Monoidal abelian envelopes

We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess–Zumino–Novikov–Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet -algebras as an example of the applications of our general construction of the braided tensor category structure.

An augmented rack is a set with a self-distributive binary operation induced by a group action, and has been extensively used in knot theory. Solutions to the Yang–Baxter equation (YBE) have been also used for knots, since the discovery of the Jones polynomial. In this paper, an interpretation of augmented racks in tensor categories is given for coalgebras that are Hopf algebra modules, and associated solutions to the YBE are constructed. Explicit constructions are given using quantum heaps and the adjoint of Hopf algebras. Furthermore, an inductive construction of Yang–Baxter solutions is given by means of the categorical augmented racks, yielding infinite families of solutions. Constructions of braided monoidal categories are also provided using categorical augmented racks.

Let G be a finite group. Given a finite G-set $\cal{X}$ and a modular tensor category $\cal{C}$ , we construct a weak G-equivariant fusion category $\cal{C}^{\cal{X}}$ , called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for $\mathbb{Z}/2$ -permutation equivariant categories, finishing thereby a program we initiated in an earlier paper.

An asymptotic cell category for cyclic groups

This paper considers the first boundary value problem for the quasilinear parabolic differential equation \[({\text {A}})\qquad L[u] = \sum_{i,j = 1}^n {a_{ij} (x,t)u_{x_i x_j } + } \sum_{i = 1}^n {b_i (x,t)u_{x_i } - u_t = - f(x,t,u)}\] on a cylindrical domain in $E_{n + 1} $. The central theorem is an existence theorem offering an iterative procedure for approximating or constructing a classical solution $u(x,t)$ for this problem. The iteration is done by successively applying a monotone integral operator with an appropriate Green’s function as its kernel. The procedure starts with either of a pair of functions $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma } (x,t) \leqq \bar \sigma (x,t)$ which satisfy \[L[\bar \sigma ] \geqq - f(x,t,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma } )\] and \[L[\bar \sigma ] \leqq - f(x,t,\bar \sigma ).\] This theorem is then used to prove an existence theorem for equation (A) on $E_{n + 1} $ by solving a sequence of problems on nested cylinders in $E_{n + 1} $.

Pairings of categories and spectra

For a given graph, the independent set problem is to find the size of a maximum set of pairwise non-adjacent its vertices. There are numerous cases of NP-hardness and polynomial-time solvability of this problem. To determine a computational status of the independent set problem, local transformations of graphs are often used. The paper considers some class of replacements of subgraphs in graphs that change the independence number in a controllable way. Every such local transform of a graph is determined by some pattern which is a subset of the power set. It is obvious that this pattern must be gradable. The paper shows that replacing subgraph exists for any gradable pattern.

We study certain classes of non-linear Hammerstein integral equations on the semi-axis and the whole line. These classes of equations arise in the theory of radiative transfer in nuclear reactors, in the kinetic theory of gases, and for travelling waves in non- linear Richer competition systems. By combining special iteration methods with the methods of construction of invariant cone segments for the appropriate non- linear operator, we are able to prove constructive existence theorems for positive solutions in various function spaces. We give illustrative examples of equations satisfying all the hypotheses of our theorems.

On the zeros of Flett's function

The work is devoted to the study and solution of one class of nonlinear integral equations with monotone and noncompact Hammerstein-type operators on whole line. Equations of this kind arise in many branches of natural science. In particular, such equations are applied in the dynamic theory of [Formula: see text]-adic open–closed strings, in the kinetic theory of gases, in the mathematical theory of the spatial-temporal spread of pandemic. A constructive theorem for existence of non-negative nontrivial and essentially bounded solution is proved. The asymptotic behavior of constructed solution at [Formula: see text] is studied. It is also proved that the difference between the limit in [Formula: see text] and the solution represents summable function on [Formula: see text] Using results obtained a non-negative integrable and bounded solution for new class of integral equations on whole line with monotone and sign-alternating nonlinearity is constructed. Under additional restrictions on nonlinearity, a uniqueness theorem for a solution in a certain class of continuous and bounded functions is also proved. At the end of the work, the specific examples of above equations, satisfying all conditions of the formulated theorem are given.

A system of singular integral equations with monotone and concave nonlinearity in the subcritical case is investigated. The specified system and its scalar analogy have direct applications in various areas of physics and biology. In particular, scalar and vector equations of this nature are encountered in the dynamic theory of p-adic strings, in the theory of radiative transfer, in the kinetic theory of gases, and in the mathematical theory of the spread of epidemic diseases. A constructive theorem of existence in the space of continuous and bounded vector functions is proved. The integral asymptotic of the constructed solution is studied. An effective iterative process for constructing an approximate solution to this system is proposed. In a certain class of bounded vector functions, the uniqueness of the solution is proved. In the class of bounded vector functions (with non-negative coordinates) with a zero limit at infinity, the absence of an identically zero solution is proved. Examples of the matrix kernel and nonlinearities are given to illustrate the importance of the obtained results. Some of the examples have applications in the above-mentioned branches of natural science.