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.     

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.     

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).     

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.     

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.     

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.     

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.     

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.     

Some descriptive set theory and core models

We analyse the trees given by sharps for Π¹₂ sets via inner core models to give a canonical decomposition of such sets when a core model is Σ¹₃ absolute. This is by way of analogy with Solovay's analysis of Π¹₁ sets into ω₁ Borel sets — Borel in codes for wellorders. We find that Π¹₂ sets are also unions of ω₁ Borel sets — but in codes for mice and wellorders. We give an application of this technique in showing that if a core model, K, is Σ¹₃ absolute thenTheorem. Every real is in K iff every Π¹₃ set of reals contains a Π¹₃ singleton.     

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.     

Tractability and Strong Tractability of Linear Multivariate Problems

Linear multivariate problems are defined as the approximation of linear operators on functions of d variables. We study the complexity of computing an ϵ-approximation in different settings. We are particularly interested in large d and/or large ϵ⁻¹. Tractability means that the complexity is bounded by c(d) K(d, ϵ), where c(d) is the cost of one information operation and K(d, ϵ) is a polynomial in d and/or in ϵ⁻¹. Strong tractability means that K(d, ϵ) is a polynomial in ϵ⁻¹, independent of d. We provide necessary and sufficient conditions for linear multivariate problems to be tractable or strongly tractable in the worst case, average case, randomized, and probabilistic settings. This is done for the class Λ^all where an information operation is defined as the computation of any continuous linear functional. We also consider the class Λ^std where an information operation is defined as the computation of a function value. We show under mild assumptions that tractability in the class Λ^all is equivalent to tractability in the class Λ^std. The proof is, however, not constructive. Finally, we consider linear multivariate problems over reproducing kernel Hilbert spaces, showing that such problems are strongly tractable even in the worst case setting.     

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.     

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).     

On the size of the Durfee square of a random integer partition

We prove a local limit theorem for the length of the side of the Durfee square in a random partition of a positive integer n as n→∞. We rely our asymptotic analysis on the power series expansion of x^m2∏_j=1^m(1−x^j)⁻² whose coefficient of xⁿ equals the number of partitions of n in which the Durfee square is m².     

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.     

Counting singular plane curves via Hilbert schemes

We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singularities for which our results suffice include the topological type with local equation x^a+y^b with ?a?3b. We work out the example of curves with the analytic type of singularity with local equation x²+yⁿ for 1<n<9.     

Homogeneous random fields and statistical mechanics

We illustrate the connection between homogeneous perturbations of homogeneous Gaussian random fields over Rⁿ or Zⁿ, with values in R^m, and classical as well as quantum statistical mechanics. In particular we construct homogeneous non-Gaussian random fields as weak limits of perturbed Gaussian random fields and study the infinite volume limit of correlation functions for a classical continuous gas of particles with inner degrees of freedom. We also exhibit the relation between quantum statistical mechanics of lattice systems (anharmonic crystals) at temperature β^?1 and homogeneous random fields over Zⁿ × S_β, where S_β is the circle of length β, which then provides a connection also with classical statistical mechanics. We obtain the infinite volume limit of real and imaginary times Green's functions and establish its properties. We also give similar results for the Gibbs state of the correspondent classical lattice systems and show that it is the limit as h → 0 of the quantum statistical Gibbs state.     

Acute triangulations of polyhedra and ℝ N

We study the problem of acute triangulations of convex polyhedra and the space ? ⁿ . Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n≥4. Further, we prove that acute triangulations of the space ? ⁿ do not exist for n≥5. In the opposite direction, in ?³, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of ?⁴ if all dihedral angles are bounded away from π/2.     

One-basedness and groups of the form G/G 00

We initiate a geometric stability study of groups of the form G/G ⁰⁰, where G is a 1-dimensional definably compact, definably connected, definable group in a real closed field M. We consider an enriched structure M?? with a predicate for G ⁰⁰ and check 1-basedness or non-1-basedness for G/G ⁰⁰, where G is an additive truncation of M, a multiplicative truncation of M, SO ₂(M) or one of its truncations; such groups G/G ⁰⁰ are now interpretable in M??. We prove that the only 1-based groups are those where G is a sufficiently ??big?? multiplicative truncation, and we relate the results obtained to valuation theory. In the last section we extend our results to ind-hyperdefinable groups constructed from those above.     

Difference sets over Galois rings with odd extension degrees and characteristic an even power of 2

We construct an infinite family of (2 ^ns , 2 ^{ns/2 -1}(2 ^ns/2?1), 2 ^{ns/2 -1}(2 ^{ns/2 -1} ?1)) difference sets over a Galois ring GR(2 ⁿ , s) with characteristic an even power n of 2 and an odd extension degree s. It makes a chain of difference sets preserving the structures when n increases and s is fixed. We introduce a new operation into GR(2 ⁿ , s). The Gauss sum associated with the multiplicative character defined by the subgroup with respect to the new operation plays an important role in the construction.