Measure of central symmetry for fields of convex figures and three-dimensional convex bodies

Let g₂³:E₂( \mathbbR³ ) ? G₂( \mathbbR³ ) \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) be the tautological vector bundle over the Grassmann manifold of 2-planes in \mathbbR³ {\mathbb{R}^3} , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of \mathbbR³ {\mathbb{R}^3} . A field of convex figures is given in γ₂³ if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ₂³ contains a figure K containing a centrally symmetric convex figure of area ( 4 + 16?2 ) \left( {4 + 16\sqrt {2} } \right) S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area ( 12?2 - 8 ) \left( {12\sqrt {2} - 8} \right) S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume ( 36?2 - 24 ) \left( {36\sqrt {2} - 24} \right) V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.     

Quantization of Lie bialgebras and shuffle algebras of Lie algebras

To any field \Bbb K \Bbb K of characteristic zero, we associate a set (\mathbbK) (\mathbb{K}) and a group G₀(\Bbb K) {\cal G}_0(\Bbb K) . Elements of (\mathbbK) (\mathbb{K}) are equivalence classes of families of Lie polynomials subject to associativity relations. Elements of G₀(\Bbb K) {\cal G}_0(\Bbb K) are universal automorphisms of the adjoint representations of Lie bialgebras over \Bbb K \Bbb K . We construct a bijection between (\mathbbK)×G₀(\Bbb K) (\mathbb{K})\times{\cal G}_0(\Bbb K) and the set of quantization functors of Lie bialgebras over \Bbb K \Bbb K . This construction involves the following steps.? 1) To each element v \varpi of (\mathbbK) (\mathbb{K}) , we associate a functor \frak a?\operatornameSh^v(\frak a) \frak a\mapsto\operatorname{Sh}^\varpi(\frak a) from the category of Lie algebras to that of Hopf algebras; \operatornameSh^v(\frak a) \operatorname{Sh}^\varpi(\frak a) contains U\frak a U\frak a .? 2) When \frak a \frak a and \frak b \frak b are Lie algebras, and r_{\frak a\frak b} ? \frak a?\frak b r_{\frak a\frak b} \in\frak a\otimes\frak b , we construct an element ?^v (r_{\frak a\frak b}) {\cal R}^{\varpi} (r_{\frak a\frak b}) of \operatornameSh^v(\frak a)?\operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak a)\otimes\operatorname{Sh}^\varpi(\frak b) satisfying quasitriangularity identities; in particular, ?^v(r_{\frak a\frak b}) {\cal R}^\varpi(r_{\frak a\frak b}) defines a Hopf algebra morphism from \operatornameSh^v(\frak a)^* \operatorname{Sh}^\varpi(\frak a)^* to \operatornameSh^v(\frak b) \operatorname{Sh}^\varpi(\frak b) .? 3) When \frak a = \frak b \frak a = \frak b and r_\frak a ? \frak a?\frak a r_\frak a\in\frak a\otimes\frak a is a solution of CYBE, we construct a series r^v(r_\frak a) \rho^\varpi(r_\frak a) such that ?^v(r^v(r_\frak a)) {\cal R}^\varpi(\rho^\varpi(r_\frak a)) is a solution of QYBE. The expression of r^v(r_\frak a) \rho^\varpi(r_\frak a) in terms of r_\frak a r_\frak a involves Lie polynomials, and we show that this expression is unique at a universal level. This step relies on vanishing statements for cohomologies arising from universal algebras for the solutions of CYBE.? 4) We define the quantization of a Lie bialgebra \frak g \frak g as the image of the morphism defined by ?^v(r^v(r)) {\cal R}^\varpi(\rho^\varpi(r)) , where r ? \mathfrakg ?\mathfrakg^* r \in \mathfrak{g} \otimes \mathfrak{g}^* .<\P>     

On the derivatives of X-ray functions

We show that for every n \geqq 4, 0 \leqq k \leqq n - 3, p ? (0, 3] n \geqq 4, 0 \leqq k \leqq n - 3, p \in (0, 3] and every origin-symmetric convex body K in \mathbbRⁿ \mathbb{R}^n , the function ||x ||^-k₂ ||x ||^-n+k+p_K \parallel x \parallel^{-k}_{2} \parallel x \parallel^{-n+k+p}_{K} represents a positive definite distribution on \mathbbRⁿ \mathbb{R}^n , where ||·||₂ \parallel \cdot \parallel_{2} is the Euclidean norm and ||·||_K \parallel \cdot \parallel_{K} is the Minkowski functional of K. We apply this fact to prove a result of Busemann-Petty type that the inequalities for the derivatives of order (n - 4) at zero of X-ray functions of two convex bodies imply the inequalities for the volume of average m-dimensional sections of these bodies for all 3 \leqq m \leqq n 3 \leqq m \leqq n . We also prove a sharp lower estimate for the maximal derivative of X-ray functions of the order (n - 4) at zero.     

Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions

Let C( \mathbbR^m ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbR^m ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm

Coincidence theorems and its applications to equilibrium problems

Given two maps h : X ×K ? \mathbbR{h : X \times K \rightarrow \mathbb{R}} and g : X → K such that, for all x ? X, h(x, g(x)) = 0{x \in X, h(x, g(x)) = 0} , we consider the equilibrium problem of finding [(x)\tilde] ? X{\tilde{x} \in X} such that h([(x)\tilde], g(x)) 3 0{h(\tilde{x}, g(x)) \geq 0} for every x ? X{x \in X} . This question is related to a coincidence problem.     

Two-dimensional variational problems with a wide range of anisotropy

We consider local minimizers u:\mathbbR² é W? \mathbbR^M u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^M} of the variational integral

From Ginzburg-Landau to Gross-Pitaevskii

(w, c) ? R², u ? L_loc³ (R^N, C)\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}(\omega, c)\in {\Op R}^2, {\upsilon} \in L_{\rm loc}^3 ({\Op R}^N, {\bf C}) and x||j||_{L^￥(R^N×R)}² ￡ max{0, 1-w+[(c²)/4]}.\font\Opr=msbm10 at 8pt \def\Op#1{\hbox{\Opr{#1}}}\Vert\varphi\Vert_{L^\infty({\Op R}^N\times{\Op R})}^2 \le \max\bigg\{0, 1-\omega+{c^2\over 4}\bigg\}.     

The behaviour of microstructures with small shears of the austenite-martensite interface in martensitic phase transformations

Let W ì \BbbR²\Omega \subset \Bbb{R}^2 denote a bounded domain whose boundary ?W\partial \Omega is Lipschitz and contains a segment G₀\Gamma_0 representing the austenite-twinned martensite interface. We prove inf_{u ? W(W)} ò_W j(?u(x,y))dxdy=0\displaystyle{\inf_{{u\in \cal W}(\Omega)} \int_\Omega \varphi(\nabla u(x,y))dxdy=0}     

On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

For open discrete mappings f:D\{ b } ? \mathbbR³ f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR³ D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR³ b \in {\mathbb{R}^3} , we prove the following statement: Let a point y ₀ belong to [`(\mathbbR³)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K _I (x, f) and outer dilatation K _O (x, f) of the mapping f at the point x satisfy certain conditions. Let B _f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y ₀, the set V ∩ f(B _f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRⁿ g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} such that A ⊂ U.     

New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]^s, t_p, q(\mathbbRⁿ) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]^-s, t_p￠, q￠(\mathbbRⁿ){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .     

