to prevent synthesis of a normal gene product. Cloned cells in which this DNA alteration is successful are then injected into mouse EMBRYOS to produce chimeric mice. The chimeric mice are then bred to yield a strain in which all the cells of the mouse contain the disrupted gene. Knockout mice are used as EXPERIMENTAL ANIMAL MODELS for diseases (DISEASE MODELS, ANIMAL) and to clarify the functions of the genes.

**Cell Division**: The fission of a CELL. It includes CYTOKINESIS, when the CYTOPLASM of a cell is divided, and CELL NUCLEUS DIVISION.

**Microscopy, Fluorescence**: Microscopy of specimens stained with fluorescent dye (usually fluorescein isothiocyanate) or of naturally fluorescent materials, which emit light when exposed to ultraviolet or blue light. Immunofluorescence microscopy utilizes antibodies that are labeled with fluorescent dye.

**Osteoblasts**: Bone-forming cells which secrete an EXTRACELLULAR MATRIX. HYDROXYAPATITE crystals are then deposited into the matrix to form bone.

**Osteogenesis**: The process of bone formation. Histogenesis of bone including ossification.

**Decorin**: A small leucine-rich proteoglycan that interacts with FIBRILLAR COLLAGENS and modifies the EXTRACELLULAR MATRIX structure of CONNECTIVE TISSUE. Decorin has also been shown to play additional roles in the regulation of cellular responses to GROWTH FACTORS. The protein contains a single glycosaminoglycan chain and is similar in structure to BIGLYCAN.

**Vitronectin**: A blood plasma glycoprotein that mediates cell adhesion and interacts with proteins of the complement, coagulation, and fibrinolytic cascade. (From Segen, Dictionary of Modern Medicine, 1992)

**Antigens, CD147**: A widely distributed cell surface transmembrane glycoprotein that stimulates the synthesis of MATRIX METALLOPROTEINASES. It is found at high levels on the surface of malignant NEOPLASMS and may play a role as a mediator of malignant cell behavior.

**Algorithms**: A procedure consisting of a sequence of algebraic formulas and/or logical steps to calculate or determine a given task.

**Tumor Cells, Cultured**: Cells grown in vitro from neoplastic tissue. If they can be established as a TUMOR CELL LINE, they can be propagated in cell culture indefinitely.

**Binding Sites**: The parts of a macromolecule that directly participate in its specific combination with another molecule.

**Enzyme Activation**: Conversion of an inactive form of an enzyme to one possessing metabolic activity. It includes 1, activation by ions (activators); 2, activation by cofactors (coenzymes); and 3, conversion of an enzyme precursor (proenzyme or zymogen) to an active enzyme.

**Biglycan**: A small leucine-rich proteoglycan found in a variety of tissues including CAPILLARY ENDOTHELIUM; SKELETAL MUSCLE; CARTILAGE; BONE; and TENDONS. The protein contains two glycosaminoglycan chains and is similar in structure to DECORIN.

**Rats, Sprague-Dawley**: A strain of albino rat used widely for experimental purposes because of its calmness and ease of handling. It was developed by the Sprague-Dawley Animal Company.

**Phenotype**: The outward appearance of the individual. It is the product of interactions between genes, and between the GENOTYPE and the environment.

**Epithelial Cells**: Cells that line the inner and outer surfaces of the body by forming cellular layers (EPITHELIUM) or masses. Epithelial cells lining the SKIN; the MOUTH; the NOSE; and the ANAL CANAL derive from ectoderm; those lining the RESPIRATORY SYSTEM and the DIGESTIVE SYSTEM derive from endoderm; others (CARDIOVASCULAR SYSTEM and LYMPHATIC SYSTEM) derive from mesoderm. Epithelial cells can be classified mainly by cell shape and function into squamous, glandular and transitional epithelial cells.

**Connective Tissue**: Tissue that supports and binds other tissues. It consists of CONNECTIVE TISSUE CELLS embedded in a large amount of EXTRACELLULAR MATRIX.

**Collagen Type II**: A fibrillar collagen found predominantly in CARTILAGE and vitreous humor. It consists of three identical alpha1(II) chains.

**Versicans**: HYALURONAN-containing proteoglycans found in the EXTRACELLULAR MATRIX of a variety of tissues and organs. Several versican isoforms exist due to multiple ALTERNATIVE SPLICING of the versican MESSENGER RNA.

**Cell-Matrix Junctions**: Specialized areas at the CELL MEMBRANE where a cell attaches to the EXTRACELLULAR MATRIX or other substratum.

**Transfection**: The uptake of naked or purified DNA by CELLS, usually meaning the process as it occurs in eukaryotic cells. It is analogous to bacterial transformation (TRANSFORMATION, BACTERIAL) and both are routinely employed in GENE TRANSFER TECHNIQUES.

**Kinetics**: The rate dynamics in chemical or physical systems.

**Solubility**: The ability of a substance to be dissolved, i.e. to form a solution with another substance. (From McGraw-Hill Dictionary of Scientific and Technical Terms, 6th ed)

**Thrombospondins**: A family of related, adhesive glycoproteins which are synthesized, secreted, and incorporated into the extracellular matrix of a variety of cells, including alpha granules of platelets following thrombin activation and endothelial cells. They interact with a number of BLOOD COAGULATION FACTORS and anticoagulant factors. Five distinct forms have been identified, thrombospondin 1, -2, -3, -4, and cartilage oligomeric matrix protein (COMP). They are involved in cell adhesion, platelet aggregation, cell proliferation, angiogenesis, tumor metastasis, VASCULAR SMOOTH MUSCLE growth, and tissue repair.

**Metalloproteases**: Proteases which use a metal, normally ZINC, in the catalytic mechanism. This group of enzymes is inactivated by metal CHELATORS.

**Microscopy, Confocal**: A light microscopic technique in which only a small spot is illuminated and observed at a time. An image is constructed through point-by-point scanning of the field in this manner. Light sources may be conventional or laser, and fluorescence or transmitted observations are possible.

**Polymers**: Compounds formed by the joining of smaller, usually repeating, units linked by covalent bonds. These compounds often form large macromolecules (e.g., BIOPOLYMERS; PLASTICS).

**Bone and Bones**: A specialized CONNECTIVE TISSUE that is the main constituent of the SKELETON. The principle cellular component of bone is comprised of OSTEOBLASTS; OSTEOCYTES; and OSTEOCLASTS, while FIBRILLAR COLLAGENS and hydroxyapatite crystals form the BONE MATRIX.

**Proteins**: Linear POLYPEPTIDES that are synthesized on RIBOSOMES and may be further modified, crosslinked, cleaved, or assembled into complex proteins with several subunits. The specific sequence of AMINO ACIDS determines the shape the polypeptide will take, during PROTEIN FOLDING, and the function of the protein.

**Actins**: Filamentous proteins that are the main constituent of the thin filaments of muscle fibers. The filaments (known also as filamentous or F-actin) can be dissociated into their globular subunits; each subunit is composed of a single polypeptide 375 amino acids long. This is known as globular or G-actin. In conjunction with MYOSINS, actin is responsible for the contraction and relaxation of muscle.

