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For every affine variety over a global function field, we show that the set of its points with coordinates in an arbitrary rank-one multiplicative subgroup of this function field is topologically dense in the set of its points with…
Conformal symmetry is taken as an attribute of theories of massless fields in manifolds with specific dimensionalities. This paper shows that this is not an absolute truth; it is a consequence of the mathematical representation used for the…
This paper shows that finitely additive measures occur naturally in very general Divergence Theorems. The main results are two such theorems. The first proves the existence of pure normal measures for sets of finite perime- ter, which yield…
Using the Observable form of Maxwell's equations, we reveal that effective parameters at materials boundaries emerge naturally as anisotropic transfer functions. The complexity of the boundary dictates the order of these functions.…
Fixed point theorems are one of the many tools used to prove existence and uniqueness of differential equations. When the data involved contains products of distributions, some of these tools may not be useful. Thus rises the necessity to…
We characterize the set of positive harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of several different chambers. We analyze the asymptotic behavior of the solutions in connection with the changes…
Singularities, such as poles and branch points, play a crucial role in investigating the analytic properties of scattering amplitudes that inform new computational techniques. In this note, we point out that scattering amplitudes can also…
Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a…
Single- and multi-valued solutions of homogeneous Maxwell equations in vacuum are considered, with ''sources'' formed by the (point- or string-like) singularities of the field strengths and, generally, irreducible to any delta-functions'…
This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point…
In this work we prove uniqueness result for an implicit discrete system defined on connected graphs. Our discrete system is motivated from a certain class of spatial segregation of reaction-diffusion equations.
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…
By relating the number of images of a function with finite domain to a certain parameter, we obtain both an upper and lower bound for the image set. Even though the arguments are elementary, the bounds are, in some sense, best possible. The…
This article presents an elementary proof of the Implicit Function Theorem for differentiable maps F(x,y), defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base point. In the…
A correspondence between arbitrary Fourier series and certain analytic functions on the unit disk of the complex plane is established. The expression of the Fourier coefficients is derived from the structure of complex analysis. The…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
In this paper we study the existence and uniqueness of fixed points of a class of mappings defined on complete, (sequentially compact) cone metric spaces, without continuity conditions and depending on another function.
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
In this paper, we prove some uniqueness theorems concerning the derivatives of meromorphic functions when they share three sets. The obtained results improve some recent existing results.
We study obstacle problems governed by two distinct types of diffusion operators involving interacting free boundaries. We obtain a somewhat surprising coupling property, leading to a comprehensive analysis of the free boundary. More…