Properties of traveling waves for integrodifference equations with nonmonotone growth functions

In this paper, we will establish some new properties of traveling waves for integrodifference equations with the nonmonotone growth functions. More precisely, for c ≥ c _*, we show that either \({\lim\limits_{\xi\rightarrow+\infty} \phi(\xi)=u*}\) or \({0 < \liminf\limits_{\xi \rightarrow + \infty} \phi(\xi) < u* < \limsup \limits_{\xi\rightarrow+\infty}\phi(\xi)\leq b,}\) that is, the wave converges to the positive equilibrium or oscillates about it at +∞. Sufficient conditions can assure that both results will arise. We can also obtain that any traveling wave with wave speed c > c* possesses exponential decay at ?∞. These results can be well applied to three types of growth functions arising from population biology. By choosing suitable parameter numbers, we can obtain the existence of oscillating waves. Our analytic results are consistent with some numerical simulations in Kot (J Math Biol 30:413–436, 1992), Li et al. (J Math Biol 58:323–338, 2009) and complement some known ones.     

A Remark on the Growth of the Denominators of Convergents

For log\frac1+?52 ￡ l_* ￡ l^* < ￥{\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty , let E(λ_*, λ^*) be the set {x ? [0,1): liminf_{n ? ￥}\fraclogq_n(x)n=l_*, limsup_{n ? ￥}\fraclogq_n(x)n=l^*}. \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}. It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that dimE(l_*, l^*) 3 \fracl_* -log\frac1+?522l^*\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}}     

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.     

A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbR^m ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β ₁ + ··· + β _n . We also provide further generalization to functions of the form (x,y₁, . . . , y_n)? g(x)^1+af₁(y₁)^-b₁ ···f_n(y_n)^-b_n{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f _k concave, positively homogeneous and nonnegative on their domains.     

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:

The noncommutative Choquet boundary II: Hyperrigidity

A (finite or countably infinite) set G of generators of an abstract C*-algebra A is called hyperrigid if for every faithful representation of A on a Hilbert space A ⊆ B(H) and every sequence of unital completely positive linear maps ϕ ₁, ϕ ₂,... from B(H) to itself,

Book Reviews

For , let E(λ_*, λ^*) be the set It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that where dim denotes the Hausdorff dimension.     

Degenerate elliptic operators in one dimension

Let H be the symmetric second-order differential operator on L ₂(R) with domain ${C_c^\infty({\bf R})}Let H be the symmetric second-order differential operator on L ₂(R) with domain C_c^￥(R){C_c^\infty({\bf R})} and action Hj = -(c j^￠)^￠{H\varphi=-(c\,\varphi^{\prime})^{\prime}} where c ? W^1,2_loc(R){ c\in W^{1,2}_{\rm loc}({\bf R})} is a real function that is strictly positive on R\{0}{{\bf R}\backslash\{0\}} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if n = n₊ún_-{\nu=\nu_+\vee\nu_-} where n_±(x)=±ò^±1_±x c^-1{\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}} then H has a unique self-adjoint extension if and only if n ? L₂(0,1){\nu\not\in L_2(0,1)} and a unique submarkovian extension if and only if n ? L_￥(0,1){\nu\not\in L_\infty(0,1)}. In both cases, the corresponding semigroup leaves L ₂(0,∞) and L ₂(−∞,0) invariant. In addition, we prove that for a general non-negative c ? W^1,￥_loc(R){ c\in W^{1,\infty}_{\rm loc}({\bf R})} the corresponding operator H has a unique submarkovian extension.     

