The Problem of the Pointwise Fourier Inversion for Piecewise Smooth Functions of Several Variables

We study the pointwise convergence problem for the inverse Fourier transform of piecewise smooth functions, i.e., whether S_rD f (\bx) ? f (\bx)S_{\rho D} f (\bx) \to f (\bx) as r? ￥\rho \to \infty . r? ￥\rho \to \infty . Here for \bx,\bxi ? \Rn\bx,\bxi \in \Rn S_rDf(\bmx)=\dsf1(2p)^n/2\intli_rD [^(f)](\bxi) e^{\dst iá\bmx,\bxi?} d\bxi . S_{\rho D}f(\bm{x})=\dsf1{(2\pi)^{n/2}}\intli_{\rho D} \widehat{f}(\bxi) e^{\dst i\langle\bm{x},\bxi\rangle} d\bxi~. is the partial sum operator using a convex and open set DD containing the origin, and rD={ r\bxi:\bxi ? D }\rho D=\left\{ \rho \bxi:\bxi\in D \right\}.     

Conjugates of Rational Equivariant Holomorphic Maps of Symmetric Domains

Let t: D ?D￠\tau: {\cal D} \rightarrow{\cal D}^\prime be an equivariant holomorphic map of symmetric domains associated to a homomorphism r: \Bbb G ?\Bbb G￠{\bf\rho}: {\Bbb G} \rightarrow{\Bbb G}^\prime of semisimple algebraic groups defined over \Bbb Q{\Bbb Q} . If G ì \Bbb G (\Bbb Q)\Gamma\subset {\Bbb G} ({\Bbb Q}) and G￠ ì \Bbb G￠(\Bbb Q)\Gamma^\prime \subset {\Bbb G}^\prime ({\Bbb Q}) are torsion-free arithmetic subgroups with r (G) ì G￠{\bf\rho} (\Gamma) \subset \Gamma^\prime , the map G\D ?G￠\D￠\Gamma\backslash {\cal D} \rightarrow\Gamma^\prime \backslash {\cal D}^\prime of arithmetic varieties and the rationality of D{\cal D} and D￠{\cal D}^\prime as well as the commensurability groups of s ? Aut (\Bbb C)\sigma \in {\rm Aut} ({\Bbb C}) determines a conjugate equivariant holomorphic map t^s: D^s ?D^￠s\tau^\sigma: {\cal D}^\sigma \rightarrow{\cal D}^{\prime\sigma} of f^s: (G\D)^s ?(G￠\D￠)^s\phi^\sigma: (\Gamma\backslash {\cal D})^\sigma \rightarrow(\Gamma^\prime \backslash {\cal D}^\prime)^\sigma of . We prove that is rational if is rational.     

On conformal minimal 2-spheres in complex Grassmann manifold <Emphasis Type="BoldItalic">G</Emphasis>(2,<Emphasis Type="BoldItalic">n</Emphasis>)

For a harmonic map f from a Riemann surface into a complex Grassmann manifold, Chern and Wolfson [4] constructed new harmonic maps ?f\partial\!f and [`(?)]f\bar{\partial}\!f through the fundamental collineations ?\partial and [`(?)]\bar{\partial} respectively. In this paper, we study the linearly full conformal minimal immersions from S ² into complex Grassmannians G(2,n), according to the relationships between the images of ?f\partial\!f and [`(?)]f\bar{\partial}\!f. We obtain various pinching theorems and existence theorems about the Gaussian curvature, K?hler angle associated to the given minimal immersions, and characterize some immersions under special conditions. Some examples are given to show that the hypotheses in our theorems are reasonable.     

Density of the polynomials in Hardy and Bergman spaces of slit domains

It is shown that for any t, 0<t<∞, there is a Jordan arc Γ with endpoints 0 and 1 such that G\{1} í \mathbbD:={z:|z| < 1}\Gamma\setminus\{1\}\subseteq\mathbb{D}:=\{z:|z|<1\} and with the property that the analytic polynomials are dense in the Bergman space \mathbbA^t(\mathbbD\G)\mathbb{A}^{t}(\mathbb{D}\setminus\Gamma) . It is also shown that one can go further in the Hardy space setting and find such a Γ that is in fact the graph of a continuous real-valued function on [0,1], where the polynomials are dense in H^t(\mathbbD\G)H^{t}(\mathbb{D}\setminus\Gamma) ; improving upon a result in an earlier paper.     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).     