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      12. A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary      Carmen Cortazar  Manuel Elgueta  Julio D. Rossi 《Annales Henri Poincare》2005,6(2):269-281 Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem $ u_t (x,t) = \int_\mathbb{R} {J\left( {\frac{{x - y}} {{u(y,t)}}} \right)dy - u(x,t){\text{ in }}\mathbb{R} \times [0,\infty )} $ u_t (x,t) = \int_\mathbb{R} {J\left( {\frac{{x - y}} {{u(y,t)}}} \right)dy - u(x,t){\text{ in }}\mathbb{R} \times [0,\infty )}       13. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      14. Multivariate Böttcher equation for polynomials with non-negative coefficients      O. D. Jones 《Aequationes Mathematicae》2002,63(3):251-265 Summary. We give conditions for the multivariate Böttcher equation b(f (x)) = b(x)l \beta(f (x)) = \beta(x)^{\lambda} to have a solution, in the case where f : \mathbbRd ? \mathbbRd f : \mathbb{R}^d \rightarrow \mathbb{R}^d is a polynomial with non-negative coefficients. The solution is constructed from the limit of the functional iterates -l-n logfn(x) -\lambda^{-n} \log f^{n}(x) .      15. Stability of a conditional Cauchy equation      Jae-Young Chung 《Aequationes Mathematicae》2012,83(3):313-320 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 {(x 1, y 1), (x 2, y 2), (x 3, y 3), . . .} a countable dense subset of \mathbb R2{\mathbb {R}^2} and let $U_d:=\bigcup\nolimits_{j=1}^{\infty} \{(x, y)\in \mathbb {R}^2:\,|x|+|y| > d,\, |x-x_j| < 1,\, |y-y_j| < 2^{-j}\}.$U_d:=\bigcup\nolimits_{j=1}^{\infty} \{(x, y)\in \mathbb {R}^2:\,|x|+|y| > d,\, |x-x_j| < 1,\, |y-y_j| < 2^{-j}\}.      16. On uniform approximation by harmonic and almost harmonic vector fields      M.?B.?DubashinskiyEmail author 《Journal of Mathematical Sciences》2012,182(5):617-629 Three- dimensional analogs of rational uniform approximation in \mathbbC \mathbb{C} are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ \mathbbR3 \mathbb{R}^3 is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ \mathbbC \mathbb{C} (when f is assumed to be in C 1 with [`(?)] f\bar \partial {\kern 1pt}f small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.      17. Linear independence of time-frequency shifts under a generalized Schrödinger representation      G. Kutyniok 《Archiv der Mathematik》2002,78(2):135-144 Let r\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e2 pi y t f(t + x) = (r\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L2(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and rG \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {rG (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L2(G), f 1 0 f \in L^2(G), f \neq 0 .      18. Idempotent multipliers for LP (\Bbb R)(\Bbb R)      G. Mockenhaupt  W. J. Ricker 《Archiv der Mathematik》2000,74(1):61-65 The algebra Bp(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, Bp(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? Bp(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? Bp(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in Bq(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.      19. Homomorphisms with respect to a function      Karim Boulabiar  Foued Gdara 《Monatshefte für Mathematik》2012,24(1):357-360 Let X be a realcompact space and H:C(X)?\mathbbR{H:C(X)\rightarrow\mathbb{R}} be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is j ? C(\mathbbR){\varphi\in C(\mathbb{R})} with ${\varphi(r)>\varphi(0)}${\varphi(r)>\varphi(0)} for all r ? \mathbbR-{0}{r\in\mathbb{R}-\{0\}} for which H°j = j°H{H\circ\varphi=\varphi\circ H} . This extends (and unifies) classical results by Hewitt and Shirota.      20. Numerical verification of condition for approximately midconvex functions      P. Spurek  Ja. Tabor 《Aequationes Mathematicae》2012,83(3):223-237 Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||)    for  x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      14. Multivariate Böttcher equation for polynomials with non-negative coefficients      O. D. Jones 《Aequationes Mathematicae》2002,63(3):251-265 Summary. We give conditions for the multivariate Böttcher equation b(f (x)) = b(x)l \beta(f (x)) = \beta(x)^{\lambda} to have a solution, in the case where f : \mathbbRd ? \mathbbRd f : \mathbb{R}^d \rightarrow \mathbb{R}^d is a polynomial with non-negative coefficients. The solution is constructed from the limit of the functional iterates -l-n logfn(x) -\lambda^{-n} \log f^{n}(x) .      15. Stability of a conditional Cauchy equation      Jae-Young Chung 《Aequationes Mathematicae》2012,83(3):313-320 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 {(x 1, y 1), (x 2, y 2), (x 3, y 3), . . .} a countable dense subset of \mathbb R2{\mathbb {R}^2} and let $U_d:=\bigcup\nolimits_{j=1}^{\infty} \{(x, y)\in \mathbb {R}^2:\,|x|+|y| > d,\, |x-x_j| < 1,\, |y-y_j| < 2^{-j}\}.$U_d:=\bigcup\nolimits_{j=1}^{\infty} \{(x, y)\in \mathbb {R}^2:\,|x|+|y| > d,\, |x-x_j| < 1,\, |y-y_j| < 2^{-j}\}.      16. On uniform approximation by harmonic and almost harmonic vector fields      M.?B.?DubashinskiyEmail author 《Journal of Mathematical Sciences》2012,182(5):617-629 Three- dimensional analogs of rational uniform approximation in \mathbbC \mathbb{C} are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ \mathbbR3 \mathbb{R}^3 is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ \mathbbC \mathbb{C} (when f is assumed to be in C 1 with [`(?)] f\bar \partial {\kern 1pt}f small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.      17. Linear independence of time-frequency shifts under a generalized Schrödinger representation      G. Kutyniok 《Archiv der Mathematik》2002,78(2):135-144 Let r\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e2 pi y t f(t + x) = (r\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L2(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and rG \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {rG (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L2(G), f 1 0 f \in L^2(G), f \neq 0 .      18. Idempotent multipliers for LP (\Bbb R)(\Bbb R)      G. Mockenhaupt  W. J. Ricker 《Archiv der Mathematik》2000,74(1):61-65 The algebra Bp(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, Bp(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? Bp(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? Bp(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in Bq(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.      19. Homomorphisms with respect to a function      Karim Boulabiar  Foued Gdara 《Monatshefte für Mathematik》2012,24(1):357-360 Let X be a realcompact space and H:C(X)?\mathbbR{H:C(X)\rightarrow\mathbb{R}} be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is j ? C(\mathbbR){\varphi\in C(\mathbb{R})} with ${\varphi(r)>\varphi(0)}${\varphi(r)>\varphi(0)} for all r ? \mathbbR-{0}{r\in\mathbb{R}-\{0\}} for which H°j = j°H{H\circ\varphi=\varphi\circ H} . This extends (and unifies) classical results by Hewitt and Shirota.      20. Numerical verification of condition for approximately midconvex functions      P. Spurek  Ja. Tabor 《Aequationes Mathematicae》2012,83(3):223-237 Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR+ ? \mathbbR+{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if f (\fracx + y2)-\fracf(x) + f(y)2 ￡ a(||x - y||)    for  x, y ? V.f \left(\frac{x + y}{2}\right)-\frac{f(x) + f(y)}{2}\leq \alpha(\|x - y\|)\quad {\rm for} \, x, y \in V.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Multivariate Böttcher equation for polynomials with non-negative coefficients