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      10. Compact Operators that Commute with a Contraction      K. Kellay  M. Zarrabi 《Integral Equations and Operator Theory》2009,65(4):543-550 Let T be a C0–contraction on a separable Hilbert space. We assume that IH − T*T is compact. For a function f holomorphic in the unit disk \mathbbD{\mathbb{D}} and continuous on [`(\mathbbD)]\overline{{\mathbb{D}}}, we show that f(T) is compact if and only if f vanishes on s(T)?\mathbbT\sigma(T)\cap{\mathbb{T}}, where σ(T) is the spectrum of T and \mathbbT{\mathbb{T}} the unit circle. If f is just a bounded holomorphic function on \mathbbD{\mathbb{D}}, we prove that f(T) is compact if and only if limn? ￥||Tnf(T)|| = 0\lim\limits_{n\rightarrow \infty}\|T^{n}f(T)\| = 0.      11. Book Reviews      Jun Wu 《Monatshefte für Mathematik》2006,149(3):259-264 For , let E(λ*, λ*) be the set It has been proved in [1] and [3] that E(λ*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that where dim denotes the Hausdorff dimension.      12. Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}      Bernhard Burgstaller 《Monatshefte für Mathematik》2009,265(4):1-11 There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x 1, x 2, ... we have limn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O2{\mathcal {O}_2} such that for an independent sequence x 1, x 2, ... of μ-distributed random elements x i we have lim infn ? ￥\mathbb P(C*(x1,?,xn) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] d ) is d.      13. On the existence of a p-adic metaplectic Tate-type $\widetilde {\gamma}$-factor      Dani Szpruch 《The Ramanujan Journal》2011,26(1):45-53 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 gy-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·gy-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)}.      14. On an asymptotically autonomous system with Tikhonov type regularizing term      Mohamed Ali Jendoubi  Ramzi May 《Archiv der Mathematik》2010,95(4):389-399 We study the asymptotic behaviour of the trajectories of the second order equation ${\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)}We study the asymptotic behaviour of the trajectories of the second order equation [(x)\ddot](t)+g[(x)\dot](t)+?f(x(t))+e(t)x(t)=g(t){\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)} , where γ > 0, g ? L1([0,+￥[;H){g \in L^1([0,+\infty[;H)}, Φ is a C 2 convex function and e{\varepsilon} is a positive nonincreasing function.      15. Smooth and strongly smooth points in symmetric spaces of measurable operators      M. M. Czerwińska  A. Kamińska  D. Kubiak 《Positivity》2012,16(1):29-51 We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? SE×{f\in S_{E^{\times}}} supporting μ(x), or s(x *) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M,t){E(\mathcal{M},\tau)}.      16. An inverse spectral theorem for Kre?n strings with a negative eigenvalue      Harald Woracek 《Monatshefte für Mathematik》2012,33(3):105-149 A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L2(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D\mathfrakmDx{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as f￠(x) + z ò0L f(y) d\mathfrakm(y) = 0,    x ? \mathbb R, f￠(0-) = 0.f^{\prime}(x) + z \int_0^L f(y)\,d\mathfrak{m}(y) = 0,\quad x \in\mathbb R,\ f^{\prime}(0-) = 0.      17. A Characteristic Property of Injective Holomorphic Germs      Emmanuel Preissmann 《Monatshefte für Mathematik》2007,45(1):233-239 Let X 0 be the germ at 0 of a complex variety and let f: X0? \Bbb Cn0f:\ X_0\rightarrow {\Bbb C}^n_0 be a holomorphic germ. We say that f is pseudoimmersive if for any g: \Bbb R0? X0g:\ {\Bbb R}_0\rightarrow X_0 such that f °g ? C￥ f \circ g \in C^{\infty} , we have g ? C￥g\in C^{\infty} . We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered.      18. Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem      V. V. Lebedev 《Functional Analysis and Its Applications》2012,46(2):121-132 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 || einf ||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 || einf ||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.      19. On the mean square of Lerch zeta-functions      R. GarunkŠtis  A. LaurinČikas  J. Steuding 《Archiv der Mathematik》2003,80(1):47-60 Using the approximate functional equation for L(l,a, s) = ?n=0￥ [(e(ln))/((n+a)s)] L(\lambda,\alpha, s) = \sum\limits_{n=0}^{\infty} {e(\lambda n)\over (n+\alpha)^s} , we prove for fixed parameters $ 0<\lambda,\alpha\leq 1 $ 0<\lambda,\alpha\leq 1 asymptotic formulas for the mean square of L(l,a,s) L(\lambda,\alpha,s) inside the critical strip. This improves earlier results of D. Klusch and of A. Laurin)ikas.      20. Exponents for three-dimensional simultaneous Diophantine approximations      Nikolay Moshchevitin 《Czechoslovak Mathematical Journal》2012,62(1):127-137 Let Θ = (θ 1,θ 2,θ 3) ∈ ℝ3. Suppose that 1, θ 1, θ 2, θ 3 are linearly independent over ℤ. For Diophantine exponents $\begin{gathered} \alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\ \beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\ \end{gathered}$\begin{gathered} \alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\}, \hfill \\ \beta (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\ \end{gathered}      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

