Bernstein Theorem Research Articles

Cantor–Bernstein Theorem for Pseudo-BCK-Algebras

We prove that if A and B are orthogonally σ-complete commutative pseudo-BCK-algebras such that A is isomorphic to a direct factor in B, and also B is isomorphic to a direct factor in A, then A and B are isomorphic. As a consequence we obtain previously known results for MV-algebras (by De Simone, Mundici and Navara), pseudo-MV-algebras (by Jakubik) and lattice-ordered groups (again by Jakubik).

Jackson and Bernstein theorems for the weight exp(−|x|) on ℝ

In 1978, Freud, Giroux and Rahman established a weightedL 1 Jackson theorem for the weight exp(−|x|) on the real line, using methods that work only inL 1. This weight is somewhat exceptional, for it sits on the boundary between weights like exp(-|x|α), α≥1, where weighted polynomials are dense, and the case α<1, where weighted polynomials are not dense. We obtain the firstL p Jackson theorem for exp(−|x|), valid in allL p , 0<p≤∞, as well as for higher order moduli of continuity. We also establish a converse Bernstein type theorem, characterizing rates of approximation in terms of smoothness of the approximated function.

A Bernstein-type theorem for Riemannian manifolds with a Killing field

The classical Bernstein theorem asserts that any complete minimal surface in Euclidean space \(\mathbb{R}^3\) that can be written as the graph of a function on \(\mathbb{R}^2\) must be a plane. In this paper, we extend Bernstein’s result to complete minimal surfaces in (may be non-complete) ambient spaces of non-negative Ricci curvature carrying a Killing field. This is done under the assumption that the sign of the angle function between a global Gauss map and the Killing field remains unchanged along the surface. In fact, our main result only requires the presence of a homothetic Killing field.

Bernstein theorems and transformations of correlation measures in statistical physics

We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products (·, ⋆) and the maps K, K−1 which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): F(γ) → F⋃(γ) = {F(α⋃β)}α,β⊂γ. For the operators F⋃(γ) and F⋂(γ), we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function F(γ).

An analogue of the Bernstein theorem for the cylinder

In 1941 S. N. Bernstein proved the following characterization theorem. Let ξ1 and ξ2 be independent random variables. If ξ1 + ξ2 and ξ1 - ξ2 are independent, then ξ1 and ξ2 are Gaussian with equal dispersion. This paper is devoted to a group analogue of the Bernstein theorem. Let X be a locally compact separable abelian group and ξ1, ξ2 be independent random X-valued variables with distributions μ1, μ2. Denote by $$ \user1{\mathbb{T}} $$ and by ∑ a the group of rotations of the circle and the a-adic solenoid, respectively. In this paper we solve the following problem: In the case where either $$ X = \user1{\mathbb{R}} \times \user1{\mathbb{T}} $$ or X = ∑ a we describe all possible distributions μ1 and μ2 under the assumption that ξ1 + ξ2 and ξ1 - ξ2 are independent.

Quantum flag varieties, equivariant quantum [formula omitted]-modules, and localization of quantum groups

Let O q ( G ) be the algebra of quantized functions on an algebraic group G and O q ( B ) its quotient algebra corresponding to a Borel subgroup B of G. We define the category of sheaves on the “quantum flag variety of G” to be the O q ( B ) -equivariant O q ( G ) -modules and prove that this is a proj-category. We construct a category of equivariant quantum D -modules on this quantized flag variety and prove the Beilinson–Bernstein's localization theorem for this category in the case when q is transcendental.

Stable constant mean curvature hypersurfaces in ℝ n+1 and ℍ n+1(−1)

In this paper, we show that all complete stable hypersurfaces in ℝ n+1(or ℍ n+1 (-1)) (n = 3, 4, 5) with constant mean curvature H > 0 (or H > 1, respectively) and finite L 2 norm of traceless second fundamental form are compact geodesic spheres. Keywords: stable hypersurface, constant mean curvature, isometric immersion, Bernstein theorem.

On the singularities and a Bernstein property of immersed stable minimal hypersurfaces

We prove: (1) a classification theorem for certain singularities of weak limits of immersed stable minimal hypersurfaces of arbitrary dimension and (2) a Bernstein theorem for complete immersed stable minimal hypersurfaces of dimension \(\leq 6\).

Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry

We consider a family of sparse polynomial systems denned by a directed graph and a bipartite graph which depend on certain parameters. A convex polyhedral cone serves as a representative of all positive solutions of the family. We study the boundary of this cone with Bernstein's second theorem and Viro's method. In particular we present new results about the parameter regions where several positive solutions appear. Since they are steady states of an underlying dynamical system of mass action kinetics, the resulting multistationarity has important implications for the dynamics of that system. Examples from applications illustrate the theoretical results.

