Classical Theorem Research Articles

Vertex-minors of graphs: A survey

For a vertex v of a graph, the local complementation at v is an operation that replaces the neighborhood of v by its complement graph. Two graphs are locally equivalent if one is obtained from the other by a sequence of local complementations. A graph H is a vertex-minor of a graph G if H is an induced subgraph of a graph locally equivalent to G. Although this concept was introduced in the 1980s, it was not widely known and except for the survey paper of Bouchet published in 1990, there is no comprehensive survey listing all the new developments. We survey classic and recent theorems and conjectures on vertex-minors and related concepts such as circle graphs, cut-rank functions, rank-width, and interlace polynomials.

Stability and localization of the Lp Bergman kernel

The aims of this paper are twofold. First, we generalize the classical Ramadanov theorem and Skwarczyński theorem for the [Formula: see text] Bergman kernels to the [Formula: see text] case, which are concerned with the compact convergence of [Formula: see text] Bergman kernels on an increasing or decreasing sequence of domains in [Formula: see text]. Second, we prove a localization principle for the [Formula: see text] Bergman kernel on bounded strongly pseudoconvex domains with smooth boundary. Utilize the localization, we study the asymptotic behavior of the [Formula: see text] Bergman kernel near strongly pseudoconvex boundary points and show that the [Formula: see text]-Schwarz content is identical to [Formula: see text] on a neighborhood of a strongly pseudoconvex boundary point.

Chen–Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples.

On weighted generator of triple sequences and its Tauberian conditions

This paper presents a new perspective on the relationship between the (N, p, q, r) method and P-convergence for triple sequences.Our primary goal is to derive Tauberian conditions that dictate the behavior of the weighted generator sequence (zlmn) in relation to the sequences (Pl), (Qm), and (Rn), with the intention of establishing a fresh interpretation. These conditions control the OL- and O-oscillatory properties and establish a connection from ( N, p, q, r) summability to P-convergence, subject to certain restrictions on the weight sequences. Furthermore, we demonstrate that specific cases, such as the OL-condition of Landau type and the O-condition of Hardy type with respect to (Pl), (Qm) and (Rn), serve as Tauberian conditions for (N, p, q, r) summability under additional conditions. Consequently, our findings encompass all the classical Tauberian theorems, including conditions related to slow decrease and slow oscillation in specific contexts.

Conditions for equality in Anderson’s theorem

The classical Anderson theorem is a well-known result relevant to multivariate statistics and probability. It establishes an integral inequality for symmetric quasiconcave functions over a shifted symmetric convex set. The aim of this note is twofold. First, we provide necessary and sufficient conditions for equality in Anderson’s theorem, extending the result of Soms (1991). Second, we propose a set of tractable requirements that guarantee strict inequality in Anderson’s theorem. Our results are used to characterize the independence of Gaussian-distributed random variables.

On Protected Quasi-Metrics

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations.

Linearized asymptotic stability for nabla Riemann-Liouville fractional difference equation

In this paper,we present a theorem about stability of nonlinear fractional difference equation with Riemann-Liouvile difference operator. The result is a version of classical theorem on linear approximation and to derive them,we prove the variation of constants formula for nabla Riemann- Liouville fractional difference equations.We also present some results concerning the existence and uniqueness of the equation under consideration.

An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

AbstractWe consider integral area-minimizing 2-dimensional currents $T$ T in $U\subset \mathbf {R}^{2+n}$ U ⊂ R 2 + n with $\partial T = Q\left [\!\![{\Gamma }\right ]\!\!]$ ∂ T = Q 〚 Γ 〛 , where $Q\in \mathbf {N} \setminus \{0\}$ Q ∈ N ∖ { 0 } and $\Gamma $ Γ is sufficiently smooth. We prove that, if $q\in \Gamma $ q ∈ Γ is a point where the density of $T$ T is strictly below $\frac{Q+1}{2}$ Q + 1 2 , then the current is regular at $q$ q . The regularity is understood in the following sense: there is a neighborhood of $q$ q in which $T$ T consists of a finite number of regular minimal submanifolds meeting transversally at $\Gamma $ Γ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$ Q = 1 . As a corollary, if $\Omega \subset \mathbf {R}^{2+n}$ Ω ⊂ R 2 + n is a bounded uniformly convex set and $\Gamma \subset \partial \Omega $ Γ ⊂ ∂ Ω a smooth 1-dimensional closed submanifold, then any area-minimizing current $T$ T with $\partial T = Q \left [\!\![{\Gamma }\right ]\!\!]$ ∂ T = Q 〚 Γ 〛 is regular in a neighborhood of $\Gamma $ Γ .

Arboreal categories and equi-resource homomorphism preservation theorems

The classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a first-order sentence φ is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence ψ. Given a notion of (syntactic) complexity of sentences, an “equi-resource” homomorphism preservation theorem improves on the classical result by ensuring that ψ can be chosen so that its complexity does not exceed that of φ.We describe an axiomatic approach to equi-resource homomorphism preservation theorems based on the notion of arboreal category. This framework is then employed to establish novel homomorphism preservation results, and improve on known ones, for various logic fragments, including first-order, guarded and modal logics.