Compact Operators that Commute with a Contraction

Let T be a C₀–contraction on a separable Hilbert space. We assume that I_H − T*T is compact. For a function f holomorphic in the unit disk \mathbbD{\mathbb{D}} and continuous on [`(\mathbbD)]\overline{{\mathbb{D}}}, we show that f(T) is compact if and only if f vanishes on s(T)?\mathbbT\sigma(T)\cap{\mathbb{T}}, where σ(T) is the spectrum of T and \mathbbT{\mathbb{T}} the unit circle. If f is just a bounded holomorphic function on \mathbbD{\mathbb{D}}, we prove that f(T) is compact if and only if lim_{n? ￥}||Tⁿf(T)|| = 0\lim\limits_{n\rightarrow \infty}\|T^{n}f(T)\| = 0.     

Book Reviews

For , let E(λ_*, λ^*) be the set It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that where dim denotes the Hausdorff dimension.     

Notes on the nuclearity of random subalgebras of O2{\mathcal {O}_2}

There exists a separable exact C*-algebra A which contains all separable exact C*-algebras as subalgebras, and for each norm-dense measure μ on A and independent μ-distributed random elements x ₁, x ₂, ... we have lim_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. Further, there exists a norm-dense non-atomic probability measure μ on the Cuntz algebra O₂{\mathcal {O}_2} such that for an independent sequence x ₁, x ₂, ... of μ-distributed random elements x _i we have lim inf_{n ? ￥}\mathbb P(C^*(x₁,?,x_n) is nuclear)=0{\rm {lim\, inf}}_{n \rightarrow \infty}\mathbb {P}(C^*(x_1,\ldots,x_n) \mbox{ is nuclear})=0. We introduce the notion of the stochastic rank for a unital C*-algebra and prove that the stochastic rank of C([0, 1] ^d ) is d.     

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 an asymptotically autonomous system with Tikhonov type regularizing term

We study the asymptotic behaviour of the trajectories of the second order equation ${\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)}We study the asymptotic behaviour of the trajectories of the second order equation [(x)\ddot](t)+g[(x)\dot](t)+?f(x(t))+e(t)x(t)=g(t){\ddot{x}(t)+\gamma \dot{x}(t)+\nabla\phi(x(t))+\varepsilon(t)x(t)=g(t)} , where γ > 0, g ? L¹([0,+￥[;H){g \in L^1([0,+\infty[;H)}, Φ is a C ² convex function and e{\varepsilon} is a positive nonincreasing function.     

Smooth and strongly smooth points in symmetric spaces of measurable operators

We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? S_E^×{f\in S_{E^{\times}}} supporting μ(x), or s(x ^*) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M,t){E(\mathcal{M},\tau)}.     

An inverse spectral theorem for Kre?n strings with a negative eigenvalue

A string is a pair (L, \mathfrakm){(L, \mathfrak{m})} where L ? [0, ￥]{L \in[0, \infty]} and \mathfrakm{\mathfrak{m}} is a positive, possibly unbounded, Borel measure supported on [0, L]; we think of L as the length of the string and of \mathfrakm{\mathfrak{m}} as its mass density. To each string a differential operator acting in the space L²(\mathfrakm){L^2(\mathfrak{m})} is associated. Namely, the Kreĭn–Feller differential operator -D_\mathfrakmD_x{-D_{\mathfrak{m}}D_x} ; its eigenvalue equation can be written, e.g., as

A Characteristic Property of Injective Holomorphic Germs

Let X ₀ be the germ at 0 of a complex variety and let f: X₀? \Bbb Cⁿ₀f:\ X_0\rightarrow {\Bbb C}^n_0 be a holomorphic germ. We say that f is pseudoimmersive if for any g: \Bbb R₀? X₀g:\ {\Bbb R}_0\rightarrow X_0 such that f °g ? C^￥ f \circ g \in C^{\infty} , we have g ? C^￥g\in C^{\infty} . We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered.     

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.     

On the mean square of Lerch zeta-functions

Using the approximate functional equation for L(l,a, s) = ?_n=0^￥ [(e(ln))/((n+a)^s)] L(\lambda,\alpha, s) = \sum\limits_{n=0}^{\infty} {e(\lambda n)\over (n+\alpha)^s} , we prove for fixed parameters $ 0<\lambda,\alpha\leq 1 $ 0<\lambda,\alpha\leq 1 asymptotic formulas for the mean square of L(l,a,s) L(\lambda,\alpha,s) inside the critical strip. This improves earlier results of D. Klusch and of A. Laurin)ikas.     

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents