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A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…
We generalize the notions of the St\"ackel transform and the coupling constant metamorphosis to quasi-exactly solvable systems. We discover that for a variety of one-dimensional and separable multidimensional quasi-exactly solvable systems,…
The Classic Howard's algorithm, a technique of resolution for discrete Hamilton-Jacobi equations, is of large use in applications for its high efficiency and good performances. A special beneficial characteristic of the method is the…
This paper introduces a deterministic algorithm for solving an instance of the Subset Sum Problem based on a new method entitled the Bipartite Synthesis Method. The algorithm is described and shown to have worst-case limiting performance…
We prove the existence of a unitary transformation that enables two arbitrarily given Hamiltonians in the same Hilbert space to be transformed into one another. The result is straightforward yet, for example, it lays the foundation to…
A detailed study of solutions to the first order partial differential equation H(x,y,z_x,z_y)=0, with special emphasis on the eikonal equation z_x^2+z_y^2=h(x,y), is made near points where the equation becomes singular in the sense that…
Metamorphism is a recently introduced integral transform, which is useful in solving partial differential equations. Basic properties of metamorphism can be verified by direct calculations. In this paper we present metamorphism as a sort of…
In this revised version, the mistake of the author confusing the weak transform and strict transform, pointed out by E. Bierstone, is corrected. It gives a self-contained proof of (embedded) resolution of singularities over a field of…
This paper introduces the theory and hardware implementation of two new algorithms for computing a single component of the discrete Fourier transform. In terms of multiplicative complexity, both algorithms are more efficient, in general,…
The local zero structure of a smooth map may qualitatively change, when the map is subjected to small perturbations. The changes may include births and/or deaths of zeros. The qualitative properties are defined as the invariances of an…
This article proposes a new approach in the treatment of the Hilbert transform and some cases of the Fourier transform whose improper integrals are principal values. This approach may be useful for teaching these issues to undergraduate…
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…
We analyze the convergence of the Conjugate Gradient (CG) method in exact arithmetic, when the coefficient matrix $A$ is symmetric positive semidefinite and the system is consistent. To do so, we diagonalize $A$ and decompose the algorithm…
In this note, we characterize the solution of a system of elliptic integro-differential equations describing a phe-notypically structured population subject to mutation, selection and migration. Generalizing an approach based on…
Verification theorems are key results to successfully employ the dynamic programming approach to optimal control problems. In this paper we introduce a new method to prove verification theorems for infinite dimensional stochastic optimal…
In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a variational problem with constraints. Then, our abstract formulation is applied to study several optimization setups including closed CMC…
A method is proposed for solving equality constrained nonlinear optimization problems involving twice continuously differentiable functions. The method employs a trust funnel approach consisting of two phases: a first phase to locate an…
The Computational Algebraic Geometry applied in Algebraic Statistics; are beginning to exploring new branches and applications; in artificial intelligence and others areas. Currently, the development of the mathematics is very extensive and…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
Here, we consider periodic homogenization for time-fractional Hamilton--Jacobi equations. By using the perturbed test function method, we establish the convergence, and give estimates on a rate of convergence. A main difficulty is the…