Alexey A. Tret’yAkov
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Alexey A. Tret’yAkov has a total of 40 co-authors. Here are the top co-authors by number of shared publications:
- Yuri G. Evtushenko
Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Moscow, Russia
16 shared publications - Agnieszka Prusińska
Siedlce University of Natural Sciences and Humanities
7 shared publications - Beata Medak
Faculty of Exact and Natural Sciences, Siedlce University of Natural Sciences and Humanities, Siedlce, Poland
7 shared publications - Vlasta Malkova
FRC CSC RAS
7 shared publications - Olga A. Brezhneva
Miami University
6 shared publications - Olga Brezhneva
Department of Mathematics, Miami University, Oxford, OH 45056, USA
5 shared publications
Latest Publications
- Open Access
- 13 August 2026
- Published by Oxford University Press (OUP) in The Quarterly Journal of Mathematics
- Vol. 77 (3) , 749-766
Nonlinear mathematical physics equations via p -regularity theory
The paper studies the existence of solutions for certain types of nonlinear heat transfer equations and nonlinear Navier–Stokes equations based on the theory of p-regularity and its most important result, namely the Generalized Lyusternik Theorem. We present this theorem using a sequence of supporting lemmas and provide a detailed procedure for finding solutions to the above equations, along with explicit formulas for these solutions.
- 13 August 2026
- Published by Oxford University Press (OUP) in The Quarterly Journal of Mathematics
- Vol. 77 (3) , 749-766
Open AccessNonlinear mathematical physics equations via p -regularity theory
The paper studies the existence of solutions for certain types of nonlinear heat transfer equations and nonlinear Navier–Stokes equations based on the theory of p-regularity and its most important result, namely the Generalized Lyusternik Theorem. We present this theorem using a sequence of supporting lemmas and provide a detailed procedure for finding solutions to the above equations, along with explicit formulas for these solutions.
- 1 April 2026
- Published by Springer Nature
- p. 3-17
A p-Factor Interpolation Polynomial for Trust-Region Frameworks Near Degenerate Solutions
- 1 April 2026
- Published by Springer Nature
- p. 3-17
A p-Factor Interpolation Polynomial for Trust-Region Frameworks Near Degenerate Solutions
- Open Access
Towards Nonlinearity: The p-Regularity Theory
We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré’s method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems.
Open AccessTowards Nonlinearity: The p-Regularity Theory
We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré’s method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems.
- 31 January 2025
- Published by Springer Nature
- p. 31-42
Singular Optimization Problems and p-Factor Approach for Their Analysis
- 31 January 2025
- Published by Springer Nature
- p. 31-42
Singular Optimization Problems and p-Factor Approach for Their Analysis
- 10 November 2023
- Published by Springer Nature
- p. 14-32
The pth-Order Karush-Kuhn-Tucker Type Optimality Conditions for Nonregular Inequality Constrained Optimization Problems
- 10 November 2023
- Published by Springer Nature
- p. 14-32
The pth-Order Karush-Kuhn-Tucker Type Optimality Conditions for Nonregular Inequality Constrained Optimization Problems
- 1 August 2023
- Published by Pleiades Publishing Ltd in Doklady Mathematics
- Vol. 108 (1) , 243-247
On Solution Existence for a Singular Nonlinear Burgers Equation with Small Parameter and p-Regularity Theory
In this paper we study a solution existence problem for the singular nonlinear Burgers equation \(F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} = f(x,t),\) where \(F:\Omega \to C([0,\pi ] \times [0,T])\) and \(\Omega = {{C}^{2}}([0,\pi ] \times [0,T]) \times \mathbb{R}\), with small parameter ε under the boundary conditions \(u(0,t) = u(\pi ,t) = 0\) and the oscillating initial condition \(u(x,0) = k\sin x.\) The first derivative of the operator F at the solution point is assumed to be degenerate, i.e., \(F{\kern 1pt} '(x{\kern 1pt} \text{*})\) is not surjective. The existence of a continuous solution to this nonlinear problem is proved by applying p-regularity theory and the Michael selection theorem.- 1 August 2023
- Published by Pleiades Publishing Ltd in Doklady Mathematics
