arXiv subjects

Pavel B. Dubovski

Publications and source records attributed to Pavel B. Dubovski.

6 recordsLinked to original sources

arXiv2026

A sharp minimum for cube-apex bodies in every dimension

We determine the minimum volume product of centrally symmetric polytopes obtained by adding facet apices to a cube while fixing their total outward height. Heights and tangential coordinates vary subject to a product condition on the polar-cap denominators. In every dimension at least two, the minimum has an explicit formula, attained by concentrating the height at one opposite pair of corner apices. For positive height in dimensions at least three, every minimizing array has this structure. The product condition strictly extends coordinatewise quadratic constraints. We also obtain two bounds without that condition: a four-dimensional Mahler bound for a specified height range, and a correction measuring volume generated jointly by two apices. The latter certifies a continuous family outside the preceding sufficient conditions. The proof combines known cap volumes with a sharp aggregate height estimate and a double-pyramid construction. Lean~4 verifies the actual-volume minimum, the necessary equality structure and both extensions.

math.MG↗

arXiv2026

"Phase Transition" in fractional differential equations

The basic result in the theory of linear ODEs is that the dimension of the fundamental set of linearly independent solutions is equal to the order of equation. This view is typically extended from integer-order to fractional differential equations. The purpose of this research is to demonstrate that this expectation is incorrect and explain why. We show that linear fractional differential equations may admit additional linearly independent solutions beyond their order. This phenomenon occurs when the coefficients of the equation cross certain threshold values. This phase transition produces a hyper-dimensional fundamental set of solutions and offers new insight into fractional differential equations and leads to the revisions of the statements of initial and boundary value problems.

math.GM↗

arXiv2022

Fractional Quasi-Bessel Equations

In this paper we consider fractional quasi-Bessel equations $$\sum_{i=1}^{m}d_i x^{α_i+p_i}D^{α_i} u(x) + (x^β- ν^2)u(x)=0$$ and construct their existence and uniqueness theory in the class of fractional series. Our methodology allows us to obtain new results for a broad class of fractional differential equations including Cauchy-Euler and constant-coefficient equations.

math.AP↗

arXiv2022

Substitution Method for Fractional Differential Equations

Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this singularity is well known. It allows to solve some equations but increases the order of the equation and sometimes leads to wrong numerical results or instability. We suggest another approach: the elimination of singularity by substitution. It does not increase the order of equation and its numerical implementation provides the opportunity to define fractional derivative as the limit of discretization. We present a sufficient condition for the substitution-generated difference approximation to be well-conditioned. We demonstrate how some equations can be solved using this method with full confidence that the solution is accurate with at least second order of approximation.

math.NA↗

arXiv2021

Mathematics of Parking: Varying Parking Rate

In the classical parking problem, unit intervals ("car lengths") are placed uniformly at random without overlapping. The process terminates at saturation, i.e. until no more unit intervals can be stowed. In this paper, we present a generalization of this problem in which the unit intervals are placed with an exponential distribution with rate parameter $λ$. We show that the mathematical expectation of the number of intervals present at saturation satisfies a certain integral equation. Using Laplace transforms and Tauberian theorems, we investigate the asymptotic behavior of this function and describe a way to compute the corresponding limits for large $λ$. Then, we derive another integral equation for the derivative of this function and use it to compute the above limits for small $λ$ with the help of some asymptotic results for integral equations. We also show that the corresponding limits converge to the uniform case as $λ$ vanishes, yielding the well-known Renyi constant. Finally, we reveal the asymptotic behavior of the variance of the intervals at saturation.

math.PR↗

arXiv2021

Analysis of solutions of some multi-term fractional Bessel equations

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space $C^p$ is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation $$ \sum_{i=1}^{m}d_i\, x^{α_i}D^{α_i} u(x) + (x^β- ν^2)u(x)=0, \ \ α_i > 0, β> 0, $$ which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients $d_i$. Our theory leads to a threshold admissible value for $ν^2$, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.

math.AP↗