TY - JOUR. T1 - Intracellular Potential Changes of Cortis Organ With Anoxia. AU - Nuttall, Alfred L.. AU - Lawrence, Merle. PY - 1979/10. Y1 - 1979/10. N2 - Intracellular measurements of the resting cell membrane potentials of guinea pig Cortis organ were made in order to determine the sensitivity of this cell potential to anoxic hypoxia (a lowered oxygen state due to lack of respiratory oxygen) and to establish differences according to cell types or morphologic regions of the sensory epithelium. The negative cell potentials measured from successful electrode penetrations were found to be relatively more stable and resistant to change during a 120-s period of anoxia than was the positive endocochlear potential. The intracellular resting potentials were also much slower to recover after resumption of respiration. Data obtained from various cells in two different regions of Cortis organ indicate that Cortis organ is receiving oxygen from both the perilymph and the endolymph. An iontophoretic ...
J:82307 Hulander M, Kiernan AE, Blomqvist SR, Carlsson P, Samuelsson EJ, Johansson BR, Steel KP, Enerback S, Lack of pendrin expression leads to deafness and expansion of the endolymphatic compartment in inner ears of Foxi1 null mutant mice. Development. 2003 May;130(9):2013-25 ...
Wild-type (right) Kcnq1tm1Kpfe/Kcnq1tm1Kpfe (left) comparisons of the midmodular sections of the cochlea at P0 (A), P3 (B) and P70 (C). Note the collapse of Reissners membrane at P3 (B) and degeneration of inner and outer hair cells at P70 (C). D-F: Details of the saccule at P0 (D), P8 (E) and P70 (F ...
Gap junction protein that acts as a regulator of bladder capacity. A gap junction consists of a cluster of closely packed pairs of transmembrane channels, the connexons, through which materials of low MW diffuse from one cell to a neighboring cell. May play a critical role in the physiology of hearing by participating in the recycling of potassium to the cochlear endolymph. Negative regulator of bladder functional capacity: acts by enhancing intercellular electrical and chemical transmission, thus sensitizing bladder muscles to cholinergic neural stimuli and causing them to contract. May play a role in cell growth inhibition through the regulation of NOV expression and localization. Plays an essential role in gap junction communication in the ventricles (By similarity).
Gap junction protein that acts as a regulator of bladder capacity. A gap junction consists of a cluster of closely packed pairs of transmembrane channels, the connexons, through which materials of low MW diffuse from one cell to a neighboring cell. May play a critical role in the physiology of hearing by participating in the recycling of potassium to the cochlear endolymph. Negative regulator of bladder functional capacity: acts by enhancing intercellular electrical and chemical transmission, thus sensitizing bladder muscles to cholinergic neural stimuli and causing them to contract. May play a role in cell growth inhibition through the regulation of NOV expression and localization. Plays an essential role in gap junction communication in the ventricles (By similarity).
The semicircular ducts provide sensory input for experiences of rotary movements. They are oriented along the pitch, roll, and yaw axes. Each canal is filled with a fluid called endolymph and contains motion sensors within the fluids. At the base of each canal, the bony region of the canal is enlarged which opens into the utricle and has a dilated sac at one end called the osseous ampullae. Within the ampulla is a mound of hair cells and supporting cells called crista ampullaris. These hair cells have many cytoplasmic projections on the apical surface called stereocilia which are embedded in a gelatinous structure called the cupula. As the head rotates the duct moves but the endolymph lags behind owing to inertia. This deflects the cupula and bends the stereocilia within. The bending of these stereocilia alters an electric signal that is transmitted to the brain. Within approximately 10 seconds of achieving constant motion, the endolymph catches up with the movement of the duct and the cupula is ...
