共查询到20条相似文献,搜索用时 720 毫秒
1.
A. R. Wadsworth 《manuscripta mathematica》2012,139(3-4):343-389
We prove formulas for SK1(E, τ), which is the unitary SK1 for a graded division algebra E finite-dimensional and semiramified over its center T with respect to a unitary involution τ on E. Every such formula yields a corresponding formula for SK1(D, ρ) where D is a division algebra tame and semiramified over a Henselian valued field and ρ is a unitary involution on D. For example, it is shown that if ${\sf{E} \sim \sf{I}_0 \otimes_{\sf{T}_0}\sf{N}}$ where I 0 is a central simple T 0-algebra split by N 0 and N is decomposably semiramified with ${\sf{N}_0 \cong L_1\otimes_{\sf{T}_0} L_2}$ with L 1, L 2 fields each cyclic Galois over T 0, then $${\rm SK}_1(\sf{E}, \tau) \,\cong\ {\rm Br}(({L_1}\otimes_{\sf{T}_0} {L_2})/\sf{T}_0;\sf{T}_0^\tau)\big/ \left[{\rm Br}({L_1}/\sf{T}_0;\sf{T}_0^\tau)\cdot {\rm Br}({L_2}/\sf{T}_0;\sf{T}_0^\tau) \cdot \langle[\sf{I}_0]\rangle\right].$$ 相似文献
2.
Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities
Avram Sidi 《Constructive Approximation》2012,36(3):331-352
In this paper, we provide the Euler?CMaclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f??C ??(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms where $\widehat{P}(y),P_{s}(y)$ and $\widehat{Q}(y),Q_{s}(y)$ are polynomials in y. The ?? s and ?? s are distinct, complex in general, and different from ?1. They also satisfy The results we obtain in this work extend the results of a recent paper [A.?Sidi, Numer. Math. 98:371?C387, 2004], which pertain to the cases in which $\widehat{P}(y)\equiv0$ and $\widehat{Q}(y)\equiv0$ . They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x??a+ and x??b?. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let $D_{\omega}=\frac{d}{d\omega}$ , h=(b?a)/n, where n is a positive integer, and define $\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)$ . Then with $\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}$ and $\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}$ , one of these results reads where ??(z) is the Riemann Zeta function and ?? i are Stieltjes constants defined via $\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]$ , i=0,1,???. 相似文献
3.
Francesca Alessio Piero Montecchiari 《Calculus of Variations and Partial Differential Equations》2013,46(3-4):591-622
We consider a class of semilinear elliptic equations of the form $$ \label{eq:abs}-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\mathbb {R}^{3},$$ where ${a:\mathbb {R} \to \mathbb {R}}$ is a periodic, positive, even function and, in the simplest case, ${W : \mathbb {R} \to \mathbb {R}}$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W^{\prime}(q(x))=0,\ x\in\mathbb {R},\quad q(x)\to\pm1\,{\rm as}\, x\to \pm\infty,$$ we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to ${(y, z) \in \mathbb {R}^{2}}$ and such that ${\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}$ in ${\mathbb {R}^{3}}$ . 相似文献
4.
Let V be a convex subset of a normed space and let a nondecreasing function α : [0, ∞) → [0, ∞) be given. A function ${f : V \rightarrow \mathbb{R}}$ is called α-midconvex if $$f\left(\frac{x+y}{2} \right)\leq \frac{f(x)+f(y)}{2}+\alpha(\|x-y\|) \quad \,{\rm for}\, x,y\in V.$$ It is known (Tabor in Control Cybern., 38/3:656–669, 2009) that if ${f : V \rightarrow \mathbb{R}}$ is α-midconvex, locally bounded above at every point of V then $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+P_\alpha(\|x-y\|) \quad \,{\rm for}\, x, y \in V,t \in [0,1],$$ where ${P_\alpha(r):=\sum_{k=0}^\infty \frac{1}{2^k} \alpha(2{\rm dist}(2^kr, \mathbb{Z}))}$ for ${r \in \mathbb{R}}$ . We show that under some additional assumptions the above estimation cannot be improved. 相似文献
5.
