Main Theorem Research Articles

On the triviality of a family of linear hyperplanes

Let k be a field, m a positive integer, V an affine subvariety of Am+3 defined by a linear relation of the form x1r1⋯xmrmy=F(x1,…,xm,z,t), A the coordinate ring of V and G=X1r1⋯XmrmY−F(X1,…,Xm,Z,T). In [13], the second author had studied the case m=1 and had obtained several necessary and sufficient conditions for V to be isomorphic to the affine 3-space and G to be a coordinate in k[X1,Y,Z,T].In this paper, we study the general higher-dimensional variety V for each m⩾1 and obtain analogous conditions for V to be isomorphic to Am+2 and G to be a coordinate in k[X1,…,Xm,Y,Z,T], under a certain hypothesis on F. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds.We also describe the isomorphism classes and automorphisms of integral domains of the type A under certain conditions. These results show that for each d⩾3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic.

Ellipsoidal and hyperbolic Radon transforms; microlocal properties and injectivity

We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, R, which defines the integrals of a compactly supported L2 function, f, over ellipsoids and hyperboloids with centers on a smooth connected surface, S. Our transform is shown to be a Fourier Integral Operator (FIO) and in our main theorem we prove that R satisfies the Bolker condition if the support of f is contained in a connected open set that is not intersected by any plane tangent to S. Under certain conditions, this is an equivalence. We give examples where our theory can be applied. Focusing specifically on a cylindrical geometry of interest in Ultrasound Reflection Tomography (URT), we prove injectivity results and investigate the visible singularities. In addition, we present example reconstructions of image phantoms in two-dimensions and validate our microlocal theory.

Space-like quantitative uniqueness for parabolic operators

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a C1 potential V. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.

Model for Coupling Scheme and its Impact on Neurons Behaviour

Collective oscillations is typical phenomena observed in the systems biology including neurons. Investigating the mechanisms for which it occurs in neural networks evokes a significant interest among neuroscientists. From mathematical point of view, the coupling schemes rule the neuron’s behaviours ranging from microscopic to macroscopic scales. This paper aims to study the impacts of coupling schemes in a minimal network of two fully coupled identical oscillators (e.g., neurons). We proceed with the research by employing the numerical approach and time-series analysis. We consider both Kuramoto-like oscillator and Leaky integrate-and-fire neuron as the objects of study. In the former case, we found the phase of two oscillators are perfectly locked and stable if their frequency are identical, as stated by the main theorem. In the latter case, the membrane potentials of two neurons are perfectly synchronized, characterized by the same firing rate, due to the effect of excitatory delta pulses.

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010).

Invariant Synchrony and Anti-synchrony Subspaces of Weighted Networks

The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks. In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.

On a System of Hadamard Fractional Differential Equations with Nonlocal Boundary Conditions on an Infinite Interval

Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem.

Characterization of Jordan two-sided centralizers and related maps on triangular rings

Abstract The main goal of this paper is to characterize Jordan two-sided centralizers, Jordan centralizers and related maps on triangular rings without identity. As an application of our main theorem, we characterize Jordan generalized derivations on triangular rings. Precisely, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. As consequences, and apart from proving the other results, many known theorems can be either generalized or deduced.

New efficient bi-parametric families of iterative methods with engineering applications and their basins of attraction

We present in this paper two new families of bi-parametric multipoint higher order iterative methods of optimal order for determining simple roots of the nonlinear equation Ω(s)=0. The proposed families of methods are derivative-free with the optimal three-point fourth order methods requiring only three function evaluations per iteration and the optimal four-point eighth order methods consuming four function evaluations per iteration. Taylor’s series expansion and divided difference techniques are employed for the formulation of the methods. Their theoretical convergence properties are thoroughly analysed through the main theorems. Numerical experiments on nonlinear functions with some engineering applications are presented and are compared with some existing methods to demonstrate the effectiveness, applicability and validity of the proposed families of methods. Finally, graphical comparison is made through the basins of attraction which further provide useful information about their dynamical behaviour in the complex plane.