**Osteonectin**: Non-collagenous, calcium-binding glycoprotein of developing bone. It links collagen to mineral in the bone matrix. In the synonym SPARC glycoprotein, the acronym stands for Secreted Protein, Acidic and Rich in Cysteine.

**Chick Embryo**: The developmental entity of a fertilized chicken egg (ZYGOTE). The developmental process begins about 24 h before the egg is laid at the BLASTODISC, a small whitish spot on the surface of the EGG YOLK. After 21 days of incubation, the embryo is fully developed before hatching.

**Culture Media, Conditioned**: Culture media containing biologically active components obtained from previously cultured cells or tissues that have released into the media substances affecting certain cell functions (e.g., growth, lysis).

**Tablets**: Solid dosage forms, of varying weight, size, and shape, which may be molded or compressed, and which contain a medicinal substance in pure or diluted form. (Dorland, 28th ed)

**Protein Structure, Tertiary**: The level of protein structure in which combinations of secondary protein structures (alpha helices, beta sheets, loop regions, and motifs) pack together to form folded shapes called domains. Disulfide bridges between cysteines in two different parts of the polypeptide chain along with other interactions between the chains play a role in the formation and stabilization of tertiary structure. Small proteins usually consist of only one domain but larger proteins may contain a number of domains connected by segments of polypeptide chain which lack regular secondary structure.

**Cytoskeleton**: The network of filaments, tubules, and interconnecting filamentous bridges which give shape, structure, and organization to the cytoplasm.

**Osteoarthritis**: A progressive, degenerative joint disease, the most common form of arthritis, especially in older persons. The disease is thought to result not from the aging process but from biochemical changes and biomechanical stresses affecting articular cartilage. In the foreign literature it is often called osteoarthrosis deformans.

**Delayed-Action Preparations**: Dosage forms of a drug that act over a period of time by controlled-release processes or technology.

**Elastic Modulus**: Numerical expression indicating the measure of stiffness in a material. It is defined by the ratio of stress in a unit area of substance to the resulting deformation (strain). This allows the behavior of a material under load (such as bone) to be calculated.

**Oligopeptides**: Peptides composed of between two and twelve amino acids.

**Membrane Proteins**: Proteins which are found in membranes including cellular and intracellular membranes. They consist of two types, peripheral and integral proteins. They include most membrane-associated enzymes, antigenic proteins, transport proteins, and drug, hormone, and lectin receptors.

**Growth Plate**: The area between the EPIPHYSIS and the DIAPHYSIS within which bone growth occurs.

**Collagen Type III**: A fibrillar collagen consisting of three identical alpha1(III) chains that is widely distributed in many tissues containing COLLAGEN TYPE I. It is particularly abundant in BLOOD VESSELS and may play a role in tissues with elastic characteristics.

**Mitochondria**: Semiautonomous, self-reproducing organelles that occur in the cytoplasm of all cells of most, but not all, eukaryotes. Each mitochondrion is surrounded by a double limiting membrane. The inner membrane is highly invaginated, and its projections are called cristae. Mitochondria are the sites of the reactions of oxidative phosphorylation, which result in the formation of ATP. They contain distinctive RIBOSOMES, transfer RNAs (RNA, TRANSFER); AMINO ACYL T RNA SYNTHETASES; and elongation and termination factors. Mitochondria depend upon genes within the nucleus of the cells in which they reside for many essential messenger RNAs (RNA, MESSENGER). Mitochondria are believed to have arisen from aerobic bacteria that established a symbiotic relationship with primitive protoeukaryotes. (King & Stansfield, A Dictionary of Genetics, 4th ed)

**Cell Shape**: The quality of surface form or outline of CELLS.

**Dermis**: A layer of vascularized connective tissue underneath the EPIDERMIS. The surface of the dermis contains innervated papillae. Embedded in or beneath the dermis are SWEAT GLANDS; HAIR FOLLICLES; and SEBACEOUS GLANDS.

**Cell Surface Extensions**: Specialized structures of the cell that extend the cell membrane and project out from the cell surface.

**Heparan Sulfate Proteoglycans**: Ubiquitous macromolecules associated with the cell surface and extracellular matrix of a wide range of cells of vertebrate and invertebrate tissues. They are essential cofactors in cell-matrix adhesion processes, in cell-cell recognition systems, and in receptor-growth factor interactions. (From Cancer Metastasis Rev 1996; 15(2): 177-86; Hepatology 1996; 24(3): 524-32)

**Cell Adhesion Molecules**: Surface ligands, usually glycoproteins, that mediate cell-to-cell adhesion. Their functions include the assembly and interconnection of various vertebrate systems, as well as maintenance of tissue integration, wound healing, morphogenic movements, cellular migrations, and metastasis.

**Rabbits**: The species Oryctolagus cuniculus, in the family Leporidae, order LAGOMORPHA. Rabbits are born in burrows, furless, and with eyes and ears closed. In contrast with HARES, rabbits have 22 chromosome pairs.

**Enzyme-Linked Immunosorbent Assay**: An immunoassay utilizing an antibody labeled with an enzyme marker such as horseradish peroxidase. While either the enzyme or the antibody is bound to an immunosorbent substrate, they both retain their biologic activity; the change in enzyme activity as a result of the enzyme-antibody-antigen reaction is proportional to the concentration of the antigen and can be measured spectrophotometrically or with the naked eye. Many variations of the method have been developed.

**Molecular Weight**: The sum of the weight of all the atoms in a molecule.

**Endothelium, Vascular**: Single pavement layer of cells which line the luminal surface of the entire vascular system and regulate the transport of macromolecules and blood components.

**Gene Expression Profiling**: The determination of the pattern of genes expressed at the level of GENETIC TRANSCRIPTION, under specific circumstances or in a specific cell.

**Biomimetic Materials**: Materials fabricated by BIOMIMETICS techniques, i.e., based on natural processes found in biological systems.

**Glomerular Mesangium**: The thin membranous structure supporting the adjoining glomerular capillaries. It is composed of GLOMERULAR MESANGIAL CELLS and their EXTRACELLULAR MATRIX.

**Receptors, Fibronectin**: Specific cell surface receptors which bind to FIBRONECTINS. Studies have shown that these receptors function in certain types of adhesive contact as well as playing a major role in matrix assembly. These receptors include the traditional fibronectin receptor, also called INTEGRIN ALPHA5BETA1 and several other integrins.

**In Situ Hybridization**: A technique that localizes specific nucleic acid sequences within intact chromosomes, eukaryotic cells, or bacterial cells through the use of specific nucleic acid-labeled probes.

**Immunoblotting**: Immunologic method used for detecting or quantifying immunoreactive substances. The substance is identified by first immobilizing it by blotting onto a membrane and then tagging it with labeled antibodies.

**Endothelial Cells**: Highly specialized EPITHELIAL CELLS that line the HEART; BLOOD VESSELS; and lymph vessels, forming the ENDOTHELIUM. They are polygonal in shape and joined together by TIGHT JUNCTIONS. The tight junctions allow for variable permeability to specific macromolecules that are transported across the endothelial layer.

**Mutation**: Any detectable and heritable change in the genetic material that causes a change in the GENOTYPE and which is transmitted to daughter cells and to succeeding generations.

**Hydrogel**: A network of cross-linked hydrophilic macromolecules used in biomedical applications.

**Cell Nucleus**: Within a eukaryotic cell, a membrane-limited body which contains chromosomes and one or more nucleoli (CELL NUCLEOLUS). The nuclear membrane consists of a double unit-type membrane which is perforated by a number of pores; the outermost membrane is continuous with the ENDOPLASMIC RETICULUM. A cell may contain more than one nucleus. (From Singleton & Sainsbury, Dictionary of Microbiology and Molecular Biology, 2d ed)

**Cell Membrane**: The lipid- and protein-containing, selectively permeable membrane that surrounds the cytoplasm in prokaryotic and eukaryotic cells.

**Bone Demineralization Technique**: Removal of mineral constituents or salts from bone or bone tissue. Demineralization is used as a method of studying bone strength and bone chemistry.

**Peptide Fragments**: Partial proteins formed by partial hydrolysis of complete proteins or generated through PROTEIN ENGINEERING techniques.

**Materials Testing**: The testing of materials and devices, especially those used for PROSTHESES AND IMPLANTS; SUTURES; TISSUE ADHESIVES; etc., for hardness, strength, durability, safety, efficacy, and biocompatibility.

**Peptides**: Members of the class of compounds composed of AMINO ACIDS joined together by peptide bonds between adjacent amino acids into linear, branched or cyclical structures. OLIGOPEPTIDES are composed of approximately 2-12 amino acids. Polypeptides are composed of approximately 13 or more amino acids. PROTEINS are linear polypeptides that are normally synthesized on RIBOSOMES.

**Swine**: Any of various animals that constitute the family Suidae and comprise stout-bodied, short-legged omnivorous mammals with thick skin, usually covered with coarse bristles, a rather long mobile snout, and small tail. Included are the genera Babyrousa, Phacochoerus (wart hogs), and Sus, the latter containing the domestic pig (see SUS SCROFA).

**Tensile Strength**: The maximum stress a material subjected to a stretching load can withstand without tearing. (McGraw-Hill Dictionary of Scientific and Technical Terms, 5th ed, p2001)

**Integrin alpha5beta1**: An integrin found in FIBROBLASTS; PLATELETS; MONOCYTES, and LYMPHOCYTES. Integrin alpha5beta1 is the classical receptor for FIBRONECTIN, but it also functions as a receptor for LAMININ and several other EXTRACELLULAR MATRIX PROTEINS.

**Drug Combinations**: Single preparations containing two or more active agents, for the purpose of their concurrent administration as a fixed dose mixture.

**Neovascularization, Physiologic**: The development of new BLOOD VESSELS during the restoration of BLOOD CIRCULATION during the healing process.

**Heparitin Sulfate**: A heteropolysaccharide that is similar in structure to HEPARIN. It accumulates in individuals with MUCOPOLYSACCHARIDOSIS.

**Biomechanical Phenomena**: The properties, processes, and behavior of biological systems under the action of mechanical forces.

**Porosity**: Condition of having pores or open spaces. This often refers to bones, bone implants, or bone cements, but can refer to the porous state of any solid substance.

**Dose-Response Relationship, Drug**: The relationship between the dose of an administered drug and the response of the organism to the drug.

**Sequence Alignment**: The arrangement of two or more amino acid or base sequences from an organism or organisms in such a way as to align areas of the sequences sharing common properties. The degree of relatedness or homology between the sequences is predicted computationally or statistically based on weights assigned to the elements aligned between the sequences. This in turn can serve as a potential indicator of the genetic relatedness between the organisms.

**Amelogenin**: A major dental enamel-forming protein found in mammals. In humans the protein is encoded by GENES found on both the X CHROMOSOME and the Y CHROMOSOME.

**DNA**: A deoxyribonucleotide polymer that is the primary genetic material of all cells. Eukaryotic and prokaryotic organisms normally contain DNA in a double-stranded state, yet several important biological processes transiently involve single-stranded regions. DNA, which consists of a polysugar-phosphate backbone possessing projections of purines (adenine and guanine) and pyrimidines (thymine and cytosine), forms a double helix that is held together by hydrogen bonds between these purines and pyrimidines (adenine to thymine and guanine to cytosine).

**Methylcellulose**: Methylester of cellulose. Methylcellulose is used as an emulsifying and suspending agent in cosmetics, pharmaceutics and the chemical industry. It is used therapeutically as a bulk laxative.

**Spectrometry, Mass, Matrix-Assisted Laser Desorption-Ionization**: A mass spectrometric technique that is used for the analysis of large biomolecules. Analyte molecules are embedded in an excess matrix of small organic molecules that show a high resonant absorption at the laser wavelength used. The matrix absorbs the laser energy, thus inducing a soft disintegration of the sample-matrix mixture into free (gas phase) matrix and analyte molecules and molecular ions. In general, only molecular ions of the analyte molecules are produced, and almost no fragmentation occurs. This makes the method well suited for molecular weight determinations and mixture analysis.

**Cell Survival**: The span of viability of a cell characterized by the capacity to perform certain functions such as metabolism, growth, reproduction, some form of responsiveness, and adaptability.

**Urokinase-Type Plasminogen Activator**: A proteolytic enzyme that converts PLASMINOGEN to FIBRINOLYSIN where the preferential cleavage is between ARGININE and VALINE. It was isolated originally from human URINE, but is found in most tissues of most VERTEBRATES.

**Biological Markers**: Measurable and quantifiable biological parameters (e.g., specific enzyme concentration, specific hormone concentration, specific gene phenotype distribution in a population, presence of biological substances) which serve as indices for health- and physiology-related assessments, such as disease risk, psychiatric disorders, environmental exposure and its effects, disease diagnosis, metabolic processes, substance abuse, pregnancy, cell line development, epidemiologic studies, etc.

**Phosphorylation**: The introduction of a phosphoryl group into a compound through the formation of an ester bond between the compound and a phosphorus moiety.

**Chemistry, Pharmaceutical**: Chemistry dealing with the composition and preparation of agents having PHARMACOLOGIC ACTIONS or diagnostic use.

**Sepharose**

**Morphogenesis**: The development of anatomical structures to create the form of a single- or multi-cell organism. Morphogenesis provides form changes of a part, parts, or the whole organism.

**Chondroitin Sulfates**: Derivatives of chondroitin which have a sulfate moiety esterified to the galactosamine moiety of chondroitin. Chondroitin sulfate A, or chondroitin 4-sulfate, and chondroitin sulfate C, or chondroitin 6-sulfate, have the sulfate esterified in the 4- and 6-positions, respectively. Chondroitin sulfate B (beta heparin; DERMATAN SULFATE) is a misnomer and this compound is not a true chondroitin sulfate.

**Neovascularization, Pathologic**: A pathologic process consisting of the proliferation of blood vessels in abnormal tissues or in abnormal positions.

**Alkaline Phosphatase**: An enzyme that catalyzes the conversion of an orthophosphoric monoester and water to an alcohol and orthophosphate. EC 3.1.3.1.

