Classical Theorem Research Articles

Interaction of stream-wise vortices generated by swirler grid

A system of stream-wise vortices has been created using a grid of swirling elements with alternating orientations (like a chessboard). The particle image velocimetry method has been used to map the velocity field in several planes perpendicular to the stream. The mesh-based Reynolds number is 1.35×104 and 2.71×104, respectively. The stream-wise development of turbulent kinetic energy (TKE) shows first an increase in a distance of x≈10M, followed by power-law decay. Individual vortices are detected in each snapshot. The radial profile of TKE transformed to a vortex coordinate system is almost constant, either with maximum as in static frame or zero as observed by previous research. The properties of detected vortices are studied statistically: the meandering amplitude expressed as the standard deviation of vortex positions grows roughly as ∼ex, i.e., faster than expected random-walk growth ∼x. Vortex circulation decays exponentially as predicted by classical Helmholtz theorem. The interaction between neighboring vortices is expressed via correlation of selected quantities. Correlation of energy develops downstream from anticorrelation to a positive correlation. The strongest correlation is observed between the first vortex circulation and the second vortex position perpendicular to their connection line. Other correlations are weak.

On application of Gaussian kernels and Laplace functions combined with Kolmogorov's theorem for approximation of functions of several variables

A special class of approximations of measurable functions of several variables on the unit coordinate cube is investigated. The class is constructed on the base of Kolmogorov's theorem (in version by Sprecher–Golubkov) stating that a continuous function $f$ of several variables can be represented as a finite superposition of continuous single-variable functions — so called outer functions (which depend of $f$) and inner one $\Psi$ (which is independent of $f$ and is monotone). In the case of continuous functions $f$ the class under study is obtained with outer functions approximated by linear combinations of quadratic exponentials (also known as Gaussian functions or Gaussian kernels) and with the inner function $\Psi$ approximated by Laplace functions. As is known, a measurable function $f$ can be approximated by a continuous one (up to a set of small measure) with the help of classical Luzin's theorem. The effectiveness of such approach is based on assertions that, firstly, the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions, and, secondly, that a continuous monotone function on such an interval can be approximated as accurately as desired by a linear combination of translations and dilations of the Laplace integral (in other words, Laplace functions). It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous and piecewise continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

A generalization of the Hawking black hole area theorem

Hawking’s black hole area theorem was proven using the null energy condition (NEC), a pointwise condition violated by quantum fields. The violation of the NEC is usually cited as the reason that black hole evaporation is allowed in the context of semiclassical gravity. Here we provide two generalizations of the classical black hole area theorem: first, a proof of the original theorem with an averaged condition, the weakest possible energy condition to prove the theorem using focusing of null geodesics. Second, a proof of an area-type result that allows for the shrinking of the black hole horizon but provides a bound on it. This bound can be translated to a bound on the black hole evaporation rate using a condition inspired from quantum energy inequalities. Finally, we show how our bound can be applied to two cases that violate classical energy conditions.

A Geometric Approach to Polynomial and Rational Approximation

Abstract We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.

A generalization of Bondy’s pancyclicity theorem

AbstractThe bipartite independence number of a graph $G$ , denoted as $\tilde \alpha (G)$ , is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$ , there is an edge between $A$ and $B$ . McDiarmid and Yolov showed that if $\delta (G)\geq \tilde \alpha (G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $\delta (G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$ . In this paper, we show that $\delta (G)\geq \tilde \alpha (G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$ , thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.

Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention

Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. Characteristic examples are nonautonomous and discontinuous systems due to the presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists of designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). Finally, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid’s diffusion.

Efficient Normalization of Linear Temporal Logic

In the mid 1980s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of Linear Temporal Logic (LTL) with past operators) is equivalent to a formula of the form \(\bigwedge _{i=1}^n {\mathbf {G}}{\mathbf {F}}\varphi _i \vee {\mathbf {F}}{\mathbf {G}}\psi _i\) , where φ i and ψ i contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalization procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present direct and purely syntactic normalization procedures for LTL, yielding a normal form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalizes the formula, translates it into a special very weak alternating automaton, and applies a simple determinization procedure, valid only for these special automata.