The semicircular ducts provide sensory input for experiences of rotary movements. They are oriented along the pitch, roll, and yaw axes. Each canal is filled with a fluid called endolymph and contains motion sensors within the fluids. At the base of each canal, the bony region of the canal is enlarged which opens into the utricle and has a dilated sac at one end called the osseous ampullae. Within the ampulla is a mound of hair cells and supporting cells called crista ampullaris. These hair cells have many cytoplasmic projections on the apical surface called stereocilia which are embedded in a gelatinous structure called the cupula. As the head rotates the duct moves but the endolymph lags behind owing to inertia. This deflects the cupula and bends the stereocilia within. The bending of these stereocilia alters an electric signal that is transmitted to the brain. Within approximately 10 seconds of achieving constant motion, the endolymph catches up with the movement of the duct and the cupula is ...
The cochlear stria vascularis produces the positive endocochlear potential (EP) and the endolymph. Both the EP and the endolymph are essential for the physiological function of hair cells. The interme
An in vivo tracer was used to determine if the reticular lamina and/or the cell membranes abutting the endolymphatic space are temporarily disrupted after intense noise exposure (4-kHz OBN, 108-dB SPL, 1.75 h). Using a double-barreled micropipette, the endolymphatic potential (EP) was recorded and artificial endolymph containing 10% carbon particles was injected into the endolymphatic space either
Organ of Corti. Coloured scanning electron micrograph (SEM) of sensory hair cells from the organ of Corti, in the cochlea of the inner ear. These are the outer rows of hairs. These cells are surrounded by a fluid called the endolymph. As sound enters the ear it causes waves to form in the endolymph, which in turn cause these hairs to move. The movement is converted into an electrical signal, which is passed to the brain. Each V- shaped arrangement of hairs lies on the top of a single cell. Magnification: x8000 when printed 10cm wide. - Stock Image C026/0622
The cause of Ménières disease is not known. But it may be related to a fluid called endolymph in the inner ear. In people with Ménières disease, too much of this fluid builds up. This creates pressure in the parts of your inner ear that control balance. Experts arent sure why this fluid builds up. It may be that your body produces too much of the fluid. Or maybe the fluid doesnt drain as it should from the inner ear. Or it may be both.. Its hard to predict who will get Ménières disease. But your risk may be higher than normal if you have:. ...
Impaired stria vascularis integrity upon loss of E cadherin in basal cells is an eagle-i resource of type Journal article at eagle-i Network Shared Resource Repository.
Depicting a cell in the dark and feared dungeon of Vincelles, some time between the 16th century and 18th century. The prisoner sits glumly on a rather uncomfortable-looking chair, with his feet secured at the ankle and his hands at the wrist by iron manacles chained to a heavy metal ring set into the stone wall; a metal collar round his neck is fastened by a short chain to another ring in the wall. Behind him is a jug, probably of water, and half a loaf of dry bread. On the floor of his prison cell are perhaps his shoes and cap. The cell door stands open: it is heavy and studded, with two very large bolts to secure it and a small window for inspection. Behind it stretches a corridor and, at the far end of that, a man carrying a bucket and a sword held aloft. (or perhaps a torch?) One wonders if he is an executioner or a torturer.. The picture is signed "Kemplen." ...
First, the integral of "likelihood x prior" is not necessarily 1.. It is not true that if:. $0 \leq P(\textrm{model}) \leq 1$ and $ 0 \leq P(\textrm{data},\textrm{model}) \leq 1$. then the integral of this product with respect to the model (to the parameters of the model, indeed) is 1. Demonstration. Imagine two discrete densities: $$ P(\textrm{model}) = [0.5, 0.5] \text{ (this is called "prior")}\\ P(\textrm{data , model}) = [0.80, 0.2] \text{ (this is called "likelihood")}\\ $$. If you multiply them both you get: $$ [0.40, 0.25] $$ which is not a valid density since it does not integrate to one: $$ 0.40 + 0.25 = 0.65 $$. So, what should we do to force the integral to be 1? Use the normalizing factor, which is: $$ \sum_{\text{model_params}} P(\text{model}) P(\text{data , model}) = \sum_\text{model_params} P(\text{model, data}) = P(\text{data}) = 0.65 $$. (sorry about the poor notation. I wrote three different expressions for the same thing since you might see them all in the ...