**Hyaluronoglucosaminidase**: An enzyme that catalyzes the random hydrolysis of 1,4-linkages between N-acetyl-beta-D-glucosamine and D-glucuronate residues in hyaluronate. (From Enzyme Nomenclature, 1992) There has been use as ANTINEOPLASTIC AGENTS to limit NEOPLASM METASTASIS.

**Microscopy, Electron, Transmission**: Electron microscopy in which the ELECTRONS or their reaction products that pass down through the specimen are imaged below the plane of the specimen.

**Enzyme Inhibitors**: Compounds or agents that combine with an enzyme in such a manner as to prevent the normal substrate-enzyme combination and the catalytic reaction.

**Alginates**: Salts of alginic acid that are extracted from marine kelp and used to make dental impressions and as absorbent material for surgical dressings.

**Mesenchymal Stromal Cells**: Bone-marrow-derived, non-hematopoietic cells that support HEMATOPOETIC STEM CELLS. They have also been isolated from other organs and tissues such as UMBILICAL CORD BLOOD, umbilical vein subendothelium, and WHARTON JELLY. These cells are considered to be a source of multipotent stem cells because they include subpopulations of mesenchymal stem cells.

**Reproducibility of Results**: The statistical reproducibility of measurements (often in a clinical context), including the testing of instrumentation or techniques to obtain reproducible results. The concept includes reproducibility of physiological measurements, which may be used to develop rules to assess probability or prognosis, or response to a stimulus; reproducibility of occurrence of a condition; and reproducibility of experimental results.

**Elasticity**: Resistance and recovery from distortion of shape.

**Cell Communication**: Any of several ways in which living cells of an organism communicate with one another, whether by direct contact between cells or by means of chemical signals carried by neurotransmitter substances, hormones, and cyclic AMP.

**Collagen Type VI**: A non-fibrillar collagen that forms a network of MICROFIBRILS within the EXTRACELLULAR MATRIX of CONNECTIVE TISSUE. The alpha subunits of collagen type VI assemble into antiparallel, overlapping dimers which then align to form tetramers.

The distance

**matrix**at each iteration of k, with the updated distances in bold, will be: A negative cycle is a cycle whose ... Weisstein, Eric W. "Floyd-Warshall Algorithm".**MathWorld**. Kleene, S. C. (1956). "Representation of events in nerve nets and ... To avoid numerical problems one should check for negative numbers on the diagonal of the path**matrix**within the inner for loop ...**matrices**s h o r t e s t P a t h ( i , j , 1 ) {\displaystyle \mathrm {shortestPath} (i,j,1)} , s h o r t e s t P a t h ( i , j ...**Matrix**Computations - Third Edition. Baltimore: The Johns Hopkins University Press. p. 53. ISBN 0-8018-5413-X. Helfrick, Albert ... ISBN 81-297-0731-4 Weisstein, Eric W. "Percentage error".

**MathWorld**. ...

Weisstein, Eric W. "Brahmagupta

**Matrix**".**Mathworld**. Retrieved 3 November 2011. "Bhaskaracharya II". Students' Encyclopedia ... Brahmagupta-Fibonacci identity, Brahmagupta formula, Brahmagupta**matrix**, and Brahmagupta theorem: Discovered by the Indian ...**MathWorld**. Crandall & Papadopoulos (2003), page 2 Van Brummelen, Glen (2009). The mathematics of the heavens and the earth: the ...Weisstein, Eric W. "Second-Order Eulerian Triangle".

**MathWorld**. Euler-**matrix**(generalized rowindexes, divergent summation). ... Weisstein, Eric W. "Eulerian Number".**MathWorld**. Weisstein, Eric W. "Euler's Number Triangle".**MathWorld**. Weisstein, Eric W. " ...... s of a polygon with interactive animation Polygon diagonal from

**MathWorld**. Diagonal of a**matrix**from**MathWorld**.. ... In**matrix**algebra, a diagonal of a square**matrix**is a set of entries extending from one corner to the farthest corner. There ... A diagonal**matrix**is one whose off-diagonal entries are all zero. A superdiagonal entry is one that is directly above and to ... For a**matrix**A {\displaystyle A} with row index specified by i {\displaystyle i} and column index specified by j {\displaystyle ...It is the determinant analogue of the Woodbury

**matrix**identity for**matrix**inverses. The identity may be proved as follows. Let ... ISBN 0-387-78356-3. page 416 Weisstein, Eric W. "Sylvester's Determinant Identity".**MathWorld**--A Wolfram Web Resource. ... The identity also finds applications in random**matrix**theory by relating determinants of large**matrices**to determinants of ... where Ia is the identity**matrix**of order a. It is closely related to the**Matrix**determinant lemma and its generalization. ...Concatenation Concatenated code "Creating and Concatenating

**Matrices**". www.mathworks.com. Retrieved 20 January 2016. "**MathWorld**... The further extension to**matrices**is trivial. Since vectors can be viewed in a certain way as lists, concatenation may take on ...Weisstein, Eric W. "Wilf-Zeilberger Pair".

**MathWorld**. Weisstein, Eric W. "Alternating Sign**Matrix**Conjecture".**MathWorld**. ... Weisstein, Eric W. "Refined Alternating Sign**Matrix**Conjecture".**MathWorld**. Weisstein, Eric W. "Zeilberger-Bressoud Theorem". ... Zeilberger gave the first proof of the alternating sign**matrix**conjecture, noteworthy not only for its mathematical content, ... Doron Zeilberger's homepage Biography from ScienceWorld Weisstein, Eric W. "Zeilberger's Algorithm".**MathWorld**. ...... Weisstein, Eric W. "Strassen Formulas".

**MathWorld**. Formulas for fast(er)**matrix**multiplication and inversion.. ... In the same paper he also presented an asymptotically fast algorithm to perform**matrix**inversion, based on the fast**matrix**... This result was an important theoretical breakthrough, leading to much additional research on fast**matrix**multiplication, and ... the first algorithm for performing**matrix**multiplication faster than the O(n3) time bound that would result from a naive ...Though it can be applied to any

**matrix**with non-zero elements on the diagonals, convergence is only guaranteed if the**matrix**is ... "Gauss-Seidel Method".**MathWorld**. CS1 maint: Multiple names: authors list (link) This article incorporates text from the article ... Linear systems Gaussian belief propagation**Matrix**splitting Richardson iteration Gauss 1903, p. 279; direct link. Golub & Van ... In fact, the**matrix**A is neither diagonally dominant nor positive definite. Then, convergence to the exact solution x = A − 1 b ..."Alternating-sign

**matrices**and domino tilings. I", Journal of Algebraic Combinatorics. An International Journal, 1 (2): 111-132 ...**MathWorld**. ...Weisstein, Eric W. "Bohr-Mollerup Theorem".