- Vol. 108 (1) , 243-247
On Solution Existence for a Singular Nonlinear Burgers Equation with Small Parameter and p-Regularity Theory
In this paper we study a solution existence problem for the singular nonlinear Burgers equation \(F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} = f(x,t),\) where \(F:\Omega \to C([0,\pi ] \times [0,T])\) and \(\Omega = {{C}^{2}}([0,\pi ] \times [0,T]) \times \mathbb{R}\), with small parameter ε under the boundary conditions \(u(0,t) = u(\pi ,t) = 0\) and the oscillating initial condition \(u(x,0) = k\sin x.\) The first derivative of the operator F at the solution point is assumed to be degenerate, i.e., \(F{\kern 1pt} '(x{\kern 1pt} \text{*})\) is not surjective. The existence of a continuous solution to this nonlinear problem is proved by applying p-regularity theory and the Michael selection theorem.- Open Access
On the Finite Complexity of Solutions in a Degenerate System of Quadratic Equations: Exact Formula
The paper describes an application of the p-regularity theory to Quadratic Programming (QP) and nonlinear equations with quadratic mappings. In the first part of the paper, a special structure of the nonlinear equation and a construction of the 2-factor operator are used to obtain an exact formula for a solution to the nonlinear equation. In the second part of the paper, the QP problem is reduced to a system of linear equations using the 2-factor operator. The solution to this system represents a local minimizer of the QP problem along with its corresponding Lagrange multiplier. An explicit formula for the solution of the linear system is provided. Additionally, the paper outlines a procedure for identifying active constraints, which plays a crucial role in constructing the linear system.
Open AccessOn the Finite Complexity of Solutions in a Degenerate System of Quadratic Equations: Exact Formula
The paper describes an application of the p-regularity theory to Quadratic Programming (QP) and nonlinear equations with quadratic mappings. In the first part of the paper, a special structure of the nonlinear equation and a construction of the 2-factor operator are used to obtain an exact formula for a solution to the nonlinear equation. In the second part of the paper, the QP problem is reduced to a system of linear equations using the 2-factor operator. The solution to this system represents a local minimizer of the QP problem along with its corresponding Lagrange multiplier. An explicit formula for the solution of the linear system is provided. Additionally, the paper outlines a procedure for identifying active constraints, which plays a crucial role in constructing the linear system.
- 1 July 2023
- Published by The Russian Academy of Sciences in Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
- Vol. 512 (1) , 5-9
ON THE APPLICATION OF THE SOLUTION OF THE DEGENERATE NONLINEAR BURGERS EQUATION WITH A SMALL PARAMETER AND THE THEORY OF p-REGULARITY
The article discusses various modifications of the nonlinear Burgers equation with small parameter and degenerate in solution of the form\(F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} - f(x,t) = 0,\) (1)where \(F:\Omega \to C([0,\pi ] \times [0,T])\), \(T 0\), \(\Omega = {{C}^{2}}([0,\pi ] \times [0,T]\,)\,\mathbb{R}\) and \(u(0,t) = u(\pi ,t) = 0\), \(u(x,0) = \varphi (x)\), \(f(x,t) \in C([0,\pi ] \times [0,T])\), \(\varphi (x) \in C[0,\pi ]\). We will be interested in the most important in applications case of a small parameter ε with oscillating initial conditions of the form \(\varphi (x) = k\sin x\), where k –some, generally speaking, constant depending on ε, and study the question of the existence of a solution in neighborhood of the trivial \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), which corresponds to \(k = k{\kern 1pt} * = 0\) and at what initial Under certain conditions on the values of k, it is possible to construct an analytical approximation of this solution for small ε.We will look for a solution in the traditional way of separation of variables on a subspace of functions of the form \(u(x,t) = v(t)u(x)\), where \(v(t) = c{{e}^{{ - t}}}\), \(u(x) \in {{\mathcal{C}}^{2}}([0,\pi ])\). In this case, the problem under consideration is degenerate at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), since \({\text{Im}}F_{u}^{'}(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) \ne Z = \mathcal{C}([0,\pi ] \times [0,T])\). This follows from the Sturm-Liouville theory. To achieve our goals, we apply the apparatus of p-regularity theory [6, 7, 15, 16] and show that the mapping \(F(u,\varepsilon )\) is 3-regular at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), т.е. p = 3.- 1 July 2023