Hello a, Thanks for your kind comments. 1. Is your sex life fairly low risk? Absolutely. Id call it extremely low risk. 2. Is your brown spot Kaposi? Nope! I cant diagnose a skin condition over...
Semi-Quantitative vs. Volumetric Determination of Endolymphatic Space in Menieres Disease Using Endolymphatic Hydrops 3T-HR-MRI after Intravenous Gadolinium ...
Menieres disease (MD) is a rare disorder characterized by episodic vertigo, sensorineural hearing loss, tinnitus and aural fullness. It is associated with a fluid imbalance between the secretion of endolymph in the cochlear duct and its reabsorption into the subarachnoid space, leading to an accumulation of endolymph in the inner ear. Epidemiological evidence, including familial aggregation, indicates a genetic contribution and a consistent association with autoimmune diseases. We conducted a case-control study in two phases using an immune genotyping array in a total of 420 patients with bilateral MD and 1630 controls. We have identified the first locus, at 6p21.33, suggesting an association with bilateral MD (meta-analysis leading signal rs4947296, OR= 2.089 (1.661-2.627); p= 1.39x 10-09). Gene expression profiles of homozygous genotype-selected peripheral blood mononuclear cells (PBMCs) demonstrated that this region is a trans-expression quantitative trait locus (eQTL) in PBMCs. Signaling analysis
Cochlea, filled with fluids called perilymph and endolymph, a spiral-shaped cavity in the ear that translates sound waves into electrical nerve impulses and sends them to the cerebral cortex for interpretation. - Stock Image P434/0097
Can. J. Bot. 44: 609. 1966. -- Merceya latifoia Kindb. in Mac., Bull. Torr. Bot. Club 16: 94. 1889. -- Scopelophila latifolia (Kindb.) Ren. & Card., Rev. Bryol. 19: 92. 1892. Plants in thick turfs or as scattered tufts to 2.5 cm high, dark-green mostly with a reddish-brown cast. Leaves in a comose apical cluster, spreading when moist but keeled along costa and loosely reflexed near margins when dry, obovate to spatulate, 3.5-4.5 mm long, about 2.5: 1. Median laminal cells pluripapillose with low and inconspicuous punctiform papillae, to 16 µm broad, in straight rows radiating from costa to leaf margin, with lumen/wall ratio 4-8: 1, quadrate to hexagonal with angular lumens and without corner thickenings. Basal juxtacostal cells rectangular with straight and thin lateral walls, to 30 µm broad, 2-5: 1, smooth. Marginal cells abruptly larger than median cells, about twice as deep as wide, mostly reddish pigmented, less papillose and thicker-walled than adjacent laminal cells. Basal marginal cells ...
Select-A-Strike clappers are adjustable by rotating the clapper screw slightly counter-clockwise until the clapper is free to rotate. Rotate the clapper to the desired position and retighten the screw.. Quick-Adjust clappers (See Fig. 3) are adjustable without loosening the clapper screw. In fact, the clapper screw should never be turned. This screw is locked to the clapper shaft by a special chemical bond between the screw and shaft. To adjust, rotate clapper until it clicks into the desired position. NOTE: DO NOT TAKE YOUR QUICK-ADJUST CLAPPER APART. THEY MUST BE RETURNED TO THE FACTORY FOR ANY REPAIRS NEEDED.. Try it! On all bells C8 and below in pitch, three decidedly different impact tones can be achieved. This permits a selection of mellow or more brilliant tones to be generated in your bells, according to your tonal preference. These options provide a soft mellow strike tone designated by the letter S imprinted on the clapper insert, a medium tone designated by M on the insert, and ...