Bounds and monotonicity for the generalized Robin problem

We consider the principal eigenvalue λ ₁^Ω(α) corresponding to Δu = λ (α) u in W, \frac?u?v = au \Omega, \frac{\partial u}{\partial v} = \alpha u on ∂Ω, with α a fixed real, and W ì Rⁿ\Omega \subset {\mathcal{R}}^n a C ^0,1 bounded domain. If α > 0 and small, we derive bounds for λ ₁^Ω(α) in terms of a Stekloff-type eigenvalue; while for α > 0 large we study the behavior of its growth in terms of maximum curvature. We analyze how domain monotonicity of the principal eigenvalue depends on the geometry of the domain, and prove that domains which exhibit domain monotonicity for every α are calibrable. We conjecture that a domain has the domain monotonicity property for some α if and only if it is calibrable.     

Numerical Ranges of Radial Toeplitz Operators on Bergman Space

A Toeplitz operator T_fT_\phi with symbol f\phi in L^￥(\mathbbD)L^{\infty}({\mathbb{D}}) on the Bergman space A²(\mathbbD)A^{2}({\mathbb{D}}), where \mathbbD\mathbb{D} denotes the open unit disc, is radial if f(z) = f(|z|)\phi(z) = \phi(|z|) a.e. on \mathbbD\mathbb{D}. In this paper, we consider the numerical ranges of such operators. It is shown that all finite line segments, convex hulls of analytic images of \mathbbD\mathbb{D} and closed convex polygonal regions in the plane are the numerical ranges of radial Toeplitz operators. On the other hand, Toeplitz operators T_fT_\phi with f\phi harmonic on \mathbbD\mathbb{D} and continuous on [`(\mathbbD)]{\overline{\mathbb{D}}} and radial Toeplitz operators are convexoid, but certain compact quasinilpotent Toeplitz operators are not.     

Inverse Scattering at Fixed Energy in de Sitter�CReissner�CNordstr?m Black Holes

In this paper, we consider massless Dirac fields propagating in the outer region of de Sitter–Reissner–Nordstr?m black holes. We show that the metric of such black holes is uniquely determined by the partial knowledge of the corresponding scattering matrix S(λ) at a fixed energy λ ≠ 0. More precisely, we consider the partial wave scattering matrices S(λ, n) (here λ ≠ 0 is the fixed energy and n ? \mathbbN^*{n \in \mathbb{N}^{*}} denotes the angular momentum) defined as the restrictions of the full scattering matrix on a well chosen basis of spin-weighted spherical harmonics. We prove that the mass M, the square of the charge Q ² and the cosmological constant Λ of a dS-RN black hole (and thus its metric) can be uniquely determined from the knowledge of either the transmission coefficients T(λ, n), or the reflexion coefficients R(λ, n) (resp. L(λ, n)), for all n ? L{n \in {\mathcal{L}}} where L{\mathcal{L}} is a subset of \mathbbN^*{\mathbb{N}^{*}} that satisfies the Müntz condition ?_{n ? L}\frac1n = +￥{\sum_{n \in{\mathcal{L}}}\frac{1}{n} = +\infty} . Our main tool consists in complexifying the angular momentum n and in studying the analytic properties of the “unphysical” scattering matrix S(λ, z) in the complex variable z. We show, in particular, that the quantities \frac1T(l,z){\frac{1}{T(\lambda,z)}}, \fracR(l,z)T(l,z){\frac{R(\lambda,z)}{T(\lambda,z)}} and \fracL(l,z)T(l,z){\frac{L(\lambda,z)}{T(\lambda,z)}} belong to the Nevanlinna class in the region ${\{z \in \mathbb{C}, Re(z) > 0 \}}${\{z \in \mathbb{C}, Re(z) > 0 \}} for which we have analytic uniqueness theorems at our disposal. Eventually, as a by-product of our method, we obtain reconstruction formulae for the surface gravities of the event and cosmological horizons of the black hole which have an important physical meaning in the Hawking effect.     

Roots of Ehrhart polynomials arising from graphs

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle |z+\fracd4| ￡ \fracd4|z+\frac{d}{4}| \le \frac{d}{4} or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip -\fracD2 ￡ Re(a) ￡ \fracD2-1-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.     

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}}     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and w_l(0) = (lj₁, l^\fracq+1p+1j₂)w_{\lambda}(0) = ({\lambda}{\varphi}_1, {\lambda}^{\frac{q+1}{p+1}}{\varphi}_2), for some nonnegative functions φ₁, φ₂ ?\in C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.     