For a given triangle T and a real number ρ we define Ceva’s triangle ${\mathcal{C}_{\rho}(T)}$ to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ: (1 – ρ). We identify the smallest interval ${\mathbb{M}_T \subset \mathbb{R}}$ such that the family ${\mathcal{C}_{\rho}(T), \rho \in \mathbb{M}_T}$ , contains all Ceva’s triangles up to similarity. We prove that the composition of operators ${\mathcal{C}_\rho, \rho \in \mathbb{R}}$ , acting on triangles is governed by a certain group structure on ${\mathbb{R}}$ . We use this structure to prove that two triangles have the same Brocard angle if and only if a congruent copy of one of them can be recovered by sufficiently many iterations of two operators ${\mathcal{C}_\rho}$ and ${\mathcal{C}_\xi}$ acting on the other triangle. 相似文献
6.
Bodo Dittmar 《Complex Analysis and Operator Theory》2013,7(4):1065-1080
Let G B (x, y) be the Green’s function of the unit ball B in ${\mathbb{R}^n, n \ge 3,}$ and ${\Gamma_B (x,y)=\int_BG_B(x, z)G_B(z, y)dz}$ the iterated Green’s function. The function $$E_x^y(\tau_B) = \frac{\Gamma_B(x, y)}{G_B(x, y)}$$ is the expectation of the lifetime of a Brownian motion starting at ${x \in \overline{B}}$ , killed on exiting B and conditioned to converge to and to be stopped at ${y \in \overline{B}}$ . The aim of the paper is to prove that $$\sup_{x \in \partial B,y \in B} E_x^y(\tau_B) = \sup_{x,y \in \partial B} E_x^y(\tau_B) = E_{x_0}^{-x_0}(\tau_B), x_0 \in\partial B$$ and that the maximum value of ${E_x^y(\tau_B)}$ occurs if and only if x, y are diametrically opposite points on the boundary of B. 相似文献
7.
In this paper we will study the stability properties of self-similar solutions of $1$ D cubic NLS equations with time-dependent coefficients of the form 0.1 $$\begin{aligned} \displaystyle { iu_t+u_{xx}+\frac{u}{2} \left(|u|^2-\frac{A}{t}\right)=0, \quad A\in \mathbb{R }. } \end{aligned}$$ The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation $$\begin{aligned} iv_t+ v_{xx} +\frac{v}{2t}(|v|^2-A)=0. \end{aligned}$$ As a by-product of our results we prove that Eq. (0.1) is well-posed in appropriate function spaces when the initial datum is given by $u(0,x)= z_0 \mathrm p.v \frac{1}{x}$ for some values of $z_0\in \mathbb{C }\setminus \{ 0\}$ , and $A$ is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution. 相似文献
8.
We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ 0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ 0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two. 相似文献
9.
10.
Francesca Alessio Piero Montecchiari 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(3):1317-1346
This paper concerns the existence and asymptotic characterization of saddle solutions in ${\mathbb {R}^{3}}$ for semilinear elliptic equations of the form $$-\Delta u + W'(u) = 0,\quad (x, y, z) \in {\mathbb {R}^{3}} \qquad\qquad\qquad (0.1)$$ where ${W \in \mathcal{C}^{3}(\mathbb {R})}$ is a double well symmetric potential, i.e. it satisfies W(?s) = W(s) for ${s \in \mathbb {R},W(s) > 0}$ for ${s \in (-1,1)}$ , ${W(\pm 1) = 0}$ and ${W''(\pm 1) > 0}$ . Denoted with ${\theta_{2}}$ the saddle planar solution of (0.1), we show the existence of a unique solution ${\theta_{3} \in {\mathcal{C}^{2}}(\mathbb {R}^{3})}$ which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies ${0 < \theta_{3}(x,y,z) < 1}$ for x, y, z > 0 and ${\theta_{3}(x, y, z) \to_{z \to + \infty} \theta_{2}(x, y)}$ uniformly with respect to ${(x, y) \in \mathbb {R}^{2}}$ . 相似文献
11.
Bart De Bruyn 《Designs, Codes and Cryptography》2012,64(1-2):81-91
Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ . 相似文献
12.
Istv��n Kov��csv?lgyi 《Aequationes Mathematicae》2011,81(1-2):55-64
Two methods to prove regularity properties of the linear functional equation $$f(x)=h_0(x,y)+\sum_{j=1}^n h_j(x,y)f(x+g_j(y)), $$ where ${(x,y) \in D \subset \mathbb{R}^r \times \mathbb{R}^s}$ , ${x \in \mathbb{R}^r}$ and ${y \in \mathbb{R}^s}$ , with few parameters i.e. allowing 1 ?? s < r are examined. It is proved that??under certain conditions, for some class of equations and in some sense??they are equivalent. 相似文献
13.
