Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

Stability of unitary SK<Subscript>1</Subscript> of Azumaya algebras

For an Azumaya algebra A with center C of rank n ² and a unitary involution τ, we study the stability of the unitary SK₁ under reduction. We show that if R = C ^τ is a Henselian ring with maximal ideal \mathfrakm{\mathfrak{m}} and 2 and n are invertible in R then SK₁(A, t) @ SK₁(A/ \mathfrakm A, overline t){{{\rm SK}_1}(A, \tau) \cong {{\rm SK}_1}(A/ \mathfrak{m} A, overline \tau)}.     

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

On the Crank-Nicolson scheme once again

Let (t_j)_{j ? \mathbbN}{\left(\tau_j\right)_{j\in\mathbb{N}}} be a sequence of strictly positive real numbers, and let A be the generator of a bounded analytic semigroup in a Banach space X. Put A_n=?_j=1ⁿ(I+\frac12 t_jA) (I-\frac12 t_jA)^-1{A_n=\prod_{j=1}^n\left(I+\frac{1}{2} \tau_jA\right) \left(I-\frac{1}{2} \tau_jA\right)^{-1}}, and let x ? X{x\in X}. Define the sequence (x_n)_{n ? \mathbbN} ì X{\left(x_n\right)_{n\in\mathbb{N}}\subset X} by the Crank–Nicolson scheme: x _n  = A _n x. In this paper, it is proved that the Crank–Nicolson scheme is stable in the sense that sup_{n ? \mathbbN}||A_nx|| < ￥{\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert<\infty}. Some convergence results are also given.     

Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

We study the well-posedness of the fractional differential equations with infinite delay (P ₂): D^a u(t)=Au(t)+ò^t_-￥a(t-s)Au(s)ds + f(t), (0 ￡ t ￡ 2p){D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}, where A is a closed operator in a Banach space ${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}${X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)} and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P ₂) on Lebesgue-Bochner spaces L^p(\mathbbT, X){L^p(\mathbb{T}, X)} and periodic Besov spaces B _p,q^s(\mathbbT, X){{B} _{p,q}^s(\mathbb{T}, X)} .     

Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

Given an isotropic random vector X with log-concave density in Euclidean space \mathbbRⁿ{\mathbb{R}^n} , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that

Empirical Processes with a Bounded Ψ 1 Diameter

We study the empirical process ${{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}We study the empirical process sup_{f ? F}|N^-1?_i=1^N f²(X_i)-\mathbbEf²|{{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}, where F is a class of mean-zero functions on a probability space (Ω, μ), and (X_i)_{i = 1}^N{(X_{i})_{i =1}^N} are selected independently according to μ.     