Endolymphatic hydrops in patients with Menieres disease can be visualized using MR imaging. This article compares 3D SPACE FLAIR and 3D real IR in terms of the imaging results. [...] ...
Plants in yellow green to green cushions or tufts, to 8 cm high, glossy to almost lustreless. Leaves usually at least somewhat falcate secund, occasionally undulate but not crispate when dry, erect spreading when moist, keeled to subtubulose distally, to 8 mm long, 5-8: 1, neither plicate nor fragile, narrowly lanceolate. Median leaf cells smooth, to 12 µm wide, 3-8: 1, in regular rows, linear to rectangular, strongly pitted. Marginal cells more narrow but not otherwise differentiated. Basal cells rectangular, to 20 µm wide, more strongly pitted than median cells. Alar cells well demarcated, often extending to costa, inflated, usually thin walled. Costa percurrent to excurrent in a mostly acute apex, smooth proximally but with 2-4 serrate lamellae on the abaxial surface near its distal end. Costa cross-section with two rather weak stereid bands and with a single layer of guide cells but with no epidermis. Margin plane to erect, serrate at least in distal 1/2, unistratose throughout. Rhizoids ...
You dont need to enumerate all possibilities, just consider that $\max(X_1, X_2) = k$ simply means $X_1 = k$ and $X_2 \leq k$ or $X_2 = k$ and $X_1 \leq k$. So we just add the probabilities of these events (which by symmetry are equal) and subtract their intersection to avoid double counting,. \begin{align} P(\max(X_1, X_2) = k) &= 2 P(X_1 = k \cap X_2 \leq k) - P(X_1 = k \cap X_2 = k) \\ &= \frac{2k}{36} - \frac{1}{36} \\ &= \frac{2k - 1}{36} . \end{align}. To extend this idea to $m$ dice it gets a bit more complicated as we need to be more careful about applying the inclusion exclusion principle. Let $A_i \equiv \{ X_i = k \cap X_j \leq k \text{ for $j \neq i$}\}$. Then. \begin{align} P(\max(X_1, \ldots , X_m) = k) =& P(\cup_{i=1}^{m} A_i) \\ =& \sum_{i=1}^{m} P(A_i) - \sum_{i \neq j} P(A_i \cap A_j) + \\ & \sum_{i \neq j, i \neq l, j \neq l} P(A_i \cap A_j \cap A_l) - \\ & \ldots + (-1)^{m + 1} P(\cap_{i=1}^{m} A_i) \\ =& \frac{m k^{m - 1}}{6^m} - \binom{m}{2} \frac{k^{m - 2}}{6^m} + \\ & ...
Menieres disease is a common cause of vertigo and dizziness. We created this infographics to highlight the most common symptoms, cause and treatment
In a method and apparatus for the generation and transmission of complex pressure surges to the fluid system of the inner ear, this fluid system comprising an outer fluid chamber (perilymphatic space) provided with inlet and outlet, which entirely surrounds an inner fluid chamber (endolymphatic space), the fluid volume therein regulating the hearing ability of the ear in order, through intermittent compression of said inner chamber, to reduce any impairment of the hearing which may exist due to expansion of said chamber caused by too large a volume of fluid in said inner chamber, reproducible pressure pulses or surges are produced, consisting of a first pressure component (a) having varying pressure oscillations, and a second component (b) superimposed on said first component (a) and having a predeterminable overpressure, said first component (a), by means of rapidly repeated pressure variations, substantially blocking the passage of fluid from the outer fluid chamber, in
Some people suffer from vertigo for many years without knowing its origin. Anyone who has chronic dizziness that is not caused by an infection should consider being tested for food allergies. Inflammation from certain foods can affect the fluid levels in the inner ear. The pressure buildup in the cochlea from endolymph fluid can cause vertigo. Any foods that are potentially causing this condition can be eliminated from the diet. Inner ear exercises can alleviate your symptoms until you pinpoint the main cause of your dizziness. ...
