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Most of the analytical studies on spin glasses are performed by using mean-field theory and renormalization group analysis. Analytical studies on finite-dimensional spin glasses are very challenging. In this short note, a possible exten-…
The classical Stein--Tomas theorem extends the theory of linear Fourier restriction estimates from smooth manifolds to fractal measures exhibiting Fourier decay. In the multilinear setting, transversality allows for Fourier extension…
In this paper we establish a general framework in which the verification of support theorems for generalized convex functions acting between an algebraic structure and an ordered algebraic structure is still possible. As for the domain…
An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically…
In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of…
In this paper, we introduce Kondratiev spaces of fractional smoothness based on their close relation to refined localization spaces. Moreover, we investigate relations to other approaches leading to extensions of the scale of Kondratiev…
The Tomas-Stein inequality or the adjoint Fourier restriction inequality for the sphere $S^1$ states that the mapping $f\mapsto \hat{f\sigma}$ is bounded from $L^2(S^1)$ to $L^6(\mathbb{R}^2)$. We prove that there exists an extremizer for…
We find the complete rational homology for the finite subset spaces of a $d$-dimensional sphere. We also determine the integral homology in top $d$ degrees and obtain a partial description of it in codimension $d$.
We provide estimates for weighted Fourier sums of integrable functions defined on the sphere when the weights originate from a multiplier operator acting on the space where the function belongs. That implies refined estimates for weighted…
A spin system on a lattice can usually be modelled at large scales by an effective quantum field theory. A key mathematical result relating the two descriptions is the quantum central limit theorem, which shows that certain spin observables…
This paper introduces a weighted generalized inverse framework for Fourier extensions, designed to suppress spurious oscillations in the extended region while maintaining high approximation accuracy on the original interval. By formulating…
The mode problem on the factored 3--sphere is applied to field theory calculations for massless fields of spin 0, 1/2 and 1. The degeneracies on the factors, including lens spaces, are neatly derived in a geometric fashion. Vacuum energies…
We study the accessory parameter problem for four-punctured spheres from the point of view of modular forms. The value of the accessory parameter giving the uniformization is characterized as the unique zero of a system of equations. This…
We study a Szemer\'edi-Trotter type theorem in finite fields. We then use this theorem to obtain an improved sum-product estimate in finite fields.
We study the Fourier transforms of indicator functions of some special high-dimensional finite type domains and obtain estimates of the associated lattice point discrepancy.
We study subsets of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$, for odd $q$, which contain either a sphere for each radius or a sphere for each first coordinate of the center. We call such sets radii spherical…
We use the Hardy spaces for Fourier integral operators to obtain bounds for spherical maximal functions in $L^{p}(\mathbb{R}^{n})$, $n\geq2$, where the radii of the spheres are restricted to a compact interval in $(0,\infty)$. These bounds…
In this paper we introduce a class of mathematical objects called \emph{extensors} and develop some aspects of their theory with considerable detail. We give special names to several particular but important cases of extensors. The…
We prove the invertibility of the relevant single and double layer potentials associated to some generalizations of the Stokes operator on bounded domains. In order to do that, we first develop an ``algebra tool kit'' to deal with limit and…
We establish the various properties as well as diverse relations of the ascent and descent spectra for bounded linear operators. We specially focus on the theory of subspectrum. Furthermore, we construct a new concept of convergence for…