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Introduce several KAM theorems for infinite dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori. Especially, introduce a KAM theorem in the paper(Cummun. Math.…

In this paper we offer some new identities associated with mock theta functions and establish new Bailey pairs related to indefinite quadratic forms. We believe our proof is instructive use of changing base of Bailey pairs, and offers new…

We prove a new fixed point theorem of Schauder-type which applies to discontinuous operators in non-compact domains. In order to do so, we present a modification of a recent Schauder-type theorem due to Pouso. We apply our result to…

Extending the symmetric framework of D'Adderio and Mellit, we establish a nonsymmetric generalization of the compositional Delta theorem. Building on Blasiak et al.'s theory of flagged LLT polynomials, we derive signed and unsigned…

This mostly expository paper centers on recently proved conjectures in two areas: A) A conjecture of A. Oppenheim on the values of real indefinite quadratic forms at integral points. B) Conjectures of Dani, Raghunathan, and Margulis on…

The quadratic term in the Taylor expansion at the origin of the backscattering transformation in odd dimensions $n\ge 3$ gives rise to a symmetric bilinear operator $B_2$ on $C_0^\infty({\mathbb R}^n)\times C_0^\infty({\mathbb R}^n)$. In…

Following an idea of Ozawa, we give a new proof of Kazhdan's property (T) for ${\rm SL}(3,\mathbb Z)$, by showing that $\Delta^2- \frac{1}{6} \Delta$ is a hermitian sum of squares in the group algebra, where $\Delta$ is the unnormalized…

We prove the Ces\`aro boundedness of eigenfunctions of the Dirac operator on the half-line with a square-summable potential. The proof is based on the theory of Krein systems and, in particular, on the continuous version of a theorem by A.…

We find conditions under which the restriction of a divergence-free vector field $B$ to an invariant toroidal surface $S$ is linearisable. The main results are similar in conclusion to Arnold's Structure Theorems but require weaker…

A quasi-product on the normed space is defined. In addition, the notions of the eigenvectors of a linear operator can be extended for the nonlinear operator. Based on the quasi-product and the generalized eigenvectors, the spectral theorems…

In this review paper we carry on our investigations on Schroedinger operators with inverse square potentials on the half-line. Depending on several parameters, such operators possess either a finite number of complex eigenvalues, or an…

We establish a real version of Turrittin's result on polynomial and formal normal forms of linear systems of ODEs with meromorphic coefficients. Both the normal forms or the transformations used have only real coefficients. In order to…

In this paper we study how to distinguish two embeddings of a finite collection of disjoint circles into the plane up to planar isotopy. We adopt the spirit of the approach by V. Turaev, Operator Invariants of Tangles, Math. USSR-Izv. 35…

Let $ K $ be a global function field of characteristic $ 2 $. For each non-trivial place $ v $ of $ K $, let $ K_{v} $ be the completion of $ K $ at $ v $. We show that if two non-degenerate quadratic forms are similar over every $ K_{v} $,…

Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint…

We prove the Hasse principle and weak approximation for varieties defined over number fields by the nonsingular intersection of pairs of quadratic forms in 8 variables. The argument develops work of Colliot-Thelene, Sansuc and…

Preliminary results of our investigations on solving indefinite qua\-dra\-tic programs by dynamical systems are given. First, dynamical systems corresponding to two fundamental DC programming algorithms to deal with indefinite quadratic…

We prove symplectic versions of Suslin's famous $n!$-theorem for algebras over quadratically closed perfect fields of characteristic $\neq 2$ and for algebras over finite fields of characteristic $\neq 2$.

We extend the well-known Katznelson-Tzafriri theorem, originally posed for power-bounded operators, to the case of Ces\`aro bounded operators of any order $\alpha>0.$ For this purpose, we use a functional calculus between a new class of…

Consider groups such as Mordell-Weil groups of abelian varieties over number fields, odd algebraic $K$-theory groups of number fields, or finitely generated subgroups of the multiplicative groups of number fields. They are all equipped with…