Home About us Contact
|
Home About us Contact | ||
|
|
||
Positive Number (positive + number)
Selected AbstractsOn the numerical computation of blowing-up solutions for semilinear parabolic equationsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2001D. Fayyad Theoretical aspects related to the approximation of the semilinear parabolic equation: $u_t=\Delta u+f(u)$\nopagenumbers\end, with a finite unknown ,blow-up' time Tb have been studied in a previous work. Specifically, for , a small positive number, we have considered coupled systems of semilinear parabolic equations, with positive solutions and ,mass control' property, such that: \def\ve{^\varepsilon}$$u_t\ve=\Delta u\ve+f(u\ve)v\ve\qquad v_t\ve=\Delta v\ve-\varepsilon f(u\ve)v\ve$$\nopagenumbers\end The solution \def\ve{^\varepsilon}$$\{u\ve,v\ve\}$$\nopagenumbers\end of such systems is known to be global. It is shown that $$\|(u^\varepsilon-u)(\, .\, ,t)\|_\infty\leq C(M_T)\varepsilon$$\nopagenumbers\end, \def\lt{\char'74}$t\leq T \lt T_b$\nopagenumbers\end where $M_T=\|u(\, .\, ,T)\|_\infty$\nopagenumbers\end and $C(M_T)$\nopagenumbers\end is given by (6). In this paper, we suggest a numerical procedure for approaching the value of the blow-up time Tb and the blow-up solution u. For this purpose, we construct a sequence $\{M_\eta\}$\nopagenumbers\end, with $\lim_{\eta\rightarrow 0}M_\eta=\infty$\nopagenumbers\end. Correspondingly, for $\varepsilon\leq1/2C(M_\eta+1)=\eta^\alpha$\nopagenumbers\end and \def\lt{\char'74}$0\lt\alpha\lt\,\!1$\nopagenumbers\end, we associate a specific sequence of times $\{T_\varepsilon\}$\nopagenumbers\end, defined by $\|u^\varepsilon(\, .\, ,T_\varepsilon)\|_\infty=M_\eta$\nopagenumbers\end. In particular, when $\varepsilon=\eta\leq\eta^\alpha$\nopagenumbers\end, the resulting sequence $\{T_\varepsilon\equiv T_\eta\}$\nopagenumbers\end, verifies, $\|(u-u^\eta)(\, .\, ,t)\|_\infty\leq{1\over2}(\eta)^{1-\alpha}$\nopagenumbers\end, \def\lt{\char'74}$0\leq t\leq T_\eta\lt T_{\rm b}$\nopagenumbers\end with $\lim_{\eta\rightarrow 0}T_\eta=T_{\rm b}$\nopagenumbers\end. The two special cases of a single-point blow-up where $f(u)=\lambda{\rm e}^u$\nopagenumbers\end and $f(u)=u^p$\nopagenumbers\end are then studied, yielding respectively sequences $\{M_\eta\}$\nopagenumbers\end of order $O(\ln|\ln(\eta)|)$\nopagenumbers\end and $O(\{|\ln(\eta)|\}^{1/p-1})$\nopagenumbers\end. The estimate $|T_\eta-T_{\rm b}|/T_{\rm b}=O(1/|\ln(\eta)|)$\nopagenumbers\end is proven to be valid in both cases. We conduct numerical simulations that confirm our theoretical results. Copyright © 2001 John Wiley & Sons, Ltd. [source] Multiresolution of quasicrystal diffraction spectraACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2009Avi Elkharrat A method for analyzing and classifying two-dimensional pure point diffraction spectra (i.e. a set of Bragg peaks) of certain self-similar structures with scaling factor , > 1, such as quasicrystals, is presented. The two-dimensional pure point diffraction spectrum , is viewed as a point set in the complex plane in which each point is assigned a positive number, its Bragg intensity. Then, by using a nested sequence of self-similar subsets called ,-lattices, we implement a multiresolution analysis of the spectrum ,. This analysis yields a partition of , simultaneously in geometry, in scale and in intensity (the `fingerprint' of the spectrum, not of the diffracting structure itself). The method is tested through numerical explorations of pure point diffraction spectra of various mathematical structures and also with the diffraction pattern of a realistic model of a quasicrystal. [source] Marginal relevance of disorder for pinning modelsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2010Giambattista Giacomin The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics. In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched models have different critical points and critical exponents) if the return probability exponent ,, a positive number that characterizes the model, is larger than ½. Weak disorder has been predicted to be irrelevant (i.e., coinciding critical points and exponents) if , < ½. Recent mathematical work has put these predictions on firm ground. In renormalization group terms, the case , = ½ is a marginal case, and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality. The question is also particularly intriguing because the case , = ½ includes the classical models of two-dimensional wetting of a rough substrate, of pinning of directed polymers on a defect line in dimension (3 + 1) or (1 + 1), and of pinning of an heteropolymer by a point potential in three-dimensional space. Here we prove disorder relevance both for the general , = ½ pinning model and for the hierarchical pinning model proposed by Derrida, Hakim, and Vannimenus, in the sense that we prove a shift of the quenched critical point with respect to the annealed one. In both cases we work with Gaussian disorder and we show that the shift is at least of order exp(,1/,4) for , small, if ,2 is the disorder variance. © 2009 Wiley Periodicals, Inc. [source] On Nonexistence of type II blowup for a supercritical nonlinear heat equationCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 11 2004Hiroshi Matano In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut = ,u + |u|p,1u either on ,N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, We prove that if ps < p < p*, then blowup is always of type I, where p* is a certain (explicitly given) positive number. More precisely, the rate of blowup in the L, norm is always the same as that for the corresponding ODE dv/dt = |v|p,1v. Because it is known that "type II" blowup (or, equivalently, "fast blowup") can occur if p > p*, the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > ps and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. We then establish useful estimates for the so-called incomplete blowup, which reveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. © 2004 Wiley Periodicals, Inc. [source] Counter-current gas-liquid wavy film flow between the vertical plates analyzed using the Navier-Stokes equationsAICHE JOURNAL, Issue 8 2010Yu. Ya. Abstract The article is devoted to a theoretical analysis of counter-current gas-liquid wavy film flow between vertical plates. We consider two-dimensional nonlinear waves on the interface over a wide variation of parameters. The main interest is to analyse the wave structure at the parameter values corresponding to the onset of flooding observed in experiments. We use the Navier-Stokes equations in their full statement to describe the liquid phase hydrodynamics. For the gas phase equations, we use two models: (1) the Navier-Stokes system and (2) the simplified Benjamin-Miles approach where the liquid phase is a small disturbance for the laminar or turbulent gas flow. With the superficial gas velocity increasing and starting from some value of the velocity, the waves demonstrate a rapid decreasing of both the minimal film thickness and the phase wave velocity. We obtain a region of the gas velocity where we have two solutions at one set of the problem parameters and where the flooding takes place. Both the phase wave velocity and the minimal film thickness are positive numbers at such values of the velocity. We calculate the flooding point dependences on the liquid Reynolds number for two different liquids. The wave regime corresponding to the flooding point demonstrates negative u- velocities in the neighbourhood of the interface near the film thickness maximum. At smaller values of the superficial gas velocity, the negative u- velocities take place in the neighbourhood of the film thickness minimum. © 2009 American Institute of Chemical Engineers AIChE J, 2010 [source] Navigator-gated three-dimensional MR angiography of the pulmonary arteries using steady-state free precession,JOURNAL OF MAGNETIC RESONANCE IMAGING, Issue 6 2005Benjamin K. Hui AB Abstract Purpose To assess the quality of a navigator-gated, free breathing, steady-state free precession (SSFP) technique in comparison to a single breathhold for pulmonary artery imaging in normal volunteers. Materials and Methods Sagittal sections of the left pulmonary arteries of 10 volunteers were obtained with a three-dimensional SSFP sequence using both a single breathhold of 30 seconds and a navigator-gated version of the same sequence. The images were compared and rated by a blinded cardiovascular radiologist for image quality, sharpness, and artifact. Results On a scale ranging from ,2 to 2, in which positive numbers denote that the navigator method was favorable compared to the single breathhold method, image quality was rated 0.7 ± 1.4, sharpness 0.6 ± 1.5, and artifact 0.1 ± 1.4. Thus, there was no statistical difference between the two methods. Conclusion The navigator-gated SSFP sequence is able to acquire images equal in quality to the breathhold sequence. This may be of clinical importance for pulmonary imaging in patients who are unable to sustain a long breathhold. J. Magn. Reson. Imaging 2005;21:831,835. © 2005 Wiley-Liss, Inc. [source] Expression of potential molecular markers in renal cell carcinoma: impact on clinicopathological outcomes in patients undergoing radical nephrectomyBJU INTERNATIONAL, Issue 7 2009Iori Sakai OBJECTIVES To evaluate the expression levels of several potential molecular markers in renal cell carcinoma (RCC), to clarify the significance of these markers as prognostic predictors in patients undergoing radical nephrectomy (RN). PATIENTS AND METHODS The study included 153 patients with clinically organ-confined RCC undergoing RN. Expression levels of 12 proteins, including Aurora-A, Bcl-2, Bcl-xL, clusterin, heat-shock protein 27 (HSP27), HSP70, HSP90, Ki-67, matrix metalloproteinase (MMP)-2 and -9, p53 and vascular endothelial growth factor, in RN specimens obtained from these 153 patients were measured by immunohistochemical staining. RESULTS Of the 12 markers, clusterin, HSP27, Ki-67, MMP-2 and -9 expression were significantly associated with several conventional prognostic factors. Univariate analysis also identified these five markers as significant predictors of disease recurrence, while mode of presentation, pathological stage, grade and microvascular invasion were also significant. Of these significant factors, Ki-67 expression, pathological stage and microvascular invasion appeared to be independently related to disease recurrence. Furthermore, there were significant differences in recurrence-free survival according to positive numbers of these three independent factors, i.e. disease recurred in four of 56 patients who were negative for risk factors (7%), 17 of 71 positive for one risk factor (24%), and 20 of 26 positive for two or three risk factors (77%). CONCLUSIONS Combined evaluation of Ki-67 expression, pathological stage and microvascular invasion would be particularly useful for further refinement of the system for predicting the outcome after RN for patients with RCC. [source] | ||||||||||||||||