Sufficient Conditions Research Articles

A New Sufficient & Necessary Condition for Testing Linear Separability Between Two Sets.

As a fundamental mathematical problem in the field of machine learning, the linear separability test still lacks a theoretically complete and computationally efficient method. This paper proposes and proves a sufficient and necessary condition for linear separability test based on a sphere model. The advantage of this test method is two-fold: (1) it provides not only a qualitative test of linear separability but also a quantitative analysis of the separability of linear separable instances; (2) it has low time cost and is more efficient than existing test methods. The proposed method is validated through a large number of experiments on benchmark datasets and artificial datasets, demonstrating both its correctness and efficiency.

Enhancing vehicular platoon stability in the presence of communication Cyberattacks: A reliable longitudinal cooperative control strategy

This study proposes a reliable longitudinal distributed control strategy for connected and automated vehicles (CAVs) in the presence of communication cyberattacks. The proposed strategy is designed to utilize dependable measurements from onboard sensors to evaluate the reliability of vehicle-to-vehicle (V2V) information and enhance the platoon string stability in an integrated manner. Specifically, this paper proposes a real-time reliability evaluation mechanism for the ego vehicle based on sensing and communication information from multiple predecessors, employing an upper-tailed chi-squared test. The mechanism is then incorporated into a distributed multi-predecessor linear feedback controller to adaptively weigh the preceding vehicles’ information, ensuring robustness against cyberattacks and maintaining the string stability of the vehicular platoon. To better understand the whole framework, sufficient conditions of true string stability and pseudo string stability are theoretically derived, unveiling the disturbance amplification jointly impacted by measurement accuracy of onboard sensors, cyberattack severity in V2V communication, and control parameters. Numerical experiments are conducted to validate the controller across various types of communication cyberattacks. The results suggest that the proposed control strategy can significantly alleviate the impact of cyberattacks and simultaneously dampen traffic oscillations effectively.

The rise (and fall) of tech clusters

Tech clusters play a growing role in knowledge-based economies by accommodating high-tech firms and providing an environment that fosters location-dependent knowledge spillovers and promote R&D investments by firms. Yet, not much is known about the economic conditions under which such entities may form in equilibrium without government interventions. This paper develops a spatial equilibrium model with a competitive final sector and a monopolistically competitive intermediate sector, which allows us to determine necessary and sufficient conditions for a tech cluster to emerge as an equilibrium outcome. We show that strongly localized knowledge spillovers, skilled labor abundance, and low commuting costs are key drivers for a tech cluster to form. With continual improvements in infrastructure and communication technology that lowers distance decay in knowledge spillover or coordination costs, tech clusters will eventually be fragmented.

Special Properties of Morita Contexts

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective.

The Monsky Matrices and Non-congruent Numbers

We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices. Many known criteria for non-congruent numbers can be viewed as special cases of our results.

Arc Connectivity and Submodular Flows in Digraphs

AbstractLet $$D=(V,A)$$ D = ( V , A ) be a digraph. For an integer $$k\ge 1$$ k ≥ 1 , a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer $$\tau \ge 1$$ τ ≥ 1 , suppose $$d_A^+(U)+(\frac{\tau }{k}-1)d_A^-(U)\ge \tau $$ d A + ( U ) + ( τ k - 1 ) d A - ( U ) ≥ τ for all $$U\subsetneq V, U\ne \emptyset $$ U ⊊ V , U ≠ ∅ , where $$d_A^+(U)$$ d A + ( U ) and $$d_A^-(U)$$ d A - ( U ) denote the number of arcs in A leaving and entering U, respectively. Let $${\mathcal {C}}$$ C be a crossing family over ground set V, and let $$f:{\mathcal {C}}\rightarrow {\mathbb {Z}}$$ f : C → Z be a crossing submodular function such that $$f(U)\ge \frac{k}{\tau }(d_A^+(U)-d_A^-(U))$$ f ( U ) ≥ k τ ( d A + ( U ) - d A - ( U ) ) for all $$U\in {\mathcal {C}}$$ U ∈ C . Then D has a k-arc-connected flip J such that $$f(U)\ge d_J^+(U)-d_J^-(U)$$ f ( U ) ≥ d J + ( U ) - d J - ( U ) for all $$U\in {\mathcal {C}}$$ U ∈ C . The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is $$\tau $$ τ -edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