On the Equivalence in ZF+BPI of the Hahn–Banach Theorem and Three Classical Theorems

On the Equivalence in ZF+BPI of the Hahn–Banach Theorem and Three Classical Theorems

A New Notion of Fuzzy Function Ideal Convergence

P.M. Pu and Y.M. Liu extended Moore-Smith’s convergence of nets to fuzzy topology and Y.M. Liu provided analogous results to J. Kelley’s classical characterization theorem of net convergence by introducing the notion of fuzzy convergence classes. In a previous paper, the authors of this study provided modified versions of this characterization by using an alternative notion of convergence of fuzzy nets, introduced by B.M.U. Afsan, named fuzzy net ideal convergence. Our main scope here is to generalize and simplify the preceding results. Specifically, we insert the concept of a fuzzy function ideal convergence class, L, on a non-empty set, X, consisting of triads (f,e,I), where f is a function from a non-empty set, D, to the set FP(X) of fuzzy points in X, which we call fuzzy function, e∈FP(X), and I is a proper ideal on D, and we provide necessary and sufficient conditions to establish the existence of a unique fuzzy topology, δ, on X, such that (f,e,I)∈L iff fI-converges to e, relative to the fuzzy topology δ.

Toric differential forms and periods of complete intersections

Let n be an even natural number. We compute the periods of any n2-dimensional complete intersection algebraic cycle inside an n-dimensional non-degenerated intersection of a projective simplicial toric variety. Using this information we determine the cycle class of such algebraic cycles. As part of the proof we develop a toric generalization of a classical theorem of Macaulay about complete intersection Artin Gorenstein rings, and we generalize an algebraic cup formula for residue forms due to Carlson and Griffiths to the toric setting.

Note on Hamiltonian Graphs in Abelian 2-Groups

We analyze a graph G whose vertices are subgroups of ℤ2k isomorphic to ℤ2 × ℤ2. Two vertices are joined if their respective subgroups have nontrivial intersection. We prove that such a graph is 6(2k−2 − 1)-regular. If a graph is regular, a classical theorem by Ore claims that a graph is Hamiltonian if the degree of any vertex is at least one half of the number of vertices. Using Ore’s theorem, we show that G is Hamiltonian for k ∈{3, 4}. Ore’s theorem cannot be applied when k ≥ 5. Nevertheless, we manage to construct a Hamiltonian cycle for k = 5. Our construction uses orbits of one ℤ24 group under an action of an automorphism of order 31. It is highly likely that this approach could be generalized for k > 5.

Classification of 𝒪_{∞}-Stable 𝒞*-Algebras

I present a proof of Kirchberg’s classification theorem: two separable, nuclear, O ∞ \mathcal O_\infty -stable C ∗ C^\ast -algebras are stably isomorphic if and only if they are ideal-related K K KK -equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.

On the zero bounds of quaternionic polynomials with restricted coefficients

The primary objective of this paper is to explore extensions of the classical Enestorm-Kakeya theorem and its diverse generalizations regarding the distribution of polynomial zeros, transitioning from the complex domain to the quaternionic context.

Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel

We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem.

A Comment on: “A Modern Gauss–Markov Theorem”

We show that Theorem 4 in Hansen (2022) applies to exactly the same class of estimators as does the classical Aitken theorem. We furthermore point out that Theorems 5–7 in Hansen (2022) contain extra assumptions not present in the classical Gauss–Markov or Aitken theorem, and thus the former theorems do not contain the latter ones as special cases.

Sequences of independent functions and structure of rearrangement invariant spaces

The main aim of the survey is to present results of the last decade on the description of subspaces spanned by independent functions in $L_p$-spaces and Orlicz spaces on the one hand, and in general rearrangement invariant spaces on the other. A new approach is proposed, which is based on a combination of results in the theory of rearrangement invariant spaces, methods of the interpolation theory of operators, and some probabilistic ideas. The problem of the uniqueness of the distribution of a function such that a sequence of its independent copies spans a given subspace is considered. A general principle is established for the comparison of the complementability of subspaces spanned by a sequence of independent functions in a rearrangement invariant space on $[0,1]$ and by pairwise disjoint copies of these functions in a certain space on the half-line $(0,\infty)$. As a consequence of this principle we obtain, in particular, the classical Dor-Starbird theorem on the complementability of subspaces spanned by independent functions in the $L_p$-spaces. Bibliography: 103 titles.

A class of definite integrals involving generalized hypergeometric functions

Recently Masjed-Jamei and Koepf obtained generalizations of various classical summation theorems for the 2F1, 3F2, 4F3, 5F4 and 6F5 generalized hypergeometric series. We evaluate a new class of integrals involving generalized hypergeometric function by employing the results given by Masjed- Jamei and Koepf and MacRobert integral, and we give several special cases.

Distributions of zeros and masses of entire and subharmonic functions with restrictions on their growth along the strip

Let $\mathrm Z$ and $\mathrm W$ be distributions of points on the complex plane $\mathbb C$. The following problem dates back to F. Carlson, T. Carleman, L. Schwartz, A. F. Leont'ev, B. Ya. Levin, J.-P. Kahane, and others. For which $\mathrm Z$ and $\mathrm W$, for an entire function $g\neq 0$ of exponential type which vanishes on $\mathrm W$, there exists an entire function $f\neq 0$ of exponential type that vanishes on $\mathrm Z$ and is such that $|f|\leqslant |g|$ on the imaginary axis? The classical Malliavin-Rubel theorem of the early 1960s completely solves this problem for "positive" $\mathrm Z$ and $\mathrm W$ (which lie only on the positive semiaxis). Several generalizations of this criterion were established by the author of the present paper in the late 1980s for "complex" $\mathrm Z \subset \mathbb C$ and $\mathrm W\subset \mathbb C$ separated by angles from the imaginary axis, with some advances in the 2020s. In this paper, we solve more involved problems in a more general subharmonic framework for distributions of masses on $\mathbb C$. All the previously mentioned results can be obtained from the main results of this paper in a much stronger form (even for the initial formulation for distributions of points $\mathrm Z$ and $\mathrm W$ and entire functions $f$ and $g$ of exponential type). Some results of the present paper are closely related to the famous Beurling-Malliavin theorems on the radius of completeness and a multiplier.