The scattering of electromagnetic waves by a perfectly conducting infinite cylinder

We consider the scattering of a plane time-harmonic electromagnetic wave by a perfectly conducting infinite cylinder with axis in the direction k , where k is the unit vector along the z axis. Suppose the incident wave propagates in a direction perpendicular to the cylinder. For a given observation angle θ, let F_D(θ, α) k be the far-field pattern of the electric field corresponding to an incident wave with direction angle α polarized perpendicular to the axis and let F_N(θ; α) k be the far-field pattern of the magnetic field corresponding to an incident wave with direction angle α polarized parallel to the z axis. Let {α_n}_n=1^∞ be a distinct set of angles in [ ? π, π] and μ a complex number. Then, necessary and sufficient conditions are given for the set {(1 ? μ)F_D(θ;α_n) + μF_N(θ;α_n)}_{n = 1}^∞ to be complete in L²[ ? π, π]. Applications, together with numerical examples, are given to the inverse scattering problem of determining the shape of the cylinder from a knowledge of the far-field data.     

Uniform Hölder Bounds for Nonlinear Schrödinger Systems with Strong Competition

For the positive solutions of the Gross–Pitaevskii system we prove that L^∞‐boundedness implies C^0,α‐boundedness for every α ? (0,1), uniformly as β → +∞. Moreover, we prove that the limiting profile as β → +∞ is Lipschitz‐continuous. The proof relies upon the blowup technique and the monotonicity formulae by Almgren and Alt, Caffarelli, and Friedman. This system arises in the Hartree‐Fock approximation theory for binary mixtures of Bose–Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given. © 2009 Wiley Periodicals, Inc.     

An entropy-theoretic proof of the Sucheston zero-one law

The purpose of this note is to provide a new proof to the following reformulation of the Sucheston zero-one law: An automorphism T of a Lebesgue space X is mixing if and only if, for every subsequence A of the sequence of natural numbers and every partition α of X having finite entropy, there exists a subsequence B = {b(j)} of A such that Λ_{r = 1}^∞ V_{j = r}^∞T^b(j)(α) is the trivial partition.     

A limit theorem for the continuous state branching process

In this paper we discuss the limit of the martingale e^-αtK_t as t→∞, where X_t is a continuous state branching process and E[X_t] = e^αt. The important case is α > 0. Necessary and sufficient conditions are given for the limit to be positive.     

Projective limits of finite decomposition systems

Special finite topological decomposition systems were used to get compactifications of topological spaces in [6]. In this paper the notion of finite decomposition systems is applied for topological measure spaces. We get two canonical topological measure spaces X_∞ and X_∞^d being projective limits of (discrete) finite decomposition systems for each topological measure space X = (X, O, A, P) and each net (A_α) α ? I of upward filtering finite σ-algebras in A. X_∞ is a compact topological measure space and the idea to construct is the same as used in [6]. The compactifications of [6] are cases of some special X_∞. Further on we obtain that each measurable set of the remainder of X_∞ has measure zero with respect to the limit measure P_∞ (Theorem 1). X_∞^d is the STONE representation space X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} A_α, hence a Boolean measure space with regular Borel measure. Some measure theoretical and topological relations between X, X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) and x(A) where x(A) is the Stone representation space of A, are given in Theorem 2. and 4. As a corollary from Theorem 2. we get a measure theoretical-topological version to the Theorem of Alexandroff Hausdorff for compact T₂ measure spaces x with regular Borel measure (Theorem 3.).     

Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α), α ∈ [2, 3)

We develop a precise analysis of J. O’Hara’s knot functionals E^(α), α ∈ [2, 3), that serve as self‐repulsive potentials on (knotted) closed curves. First we derive continuity of E^(α) on injective and regular H² curves and then we establish Fréchet differentiability of E^(α) and state several first variation formulae. Motivated by ideas of Z.‐X. He in his work on the specific functional E⁽²⁾, the so‐called Möbius Energy, we prove C^∞‐smoothness of critical points of the appropriately rescaled functionals $\tilde{E}^{(\alpha )}= {\rm length}^{\alpha -2}E^{(\alpha )}$ by means of fractional Sobolev spaces on a periodic interval and bilinear Fourier multipliers.     

On Singular Points of Meromorphic Functions Determined by Continued Fractions

It is shown that Leighton’s conjecture about singular points of meromorphic functions represented by C-fractions K^∞_n=1(a _n z ^αn /1) with exponents α₁, α₂,... tending to infinity, which was proved by Gonchar for a nondecreasing sequence of exponents, holds also for meromorphic functions represented by continued fractions K^∞_n=1(a _n A _n (z)/1), where A₁,A₂,... is a sequence of polynomials with limit distribution of zeros whose degrees tend to infinity.     

On Monge‐Ampère equations with homogenous right‐hand sides

We study the regularity and behavior at the origin of solutions to the two‐dimensional degenerate Monge‐Ampère equation det D²_u = |x|^α with α > ?2. We show that when α > 0, solutions admit only two possible behaviors near the origin, radial and nonradial, which in turn implies C^{2, δ}‐regularity. We also show that the radial behavior is unstable. For α < 0 we prove that solutions admit only the radial behavior near the origin. © 2008 Wiley Periodicals, Inc.     

Weak-1 generators for a class of subalgebras of l1

Let α ? 0 and let $\text{D(}\text{α}\text{) = {f(z) = ∑}{}_0{}^{\mathrm{\infty }}\text{α}{}_{\mathrm{n}}\text{z}{}^{\mathrm{n}}\text{¦ ∑}{}_0{}^{\mathrm{\infty }}\text{(n + 1)}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{¦ a}{}_{\mathrm{n}}\text{¦ < ∞}}$. Then D(α) is a subalgebra of l¹. We discuss the weak-1 generators of D(α). We use some of our techniques to prove that if ? is a weak-1 generator of H^∞ and ∥ ? ∥_∞ ? 1, then the composition operator C_? on the Dirichlet space has dense range.     

Classes of interval graphs under expanding length restrictions

Let C(α) denote the finite interval graphs representable as intersection graphs of closed real intervals with lengths in [1, α]. The points of increase for C are the rational α ≥ 1. The set D(α) = [∩_β>αC(β)]\C(α) of graphs that appear as soon as we go past α is characterized up to isomorphism on the basis of finite sets E(α) of irreducible graphs for each rational α. With α = p/q and p and q relatively prime, ∣E(α)∣ is computed for all (p,q) with q ? 2 and p = q + 1. When q = 1, E(p) contains only the bipartite star K¹, p_+2. A lowr bound on ∣E(α)∣ is given for all rational α.     

Multipliers of the Hölder classes

Following the lines of [13] we introduce the classes of mixed smoothness $\text{L}$α_p(Rⁿ), $\text{L}$α_p(Zⁿ) for a multi-index α = (α₁,…, α_n). Such classes are naturally tied up with the study of semi-elliptic differential and difference equations.Besides a brief presentation of such classes, we concentrate our research on the study of mixed homogeneous multipliers with homogeneity β = (β₁, …, β_n) and their preservation of mixed homogeneous Hölder classes $\text{L}$_α^∞, for a different multi-index α.In the last paragraph we apply the results to produce various improvements of the classical Schauder's estimates, for differential and difference equations, in the parabolic and elliptic case.     

Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γ_p,w = {f: ∫₀^α (f **)^p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γ_p,w , and the necessary and sufficient conditions for which a spherical element of Γ_p,w is a smooth point of the unit ball in Γ_p,w . We show that strict convexity of the Lorentz spaces Γ_p,w is equivalent to 1 < p < ∞ and to the condition ∫₀^∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γ_p,w from any convex set K ? Γ_p,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The inducibility of complete bipartite graphs

For any s ≥ 1 and t ≥ (^S₂), we prove that among all graphs with n vertices the graph that contains the maximal number of induced copies of K_{t, t+s} for any fixed s ≥ 1 and t ≥ (^s₂) is K_{(n/2)+α(n/2)-α} for some function α = o(n). We show that this is not valid for t < (^s₂). Analogous results for complete multipartite graphs are also obtained.     

Differential Inequalities and Starlikeness

We study some differential inequalities in the unit disc which imply starlikeness: for example if ? (z) = z + Σ^∞ _n=2 a_n (?)zⁿ is analytic in D = {z | |z| < 1} and |z(?′′(z) ? 1)| ≤ 1 ? α, z?D for some α ] [0, 1), then ? is one-to-one on D and ? (D) is a starlike domain with respect to the origin.     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.     