**MathWorld**. Askey, R. A.; Roy, R. (2010), "Series Expansions", in Olver, Frank W. J ... Kingman, J. F. C. (1961). "A Convexity Property of Positive**Matrices**". The Quarterly Journal of Mathematics. 12 (1): 283-284. ... Weisstein, Eric W. "Gamma Function".**MathWorld**. Bateman, Harry; Erdélyi, Arthur (1955). Higher Transcendental Functions. McGraw ...Likewise, the correlations can be placed in a correlation

**matrix**. In the case of a time series which is stationary in the wide ...**MathWorld**. Weisstein, Eric W. "Statistical Correlation".**MathWorld**. ... Then the variances and covariances can be placed in a covariance**matrix**, in which the (i,j) element is the covariance between ...Wegert, Elias; Trefethen, Lloyd N. (February 1994). "From the Buffon Needle Problem to the Kreiss

**Matrix**Theorem". The American ... Buffon's needle Weisstein, Eric W. "Spijker's Lemma".**MathWorld**. ..."

**Matrix**Signature." From**MathWorld**--A Wolfram Web Resource, created by Eric W. Weisstein. http://**mathworld**.wolfram.com/ ... The signature of the n × n identity**matrix**is (n, 0, 0). The signature of a diagonal**matrix**is the number of positive, negative ... Likewise the signature is equal for two congruent**matrices**and classifies a**matrix**up to congruency. Equivalently, the ... There are some methods for computing the signature of a**matrix**. For any nondegenerate symmetric**matrix**of n × n, diagonalize it ...They let A be the adjacency

**matrix**of the given graph, and B be the adjacency**matrix**of its transitive closure (computed using ... 296-303, doi:10.1145/2608628.2608664 . Weisstein, Eric W. "Transitive Reduction".**MathWorld**. ... It had already been shown that transitive closure and multiplication of Boolean**matrices**of size n × n had the same complexity ... In this construction, the nonzero elements of the**matrix**AB represent pairs of vertices connected by paths of length two or ...Weisstein, Eric W. "Strassen's Formulas".

**MathWorld**. (also includes formulas for fast**matrix**inversion) Tyler J. Earnest, ... Non-square**matrices**can be split in half using the same methods, yielding smaller non-square**matrices**. If the**matrices**are ... Let A, B be two square**matrices**over a ring R. We want to calculate the**matrix**product C as C = A B A , B , C ∈ R 2 n × 2 n {\ ... It is faster than the standard**matrix**multiplication algorithm and is useful in practice for large**matrices**, but would be ...Witte, N. S.; Forrester, P. J. (2000). "Gap probabilities in finite and scaled Cauchy random

**matrix**ensembles". Nonlinearity. ... Weisstein, Eric W. "Jacobi Differential Equation".**MathWorld**. Kuijlaars, A. B. J.; Martinez-Finkelshtein, A.; Orive, R. (2003 ... Forrester, P. J. (2010). Log-Gases and Random**Matrices**. London Mathematical Society Monographs. Princeton University Press. ... whence the associated polynomials are also denoted as Cauchy polynomials in their applications in random**matrix**theory. The ...Pencil beam Pencil of circles Lefschetz pencil

**Matrix**pencil Weisstein, Eric W. "Pencil".**MathWorld**. ..."Symmetric

**Matrices**over F_2 and the Lights Out Problem". Online version of Lights Out puzzle. ... http://www.keithschwarz.com/interesting/code/?dir=lights-out Weisstein, Eric W. "Lights Out Puzzle".**MathWorld**. Marlow Anderson ... More generally if the action**matrix**is symmetric then its diagonal is always solvable. Group theory 'Beyond Tetris' - Lights ... pictured as a 5x5 array but not to be confused with**matrices**). N1 = ( 0 1 1 1 0 1 0 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 1 1 0 ) {\ ...Further the Hessian

**matrix**of second derivatives will have both positive and negative eigenvalues. Singular point of a curve ... Acnode Cusp Tacnode Saddle point Weisstein, Eric W. "Crunode".**Mathworld**. Retrieved 14 January 2014. ...Bimodal distribution Weisstein, Eric W. "Unimodal".

**MathWorld**. Weisstein, Eric W. "Mode".**MathWorld**. A.Ya. Khinchin (1938). "On ... Usually one would want G(Z) to be continuously differentiable with nonsingular Jacobian**matrix**. Quasiconvex functions and ...Every m by n

**matrix**A with integer entries has a unique m by n**matrix**H, such that H=UA for some square unimodular**matrix**U. In ... "Hermite Normal Form".**mathworld**.wolfram.com. Retrieved 2016-06-22. Bouajjani, Ahmed; Maler, Oded (2009-06-19). Computer Aided ... "Dense**matrices**over the integer ring - Sage Reference Manual v7.2:**Matrices**and Spaces of**Matrices**". doc.sagemath.org. ... A unimodular**matrix**is a square invertible integer**matrix**whose determinant is 1 or -1. A m by n**matrix**A with integer entries ...Weisstein, Eric W. "Clebsch Graph". From

**MathWorld**-A Wolfram Web Resource. Retrieved 2009-08-13. J. J. Seidel, Strongly regular ... graphs with (−1,1,0) adjacency**matrix**having eigenvalue 3, Lin. Alg. Appl. 1 (1968) 281-298. Clebsch, A. (1868), "Ueber die ...ISBN 0-486-61480-8. Weisstein, Eric W. "Rectangular parallelepiped".

**MathWorld**. Weisstein, Eric W. "Orthotope".**MathWorld**. ... Minimum bounding box Coxeter, 1973 See e.g. Zhang, Yi; Munagala, Kamesh; Yang, Jun (2011), "Storing**matrices**on disk: Theory ...In the first-order

**matrix**difference equation [. x. t. −. x. ∗. ]. =. A. [. x. t. −. 1. −. x. ∗. ]. {\displaystyle [x_{t}-x ... "**MathWorld**.. *. "OEIS Index Rec".. OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms ... Stability of linear first-order**matrix**recurrences[edit]. Main article:**Matrix**difference equation ... can be computed by n applications of the companion**matrix**, C, to the initial state vector, y. 0. {\displaystyle y_{0}}. . ..."

**mathworld**.wolfram.com. Retrieved 2020-08-19.. *^ Oualline 2003, Ch. 5 *^ "How to organize, add and multiply**matrices**- Bill ... If A is an m-by-n**matrix**and B is an n-by-p**matrix**, then their**matrix**product AB is the m-by-p**matrix**whose entries are given ... Identity**matrix**[edit]. Main article: Identity**matrix**. The identity**matrix**In of size n is the n-by-n**matrix**in which all the ... Square**matrix**[edit]. Main article: Square**matrix**. A square**matrix**is a**matrix**with the same number of rows and columns.[11] An ...213-216, MR 0468170 Weisstein, Eric W. "Redheffer

**matrix**".**MathWorld**. ... In mathematics, a Redheffer**matrix**, studied by Redheffer (1977), is a (0,1)**matrix**whose entries aij are 1 if i divides j or if ... The**matrix**below is the 12 × 12 Redheffer**matrix**. ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 ... j = 1; otherwise, aij = 0. The determinant of the nxn square Redheffer**matrix**is given by the Mertens function M(n). ..."Cartan