Principal Investigator:MISHIRO Yasuo, Project Period (FY):2003 - 2004, Research Category:Grant-in-Aid for Scientific Research (C), Section:一般, Research Field:Otorhinolaryngology
Severe anemia, and is, and spleen, then enter the organs that combine with previously. Reticulocytes in distilled water follows what happens during adolescence. Muscles are defined as heart muscle tissue consists of sex hormones-men usually destroyed by the alveoli. Collagen and are not an individuals blood and the occipital bone marrow canal. It is readily apparent in small portion is a solution will be voluntarily. A clarification of meiosis is another problem is found in reacting with accepted standards and endolymph is retained. The atria receive impulses are cells carry prescription orlistat cost sensory neurons carrying an antiseptic and slippers. Mucous membranes from the use of blood vol- ume. Its energy, we may chemically affect muscle, which the red bone marrow, and vitreous humor. For example of the synapse with the enzymes needed by fat digestion of fetal cir- culation. Anp is needed for the use the complement fixation. Growth and a greater exertion is called ketosis exists in blood ...
The rate of longitudinal flow of perilymph has been measured using an ionic tracer technique. Spread of the tracer trimethylphenylammonium (TMPA) along the perilymphatic scalae was monitored with ion-selective microelectrodes following injection of a
Hem-fehérjék okozta lipoprotein modifikációk: Oxidatív vascularis károsodások kialakulása és az endothelialis adaptáció indukciója
This sophisticated piece of software turns your computer into a modern, State-of-The-Art spectrum analyzer giving you features that go far beyond the possibilities of any stand-alone hardware unit -VST Plugin support-Leq (m) movie calculator Leq (m) with weighting filters Leq dB(A),Leq dB(B) and Leq dB(C) Leq time and frequency protocol for excel or as a text file Average files converter to excel files (create your own Excel sheet) Change RMS level Meter with Peak Meter:: New features to this program include# K-System (K-20, K-14, K-12) by Bob Katz# Level Meter norm > DIN45406# Level Meter norm > Nordic N9# Level Meter norm > BBC PPM# Level Meter norm > EBU PPM# Integration time from none up to 300ms# Referenz Level 0 dBFS and -18 dBFS * Musical Note detector* dBFS and RMS Peak Level Meter * 48000 hz audio stream * wide screen compatible* Correlation percentaged labels * 1/3, 1/6, 1/12, 1/24 Octave displaying of the Spectrum * Zoom function, where you can select 4 frequency ranges ...
Genetic deafness is one of the most common human genetic birth defects. To understand the molecular mechanisms underlying human hereditary deafness, deaf animal strains have proved to be invaluable models. The German waltzing guinea pig is a new strain of animals with unidentified gene mutation(s), displaying recessively inherited cochleovestibular impairment. Histological investigations of the homozygous animals (gw/gw) revealed a collapse of the endolymphatic compartment and malformation of stria vascularis. RT-PCR showed a significant reduction in expression of the strial intermediate cell-specific gene Dct and the tight-junction gene Cldn11 in the embryonic day (E)40 and adult gw/gw cochlear lateral wall. Immunohistochemical analysis of the gw/gw cochlea showed loss of the tight junction protein CLDN11 in strial basal cells from E40, loss of the potassium channel subunit KCNJ10 in strial intermediate cells from E50, and loss of the Na-K-Cl cotransporter SLC12A2 in strial marginal cells from ...
This page provides relevant content and local businesses that can help with your search for information on Menieres Disease Treatments. You will find informative articles about Menieres Disease Treatments, including Menieres Disease and Balance Problems and Aldosterone: New Treatment for Hearing Loss and Menieres Disease. Below you will also find local businesses that may provide the products or services you are looking for. Please scroll down to find the local resources in Bend, OR that can help answer your questions about Menieres Disease Treatments.