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      9. The mean value of the zeta-function on <Emphasis Type="Italic">σ</Emphasis>=1      Aleksandar Ivić 《The Ramanujan Journal》2011,26(2):209-227 Let R(T): = ò1T|z(1+it)|2 dt - z(2)T+plogTR(T):= \int_{1}^{T}|\zeta (1+it)|^{2}\,\mathrm{d}t - \zeta (2)T+\pi\log T. We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for ò1TR(t) dt\int_{1}^{T}R(t)\,\mathrm{d}t and ò1TR2(t) dt\int_{1}^{T}R^{2}(t)\,\mathrm{d}t. These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.      10. Properties of Riesz fractional derivatives for Korteweg–de Vries solitons      Vladimir Varlamov 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,138(6):1017-1031 Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (?Δ) α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D α  = (−Δ) α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u 0(X) with X = x − 4t, these derivatives, u α (X) = D α u 0(X), and their Hilbert transforms, v α (X) = −HD α u 0(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ+(s, a) and ζ−(s, a), respectively. New properties are established for u α (X) and v α (X). It is proved that the functions w α (X) = u α (X) + iv α (X) with α > −1 are solutions of the differential equation -\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      11. Lebesgue property for convex risk measures on Orlicz spaces      J. Orihuela  M. Ruiz Galán 《Mathematics and Financial Economics》2012,6(1):15-35 We present a robust representation theorem for monetary convex risk measures r: X ? \mathbbR{\rho : \mathcal{X} \rightarrow \mathbb{R}} such that limnr(Xn) = r(X) whenever (Xn) almost surely converges to X,\lim_n\rho(X_n) = \rho(X)\,{\rm whenever}\,(X_n)\,{\rm almost\,surely\,converges\,to}\,X,      12. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      13. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      14. Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function      Leokadia Białas-Cież  Marta Kosek 《Potential Analysis》2011,34(3):207-221 A compact set K ì \mathbbCN{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that VK(z) 3 B(dist(z,K))b       if    dist(z,K) ￡ 1, \labelLS V_K(z)\geq B(\rm{dist}(z,K))^\beta\qquad \rm{ if}\quad \rm{ dist}(z,K)\leq 1, \label{LS}      15. Multiplicative type functional equations arising from characterization problems      Károly Lajkó  Fruzsina Mészáros 《Aequationes Mathematicae》2012,83(3):199-208 We give the general and the so-called density function solutions of equation lllfU(x)fV(y)=fX(\frac1-y1-xy ) fY (1-xy) \fracy1-xy        ( (x, y) ? (0,1)2 )\begin{array}{lll}f_{U}(x)f_{V}(y)=f_{X}\left(\frac{1-y}{1-xy} \right) f_{Y} (1-xy) \frac{y}{1-xy} \qquad \left( (x, y) \in (0,1)^2 \right)\end{array}      16. On one class of extreme extensions of a measure      M. T. Tarashchanskii 《Ukrainian Mathematical Journal》2011,62(9):1476-1486 We consider a relationship between two sets of extensions of a finite finitely additive measure μ defined on an algebra \mathfrakB \mathfrak{B} of sets to a broader algebra \mathfrakA \mathfrak{A} . These sets are the set ex S μ of all extreme extensions of the measure μ and the set H μ of all extensions defined as l(A) = [^(m)]( h(A) ),   A ? \mathfrakA \lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A} , where [^(m)] \hat{\mu } is a quotient measure on the algebra \mathfrakB / m {{\mathfrak{B}} \left/ {\mu } \right.} of the classes of μ-equivalence and h:\mathfrakA ? \mathfrakB / m h:\mathfrak{A} \to {{\mathfrak{B}} \left/ {\mu } \right.} is a homomorphism extending the canonical homomorphism \mathfrakB \mathfrak{B} to \mathfrakB / m {{\mathfrak{B}} \left/ {\mu } \right.} . We study the properties of extensions from H μ and present necessary and sufficient conditions for the existence of these extensions, as well as the conditions under which the sets ex S μ and H μ coincide.      17. Blow up of subcritical quantities at the first singular time of the mean curvature flow      Nam Q. Le 《Geometriae Dedicata》2011,151(1):361-371 Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.      18. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      19. A New Recursion in the Theory of Macdonald Polynomials      A. M. Garsia  J. Haglund 《Annals of Combinatorics》2012,16(1):77-106 The bigraded Frobenius characteristic of the Garsia-Haiman module M μ is known [7, 10] to be given by the modified Macdonald polynomial [(H)\tilde]m[X; q, t]{\tilde{H}_{\mu}[X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?p1 [(H)\tilde]m[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]} is the bigraded Frobenius characteristic of the restriction of M μ from S n to S n-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c μ v occurring in the expansion ?p1 [(H)\tilde]m[X; q, t] = ?v ? mcmv [(H)\tilde]v[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}. In particular, it follows from this formula that the bigraded Hilbert series F μ (q, t) of M μ may be calculated from the recursion Fm (q, t) = ?v ? mcmv Fv (q, t){F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fm(q, t) ? N[q, t]{F\mu (q, t) \in \mathbf{N}[q, t]}. This difficulty arises from the fact that the c μ v have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F μ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].      20. Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients      Oliver Knill  John Lesieutre 《Complex Analysis and Operator Theory》2012,6(1):237-255 We consider Dirichlet series zg,a(s)=?n=1￥ g(na) e-ln s{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?n=1￥ g(na) zn{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series zg,a(s) = ?n=1￥ g(n a)/ns{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ 0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ 0 satisfies σ 0 ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ g,α (s) has an analytic continuation to the entire complex plane.      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

The mean value of the zeta-function on <Emphasis Type="Italic">σ</Emphasis>=1

Let R(T): = ò₁^T|z(1+it)|² dt - z(2)T+plogTR(T):= \int_{1}^{T}|\zeta (1+it)|^{2}\,\mathrm{d}t - \zeta (2)T+\pi\log T. We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for ò₁^TR(t) dt\int_{1}^{T}R(t)\,\mathrm{d}t and ò₁^TR²(t) dt\int_{1}^{T}R^{2}(t)\,\mathrm{d}t. These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.     