On the failure of Ornstein theory in the finitary category

We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai’s factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [Finitary isomorphism of m m -dependent processes, Amer. Math. Soc., Birkhauser, Providence, RI, 1992, pp. 373–376], which was first suggested by D. Rudolph [A characterization of those processes finitarily isomorphic to a Bernoulli shift, Birkhäuser, Boston, Mass., 1981, pp. 1–64]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai’s factor theorem for ergodic processes. Another consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [Math. Inst. Hautes Études Sci. 128 (2018), 1–119].

On the Coincidence Theorem

We are proving Coincidence theorem due to Walsh for the case when the total degree of a polynomial is less than the number of arguments. Also, the following result has been proven: if $$p(z)$$ is a complex polynomial of degree $$n$$, then closed disk D that contains at least $$n-1$$ of its zeros (counting multiplicity) contains at least $$\left[\frac{n-2k+1}{2} \right]$$ zeros of its $$k$$-th derivative, provided that the arithmetical mean of these zeros is also centre of D. We also prove a variation of the classical composition theorem due to Szegö.

Computational approach in obtaining analytic solutions of a generalized nonlinear breaking soliton equation with applications in engineering and physics

Higher-order nonlinear wave models have been a source of attraction to a huge number of researchers in recent times as a result of their significance in mathematical physics, other nonlinear sciences as well as engineering. In consequence, we outline in this paper the analytical studies entrenched on a generalized structure of a nonlinear breaking soliton equation with higher-order nonlinearity in four variables which have applications in science as well as engineering. Lie group theory is utilized to generate an 11-dimensional Lie algebra associated with the equation under consideration and in addition one parametric group of transformations related to the algebra is calculated. Besides, the technique is further invoked in performing reductions of the various subalgebras of the understudy model. Moreover, in conjunction with the theory, the direct integration technique is engaged to secure an analytic solution of the equation and as a result, a general analytic solution with regard to the second-kind elliptic-integral function is furnished. Moreover, we engage the novel simplest equation technique to gain more general solutions to the equation. In consequence, solitonic solutions comprising periodic, dark-bright, topological kink as well as singular solutions are achieved. We complemented that by exhibiting the dynamics of the secured solutions with the aid of graphical representations. In conclusion, we calculate conserved quantities associated with the aforementioned equation by invoking the well-celebrated classical Noether theorem technique.

Iteration with Bisection to Approximate the Solution of a Boundary Value Problem

Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green’s function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point problem into multiple fixed-point problems that one can easily iterate to approximate a solution of a differential equation satisfying one boundary condition, then apply a bisection method in an intermediate value theorem argument to meet a second boundary condition. Error estimates on the iterates are also established. The technique will be illustrated on a second-order right focal boundary value problem, with an example provided showing how to apply the results.

Non-parametric Representation Learning with Kernels

Unsupervised and self-supervised representation learning has become popular in recent years for learning useful features from unlabelled data. Representation learning has been mostly developed in the neural network literature, and other models for representation learning are surprisingly unexplored. In this work, we introduce and analyze several kernel-based representation learning approaches: Firstly, we define two kernel Self-Supervised Learning (SSL) models using contrastive loss functions and secondly, a Kernel Autoencoder (AE) model based on the idea of embedding and reconstructing data. We argue that the classical representer theorems for supervised kernel machines are not always applicable for (self-supervised) representation learning, and present new representer theorems, which show that the representations learned by our kernel models can be expressed in terms of kernel matrices. We further derive generalisation error bounds for representation learning with kernel SSL and AE, and empirically evaluate the performance of these methods in both small data regimes as well as in comparison with neural network based models.

Polynomial extension of some symmetric partition regular structures

The study of symmetric structures is a new trend in combinatorial number theory. Recently in [9], Di Nasso proved symmetrized versions of some classical theorems like Hindman's theorem, Van der Waerden's theorem and Deuber's theorem. This opens the question of which other classical theorems can be symmetrized, as well as if other symmetric operations allow for such generalizations. Here we give positive answers to both questions, by showing that symmetrization of the polynomial extension of Van der Waerden's and Deuber's Theorem is possible.

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.