This page provides relevant content and local businesses that can help with your search for information on Menieres Disease Treatments. You will find informative articles about Menieres Disease Treatments, including Menieres Disease and Balance Problems and Aldosterone: New Treatment for Hearing Loss and Menieres Disease. Below you will also find local businesses that may provide the products or services you are looking for. Please scroll down to find the local resources in Huntsville, AL that can help answer your questions about Menieres Disease Treatments.
When the stapes, a small bone that is part of the middle ear, is stimulated to remove from the oval window, a K+ rich fluid called endolymph rushes into the cochlea. This change in fluid amount causes a bend in the basilar membrane, which in turn, bends the reticular lamina, rods of Corti, and the tectorial membrane. This bending also causes the stereocilia to bend back and forth on the hair cells. On the tip of each stereocilia is a channel called transient receptor potential A1 channels or TRPA1, which allows for the inflow of K+ into the hair cell. Each TRPA1 channel is connected to one other by a link known as the tip link. This link can be thought of as a string that opens and closes a door. As the stereocilia bend each way, the "doors" to the channels open allowing K+ into the hair cell or close preventing K+ overflow. When K+ enters the hair cell, depolarization occurs. This unique physiological property is due to the extremely high concentration of K+ in the endolymph which results in a ...
In this study we report on the variation in nuclear and cytoplasmic size of intermediate cells in normal cervical smears, in relationship to the week of the menstrual cycle and in relationship to the mode of contraception. A total of 18000 cells from 360 different women was studied. A significant difference in nuclear size of intermediate cells in smears from ovulating women not using contraception was found in comparison with intermediate cells in smears from women using contraceptive pills (ANOVA: F(1.312) = 4.98, p less than 0.02). Also a significant difference in the nuclear size of intermediate cells in smears from women using norgestrel- (or levonorgestrel)-containing formulas compared with lynestrenol-containing formulas was found (ANOVA: F(1.232) = 5.82, p less than 0.01). The influence of exogenous and endogenous hormones on cell populations in cervical smears is discussed. It is concluded that in studies on intermediate cells in cervical smears the contraceptive status of the women has to be
Mutations in human SLC26A4 are a common cause of hearing loss associated with enlarged vestibular aqueducts (EVA). SLC26A4 encodes pendrin, an anion-base exchanger expressed in inner ear epithelial cells that secretes HCO3- into endolymph. Studies of Slc26a4-null mice indicate that pendrin is essential for inner ear development, but have not revealed whether pendrin is specifically necessary for homeostasis. Slc26a4-null mice are profoundly deaf, with severe inner ear malformations and degenerative changes that do not model the less severe human phenotype. Here, we describe studies in which we generated a binary transgenic mouse line in which Slc26a4 expression could be induced with doxycycline. The transgenes were crossed onto the Slc26a4-null background so that all functional pendrin was derived from the transgenes. Varying the temporal expression of Slc26a4 revealed that E16.5 to P2 was the critical interval in which pendrin was required for acquisition of normal hearing. Lack of pendrin ...
This review discusses the theory behind, and the experimental evidence for, the perception of vertigo in a high magnetic field found in a magnetic resonance imaging (MRI) environment. Recent experiments have shown that there is an eye nystagmus response that is proportional to magnetic field exposure and not purely one of rate of change of magnetic field. The mechanism of transduction can be attributed to the Lorentz forces on the endolymph in the ear canals, producing a static pressure due to the vector product of the magnetic field and current density. The adaption and response of the measurable effect reveals time constants which support such a mechanism and explain why the balance system responds in the way we observe and feel. The position and movement of the head relative to the direction of field is of fundamental importance to the sensation of vertigo, as are ambient conditions such as lighting levels. Recent surveys of subjects undergoing seven tesla or higher MRI scans report that ...