Summary. We give conditions for the multivariate Böttcher equation b(f (x)) = b(x)^l \beta(f (x)) = \beta(x)^{\lambda} to have a solution, in the case where f : \mathbbR^d ? \mathbbR^d f : \mathbb{R}^d \rightarrow \mathbb{R}^d is a polynomial with non-negative coefficients. The solution is constructed from the limit of the functional iterates -l^-n logfⁿ(x) -\lambda^{-n} \log f^{n}(x) .     

Stability of a conditional Cauchy equation

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 {(x ₁, y ₁), (x ₂, y ₂), (x ₃, y ₃), . . .} a countable dense subset of \mathbb R²{\mathbb {R}^2} and let

On uniform approximation by harmonic and almost harmonic vector fields

Three- dimensional analogs of rational uniform approximation in \mathbbC \mathbb{C} are considered. These analogs are related to approximation properties of harmonic (i. e., curl-free and solenoidal) vector fields. The usual uniform approximation by fields harmonic near a given compact set K ⊂ \mathbbR³ \mathbb{R}^3 is compared with the uniform approximation by smooth fields whose curls and divergences tends to zero uniformly on K. A similar two-dimensional modification of the uniform approximation by functions f that are complex analytic near a given compact set K ⊂ \mathbbC \mathbb{C} (when f is assumed to be in C ¹ with [`(?)] f\bar \partial {\kern 1pt}f small on K) results in a problem equivalent to the original one. In the three-dimensional settings, the two problems (of harmonic and of almost harmonic approximation) are different. The first problem is nonlocal whereas the second one is local (i. e., an analog of the Bishop theorem on the locality of R(K) is still valid for almost harmonic approximation). Almost curl-free approximation is also considered. Bibliography: 7 titles.     

Linear independence of time-frequency shifts under a generalized Schrödinger representation

Let r_\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of \mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions {t ? e^{2 pi y t} f(t + x) = (r_\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all f ? L²(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and r_G \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {r_G (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L²(G), f 1 0 f \in L^2(G), f \neq 0 .     

Idempotent multipliers for LP (\Bbb R)(\Bbb R)

The algebra B_p(\Bbb R){\cal B}_p({\Bbb R}), p ? (1,￥)\{2}p\in (1,\infty )\setminus \{2\}, consisting of all measurable sets in \Bbb R{\Bbb R} whose characteristic function is a Fourier p-multiplier, forms an algebra of sets containing many interesting and non-trivial elements (e.g. all intervals and their finite unions, certain periodic sets, arbitrary countable unions of dyadic intervals, etc.). However, B_p(\Bbb R){\cal B}_p({\Bbb R}) fails to be a s\sigma -algebra. It has been shown by V. Lebedev and A. Olevskii [4] that if E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}), then E must coincide a.e. with an open set, a remarkable topological constraint on E. In this note we show if $2 < p < \infty $2 < p < \infty , then there exists E ? B_p(\Bbb R)E\in {\cal B}_p({\Bbb R}) which is not in B_q(\Bbb R){\cal B}_q({\Bbb R}) for any q > pq>p.     

Homomorphisms with respect to a function

Let X be a realcompact space and H:C(X)?\mathbbR{H:C(X)\rightarrow\mathbb{R}} be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is j ? C(\mathbbR){\varphi\in C(\mathbb{R})} with ${\varphi(r)>\varphi(0)}${\varphi(r)>\varphi(0)} for all r ? \mathbbR-{0}{r\in\mathbb{R}-\{0\}} for which H°j = j°H{H\circ\varphi=\varphi\circ H} . This extends (and unifies) classical results by Hewitt and Shirota.     

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if