A variational problem with lack of compactness for H(S;S) maps of prescribed degree

We consider, for maps in H^1/2(S¹;S¹), a family of (semi)norms equivalent to the standard one. We ask whether, for such a norm, there is some map in H^1/2(S¹;S¹) of prescribed topological degree equal to 1 and minimal norm. In general, the answer is no, due to concentration phenomena. The existence of a minimal map is sensitive to small perturbations of the norm. We derive a sufficient condition for the existence of minimal maps. In particular, we prove that, for every given norm, there are arbitrarily small perturbations of it for which the minimum is attained. In case there is no minimizer, we determine the asymptotic behavior of minimizing sequences. We prove that, for such minimizing sequences, the energy concentrates near a point of S¹. We describe this concentration in terms of bubbling-off of circles.     

Restrictions on free actions of the alternating groupA 6 on products of spheres

We prove that the alternating groupA ⁶ cannot freely act on (S ⁿ)⁵ We give an example of free action of the alternating groupA ⁴ on (S ⁿ)³.     

Oscillation of solutions of second-order nonlinear differential equations of Euler type

We consider the nonlinear Euler differential equation t²x^″+g(x)=0. Here g(x) satisfies xg(x)>0 for x≠0, but is not assumed to be sublinear or superlinear. We present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory. Also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both. We derive explicit conditions and improve the results presented in the previous literature. We extend our results to the extended equation t²x^″+a(t)g(x)=0.     

Derivations for a class of matrix function algebras

We study a class of matrix function algebras, here denoted T⁺(C_n). We introduce a notion of point derivations, and classify the point derivations for certain finite dimensional representations of T⁺(C_n). We use point derivations and information about n×n matrices to show that every T⁺(C_n)-valued derivation on T⁺(C_n) is inner.     

Minorations pour l'équation de Catalan

We consider Catalan's equation x^p − y^q = 1 (with p, q prime and |x|, |y| > 1). We show that, besides the obvious solution 3² − 2³ = 1, min {p; q} > 10⁵ and max {p; q} > 10⁶.     

Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant

We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L ²(?) when the translations and modulations of the window are associated with certain non-separable lattices in ?² which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ?²ⁿ with the property that the short-time Fourier transform defines an isometric embedding from L ²(? ⁿ ) to L _μ ² (?²ⁿ ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point.     

The two-dimensional eikonal equation

We study the two-dimensional eikonal equation ψ _x ² + ψ _y ² = 1/v ²(x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds ² = [dx ² + dy ²]/v ²(x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

A New Strong Laplacian on Differential Forms

We construct a strong Laplacian D ^* D by using the third operator in the basis {d,d ^*,D} of the space of natural first-order operators acting on the differential forms of a Riemannian manifold (M,g). We study the properties of the Laplacian D ^* D and obtain Weitzenbock's formula relating the three strong Laplacians dd ^*, d ^* d, and D ^* D to the curvature of the manifold (M,g).     

Decomposing Rd into finitely many semigroups

We determine the precise structure of those (additive) semigroups in R^d which belong to at least one partition of R^d into finitely many disjoint Borel measurable semigroups.We also find the structure of the convex sets in R^d which belong to some partition of R^d into finitely many disjoint convex sets.     

Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations

We study nonlinear Schrödinger equations, posed on a three dimensional Riemannian manifold M. We prove global existence of strong H¹ solutions on M=S³ and M=S²×S¹ as far as the nonlinearity is defocusing and sub-quintic and thus we extend results of Ginibre, Velo and Bourgain who treated the cases of the Euclidean space R³ and the torus T³=R³/Z³ respectively. The main ingredient in our argument is a new set of multilinear estimates for spherical harmonics.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

Sums and commutators of generators of noncommuting strongly continuous groups

We consider two infinitesimal generators A, B of strongly continuous groups in a general Banach space X, such that either the commutator between A and B commutes both with e^tA and with e^tB, or the commutator is a multiple of A. We prove that under suitable assumptions the sum A+B and the commutator [A,B] are closable, and their closures generate strongly continuous groups. We give explicit representation formulas for such groups, in terms of e^tA and e^tB.     

Directional complexity of the hypercubic billiard

We consider a minimal rotation on the torus T^d of direction ω. A natural cellular decomposition of the torus is associated to this map. We consider an infinite orbit for this map. We compute the complexity of the associated word. Under some hypothesis on the direction, we obtain an exact formula which shows that the order of magnitude is n^d. This result is related to the billiard map inside a hypercube of R^d+1.     

A Near-Commutativity Property for Rings

We revisit rings with the property that |A ²| ≤ 3 for each 2-subset A. We then investigate commutativity of rings R with the property that for each pair a, b of noncommuting elements of R, there exists an integer n > 1 for which a ⁿ = b ⁿ .     

Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

Angular momentum and gravimagnetization of the N=2 supersymmetric vacuum

We consider the gravimagnetization of the N=2 supersymmetric vacuum in the presence of the ??-deformation. We argue that the Seiberg-Witten prepotential is related to the vacuum density of the angular momentum in the Euclidean space ? ⁴ . We conjecture the possible role of the dyonic instantons as the microscopic angular momentum carriers that could yield a spontaneous vacuum gravimagnetization. We interpret the dyonic instanton as an analogue of the Euclidean bounce in ? ⁴ . Such a bounce is related to the Schwinger pair production. We also briefly discuss the induced angular momentum in ? ⁴ in the dual Liouville formulation of the SU(2) theory in terms of the hypothesis of the Alday-Gaiotto-Tachikawa correspondence.     

Catalan and Apéry numbers in residue classes

We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p^13/2(logp)⁶ elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring p^O(p) elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apéry numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p.     

Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

We develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space L_m(C^n×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in L_m(C^n×n). We present a family of natural norms on the space L_m(C^n×n) and show that the norms on the spaces C^m+1 and C^n×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space L_m(C^n×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space L_m(C^n×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.     

Saturated boolean algebras and their stone spaces

For an infinite cardinal κ, we call a compact zero-dimensional space a κ-Parovicenko space if its boolean algebra of clopen sets is κ-saturated and has cardinality κ^<κ. We answer some questions about these spaces which were posed in [14]. For instance, it is shown that a κ⁺-Parovicenko spacé can be a Stone-Cech remainder in a natural way. We show that some of the results in [14] which used the assumption κ^<κ=κ, do indeed require this assumption. We also show that if 2^κ=κ⁺ then each compact F_κ⁺-space with weight κ⁺ can be embedded into a κ⁺-Parovicenko space (and so into an extremally disconnected space).