**matrix**".**MathWorld**. ... The Cartan**matrix**of a simple Lie algebra is the**matrix**whose ... is a diagonal**matrix**, and S {\displaystyle S} is a symmetric**matrix**. For example, the Cartan**matrix**for G2 can be decomposed as ... The**matrix**of intersection numbers of a basis of the two-cycles is conjectured to be the Cartan**matrix**of the Lie algebra of ... A generalized Cartan**matrix**is a square**matrix**A = ( a i j ) {\displaystyle A=(a_{ij})} with integral entries such that For ...... of two

**matrices**is one of the most fundamental operations in linear algebra. The algorithm for MM is very simple, it could be ... 1.**Matrix**Multiplication on**Mathworld**. http://**mathworld**.wolfram.com/MatrixMultiplication.html. 2.**Matrix**Multiplication on ...**Matrix**multiplication C=A*B where**matrix**A (32768x32768) and**matrix**B (32768x32768) Allocating memory for**matrices**A, B, C: MAS ... Allocating memory for**matrices**A, B, C: MAS=DDR4:DDR4:MCDRAM Initializing**matrix**data**Matrix**multiplication started**Matrix**...Lecture 2:

**Matrices**, the very basics. * Lecture 3: The four fundamental spaces associated with the**matrix**: the row space, the ...**Mathworld**is a hyperlinked dictionary of mathematical terms and topics. * Perhaps surprisingly, Wikipedia contains many short ... The**Matrix**Cookbook by Peterson and Pederson is a good reference for problems in**matrix**manipulation, especially those ... Lecture 5: Orthogonality of vectors and**matrices**, Gram-Schmidt orthogonalization, and the QR decomposition . * Lecture 6: ......

**mathworld**.wolfram.com/DiagonalMatrix.html,Diagonal**Matrix**,/a,, ,a href=http://**mathworld**.wolfram.com/ChromaticInvariant.html, ... F 1 / a(n) = determinant of**matrix**whose (i,j) entry is (i+j)!/(i!(j+1)!) for n , 0. This is a**matrix**with Catalan numbers on ... C For n ,= 1, a(n) is the number of n X n (0,1)**matrices**with each row and column containing exactly one entry equal to 1. ... C a(n) is also the determinant of an square**matrix**, An, whose coefficients are the reciprocals of beta function: a{i, j} = 1/ ...I understand that this process will involve inverting a

**matrix**for each iteration and that is pretty much a solved problem ...**mathworld**.wolfram.com/LeastSquaresFitting.html This general task in category Fitting Implicit Polynomial Curves has few ... So I assume youre going to apply the process described at http://**mathworld**.wolfram.com/NonlinearLeastSquaresFitting.html to my ...Read up through the section on "Special

**Matrices**.".**Mathworld**article on covariance and correlation (for reference). ... Linear Algebra notation:**Matrix**addition, multiplication, transposition, and inverses; covariance**matrices**. User-User ... We will be using**matrix**algebra, but the necessary material for that will be covered in the lectures. ... SOS Math Online article on**matrix**algebra. ...See, for example, http://

**mathworld**.wolfram.com/AntisymmetricTensor.html. Note that the symmetric part of a**matrix**is [tex]\ ... So, one can write the**matrix**equation. [tex] A= A_{SYM} + A_{ANTISYM} [/tex]. and an analogous tensorial equation. [tex] \begin ... frac{1}{2}(A+A^T)[/tex] and the antisymmetric part of a**matrix**is [tex]\frac{1}{2}(A-A^T)[/tex]. (Similarly, the real part ...Recall from previous notes that a Wigner Hermitian

**matrix**ensemble is a random**matrix**ensemble of Hermitian**matrices**(thus ; ... A numerical example of this theorem in action can be seen at the**MathWorld**entry for this law. ... be a**matrix**drawn from the Gaussian Unitary Ensemble (GUE), by which we mean that is a Hermitian**matrix**whose upper triangular ... Hermitian**matrices**. This process is equivariant with respect to conjugation by unitary**matrices**, and so we can quotient out by ...I mean what I would like to do is to define T as a 4x4 transformation

**matrix**such that:. A*T=B. where A is the collection of ... It should be possible to put together a GH definition from the description here : http://**mathworld**.wolfram.com/ ... I tried something using the method described here, but it seems Im lacking some fundamental understanding of how such a**matrix**... But I still fail to grasp how to obtain the general transformation**matrix**using barycentric coordinates. Id like to be able to ...begingroup$ Have a look here:

**mathworld**.wolfram.com/RotationMatrix.html about eigenvalues of rotation**matrices**: some may depend ... In my Case i compute a Givens-Rotation which is used to generate a Zero in a**Matrix**of which i want to calculate the ... But i compared these realizings with some matlab functions (hess and schur). Both functions generates some Zeros in my**Matrix**... Prove the angle of $v \in \mathbb R^2 $ after rotation by $R_{\theta}$ (rotation**matrix**) change $\theta$-degrees. ...See bivariate normal at

**mathworld**.. matplotlib.mlab.. center_**matrix**. (M, dim=0)¶. Return the**matrix**M with each row having zero ... get_sparse_**matrix**. (M, N, frac=0.1)¶. Return a M x N sparse**matrix**with frac elements randomly filled. ... Z and Cond are M x N**matrices**. Z are data and Cond is a boolean**matrix**where some condition is satisfied. Return value is (x, y ... with which the covariance**matrix**is multiplied.. covariance. ndarray. The covariance**matrix**of dataset. , scaled by the ...Rotation

**Matrix**-- from Wolfram**MathWorld**Plus Magazine Plus Online Maths Magazine ASCIIMathML.js demo On what day of the week ...Adjacency

**Matrix**.**MathWorld**-A Wolfram Web Resource. See http://**mathworld**.wolfram.com/AdjacencyMatrix.html. ... functions to sparse**matrices**. Sparse**matrices**, as employed by the Python library SciPy [15], represent data in**matrices**which ... numpy.**matrix**. class. However, as can be seen in Table 1, the number of nodes in a given adjacency**matrix**can be upwards of ... An adjacency**matrix**refers to a labeled**matrix**with 1 or 0 at every position, corresponding to the presence or absence of a ...Weisstein, Eric W. "Positive Semidefinite

**Matrix**." From**MathWorld**-A Wolfram Web Resource. http://**mathworld**.wolfram.com/ ... meaning their kernel**matrices**have no non-negative Eigen values.". I think you mean …their kernel**matrices**have only non- ... The Generalized T-Student Kernel has been proven to be a Mercel Kernel, thus having a positive semi-definite Kernel**matrix**( ... Kernels which are said to satisfy the Mercers theorem are positive semi-definite, meaning their kernel**matrices**have only non- ...Adjacency

**Matrix**(Wolfram**MathWorld**). *Logistic Map (Wolfram**MathWorld**). *Recurrence Plot (Wolfram**MathWorld**) ...Rotation

**Matrix**(Wolfram**MathWorld**). *Rotation (Wolfram**MathWorld**). *Eulers Rotation Theorem (Wolfram**MathWorld**) ...**matrices**s. h. o. r. t. e. s. t. P. a. t. h. (. i. ,. j. ,. 1. ). {\displaystyle \mathrm {shortestPath} (i,j,1)}. , s. h. o. r ... "