ENSURING DATA UNIQUENESS IN SEMANTIC NETWORKS

The article gives a brief description of knowledge representation models. Atoms of meaning (basic, minimal informational units) combined with each other to express a common meaning represent knowledge (data about data, metadata). It is shown that most of the existing knowledge representation models are based on the network representation model. A method is proposed to ensure the uniqueness (originality) of a set of data underlying the network model of knowledge representation. To ensure the uniqueness of knowledge representation by a set of data, the article proposes to use the main theorem of arithmetic: data are denoted by simple numbers (identifiers); when multiplying among themselves several prime numbers (data identifiers) in their totality conveying the aggregate semantic meaning, a natural number is obtained, which is an identifier of knowledge. The use of the basic theorem of arithmetic also provides a formalization of important data properties: internal interpretability, structuredness, connectivity, semantic metrics and activity. The presence of these properties in the data indicates that it is already data over data, or in other words, knowledge. In certain cases, this can reduce the computational complexity of the algorithm of the linguistic processor.

Sobolev inequalities and convergence for Riemannian metrics and distance functions

If one thinks of a Riemannian metric, $$g_1$$ , analogously as the gradient of the corresponding distance function, $$d_1$$ , with respect to a background Riemannian metric, $$g_0$$ , then a natural question arises as to whether a corresponding theory of Sobolev inequalities exists between the Riemannian metric and its distance function. In this paper, we study the sub-critical case $$p < \frac{m}{2}$$ where we show a Sobolev inequality exists between a Riemannian metric and its distance function. In particular, we show that an $$L^{\frac{p}{2}}$$ bound on a Riemannian metric implies an $$L^q$$ bound on its corresponding distance function. We then use this result to state a convergence theorem and show how this theorem can be useful to prove geometric stability results by proving a version of Gromov’s conjecture for tori with almost non-negative scalar curvature in the conformal case. Examples are given to show that the hypotheses of the main theorems are necessary.

The homotopy Lie algebra of a Tor-independent tensor product

In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S$, denoted $\pi(S)$, in terms of those of $R,S_1$ and $S_2$, where $S=S_1\otimes_R S_2$. Namely, $\pi(S)$ is the pullback of (restricted) Lie algebras along the maps $\pi(S_i)\to \pi(R)$ in a wide variety cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on Andre-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincare series of the common residue field of $R,S_1,S_2$ and $S$, and that the map $R\to S$ can never be Golod.

Numerical Analysis of a Structure-Preserving Space-Discretization for an Anisotropic and Heterogeneous Boundary Controlled $N$-Dimensional Wave Equation As a Port-Hamiltonian System

The anisotropic and heterogeneous N-dimensional wave equation, controlled and observed at the boundary, is considered as a port-Hamiltonian system. The recent structure-preserving Partitioned Finite Element Method is applied, leading directly to a finite-dimensional port-Hamiltonian system, and its numerical analysis is done in a general framework, under usual assumptions for finite element. Compatibility conditions are then exhibited to reach the best trade off between the convergence rate and the number of degrees of freedom for both the state error and the Hamiltonian error. Numerical simulations in 2D are performed to illustrate the optimality of the main theorems among several choices of classical finite element families.

Vector-valued Ruelle operator for weakly contractive IFS and Dini matrix potentials

The (scalar) Ruelle operator theory is well-known both in fractal geometry and dynamical systems. In this paper we consider vector-valued Ruelle operators for weakly contractive iterative function systems associated with Dini matrix potentials. We generalize the result in the paper (J. Math. Anal. Appl. 299 341–56) to weakly contractive iterated function system. More exactly, our main theorem gives a sufficient condition for the vector-valued Ruelle operator with the Perron–Frobenius property.