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      11. Lebesgue property for convex risk measures on Orlicz spaces      J. Orihuela  M. Ruiz Galán 《Mathematics and Financial Economics》2012,6(1):15-35 We present a robust representation theorem for monetary convex risk measures r: X ? \mathbbR{\rho : \mathcal{X} \rightarrow \mathbb{R}} such that limnr(Xn) = r(X) whenever (Xn) almost surely converges to X,\lim_n\rho(X_n) = \rho(X)\,{\rm whenever}\,(X_n)\,{\rm almost\,surely\,converges\,to}\,X,      12. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      13. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      14. Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function      Leokadia Białas-Cież  Marta Kosek 《Potential Analysis》2011,34(3):207-221 A compact set K ì \mathbbCN{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that VK(z) 3 B(dist(z,K))b       if    dist(z,K) ￡ 1, \labelLS V_K(z)\geq B(\rm{dist}(z,K))^\beta\qquad \rm{ if}\quad \rm{ dist}(z,K)\leq 1, \label{LS}      15. Multiplicative type functional equations arising from characterization problems      Károly Lajkó  Fruzsina Mészáros 《Aequationes Mathematicae》2012,83(3):199-208 We give the general and the so-called density function solutions of equation lllfU(x)fV(y)=fX(\frac1-y1-xy ) fY (1-xy) \fracy1-xy        ( (x, y) ? (0,1)2 )\begin{array}{lll}f_{U}(x)f_{V}(y)=f_{X}\left(\frac{1-y}{1-xy} \right) f_{Y} (1-xy) \frac{y}{1-xy} \qquad \left( (x, y) \in (0,1)^2 \right)\end{array}      16. On one class of extreme extensions of a measure      M. T. Tarashchanskii 《Ukrainian Mathematical Journal》2011,62(9):1476-1486 We consider a relationship between two sets of extensions of a finite finitely additive measure μ defined on an algebra \mathfrakB \mathfrak{B} of sets to a broader algebra \mathfrakA \mathfrak{A} . These sets are the set ex S μ of all extreme extensions of the measure μ and the set H μ of all extensions defined as l(A) = [^(m)]( h(A) ),   A ? \mathfrakA \lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A} , where [^(m)] \hat{\mu } is a quotient measure on the algebra \mathfrakB / m {{\mathfrak{B}} \left/ {\mu } \right.} of the classes of μ-equivalence and h:\mathfrakA ? \mathfrakB / m h:\mathfrak{A} \to {{\mathfrak{B}} \left/ {\mu } \right.} is a homomorphism extending the canonical homomorphism \mathfrakB \mathfrak{B} to \mathfrakB / m {{\mathfrak{B}} \left/ {\mu } \right.} . We study the properties of extensions from H μ and present necessary and sufficient conditions for the existence of these extensions, as well as the conditions under which the sets ex S μ and H μ coincide.      17. Blow up of subcritical quantities at the first singular time of the mean curvature flow      Nam Q. Le 《Geometriae Dedicata》2011,151(1):361-371 Consider a family of smooth immersions F(·,t) : Mn? \mathbbRn+1{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò0tòMs\frac|A|n + 2 log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.      18. Regularity of weak solutions to the Navier–Stokes equations in exterior domains      Reinhard Farwig  Christian Komo 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(3):303-321 Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR3{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u 0, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u 0 and f or on the solution u itself such as Serrin’s condition || u ||Ls(0,T; Lq(W)) < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||Ls￠(t-d,t; Lq(W)){\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||22{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.      19. A New Recursion in the Theory of Macdonald Polynomials      A. M. Garsia  J. Haglund 《Annals of Combinatorics》2012,16(1):77-106 The bigraded Frobenius characteristic of the Garsia-Haiman module M μ is known [7, 10] to be given by the modified Macdonald polynomial [(H)\tilde]m[X; q, t]{\tilde{H}_{\mu}[X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?p1 [(H)\tilde]m[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]} is the bigraded Frobenius characteristic of the restriction of M μ from S n to S n-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c μ v occurring in the expansion ?p1 [(H)\tilde]m[X; q, t] = ?v ? mcmv [(H)\tilde]v[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}. In particular, it follows from this formula that the bigraded Hilbert series F μ (q, t) of M μ may be calculated from the recursion Fm (q, t) = ?v ? mcmv Fv (q, t){F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fm(q, t) ? N[q, t]{F\mu (q, t) \in \mathbf{N}[q, t]}. This difficulty arises from the fact that the c μ v have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F μ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].      20. Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients      Oliver Knill  John Lesieutre 《Complex Analysis and Operator Theory》2012,6(1):237-255 We consider Dirichlet series zg,a(s)=?n=1￥ g(na) e-ln s{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?n=1￥ g(na) zn{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series zg,a(s) = ?n=1￥ g(n a)/ns{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ 0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ 0 satisfies σ 0 ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ g,α (s) has an analytic continuation to the entire complex plane.      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

