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Classifications of irreducible components of the set of polynomial differential equations with a fixed degree and with at least one center singularity lead to some other new problems on Picard-Lefschetz theory and Brieskorn modules of…

Linear systems often involve, as a basic building block, solutions of equations of the form \begin{align*} A_Sx_S&+A_Px_P =0\\ A'_Sx_S & =0, \end{align*} where our primary interest might be in the vector variable $x_P.$ Usually, neither…

Discrete differential equations appear most prominently in planar map and lattice path enumeration. In this work we consider discrete differential equations with an additional parameter $x$, where the order of the equation is $1$ for $x=0$…

A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) formulation. The method is based on a newly established…

For the multiple differential algebra of iterated differential forms (see math.DG/0605113 and math.DG/0609287) on a diffiety (O,C) an analogue of C-spectral sequence is constructed. The first term of it is naturally interpreted as the…

Using an elementary argument, we prove new fixed point theorems for classical elliptic complexes. We obtain new results for conformal relations and coisotropic intersections. We obtain theorems for the average intersections of families of…

We investigate the problem of the existence of first integrals for multidimensional and ordinary linear differential systems with constant coefficients. The spectral method of the first integrals basis construction for these systems of…

In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calder\'{o}n's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes.

We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…

Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms…

Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…

Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…

Inverse spectral problems are studied for first-order integro-differential operators on a finite interval. These problems consist in recovering some components of the kernel from one or multiple spectra. Uniqueness theorems are proved for…

Based on previous work we consturct an equation (Lagrange equation) and relate it with a system of generalized integrals and differential equations in such a way to provide useful evaluations and connections between them.

The exact solutions of the first order differential equation with delay are derived. The equation has been introduced as a model of traffic flow. The solution describes the traveling cluster of jam, which is characterized by Jacobi's…

This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.

In [8, 9] M. G. Corrales Garcia, D. M. Barquero, C. Martin Gonzalez, M. Siles Molina, J. F Solanilla Hernandez described the center of a Leavitt path algebra and characterized it in terms of the underlying graph. We offer a different…

We give a leisurely introduction into mathematical diffraction theory with a focus on pure point diffraction. In particular, we discuss various characterisations of pure point diffraction and common models arising from cut and project…

This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…

In this paper we compute the center and, in several cases, central subalgebras of double Ore extensions of type (14641) under suitable restrictions on the defining parameters. Part of the analysis is supported by computations in SageMath.…