A New Plurisubharmonic Capacity and Functions Holomorphic Along Holomorphic Vector Fields

The main purpose of this article is to present a generalization of Forelli’s theorem for functions holomorphic along a suspension of integral curves of a diagonalizable vector field of aligned type. For this purpose, we develop a new capacity theory that generalizes the theory of projective capacity introduced by Siciak (Sophia Kokyuroku Math 14:1–96, 1982). Our main theorem improves the results of Kim et al. (J Geom Anal 19(3):655–666, 2009) and Cho (Complex Var. Elliptic Equ., 2022) as well as the original Forelli’s theorem.

Submixing and shift-invariant stochastic games

We study optimal strategies in two-player stochastic games that are played on a finite graph, equipped with a general payoff function. The existence of optimal strategies that do not make use of memory and randomisation is a desirable property that vastly simplifies the algorithmic analysis of such games. Our main theorem gives a sufficient condition for the maximizer to possess such a simple optimal strategy. The condition is imposed on the payoff function, saying the payoff does not depend on any finite prefix (shift-invariant) and combining two trajectories does not give higher payoff than the payoff of the parts (submixing). The core technical property that enables the proof of the main theorem is that of the existence of ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon$$\\end{document}-subgame-perfect strategies when the payoff function is shift-invariant. Furthermore, the same techniques can be used to prove a finite-memory transfer-type theorem: namely that for shift-invariant and submixing payoff functions, the existence of optimal finite-memory strategies in one-player games for the minimizer implies the existence of the same in two-player games. We show that numerous classical payoff functions are submixing and shift-invariant.

A note on a result of Aseeri and Kaspczyk about p-supersolvability of finite groups

Let G be a p-solvable finite group, where p is a prime dividing the order of G, and P be a Sylow p-subgroup of G. In this short note, we prove that the p-length of G is 1 if there is a subgroup H of P with such that H is s-semipermutable or s-permutably embedded in G. Our result not only simplifies, but also generalizes some main theorems of Aseeri and Kaspczyk [A result on s-semipermutable subgroups of finite groups and some applications, Commun. Algebra (2023)].

Stability analysis based on a control adjuster for switched neural networks by trajectory similarity

This paper focuses on a class of exponential stability control problem for the stochastic neural networks (SNNs) with time‐varying delays and Markov jump (TVDMJ). As a prerequisite to the main theorem, the existence and uniqueness of the solution for the main system are proved via contraction map principle. For the stability control of the system, we design a controller which can be adjusted the parameters to make the system stabilization. We use this controller to actively control the stability behavior of the system, rather than the papers by setting the appropriate parameters to make the system achieve a stable state. Based on the intermittent control, we design a novel trajectory similarity process control and obtain exponential stability results, which are less conservative than the existing theoretical results. Two examples based on some numerical calculation and simulation diagrams are presented to validate the effectiveness for the utilized techniques.

Min–max multi-step barrier options and their variants

This paper studies a new type of barrier option, min–max multi-step barrier options with diverse multiple up or down barrier levels placed in the sub-periods of the option’s lifetime. We develop the explicit pricing formula of this type of option under the Black–Scholes model and explore its applications and possible extensions. In particular, the min–max multi-step barrier option pricing formula can be used to approximate double barrier option prices and compute prices of complex barrier options such as discrete geometric Asian barrier options. As a practical example of directly applying the pricing formula, we introduce and evaluate a re-bouncing equity-linked security. The main theorem of this work is capable of handling the general payoff function, from which we obtain the pricing formulas of various min–max multi-step barrier options. The min–max multi-step reflection principle, the boundary-crossing probability of min–max multi-step barriers with icicles, is also derived.

Expanders on matrices over a finite chain ring, I

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is shown that [Formula: see text] is a moderate expander on [Formula: see text] matrices with exponent [Formula: see text]. These results generalize the main theorems in the recent work [29] of Xie and Ge. The proofs use spectral graph theory and elementary divisor theory.