Lebesgue property for convex risk measures on Orlicz spaces

We present a robust representation theorem for monetary convex risk measures r: X ? \mathbbR{\rho : \mathcal{X} \rightarrow \mathbb{R}} such that

Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

The Hermite Property of a Causal Wiener Algebra Used in Control Theory

Let \mathbbC₊ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra

Iterated Function Systems and Łojasiewicz–Siciak Condition of Green’s Function

A compact set K ì \mathbbC^N{K \subset \mathbb{C}}^{N} satisfies (ŁS) if it is polynomially convex and there exist constants B,β > 0 such that

Multiplicative type functional equations arising from characterization problems

We give the general and the so-called density function solutions of equation

On one class of extreme extensions of a measure

We consider a relationship between two sets of extensions of a finite finitely additive measure μ defined on an algebra \mathfrakB \mathfrak{B} of sets to a broader algebra \mathfrakA \mathfrak{A} . These sets are the set ex S _μ of all extreme extensions of the measure μ and the set H _μ of all extensions defined as l(A) = [^(m)]( h(A) ),   A ? \mathfrakA \lambda (A) = \hat{\mu }\left( {h(A)} \right),\,\,\,A \in \mathfrak{A} , where [^(m)] \hat{\mu } is a quotient measure on the algebra \mathfrakB

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions F(·,t) : Mⁿ? \mathbbRⁿ⁺¹{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò₀^tò_Ms\frac|A|^{n + 2} log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

A New Recursion in the Theory of Macdonald Polynomials

The bigraded Frobenius characteristic of the Garsia-Haiman module M _μ is known [7, 10] to be given by the modified Macdonald polynomial [(H)\tilde]_m[X; q, t]{\tilde{H}_{\mu}[X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?_p1 [(H)\tilde]_m[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]} is the bigraded Frobenius characteristic of the restriction of M _μ from S _n to S _n-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c _{μ v} occurring in the expansion ?_p1 [(H)\tilde]_m[X; q, t] = ?_{v ? m}c_mv [(H)\tilde]_v[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}. In particular, it follows from this formula that the bigraded Hilbert series F μ (q, t) of M _μ may be calculated from the recursion F_m (q, t) = ?_{v ? m}c_mv F_v (q, t){F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fm(q, t) ? N[q, t]{F\mu (q, t) \in \mathbf{N}[q, t]}. This difficulty arises from the fact that the c _{μ v} have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F _μ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].     

Analytic Continuation of Dirichlet Series with Almost Periodic Coefficients

We consider Dirichlet series z_g,a(s)=?_n=1^￥ g(na) e^{-l_n s}{\zeta_{g,\alpha}(s)=\sum_{n=1}^\infty g(n\alpha) e^{-\lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ _n  = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ?_n=1^￥ g(na) zⁿ{\sum_{n=1}^{\infty} g(n\alpha) z^n}. We prove that a Dirichlet series z_g,a(s) = ?_n=1^￥ g(n a)/n^s{\zeta_{g,\alpha}(s) = \sum_{n=1}^{\infty} g(n \alpha)/n^s} has an abscissa of convergence σ ₀ = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ ₀ satisfies σ ₀ ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ _g,α (s) has an analytic continuation to the entire complex plane.