We study the asymptotics of the spectrum of the boundary-value problem $$ - y'' - \lambda \rho y = 0,y(0) = y(1) = 0 $$ for the case in which the weight ρ ∈ W? 2 ?1 [0, 1] is the generalized (in the sense of distributions) derivative of a self-similar function P ∈ L 2[0, 1] of zero spectral order. 相似文献
14.
Davide Ascoli Steave C. Selvaduray 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(2):263-287
In this paper wellposedness is proved for a diagonal quasilinear hyperbolic system containing integral quadratic and Lipschitz continuous terms which prevent from looking for classical solutions in Sobolev spaces. It is the hyperbolic part of the system introduced in [Selvaduray and Fujita Yashima on Atti dell’Accademia delle Scienze di Torino 2011] as a model for air motion in ${\mathbf{R}^3}$ including water phase transitions. Unknown functions are: the densities ρ of dry air, π of water vapor, σ and ν of water in the liquid and solid state, dependent also on the mass m of the droplets or ice particles. Air velocity v and temperature T are assumed to be known. Solutions (ρ, π, σ, ν) lie in ${L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega))^2 \times L^\infty(]0,\tau^*[; W^{1,\infty}(\Omega^+))^2}$ , where ${\Omega^+ = \Omega \times]0, +\infty[,\Omega \subset \mathbf{R}^3}$ is open and bounded, and τ* is sufficiently small; they depend continuously on initial data, temperature and velocities, which are tangent to ${\partial\Omega}$ ; they lie also in ${W^{1,q}(]0,\tau^*[;L^\infty(\Omega))^2 \times\,W^{1,q}(]0,\tau^*[;L^\infty(\Omega^+))^2}$ , where ${q \in [1, \infty]}$ . 相似文献
15.
Eberhard M. Schröder 《Journal of Geometry》1995,53(1-2):167-178
Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques. 相似文献
16.
Jae-Young Chung 《Aequationes Mathematicae》2014,87(3):391-400
In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\) , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\) , then the equation holds for all \({x, y \in \mathbb{R}}\) . Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) . 相似文献
17.
V. N. Potapov 《Mathematical Notes》2013,93(3-4):479-486
Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$ -quasigroup (if $\left| \mathbb{A} \right| = n$ , then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$ -quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$ . It is shown that the quasigroups defined on Σ?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1]. 相似文献
18.
Christian Le Merdy 《Journal of Evolution Equations》2012,12(4):789-800
Let (T t ) t?≥ 0 be a bounded analytic semigroup on L p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A * both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ . 相似文献
19.
Extending a result of Meyer and Reisner (Monatsh Math 125:219–227, 1998), we prove that if ${g: \mathbb{R}\to \mathbb{R}_+}$ is a function which is concave on its support, then for every m > 0 and every ${z\in\mathbb{R}}$ such that g(z) > 0, one has $$ \int\limits_{\mathbb{R}} g(x)^mdx\int\limits_{\mathbb{R}} (g^{*z}(y))^m dy\ge \frac{(m+2)^{m+2}}{(m+1)^{m+3}},$$ where for ${y\in \mathbb{R}}$ , ${g^{*z}(y)=\inf_x \frac{(1-(x-z)y)_+}{g(x)}}$ . It is shown how this inequality is related to a special case of Mahler’s conjecture (or inverse Santaló inequality) for convex bodies. The same ideas are applied to give a new (and simple) proof of the exact estimate of the functional inverse Santaló inequality in dimension 1 given in Fradelizi and Meyer (Adv Math 218:1430–1452, 2008). Namely, if ${\phi:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}}$ is a convex function such that ${0 < \int e^{-\phi} < +\infty}$ then, for every ${z\in\mathbb{R}}$ such that ${\phi(z) < +\infty}$ , one has $$ \int\limits_{\mathbb{R}}e^{-\phi}\int\limits_{\mathbb{R}} e^{-\mathcal{L}^z\phi}\ge e,$$ where ${\mathcal {L}^z\phi}$ is the Legendre transform of ${\phi}$ with respect to z. 相似文献