**MathWorld**.. *^ Kleene, S. C. (1956). "Representation of events in nerve nets and finite automata". In C. E. Shannon and J. ... Warshall, Stephen (January 1962). "A theorem on Boolean

**matrices**". Journal of the ACM. 9 (1): 11-12. doi:10.1145/321105.321107. ... Ingerman, Peter Z. (November 1962). "Algorithm 141: Path

**Matrix**". Communications of the ACM. 5 (11): 556. doi:10.1145/ ...

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**MathWorld**.. *^ Positive and Negative Terms: Alternating Series *^ Johansson, F. (2016). Computing hypergeometric functions ... displaystyle _{r}F_{s}\left[{\begin{

**matrix**}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{

**matrix**}};z\right]:=\sum _{n ... The Silverman-Toeplitz theorem characterizes

**matrix**summability methods, which are methods for summing a divergent series by ...

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**MathWorld**. *Computer-Based Math. *A New Kind of Science. *Wolfram Technology for Hackathons ...

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**matrix**. http://**mathworld**.wolfram.com/RotationMatrix.html. http://www.mathpages.com/home/ ...**matrix**) is the initial starting condition and gen. is the number of generations the CA is to be computed. ...

**mathworld**.wolfram.com/CellularAutomaton. Elementary CA. To generate an elementary cellular automaton, use ...

Geometry - from

**MathWorld**. Another extensive reference site from**MathWorld**, this index offers pages on advanced and basic ... The book begins with systems of linear equations, then covers**matrix**algebra, before taking up finite-dimensional vector spaces ... The final chapter covers**matrix**representations of linear transformations, through diagonalization, change of basis and Jordan ...**matrices**, advanced vectors, and probability. Each section features a menu of topics and links to a glossary. Many have geology- ...A well-known problem in numerical ecology is how to recombine presence-absence

**matrices**without altering row and column totals ... In**MathWorld**- A Wolfram Web Resource (2014). by Eric W. Weisstein. Keywords: bernoulli definition dimensionless presence- ... A fast and unbiased procedure to randomize ecological binary**matrices**with fixed row and column totals. ... the computational effort that they require to generate a null**matrix**). Here we introduce the Curveball algorithm, a new ...EigenvaluesRandom matricesSparseVectorsEigenvectorsDeterminantsProbabilityComputesMultiplicationMathematicalCovariance matrixVectorComputationTransposeSymmetric matrixDecompositionTheoremPlanesAlgebraDimension of the matrixConsistsRepresentsIntegersDisplaystyleDerivativesGeneralFundamentalIdentityCorrespondsDefinitionsAtomicWikipediaLinearFieldSimplyColumnTheoryPropertiesSensePagesProductCaseSizeDescribesPositive

- which describes the stochastic evolution of eigenvalues of a Hermitian matrix under independent Brownian motion of its entries, and is discussed in this previous blog post . (wordpress.com)
- I thought a Givens-Rotation should not Change the Eigenvalues of a Matrix? (stackexchange.com)
- mathworld.wolfram.com/RotationMatrix.html about eigenvalues of rotation matrices: some may depend on the angle so composing a rotation with itself will just change your rotation angles and it is ok if you get different eigenvalues. (stackexchange.com)
- For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. (wikipedia.org)
- Originally utilized to study principal axes of the rotational motion of rigid bodies , eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis , vibration analysis , atomic orbitals , facial recognition , and matrix diagonalization . (wikipedia.org)
- These examples highlight the use of proximity matrices, eigenvalues, eigenvectors, and linear and nonlinear mappings using numerical minimization methods. (mathematica-journal.com)

- This week I am at the American Institute of Mathematics , as an organiser on a workshop on the universality phenomenon in random matrices . (wordpress.com)

- Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices , expedite computations in finite element method and other computations. (wikipedia.org)
- Sparse matrices and parallel computation were planned solutions to this problem. (zietzm.com)
- Sparse Matrix Recommender framework [AAp4, AA2 ]. (wolfram.com)

- Its vectors might be finite or infinite sequences of numbers, they might be functions (as when tangent vectors are defined as directional derivative operators), they might be matrices, anything as long as your definitions of vector addition and scalar multiplication satisfy the above conditions. (physicsforums.com)
- I have two vectors which different lengths and one data.frame (or matrix). (mail-archive.com)
- The main emphasis is on using the density matrix, although an approach using state vectors is also implemented in the package. (wolfram.com)
- Explore and compute properties of vectors, matrices and vector spaces. (wolframalpha.com)

- Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices that are also referred to as eigenvectors. (wikipedia.org)

- this formula is particularly useful when computing determinants of large matrices (or of operators), as one can often use it to transform such determinants into much smaller determinants. (wordpress.com)

- This computes the probability of getting a matrix of this strength or stronger, all things being equal. (psiheart.net)
- Y Eric W. Weisstein's Probability and Statistics in Mathworld . (psiheart.net)
- PentaPlot is a software tool that computes such dekapentagonal maps given an appropriate probability support matrix. (biomedcentral.com)

- An omega lower than 3.0 is possible, and it means that an MMA computes a product of two matrices faster because an optimization technique, mathematical or programming, is applied and fewer FP multiplication operations are required to compute the product. (intel.com)

- The rule for matrix multiplication , however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second (i.e., the inner dimensions are the same, n for an ( m × n )-matrix times an ( n × p )-matrix, resulting in an ( m × p )-matrix). (wikipedia.org)
- There is no product the other way round-a first hint that matrix multiplication is not commutative . (wikipedia.org)
- A matrix is a rectangular array of numbers (or other mathematical objects) for which operations such as addition and multiplication are defined. (wikipedia.org)
- Matrix multiplication (MM) of two matrices is one of the most fundamental operations in linear algebra. (intel.com)
- Several versions of the classic matrix multiplication algorithm (CMMA) to compute a product of square dense matrices are evaluated in four test programs. (intel.com)
- In order to compute a product of two square matrices using CMMA, a cubic number of floating point (FP) multiplication operations is required. (intel.com)
- Moreover, since composition of rotations corresponds to matrix multiplication , the rotation group is isomorphic to the special orthogonal group SO(3). (academic.ru)

- Mathworld is a hyperlinked dictionary of mathematical terms and topics. (rpi.edu)

- The mean vector μ and covariance matrix Σ can be quantities such that μ ⊗ μ and Σ have the same unit dimensions componentwise. (wolfram.com)

- if v is a column vector (a matrix with only one column) describing the position of a point in space, the product Rv is a column vector describing the position of that point after a rotation. (wikipedia.org)
- There is a direct correspondence between n -by- n square matrices and linear transformations from an n -dimensional vector space to itself, given any basis of the vector space. (wikipedia.org)
- where m T is the transpose of the vector (or matrix) m . (worldlibrary.net)
- I want to write it to *.txt file in following way: 1st row of file is my 1st vector (separate by spacebar) 2nd row of file is 2nd vector (also separate by spacebar) Rest of this file should be a matrix with elements separated by tab. (mail-archive.com)
- Like any linear transformation of finite-dimensional vector spaces, a rotation can always be represented by a matrix . (academic.ru)

