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For earlier considered our sequence A166944 in [4] we prove three statements of its connection with twin primes. We also give a sufficient condition for the infinity of twin primes and pose several new conjectures; among them we propose a…

We continue the~study of embeddings between different classes of Sobolev spaces of differential forms started in 2006 in a~paper by Gol$'$dshtein and Troyanov. As in this paper, our study is based on relations between $L_{q,p}$-cohomology…

We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations), in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple…

We review the current status of the singularity problem in string theory for non-experts. After the problem is discussed from the point of view of supergravity, we discuss classic examples and recent examples of singularity resolution in…

The role of integrable systems in string theory is discussed. We remind old examples of the correspondence between stringy partition functions or effective actions and integrable equations, based on effective application of the matrix model…

Recently a significant progress in matching the anomalous dimensions of certain class of operators in N=4 SYM theory and rotating strings was made. The correspondence was established mainly using Bethe ansatz technique applied to the spin s…

This is a transcript of lectures given at the Sixth Jorge Andre Swieca Summer School in Theoretical Physics. The subject of these lectures is soliton solutions of string theory. We construct a class of exact conformal field theories…

The phenomenological aspects of string theory are briefly reviewed. Emphasis is given to the status of 4D string model building, effective Lagrangians, model independent results, supersymmetry breaking and duality symmetries.

Further properties of a recently proposed higher order infinite spin particle model are derived. Infinitely many classically equivalent but different Hamiltonian formulations are shown to exist. This leads to a condition of uniqueness in…

An intrinsic-state formalism for IBM-4 is presented. A basis of deformed bosons is introduced which allows the construction of a general trial wave function which has Wigner's spin-isospin SU(4) symmetry as a particular limit.…

This is the fourth article in a series where we succeed in enlarging the class of exactly solvable quantum systems. We do that by working in a complete set of square integrable basis that carries a tridiagonal matrix representation for the…

The eigenpair here means the twins consist of eigenvalue and its eigenvector. This paper introduces the three steps of our study on computing the maximal eigenpair. In the first two steps, we construct efficient initials for a known but…

It is discussed and surveyed a numerical method proposed before, that alternative to the usual compression method, provides an approximation to the algorithmic (Kolmogorov) complexity, particularly useful for short strings for which…

We consider the group theoretical properties of R--R scalars of string theories in the low-energy supergravity limit and relate them to the solvable Lie subalgebra $\IG_s\subset U$ of the U--duality algebra that generates the scalar…

1. The 2-Toda lattice and its generic symmetries 2. A Larger class of symmetries for special initial conditions 3. Borel decomposition of Moment matrices, tau-functions and string-orthogonal polynomials 4. From string-orthogonal Polynomials…

In these lectures I discuss various unsolved problems of string theory and their relations to quantum gravity, 3d Ising model, large N QCD, and quantum cosmology. No solutions are presented but some new and perhaps useful approaches are…

In this paper we consider new solutions for pulsating strings. For this purpose we use tha idea of the generalized ansatz for folded and circular strings in hep-th/0311004. We find the solutions to the resulting Neumann-Rosochatius…

This contribution aims to obtain several connection formulae for the polynomial sequence, which is orthogonal with respect to the discrete Sobolev inner product \[ \langle f, g\rangle_n=\langle {\bf u}, fg\rangle+ \sum_{j=1}^M \mu_{j}…

I present a set of remarks related to joint works \cite{paper1},\cite{paper2},\cite{paper3},\cite{MMNO}. These are remarks about polynomials solutions and vertex operators, eigenproblem for polynomials and a remark related to the the…

At the present time, string theory (and its generalizations) remain relatively abstruse subjects to the particle phenomenologist and experimentalist. Yet, striking developments of the last two years offer hope that a fundamental…