Proper maps which are Lipschitz α up to the boundary

In this paper we shall construct proper holomorphic mappings from strictly pseudoconvex domains in Cⁿ into the unit ball in C^N which satisfy some regularity conditions up to the boundary. If we only require continuity of the map, but not more, then there is a large class of such maps (see [2], [3], and [5]). On the other hand, if F is C^k on the closure, k > N ? n + 1, then there is a very small class of such maps. In fact such F must be holomorphic across the boundary (see [1] and [4]). We are interested in maps F that are less than C^N ? n + 1, but more than continuous on the closure. Namely, we want to find out if this is a very small or a large class. Our main result is as follows. Theorem, (a) Let ga < 1/6; then there exists an N = N(α, n) such that we can find a map F: Bⁿ → B^N that is proper, holomorphic, and Lipschitz α up to the boundary, but F is not holomorphic across the boundary. (b) If D is a general strictly pseudoconvex domain with C^∞ -boundary in Cⁿ, then we can find a map F: D → B^N, N = N(α, n), that is proper, holomorphic, and Lipschitz α up to the boundary of D. To do part (a) of the theorem we only need to show that we can find a proper holomorphic map F = (f₁, …, F_N): Bⁿ → B^N that is Lipschitz α and f_N(z) = c(1 - Z₁)^1/6 for some constant c > 0. With this we can in fact ensure that the map in (a) is at most Lipschitz 1/6 on the closure of Bⁿ.     

Kolmogorov-type inequalities for norms of Riesz derivatives of functions of several variables with Laplacian bounded in L ∞ and related problems

Let L _∞,∞ ^Δ (? ^m ) be the space of functions f ∈ L _∞(? ^m ) such that Δf ∈ L _∞(? ^m ). We obtain new sharp Kolmogorov-type inequalities for the L _∞-norms of the Riesz derivatives D ^α f of the functions f ∈ L _∞,∞ ^Δ (? ^m ) and solve the Stechkin problem of approximating an unbounded operator D ^α by bounded operators on the class f ∈ L _∞(? ^m ) such that ‖Δf‖_∞ ≤ 1, and also the problem of the best recovery of the operator D ^α from elements of this class given with error δ.     

Prescribing scalar curvature on Sn and related problems,part II: Existence and compactness

This is a sequel to [30], which studies the prescribing scalar curvature problem on Sⁿ. First we present some existence and compactness results for n = 4. The existence result extends that of Bahri and Coron [4], Benayed, Chen, Chtioui, and Hammami [6], and Zhang [39]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions. This counting formula, together with the compactness results, completely describes when and where blowups occur. It follows from our results that solutions to the problem may have multiple blowup points. This phenomena is new and very different from the lower-dimensional cases n = 2, 3. Next we study the problem for n ≥ 3. Some existence and compactness results have been given in [30] when the order of flatness at critical points of the prescribed scalar curvature functions K(x) is β ϵ (n − 2, n). The key point there is that for the class of K mentioned above we have completed L^∞ apriori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of K(x) is β = n − 2, the L^∞ estimates for solutions fail in general. In fact, two or more blowup points occur. On the other hand, we provide some existence and compactness results when the order of flatness at critical points of K(x) is β ϵ [n − 2,n). With this result, we can easily deduce that C^∞ scalar curvature functions are dense in C^1,α (0 < α < 1) norm among positive functions, although this is generally not true in the C² norm. We also give a simpler proof to a Sobolev-Aubin-type inequality established in [16]. Some of the results in this paper as well as that of [30] have been announced in [29]. © 1996 John Wiley & Sons, Inc.     

The distribution of reduced numbers in an ideal of a real cubic number field

Let K be a real but not totally real field of degree three over Q, and let A be an ideal in K. It is proved that the reduced numbers in A (i.e., numbers α with α > 1 and ?1 < Re α^(j) < 0 for all conjugates α^(j) ≠ α) are dense in a set of intervals of constant length, and no reduced numbers in A occur in the gaps between these intervals. In fact, the intervals are determined explicitly, and criteria are given for when the reduced numbers in A actually are dense in the whole of [1, ∞).     

Blow‐up analysis,existence and qualitative properties of solutions for the two‐dimensional Emden–Fowler equation with singular potential

Motivated by the study of a two‐dimensional point vortex model, we analyse the following Emden–Fowler type problem with singular potential: where V(x) = K(x)/|x|^2α with α∈(0, 1), 0<a?K(x)?b< + ∞, ?x∈Ω and ∥?K∥_∞?C. We first extend various results, already known in case α?0, to cover the case α∈(0, 1). In particular, we study the concentration‐compactness problem and the mass quantization properties, obtaining some existence results. Then, by a special choice of K, we include the effect of the angular momentum in the system and obtain the existence of axially symmetric one peak non‐radial blow‐up solutions. Copyright © 2007 John Wiley & Sons, Ltd.