Samples of biological tissue are modelled as inhomogeneous fluids with density ?(X) and sound speed c(x) at point x. The samples are contained in the sphere |x| ? δ and it is assumed that ?(x) ? ?0 = 1 and c(x) ? c0 = 1 for |x| ? δ, and |γn(x)| ? 1 and |?γ?(x)| ? 1 where γ?(x) = ?(x) ? 1 and γn(x) = c?2(x) ? 1. The samples are insonified by plane pulses s(x · θ0 – t) where x = |θ0| = 1 and the scattered pulse is shown to have the form |x|?1es(|x| – t, θ, θ0) in the far field, where x = |x| θ. The response es(τ, θ, θ0) is measurable. The goal of the work is to construct the sample parameters γn and γ? from es(τ, θ, θ0) for suitable choiches of s, θ and θ0. In the limiting case of constant density: γ?(x)? 0 it is shown that Where δ represents the Dirac δ and S2 is the unit sphere |θ| = 1. Analogous formulas, based on two sets of measurements, are derived for the case of variable c(x) and ?(x). 相似文献
We study the large time behaviour of nonnegative solutions of the Cauchy problemut=Δum −up,u(x, 0)=φ(x). Specifically we study the influence of the rate of decay ofφ(x) for large |x|, and the competition between the diffusion and the absorption term. 相似文献
Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a0 + a1x + ... + akxk is a k-degree polynomial with integral coefficients such that (p, a0, a1, ..., ak) = 1, for any integer m, we study the asymptotic property of
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝN. Our attention is focused on two cases when , where m(x) = max{p1(x), p2(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized
Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods. 相似文献
Let a1,a2, . . . ,am ∈ ℝ2, 2≤f ∈ C([0,∞)), gi ∈ C([0,∞)) be such that 0≤gi(t)≤2 on [0,∞) ∀i=1, . . . ,m. For any p>1, we prove the existence and uniqueness of solutions of the equation ut=Δ(logu), u>0, in satisfying and logu(x,t)/log|x|→−f(t) as |x|→∞, logu(x,t)/log|x−ai|→−gi(t) as |x−ai|→0, uniformly on every compact subset of (0,T) for any i=1, . . . ,m under a mild assumption on u0 where We also obtain similar existence and uniqueness of solutions of the above equation in bounded smooth convex domains of ℝ2 with prescribed singularities at a finite number of points in the domain. 相似文献
For any integersa1,a2,a3,a4 andc witha1a2a3a4≢0(modp), this paper shows that there exists a solutionX=(x1,x2,x3,x4) ∈Z4 of the congruencea1x12
+a2x22
+a3x32
+a4x42
≡c(modp) such that
Research of Zheng Zhiyong is supported by NNSF Grant of China. He would also like to thank the first author and the Mathematics
Department of Kansas, State University for their hospitality and support. 相似文献
The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X1, X2, …, Xn) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ1, Θ2, …, Θn). Let θx* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θx* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θx* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X1, X2, …, Xn is considered for a location vector Θ = (Θ1, Θ2, …, Θn) as x gets large along a path, γ, in Rn. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞. 相似文献
Let \mathbb R{\mathbb R} be the set of real numbers, f : \mathbb R ? \mathbb R{f : \mathbb {R} \to \mathbb {R}}, e 3 0{\epsilon \ge 0} and d > 0. We denote by {(x1, y1), (x2, y2), (x3, y3), . . .} a countable dense subset of \mathbb R2{\mathbb {R}^2} and let
LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤn;F(x)=0, |x|⩽X}, where
. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪Xn−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields.
However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process.
Dedicated to the memory of Professor K G Ramanathan 相似文献
We consider the following singularly perturbed boundary-value problem:
on the interval 0 ≤x ≤ 1. We study the existence and uniqueness of its solutionu(x, ε) having the following properties:u(x, ε) →u0(x) asε → 0 uniformly inx ε [0, 1], whereu0(x) εC∞ [0, 1] is a solution of the degenerate equationf(x, u, u′)=0; there exists a pointx0 ε (0, 1) such thata(x0)=0,a′(x0) > 0,a(x) < 0 for 0 ≤x <x0, anda(x) > 0 forx0 <x ≤ 1, wherea(x)=f′v(x,u0(x),u′0(x)).
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 520–524, April, 2000. 相似文献
Approximation to the function |x| plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals |x| if α = 1. We construct a Newman Type Operator rn(x) and prove max |x|≤1|xαsgn x-rn(x)|~Cn1/4e-π1/2(1/2)αn. 相似文献
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in with smooth boundary, λ is a positive real number, and the continuous functions p1, p2, and q satisfy 1 < p2(x) < q(x) < p1(x) < N and for any . The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any is an eigenvalue, while any is not an eigenvalue of the above problem. 相似文献
We consider nonnegative solutions of initial-boundary value problems for parabolic equationsut=uxx, ut=(um)xxand
(m>1) forx>0,t>0 with nonlinear boundary conditions−ux=up,−(um)x=upand
forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove
that for each problem there exist positive critical valuesp0,pc(withp0<pc)such that forp∃(0,p0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>pcsmall data solutions exist globally in time while large data solutions are nonglobal. We havepc=2,pc=m+1 andpc=2m for each problem, whilep0=1,p0=1/2(m+1) andp0=2m/(m+1) respectively.
This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications
at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210. 相似文献
We show that the following nonlinear system of difference equations where parameters a,b,c,d and initial values x−1,x0,y−1,y0 are real numbers, is solvable in closed form, considerably generalizing some recent results. To do this, we use the method of transformation along with several tricks, transforming the system to some known solvable difference equations, by use of which we obtain some closed-form formulas for general solution to the system. The following five cases are considered separately: (1) c=0; (2) d=0; (3) a=0; (4) b=0; and (5) abcd≠0. 相似文献
The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in
the unit ball B1 ⊂ ℝN, N ≥ 3, ut = Δu + in B1 × (0,T), u|∂B1 = 0, in the supercritical range c > cHardy = does not have a solution for any nontrivial L1 initial data u0(x) ≥ 0 in B1 (or for a positive measure u0). More precisely, it was proved that a regular approximation of a possible solution by a sequence {un(x,t)} of classical solutions corresponding to truncated bounded potentials given by V(x) = ↦ Vn(x) = min{, n} (n ≥ 1) diverges; i.e., as n → ∞, un(x,t) → + ∞ in B1 × (0, T). Similar features of “nonexistence via approximation” for semilinear heat PDEs were inherent in related results by Brezis-Friedman
(1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and
remains true without the positivity assumption on data u0(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular
nonlinear parabolic problems as well as for higher order PDEs including, e.g., ut = , and , N > 4.
Dedicated to Professor S.I. Pohozaev on the occasion of his 70th birthday 相似文献
Letx1,...,xm be points in the solid unit sphere ofEn and letx belong to the convex hull ofx1,...,xm. Then
. This implies that all such products are bounded by (2/m)m(m −1)m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z0)| where
andp′(z0)=0. 相似文献
We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ut + Lu + a(x) |u|q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain
W ì \mathbb RN{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and
ò01s-1 (meas\nolimits {x ? W: |a(x)| £ s })q ds < ¥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}. 相似文献