Bochner and Bernstein theorems via the nuclear integral representation theorem

We show how, using the nuclear integral representation theorem, the Bernstein–Choquet theorem and Bochner–Schwartz theorem may be derived. In the case of the Bernstein–Choquet theorem we give an example, determining the representing measure explicitly.

On stable complete hypersurfaces with vanishing {$r$}-mean curvature

A form of Bernstein theorem states that a complete stable minimal surface in euclidean space is a plane. A generalization of this statement is that there exists no com- plete stable hypersurface of an n-euclidean space with vanishing (n − 1)-mean curvature and nowhere zero Gauss-Kronecker curvature. We show that this is the case, provided the immer- sion is proper and the total curvature is finite.

A Bernstein theorem for complete spacelike hypersurfaces of constant mean curvature in Minkowski space

In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).

Bernstein's 'Lethargy' Theorems in SF-Spaces

We prove a version of Bernstein's 'lethargy' theorem for cones in the class of SF-spaces. Moreover, an analogue of the Bernstein theorem for linear projections onto closed subspaces of an SF-space is obtained. This extends results given by G. Lewicki.

Spherical t-designs and the Bernstein Theorem

In this paper we study the construction problem of spherical t-designs as a generalization of the Chebyshev problem of Gauss type quadrature formula. It is known that spherical t-designs can be constructed if we can construct interval t-designs for the weight function (1−x)d−22. We shall show the exsitence and non-existence theorems of some interval t-desings.

On Isomorphisms of Relations Embedded into Each Other

In this paper, we consider analogs of the Cantor--Bernstein theorem for sets with binary relations. In Sec. 1, we prove an analog of this theorem for arbitrary binary relations; in Sec. 2, we consider an application; in Sec. 3, we study a class of relations with the “Cantor-- Bernstein property” and a class of exact relations, and prove that these classes are closed under certain operations.

Amoebas, Monge-Ampàre measures, and triangulations of the Newton polytope

The amoeba of a holomorphic function f is, by definition, the image in R n of the zero locus of f under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function N f with the property that its Monge-Ampére mass is concentrated to the amoeba of f We obtain results of two kinds; by approximating N f with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of f; by computing the Monge-Ampére measure, we find sharp bounds for the area of amoebas in ℝ . We also consider systems of functions f 1 ,…, f n and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.

Dirichlet series and simultaneous observability: two problems solved by the same approach

In a classical paper, Ingham gave a simple proof of an important theorem of Pólya on singular points of Dirichlet series under a uniform gap assumption on the exponents. Bernstein generalized Pólya's theorem by weakening this gap condition. We give a simpler proof of Bernstein's theorem by applying a recent generalization of Ingham's theorem. Furthermore, we also solve a simultaneous observability problem by using this theory.

Some metric properties of α-continued fractions

The α-continued fraction is a modification of the nearest integer continued fractions taking n as the integer part of y when y∈[ n−1+ α, n+ α), instead of the nearest integer. For x∈[ α−1, α), we have the following α-continued fraction expansion: x= ε 1| |c 1 + ε 2| |c 2 + ε 3| |c 3 +⋯ with c n ⩾1 and ε n =±1 for n⩾1. We prove the Borel–Bernstein theorem for α-continued fractions and also discuss some metrical properties related to max 1⩽n⩽N c n . Indeed, we prove that lim inf N→∞ log log N max 1⩽n⩽N c n N and lim N→∞ ∑ n=1 N c n− max 1⩽n⩽N c n N log N exist and have the same constant for almost every x.

A Bernstein theorem for special Lagrangian graphs

We obtain a Bernstein theorem for special Lagrangian graphs in \({\mathbb C}^n ={\mathbb R}^{2n} \) for arbitrary n only assuming bounded slope but no quantitative restriction.

COMPACTNESS AND GENERALIZED APPROXIMATION SPACES

ABSTRACT We show that generalized approximation spaces can be used to describe the relatively compact sets of Banach spaces. This leads to compactness and convergence criteria in the approximation spaces themselves. If these spaces can be described with the help of moduli of smoothness, then the criteria can be formulated in terms of the moduli. As applications we give a generalization of Bernstein's theorem about existence of elements with prescribed best approximation errors, compactness criteria for operators, a criterion for compactness in Sobolev type spaces, and a generalization of Simon's compactness criterion for subsets of Lp -spaces of Banach-space-valued functions.