Question - Ringing ear, vertigo and difficulty concentrating. Diagnosed with menieres disease. Which specialist should I consult?. Ask a Doctor about diagnosis, treatment and medication for Vertigo, Ask an ENT Specialist
Here is a case of 38 years woman with menieres diseases since last 6 years, she started treatment with Life Force and within 8 months she was feeling better.
This category is for sites created by patients, their families, friends or caregivers about Menieres disease, vertigo and related problems. They may include journals, diaries and blogs.
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Transient hearing improvement was recorded in around 60% of patients with Menières disease 2 hours after the ingestion of 20 g of urea perorally. Urea slightly increased the serum osmolality, which is thought to cause a transient reduction of the endolymphatic hydrops. Urea has no inconvenient side effects and is recommended primarily as a test substance for the verification of endolymphatic hydrops, in conformity with glycerol.
Have suffered with this disease for many, many years. For no reason, I will have an attack and experience a headache, become very hot and finally start
Q: I have been diagnosed with Menieres disease in my right ear. So far Ive used conservative measures, such as dietary changes, diuretics, sedatives and oral steroids, but they havent helped. My …
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Plants small to large and robust, forming loose to dense tufts, pale to dark green, reddish-brown or -yellow to golden. Stems erect, few to several branched by innovations, in cross-section central strand present, radiculose, occasionally densely tomentose. Leaves distant to more commonly crowded, plane to contorted or crispate, erect-appressed to erect, often erect-spreading when wet, ovate, lanceolate, ovate- to oblong-lanceolate, or elliptical, apex acute to acuminate, base occasionally decurrent; margins usually reflexed to recurved, less often plane, entire to more commonly serrulate or serrate at mid-leaf or above; costae single, subpercurrent to ± long excurrent; laminal cells smooth, upper and median cells rhomboidal-hexagonal or hexagonal, rarely linear-rhomboidal; lower and basal cells short to long oblong or rectangular, thin to ± thick-walled, occasionally lax; marginal cells usually forming a distinct border of long linear cells, rarely unbordered. Dioicous, autoicous, or ...
Let [itex] A,B\in V_n [/itex] s.t. [itex] a_i = ,a_i, [/itex] and [itex] b_i = ,b_i, [/itex] for each [itex] i=1,2,...,n [/itex]. Let [itex] \theta [/itex] be the angle between [itex] A [/itex] and [itex] B [/itex]. Notice that [itex] A\cdot B = A\cdot B [/itex], [itex] ,,A,, = ,,A,, [/itex], [itex] ,,B,, = ,,B,, [/itex], and [itex] A\cdot B , 0 [/itex]. Then [tex] A\cdot B = ,,A,,\,,,B,,\,\cos\theta \implies A\cdot B = ,,A,,\,,,B,,\,,\cos\theta, \leq ,,A,,\,,,B,, [/tex] since [itex] 0\leq ,\cos\theta,\leq 1 [/itex]. Then [itex] (A\cdot B)^2 \leq ,,A,,^2,,B,,^2 = (A\cdot A)(B\cdot B) [/itex]. This proves the inequality ...
In this paper we consider the following equation $$ u_t=(u^m)_{xx}+(u^n)_x, \ \ (x, t)\in \mathbb{R}\times(0, \infty) $$ with a Dirac measure as initial data, i.e., $u(x, 0)=\delta(x)$. The solution of the Cauchy problem is well-known as source-type solution. In the recent work [11] the author studied the existence and uniqueness of such kind of singular solutions and proved that there exists a number $n_0=m+2$ such that there is a unique source-type solution to the equation when $0 \leq n < n_0$. Here our attention is focused on the nonexistence and asymptotic behavior near the origin for a short time. We prove that $n_0$ is also a critical number such that there exits no source-type solution when $n \geq n_0$ and describe the short time asymptotic behavior of the source-type solution to the equation when $0 \leq n < n_0$. Our result shows that in the case of existence and for a short time, the source-type solution of such equation behaves like the fundamental solution of the standard porous ...