The L p -Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results

Assume that the elliptic operator L=div (A(x)∇) is L ^p -resolutive, p>1, on the unit disc \mathbbD ì \mathbb R²\mathbb{D}\subset \mathbb {R}^{2} . This means that the Dirichlet problem

A New Class of Inner Functions Uniformly Approximable by Interpolating Blaschke Products

We study the class of inner functions Q{\Theta} whose zero set Z(Q){Z(\Theta)} stays hyperbolically close to [`(Z_\mathbbD(Q))]{\overline{Z_\mathbb{D}(\Theta)}} on the corona of H ^∞ and show that these functions are uniformly approximable by interpolating Blaschke products.     

Domains with Prescribed Automorphism Group

Let G be the automorphism group of a bounded strictly pseudoconvex domain D⊂ℂ ^N with a smooth ( C^￥\mathcal{C}^{\infty} ) boundary. Let H be a closed subgroup of G. Pertaining to the question whether it is possible to realize H as the automorphism group of a strictly pseudoconvex domain D′ which is an arbitrarily small perturbation of D in C^￥\mathcal{C}^{\infty} topology, we give a partial answer by describing sufficient conditions for D and G.     

The Sum of a Maximally Monotone Linear Relation and the Subdifferential of a Proper Lower Semicontinuous Convex Function is Maximally Monotone

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A+?fA+\partial f provided that A is a maximally monotone linear relation, and f is a proper lower semicontinuous convex function satisfying \operatornamedom A?\operatornameint\operatornamedom ?f 1 \varnothing\operatorname{dom} A\cap\operatorname{int}\operatorname{dom} \partial f\neq\varnothing. Moreover, A+?fA+\partial f is of type (FPV). The maximal monotonicity of A+?fA+\partial f when \operatornameint\operatornamedom A?\operatornamedom ?f 1 \varnothing{\operatorname{int}\operatorname{dom}}\, A\cap\operatorname{dom} \partial f\neq\varnothing follows from a result by Verona and Verona, which the present work complements.     

On the subordinate killed B.M in bounded domains and existence results for nonlinear fractional Dirichlet problems

We take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional differential equation based on the fractional Laplacian (-D_|D)^\fraca2{(-\Delta _{|D})^{\frac{\alpha }{2}}} associated to the subordinate killed Brownian motion process Z_a^D{Z_{\alpha }^{D}} in a bounded C ^1,1 domain D. Our arguments are based on potential theory tools on Z_a^D{Z_{\alpha }^{D}} and properties of an appropriate Kato class of functions K _α (D).     

Boxicity of Circular Arc Graphs

A k-dimensional box is a Cartesian product R ₁ × · · · × R _k where each R _i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some a ? \mathbbN _{3 2}{\alpha\in\mathbb{N}_{\geq 2}}, then box(G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree D < ?\fracn(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha \in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. We also demonstrate a graph having box(G) > α but with D = n\frac(a-1)2a+ \fracn2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if D < ?\fracn(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha\in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box(G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box(G) ≤ 2. We also show that both these bounds are tight.     

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.     

Resonant destabilization of a floating homogeneous ice layer

We show that a homogeneous elastic ice layer of finite thickness and infinite horizontal extension floating on the surface of a homogeneous water layer of finite depth possesses a countable unbounded set of of resonant frequencies. The water is assumed to be compressible, the viscous effects are neglected in the model. Responses of this water-ice system to spatially localized harmonic in time perturbations with the resonant frequencies grow at least as ?t\sqrt{t} in the two-dimensional (2-D) case and at least as lnt in the three-dimensional (3-D) case, when time t?￥.t\to\infty. The analysis is based on treating the 3-D linear stability problem by applying the Laplace-Fourier transform and reducing the consideration to the 2-D case. The dispersion relation for the 2-D problem D(k,w) = 0,{D}(k,\omega) = 0, obtained previously by Brevdo and Il'ichev [10], is treated analytically and also computed numerically. Here k is a wavenumber, and w\omega is a frequency. It is proved that the system D(k,w) = 0, D_k(k,w) = 0{D}(k,\omega) = 0, {D}_k(k,\omega) = 0 possesses a countable unbounded set of roots (k, w) = (0,w_n), n ? \Bbb Z(k, \omega) = (0,\omega_n), n\in\Bbb Z with Im w_n = 0.\rm{Im}\ \omega_n = 0. Then the analysis of Brevdo [6], [7], [8], [9], which showed the existence of resonances in a homogeneous elastic waveguide, is applied to show that similar resonances exist in the present water-ice model. We propose a resonant mechanism for ice-breaking. It is based on destabilizing the floating ice layer by applying localized harmonic perturbations, with a moderate amplitude and at a resonant frequency.