Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

Schlicht envelopes of holomorphy and topology

Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC²{\mathbb{C}^{2}}, and let p: [(W)\tilde]? \mathbbC²{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i_*:p₁(W) ? p₁(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.     

Closed-Range Composition Operators on Weighted Bergman Spaces

For analytic self-maps φ of the unit disk, we show that recently given conditions that are both necessary and sufficient for the composition operator C _φ to be closed-range on the classical Bergman space \mathbbA²{\mathbb{A}^2} are actually necessary and sufficient in the general setting of the weighted Bergman spaces \mathbbA^p_a{\mathbb{A}^p_{\alpha}} , for 1 ≤ p < ∞ and weights of the form (1 - |z|²)^a{(1 - |z|^2)^{\alpha}} ; where α > −1. Along the way, we show how work done in a paper of J. Akeroyd, P. Ghatage and M. Tjani can be used to improve upon the most definitive of these conditions, and we give applications.     

On Representations of Integers in Thin Subgroups of {{\rm SL}_2({\mathbb {Z}})}

Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}Let G < SL(2, \mathbb Z){\Gamma < {\rm SL}(2, {\mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀, w₀ ? \mathbb Z²  \  {0}{v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}. We consider the set S{\mathcal {S}} of all integers occurring in áv₀g, w₀?{\langle v_{0}\gamma, w_{0}\rangle}, for g ? G{\gamma \in \Gamma} and the usual inner product on \mathbb R²{\mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{\mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set \mathfrak E(N){\mathfrak {E}(N)} of integers |n| < N which are locally admissible (n ? S   (mod  q)   for all   q 3 1){(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)} but fail to be globally represented, n ? S{n \notin \mathcal {S}}, has a power savings, |\mathfrak E(N)| << N^1-e₀{|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}} for some ${\varepsilon_{0} > 0}${\varepsilon_{0} > 0}, as N → ∞.     

On the existence of a p-adic metaplectic Tate-type $\widetilde {\gamma}$-factor

Let \mathbbF\mathbb{F} be a p-adic field, let χ be a character of \mathbbF^*\mathbb{F}^{*}, let ψ be a character of \mathbbF\mathbb{F} and let g_y^-1\gamma_{\psi}^{-1} be the normalized Weil factor associated with a character of second degree. We prove here that one can define a meromorphic function [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi ,s,\psi) via a similar functional equation to the one used for the definition of the Tate γ-factor replacing the role of the Fourier transform with an integration against y·g_y^-1\psi\cdot\gamma_{\psi}^{-1}. It turns out that γ and [(g)\tilde]\widetilde{\gamma} have similar integral representations. Furthermore, [(g)\tilde]\widetilde{\gamma} has a relation to Shahidi‘s metaplectic local coefficient which is similar to the relation γ has with (the non-metalpectic) Shahidi‘s local coefficient. Up to an exponential factor, [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi,s,\psi) is equal to the ratio \fracg(c²,2s,y)g(c,s+\frac12,y)\frac{\gamma(\chi^{2},2s,\psi)}{\gamma(\chi,s+\frac{1}{2},\psi)}.     

On the sets of branch points of mappings more general than quasiregular

It is shown that if a point x ₀ ∊ ℝ ⁿ , n ≥ 3, is an essential isolated singularity of an open discrete Q-mapping f : D → [`(\mathbb R<sup>n</sup>)] \overline {\mathbb {R}^n} , B <sub>f</sub> is the set of branch points of f in D; and a point z <sub>0</sub> ∊ [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} is an asymptotic limit of f at the point x ₀; then, for any neighborhood U containing the point x ₀; the point z ₀ ∊ [`(f( B<sub>f</sub> ?U ))] \overline {f\left( {B_f \cap U} \right)} provided that the function Q has either a finite mean oscillation at the point x <sub>0</sub> or a logarithmic singularity whose order does not exceed n − 1: Moreover, for n ≥ 2; under the indicated conditions imposed on the function Q; every point of the set [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} \ f(D) is an asymptotic limit of f at the point x ₀. For n ≥ 3, the following relation is true: [`(\mathbbR<sup>n</sup> )] \f( D ) ì [`(f B_f )] \overline {\mathbb{R}^n } \backslash f\left( D \right) \subset \overline {f\,B_f } . In addition, if ￥ ? f( D ) \infty \notin f\left( D \right) , then the set f B _f is infinite and x₀ ? [`(B_f )] x_0 \in \overline {B_f } .     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Classical operators on BLOCH spaces

Let \mathbbD \mathbb{D} ⁿ denote the unit polydisk and let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥1. We study weighted composition operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } ( \mathbbD \mathbb{D} ⁿ ), α > 1. We also study Cesàro type operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } (B _n ), α > 0. Bibliography: 15 titles.     

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbR^N{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      11. On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L 2(?)      Nikolaos D. Atreas 《Advances in Computational Mathematics》2012,36(1):21-38 Let ϕ be a function in the Wiener amalgam space W￥(L1)\emph{W}_{\infty}(L_1) with a non-vanishing property in a neighborhood of the origin for its Fourier transform [^(f)]\widehat{\phi}, t={tn}n ? \mathbb Z{\bf \tau}=\{\tau_n\}_{n\in {{\mathbb Z}}} be a sampling set on ℝ and VftV_\phi^{\bf \tau} be a closed subspace of L2(\mathbbR)L_2(\hbox{\ensuremath{\mathbb{R}}}) containing all linear combinations of τ-translates of ϕ. In this paper we prove that every function f ? Vftf\in V_\phi^{\bf \tau} is uniquely determined by and stably reconstructed from the sample set Lft(f)={ò\mathbbR f(t)[`(f(t-tn))] dt}n ? \mathbb ZL_\phi^{\bf \tau}(f)=\Big\{\int_{\hbox{\ensuremath{\mathbb{R}}}} f(t) \overline{\phi(t-\tau_n)} dt\Big\}_{n\in {{\mathbb Z}}}. As our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of ϕ we provide an estimate to the corresponding error.      12. A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}      Bart De Bruyn 《Annals of Combinatorics》2010,14(3):307-318 Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2n - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2n - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.      13. Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators      Renjin Jiang  Dachun Yang 《Integral Equations and Operator Theory》2010,67(1):123-149 Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t ?1/ω ?1(t ?1) for ${t\in (0,\infty).}Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for t ? (0,￥).{t\in (0,\infty).} In this paper, the authors introduce the generalized VMO spaces VMOr, L(\mathbb Rn){{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}} associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOr,L (\mathbb Rn))*=Bw,L*(\mathbb Rn){({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}, where L * denotes the adjoint operator of L in L2(\mathbb Rn){L^2({\mathbb R}^n)} and Bw,L*(\mathbb Rn){B_{\omega,L^\ast}({\mathbb R}^n)} the Banach completion of the Orlicz–Hardy space Hw,L*(\mathbb Rn){H_{\omega,L^\ast}({\mathbb R}^n)}. Notice that ω(t) = t p for all t ? (0,￥){t\in (0,\infty)} and p ? (0,1]{p\in (0,1]} is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1, L(\mathbb Rn))*=HL*1(\mathbb Rn){({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}, where HL*1(\mathbb Rn){H_{L^\ast}^1({\mathbb R}^n)} was the Hardy space introduced by Hofmann and Mayboroda.      14. From Ginzburg-Landau to Gross-Pitaevskii      Alberto Farina 《Monatshefte für Mathematik》2003,179(2):265-269 (w, c) ? R2, u ? Lloc3 (RN, 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￥(RN×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\}.      15. Smooth Universal Taylor Series      Ch. Kariofillis  Ch. Konstadilaki  V. Nestoridis 《Monatshefte für Mathematik》2006,87(5):249-257 Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote SN (g,z)(z) = ?Nl=0\fracg(l) (z)l ! (z-z)lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A∞(Ω), such that the following hold: i)  There exists a strictly increasing sequence μn ∈ {0, 1, 2, …}, n = 1, 2, …, such that, for every pair of compact sets Γ, Δ ⊂ [`(W)]\bar{\Omega} and every l ∈ {0, 1, 2, …} we have supz ? G supw ? D \frac?l?wl Smn ( f,z) (w)-f(l)(w) ? 0,    as n ? + ￥    and\sup_{\zeta \in \Gamma} \sup_{w \in \Delta} \frac{\partial^l}{\partial w^l} S_{\mu_ n} (\,f,\zeta) (w)-f^{(l)}(w) \rightarrow 0, \hskip 7.8pt {\rm as}\,n \rightarrow + \infty \quad {\rm and} ii)  For every compact set K ì \Bbb CK \subset {\Bbb C} with K?[`(W)] = ?K\cap \bar{\Omega} =\emptyset and Kc connected and every function h: K? \Bbb Ch: K\rightarrow {\Bbb C} continuous on K and holomorphic in K0, there exists a subsequence { m￠n }￥n=1\{ \mu^\prime _n \}^{\infty}_{n=1} of {mn }￥n=1\{\mu_n \}^{\infty}_{n=1} , such that, for every compact set L ì [`(W)]L \subset \bar{\Omega} we have supz ? L supz ? K Sm￠n ( f,z)(z)-h(z) ? 0,    as  n? + ￥.\sup_{\zeta \in L} \sup_{z\in K} S_{\mu^\prime _n} (\,f,\zeta )(z)-h(z) \rightarrow 0, \hskip 7.8pt {\rm as} \, n\rightarrow + \infty .       16. On the Quaternionic B 2-Slant Helices in the Euclidean Space E 4      ?smail G?k  O. Zeki Okuyucu  Ferda? Kahraman  H. Hilmi Hacisaliho?lu 《Advances in Applied Clifford Algebras》2011,21(4):707-719 In this paper we give a new definition of harmonic curvature functions in terms of B 2 and we define a new kind of slant helix which we call quaternionic B 2–slant helix in 4–dimensional Euclidean space E 4 by using the new harmonic curvature functions. Also we define a vector field D which we call Darboux quaternion of the real quaternionic B 2–slant helix in 4–dimensional Euclidean space E 4 and we give a new characterization such as: "a: I ì \mathbb R ? E4{``\alpha : I \subset {\mathbb R} \rightarrow E^4} is a quaternionic B 2–slant helix ${\Leftrightarrow H^\prime_2 -KH_{1} = 0"}${\Leftrightarrow H^\prime_2 -KH_{1} = 0"} where H 2, H 1 are harmonic curvature functions and K is the principal curvature function of the curve α.      17. Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth      Belgacem Rebiai  Saïd Benachour 《Journal of Evolution Equations》2010,10(3):511-527 The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C 1 in \mathbbRn{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)eahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and limh? +￥hb-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.      18. On Integral Equations Related to Weighted Toeplitz Operators      Carme Cascante  Joan Fàbrega  Daniel Pascuas 《Integral Equations and Operator Theory》2011,69(3):373-391 For weighted Toeplitz operators TNj{{\mathcal T}^N_\varphi} defined on spaces of holomorphic functions in the unit ball, we derive regularity properties of the solutions f to the equation TNj(f)=h{{\mathcal T}^N_\varphi(f)=h} in terms of the regularity of the symbol φ and the data h. As an application, we deduce that if f\not o 0{f\not\equiv0} is a function in the Hardy space H 1 such that its argument [`(f)]/f{\bar f/f} is in a Lipschitz space on the unit sphere \mathbb S{{\mathbb S}}, then f is also in the same Lipschitz space, extending a result of Dyakonov to several complex variables.      19. Toeplitz Operators on Generalized Bergman Spaces      Kamthorn Chailuek  Brian C. Hall 《Integral Equations and Operator Theory》2010,66(1):53-77 We consider the weighted Bergman spaces HL2(\mathbb Bd, ml){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dml(z) = cl(1-|z|2)l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.      20. Sharp Kolmogorov-type inequalities for norms of fractional derivatives of multivariate functions      V. F. Babenko  N.V. Parfinovych  S. A. Pichugov 《Ukrainian Mathematical Journal》2010,62(3):343-357 Let C( \mathbbRm ) C\left( {{\mathbb{R}^m}} \right) be the space of bounded and continuous functions x:\mathbbRm ? \mathbbR x:{\mathbb{R}^m} \to \mathbb{R} equipped with the norm || x ||C = || x ||C( \mathbbRm ): = sup{ | x(t) |:t ? \mathbbRm } \left\| x \right\|C = {\left\| x \right\|_{C\left( {{\mathbb{R}^m}} \right)}}: = \sup \left\{ {\left| {x(t)} \right|:t \in {\mathbb{R}^m}} \right\}

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

On a class of non-uniform average sampling expansions and partial reconstruction in subspaces of L 2(?)

Let ϕ be a function in the Wiener amalgam space W_￥(L₁)\emph{W}_{\infty}(L_1) with a non-vanishing property in a neighborhood of the origin for its Fourier transform [^(f)]\widehat{\phi}, t={t_n}_{n ? \mathbb Z}{\bf \tau}=\{\tau_n\}_{n\in {{\mathbb Z}}} be a sampling set on ℝ and V_f^tV_\phi^{\bf \tau} be a closed subspace of L₂(\mathbbR)L_2(\hbox{\ensuremath{\mathbb{R}}}) containing all linear combinations of τ-translates of ϕ. In this paper we prove that every function f ? V_f^tf\in V_\phi^{\bf \tau} is uniquely determined by and stably reconstructed from the sample set L_f^t(f)={ò_\mathbbR f(t)[`(f(t-t_n))] dt}_{n ? \mathbb Z}L_\phi^{\bf \tau}(f)=\Big\{\int_{\hbox{\ensuremath{\mathbb{R}}}} f(t) \overline{\phi(t-\tau_n)} dt\Big\}_{n\in {{\mathbb Z}}}. As our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of ϕ we provide an estimate to the corresponding error.     

A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.     

Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t ?1/ω ?1(t ?1) for ${t\in (0,\infty).}Let L be a divergence form elliptic operator with complex bounded measurable coefficients, ω a positive concave function on (0, ∞) of strictly critical lower type p ω ∈(0, 1] and ρ(t) = t −1/ω −1(t −1) for t ? (0,￥).{t\in (0,\infty).} In this paper, the authors introduce the generalized VMO spaces VMOr, L(\mathbb Rn){{\mathop{\rm VMO}_ {\rho, L}({\mathbb R}^n)}} associated with L, and characterize them via tent spaces. As applications, the authors show that (VMOr,L (\mathbb Rn))*=Bw,L*(\mathbb Rn){({\rm VMO}_{\rho,L} ({\mathbb R}^n))^\ast=B_{\omega,L^\ast}({\mathbb R}^n)}, where L * denotes the adjoint operator of L in L2(\mathbb Rn){L^2({\mathbb R}^n)} and Bw,L*(\mathbb Rn){B_{\omega,L^\ast}({\mathbb R}^n)} the Banach completion of the Orlicz–Hardy space Hw,L*(\mathbb Rn){H_{\omega,L^\ast}({\mathbb R}^n)}. Notice that ω(t) = t p for all t ? (0,￥){t\in (0,\infty)} and p ? (0,1]{p\in (0,1]} is a typical example of positive concave functions satisfying the assumptions. In particular, when p = 1, then ρ(t) ≡ 1 and (VMO1, L(\mathbb Rn))*=HL*1(\mathbb Rn){({\mathop{\rm VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)}, where HL*1(\mathbb Rn){H_{L^\ast}^1({\mathbb R}^n)} was the Hardy space introduced by Hofmann and Mayboroda.     

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\}.     

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

On the Quaternionic B 2-Slant Helices in the Euclidean Space E 4

In this paper we give a new definition of harmonic curvature functions in terms of B ₂ and we define a new kind of slant helix which we call quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ by using the new harmonic curvature functions. Also we define a vector field D which we call Darboux quaternion of the real quaternionic B ₂–slant helix in 4–dimensional Euclidean space E ⁴ and we give a new characterization such as: "a: I ì \mathbb R ? E⁴{``\alpha : I \subset {\mathbb R} \rightarrow E^4} is a quaternionic B ₂–slant helix ${\Leftrightarrow H^\prime_2 -KH_{1} = 0"}${\Leftrightarrow H^\prime_2 -KH_{1} = 0"} where H ₂, H ₁ are harmonic curvature functions and K is the principal curvature function of the curve α.     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

On Integral Equations Related to Weighted Toeplitz Operators

For weighted Toeplitz operators T^N_j{{\mathcal T}^N_\varphi} defined on spaces of holomorphic functions in the unit ball, we derive regularity properties of the solutions f to the equation T^N_j(f)=h{{\mathcal T}^N_\varphi(f)=h} in terms of the regularity of the symbol φ and the data h. As an application, we deduce that if f\not o 0{f\not\equiv0} is a function in the Hardy space H ¹ such that its argument [`(f)]/f{\bar f/f} is in a Lipschitz space on the unit sphere \mathbb S{{\mathbb S}}, then f is also in the same Lipschitz space, extending a result of Dyakonov to several complex variables.     

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

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