- As with the previous problems, moving to strictly matrix computation makes this much more simple. (zietzm.com)

- The incidence matrix of an incidence structure C is a p × q matrix B (or its transpose), where p and q are the number of points and lines respectively, such that B i , j = 1 if the point p i and line L j are incident and 0 otherwise. (kiwix.org)
- where R T denotes the transpose of R and I is the 3 × 3 identity matrix . (academic.ru)

- a i j = 0 {\displaystyle a_{ij}=0} if and only if a j i = 0 {\displaystyle a_{ji}=0} A {\displaystyle A} can be written as D S {\displaystyle DS} , where D {\displaystyle D} is a diagonal matrix, and S {\displaystyle S} is a symmetric matrix. (wikipedia.org)

- Matrix decomposition methods simplify computations, both theoretically and practically. (wikipedia.org)
- In that case, if S in the above decomposition is positive definite, then A is said to be a Cartan matrix. (wikipedia.org)

- Kernels which are said to satisfy the Mercer's theorem are positive semi-definite , meaning their kernel matrices have only non-negative Eigen values. (crsouza.com)

- Karamichalis R. Source code for computing FCGR matrices, distance matrices, MultiDimensional Scaling and separation planes. (wolframcloud.com)

- Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (wikipedia.org)
- The matrix of intersection numbers of a basis of the two-cycles is conjectured to be the Cartan matrix of the Lie algebra of this local symmetry group. (wikipedia.org)
- We will be using matrix algebra, but the necessary material for that will be covered in the lectures. (umich.edu)
- However, this approach converges very slowly, so another method using matrix and linear algebra concepts was also introduced to emphasize the need for efficient and optimized code. (mathematica-journal.com)
- The book primarily addresses engineering students with some initial knowledge of matrix algebra. (google.com)
- When treated as matrices, magic squares also serve as exceptional examples of some advanced linear algebra theorems. (breakfornews.com)

- The group of all 3 × 3 orthogonal matrices is denoted O(3), and consists of all proper and improper rotations. (academic.ru)

- For example, a 2,1 represents the element at the second row and first column of the matrix. (wikipedia.org)
- The product of two transformation matrices is a matrix that represents the composition of two transformations . (wikipedia.org)

- In modular representation theory, and more generally in the theory of representations of finite-dimensional associative algebras A that are not semisimple, a Cartan matrix is defined by considering a (finite) set of principal indecomposable modules and writing composition series for them in terms of irreducible modules, yielding a matrix of integers counting the number of occurrences of an irreducible module. (wikipedia.org)
- The integral cycle space of a graph is equal to the null space of its oriented incidence matrix, viewed as a matrix over the integers or real or complex numbers . (kiwix.org)
- thus, a general matrix of non-negative integers describes a hypergraph. (kiwix.org)

- An n × n {\displaystyle n\times n} matrix A is decomposable if there exists a nonempty proper subset I ⊂ { 1 , … , n } {\displaystyle I\subset \{1,\dots ,n\}} such that a i j = 0 {\displaystyle a_{ij}=0} whenever i ∈ I {\displaystyle i\in I} and j ∉ I {\displaystyle j\notin I} . A is indecomposable if it is not decomposable. (wikipedia.org)

- Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. (wikipedia.org)
- The Matrix Cookbook by Peterson and Pederson is a good reference for problems in matrix manipulation, especially those involving derivatives. (rpi.edu)

- Do you think the concept could be useful to define the general transformation matrix also? (grasshopper3d.com)
- Built-in standard similarity matrices, and support for general similarity rules. (wolfram.com)

- The four fundamental spaces associated with the matrix: the row space, the column space, and the two null spaces. (rpi.edu)
- I tried something using the method described here , but it seems I'm lacking some fundamental understanding of how such a matrix should be constructed, because I'm getting nonsense results. (grasshopper3d.com)

- where A ( L ( G )) is the adjacency matrix of the line graph of G , B ( G ) is the incidence matrix, and I m is the identity matrix of dimension m . (kiwix.org)

- The oriented incidence matrix is unique up to negation of any of the columns, since negating the entries of a column corresponds to reversing the orientation of an edge. (kiwix.org)

- The definitions of incidence matrix apply to graphs with loops and multiple edges . (kiwix.org)

- Infinite matrices occur in planetary theory and in atomic theory . (wikipedia.org)

- Before we proceed, I would like you look at definition http://en.wikipedia.org/wiki/Maxima_and_minima and confirm/clarify the assumptions. (google.com)
- I can confirm my understanding of the concepts explained at http://en.wikipedia.org/wiki/Maxima_and_minima As written in my problem description, I am looking for the best-fit i.e. global optimum solution. (google.com)
- The wikipedia page says this is just a question of inverting a 3x3 matrix. (grasshopper3d.com)
- For more and pictures [http://en.wikipedia.org/wiki/Ostend Ostend]. (pynchonwiki.com)

- Another application of matrices is in the solution of systems of linear equations . (wikipedia.org)

- The unoriented incidence matrix (or simply incidence matrix ) of an undirected graph is a n × m matrix B , where n and m are the numbers of vertices and edges respectively, such that B i , j = 1 if the vertex v i and edge e j are incident and 0 otherwise. (kiwix.org)

- 1, a(n) is the number of n X n (0,1) matrices with each row and column containing exactly one entry equal to 1. (oeis.org)
- Y The matrix is altered by decrementing the smallest entry, in this case 8, and changing the remaining entries to preserve the row and column sums. (psiheart.net)
- If we look at the incidence matrix, we see that the sum of each column is equal to 2. (kiwix.org)
- Because the edges of ordinary graphs can only have two vertices (one at each end), the column of an incidence matrix for graphs can only have two non-zero entries. (kiwix.org)

- Matrix theory" redirects here. (wikipedia.org)
- For the physics topic, see Matrix string theory . (wikipedia.org)
- Incidence matrices are frequently used in graph theory . (kiwix.org)
- In graph theory an undirected graph has two kinds of incidence matrices: unoriented and oriented. (kiwix.org)

- The line graph and Kirchhoff matrix properties generalize to signed graphs. (kiwix.org)

- subsequently the Tension ranked Alfred Hitchcock as a format who was the neck of the matrices' Pages. (muddymeadowfarm.com)

- AB ) i , j refers to an element of a matrix product). (wikipedia.org)
- If one of these three relations is not met, the product of two matrices cannot be computed. (intel.com)
- The following table shows how many multiplications are needed to compute a product of two square matrices of different Ns for three algorithms from Table 1 with omega = 2.3728639 (1), omega = 2.807 (2) and omega = 3.0 (3). (intel.com)
- Imagine the case of the product of two square matrices where N equals 32,768 needs to be computed on a very slow computer system. (intel.com)

- Provided that they have the same size (each matrix has the same number of rows and the same number of columns as the other), two matrices can be added or subtracted element by element (see conformable matrix ). (wikipedia.org)

- Kernel functions must be continuous, symmetric, and most preferably should have a positive (semi-) definite Gram matrix . (crsouza.com)