- Published by The Russian Academy of Sciences in Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
- Vol. 512 (1) , 5-9
ON THE APPLICATION OF THE SOLUTION OF THE DEGENERATE NONLINEAR BURGERS EQUATION WITH A SMALL PARAMETER AND THE THEORY OF p-REGULARITY
The article discusses various modifications of the nonlinear Burgers equation with small parameter and degenerate in solution of the form\(F(u,\varepsilon ) = {{u}_{t}} - {{u}_{{xx}}} + u{{u}_{x}} + \varepsilon {{u}^{2}} - f(x,t) = 0,\) (1)where \(F:\Omega \to C([0,\pi ] \times [0,T])\), \(T 0\), \(\Omega = {{C}^{2}}([0,\pi ] \times [0,T]\,)\,\mathbb{R}\) and \(u(0,t) = u(\pi ,t) = 0\), \(u(x,0) = \varphi (x)\), \(f(x,t) \in C([0,\pi ] \times [0,T])\), \(\varphi (x) \in C[0,\pi ]\). We will be interested in the most important in applications case of a small parameter ε with oscillating initial conditions of the form \(\varphi (x) = k\sin x\), where k –some, generally speaking, constant depending on ε, and study the question of the existence of a solution in neighborhood of the trivial \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), which corresponds to \(k = k{\kern 1pt} * = 0\) and at what initial Under certain conditions on the values of k, it is possible to construct an analytical approximation of this solution for small ε.We will look for a solution in the traditional way of separation of variables on a subspace of functions of the form \(u(x,t) = v(t)u(x)\), where \(v(t) = c{{e}^{{ - t}}}\), \(u(x) \in {{\mathcal{C}}^{2}}([0,\pi ])\). In this case, the problem under consideration is degenerate at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), since \({\text{Im}}F_{u}^{'}(u{\kern 1pt} *,\varepsilon {\kern 1pt} *) \ne Z = \mathcal{C}([0,\pi ] \times [0,T])\). This follows from the Sturm-Liouville theory. To achieve our goals, we apply the apparatus of p-regularity theory [6, 7, 15, 16] and show that the mapping \(F(u,\varepsilon )\) is 3-regular at the point \((u{\kern 1pt} *,\varepsilon {\kern 1pt} *) = (0,0)\), т.е. p = 3.- Open Access
- 25 October 2022
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 112 (5) , 1-16
The nonlinear singular Burgers equation with small parameter and p-regularity theory
In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation $$\begin{aligned} F(u,\varepsilon )=u_{t}'-u_{xx}''+uu_{x}'+\varepsilon u^{2}=f(x,t), \end{aligned}$$ where $$F: \Omega \rightarrow \mathcal {C}([0,\pi ]\times [0,\infty ))$$ , $$\Omega = \mathcal {C}^{2}([0,\pi ]\times [0,\infty ))\times \mathbb {R}$$ , $$u(0,t)=u(\pi ,t) =0$$ , $$u(x,0)=g(x)$$ , and F, f(x, t), g(x) will be describe in the text. The first derivative of operator F at the solution point is degenerate. By virtue of p-regularity theory and Michael selection theorem, we prove the existence of continuous solution for this nonlinear problem.- 25 October 2022
- Published by Springer Nature in Letters in Mathematical Physics
- Vol. 112 (5) , 1-16
Open AccessThe nonlinear singular Burgers equation with small parameter and p-regularity theory
In this paper, we study a solutions existence problem of the following nonlinear singular Burgers equation $$\begin{aligned} F(u,\varepsilon )=u_{t}'-u_{xx}''+uu_{x}'+\varepsilon u^{2}=f(x,t), \end{aligned}$$ where $$F: \Omega \rightarrow \mathcal {C}([0,\pi ]\times [0,\infty ))$$ , $$\Omega = \mathcal {C}^{2}([0,\pi ]\times [0,\infty ))\times \mathbb {R}$$ , $$u(0,t)=u(\pi ,t) =0$$ , $$u(x,0)=g(x)$$ , and F, f(x, t), g(x) will be describe in the text. The first derivative of operator F at the solution point is degenerate. By virtue of p-regularity theory and Michael selection theorem, we prove the existence of continuous solution for this nonlinear problem. - 1 January 2022
- Published by Springer Nature
- p. 18-33
Degenerate Equality Constrained Optimization Problems and P-Regularity Theory
- 1 January 2022
- Published by Springer Nature
- p. 18-33
Degenerate Equality Constrained Optimization Problems and P-Regularity Theory