The stochastic model for the description of the so-called fragmentation process in frameworks of Kolmogorov approach is proposed. This model is represented as the branching process with continuum set $\left(0,\infty \right)$ of particle types. Each type $r\in \left(0,\infty \right)$ corresponds to the set of fragments having the size $r$. It is proved that the branching condition of this process represents the basic equation of the Kolmogorov theory.

We derive necessary and sufficient conditions for global-in-time existence of solutions of ordinary differential, stochastic differential, and parabolic equations. The conditions are formulated in terms of complete Riemannian metrics on extended phase spaces (conditions with two-sided estimates) or in terms of derivatives of proper functions on extended phase spaces (conditions with one-sided estimates).

The paper is devoted to the calculation of the index of a zero and the asymptotic index of a linear completely continuous nonnegative operator. Also the case of a nonlinear completely continuous operator $A$ whose domain and image are situated in a closed convex set $Q$ of a Banach space is considered. For this case, we formulate the rules for calculating the index of an arbitrary fixed point and the asymptotic index under the assumption that the corresponding linearizations exist and the operators of derivative do not have eigenvectors with eigenvalue $1$ in some wedges.

In this paper we give a complete and improved proof of the “Theorem on the union of two $\left(n-1\right)$-cells.” First time it was proved by the author in the form of reduction to the earlier author's technique. Then the same reduction by the same method was carried out by Kirby. The proof presented here gives a more clear reduction. We also present here the exposition of this technique in application to the given task. Besides, we use a modification of the method, connected with cyclic ramified coverings, that allows us to bypass referring to the engulfing lemma as well as to other multidimensional results, and so the theorem is proved also for spaces of any dimension. Thus, our exposition is complete and does not require references to other works for the needed technique.

A variant of the Bourgin-Yang theorem for $a$-compact perturbations of a closed linear operator (in general, unbounded and having an infinite-dimensional kernel) is proved. An application to integrodifferential equations is discussed.

The existence of the second (according to the module) eigenvalue ${\lambda }_2$ of a completely continuous nonnegative operator $A$ is proved under the conditions that $A$ acts in the space $L_p\left(\Omega \right)$ or $C\left(\Omega \right)$ and its exterior square $A\wedge A$ is also nonnegative. For the case when the operators $A$ and $A\wedge A$ are indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity of $A$ and $A\wedge A$ is examined. For the case when $A$ and $A\wedge A$ are primitive, the difference (according to the module) of ${\lambda }_1$ and ${\lambda }_2$ from each other and from other eigenvalues is proved.

We suggest the construction of an oriented coincidence index for nonlinear Fredholm operators of zero index and approximable multivalued maps of compact and condensing type. We describe the main properties of this characteristic, including applications to coincidence points. An example arising in the study of a mixed system, consisting of a first-order implicit differential equation and a differential inclusion, is given.

We study a mathematical model describing flows of electrorheological fluids. A theorem of existence of a weak solution is proved. For this purpose the approximating-topological method is used.

The integral limit theorem as to the probability distribution of the random number ${\nu }_m$ of summands in the sum ${\displaystyle {\sum }_{k=1}^{{\nu }_m}{\xi }_k}$ is proved. Here, ${\xi }_1,{\xi }_2,\dots$ are some nonnegative, mutually independent, lattice random variables being equally distributed and ${\nu }_m$ is defined by the condition that the sum value exceeds at the first time the given level $m\in \mathbb{N}$ when the number of terms is equal to ${\nu }_m$.

Many analytical problems can be reduced to determining the number of roots of a polynomial in a given disc. In turn, the latter problem admits further reduction to the generalized Rauss-Hurwitz problem of determining the number of roots of a polynomial in a semiplane. However, this procedure requires complicated coefficient transformations. In the present paper we suggest a direct method to evaluate the number of roots of a polynomial with complex coefficients in a disc, based on studying a certain equation in the algebra of polynomials. An application for computing the rotation of plane polynomial vector fields is also given.

The maps of the form $f\left(x\right)={\displaystyle {\sum }_{i=1}^n{a}_i\cdot x\cdot b_i}$, called 1-degree maps, are introduced and investigated. For noncommutative algebras and modules over them 1-degree maps give an analogy of linear maps and differentials. Under some conditions on the algebra $𝒜$, contractibility of the group of 1-degree isomorphisms is proved for the module $l_2\left(𝒜\right)$. It is shown that these conditions are fulfilled for the algebra of linear maps of a finite-dimensional linear space. The notion of 1-degree map gives a possibility to define a nonlinear Fredholm map of $l_2\left(𝒜\right)$ and a Fredholm manifold modelled by $l_2\left(𝒜\right)$. 1-degree maps are also applied to some problems of Markov chains.

This is a survey article featuring the general index locality principle introduced by the authors, which can be used to obtain index formulas for elliptic operators and Fourier integral operators in various situations, including operators on stratified manifolds and manifolds with singularities.