Optimality conditions for differentiable linearly constrained pseudoconvex programs

AbstractThe aim of this paper is to study optimality conditions for differentiable linearly constrained pseudoconvex programs. The stated results are based on new transversality conditions which can be used instead of complementarity ones. Necessary and sufficient optimality conditions are stated under suitable generalized convexity properties. Moreover, two different pairs of dual problems are proposed and weak and strong duality results proved. Finally, it is shown how transversality conditions can be applied to characterize optimality of convex quadratic problems and to efficiently solve a particular class of Max-Min problems

Existence and nonexistence of solutions for the mean curvature equation with weights

AbstractIn this paper we study existence and nonexistence of positive radial solutions of a Dirichlet problem for the prescribed mean curvature operator with weights in a ball with a suitable radius. Because of the presence of different weights, possibly singular or degenerate, the problem under consideration appears rather delicate, it requires an accurate qualitative analysis of the solutions, as well as the use of Liouville type results based on an appropriate Pohozaev type identity. In addition, sufficient conditions for global solutions to be oscillatory are given.

Endowments, patience types, and uniqueness in two-good HARA utility economies

AbstractThis paper establishes a link between endowments, patience types, and the parameters of the HARA Bernoulli utility function that ensure equilibrium uniqueness in an economy with two goods and two impatience types with additive separable preferences. We provide sufficient conditions that guarantee uniqueness of equilibrium for any possible value of $$\gamma $$ γ in the HARA utility function $$\frac{\gamma }{1-\gamma }\left( b+\frac{a}{\gamma }x\right) ^{1-\gamma }$$ γ 1 - γ b + a γ x 1 - γ . The analysis contributes to the literature on uniqueness in pure exchange economies with two-goods and two agent types and extends the result in Loi and Matta (2022).

Optimality and Duality for Robust Optimization Problems Involving Intersection of Closed Sets

AbstractIn this paper, we study a robust optimization problem whose constraints include nonsmooth and nonconvex functions and the intersection of closed sets. Using advanced variational analysis tools, we first provide necessary conditions for the optimality of the robust optimization problem. We then establish sufficient conditions for the optimality of the considered problem under the assumption of generalized convexity. In addition, we present a dual problem to the primal robust optimization problem and examine duality relations.

Sufficient Conditions for Linear Operators Related to Confluent Hypergeometric Function and Generalized Bessel Function of the First Kind to Belong to a Certain Class of Analytic Functions

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases.

CRAMER’S RULE IN MIN-PLUS ALGEBRA

Cramer’s rule is one of a method for solving a system of linear equations in conventional algebra. The system of linear equation can be solved using Cramer’s rule if . Max-plus algebra is a set where is a set of real numbers, equipped with biner operations and where and . Min-plus Algebra is a set where is a set of real numbers, equipped with biner operations and where and . In max-plus algebra has been formulated Cramer’s rule to solve a system of linear equations. Because max-plus algebra is isomorphic to min-plus algebra, Cramer’s rule can be formulated into min-plus algebra. The purpose of this research is to determine the sufficient conditions for a system of linear equations can be solved using Cramer’s rule. The method used in this research is a literature study that reviews previous research related to min-plus algebra, max-plus algebra, and Cramer’s rule in max-plus algebra. By using the appropriate analogy in max-plus algebra, we can determine the sufficient conditions so that a system of linear equations in min-plus algebra can be solved using Cramer’s rule. Based on the research, the sufficient conditions for a system of linear equations can be solved using Cramer’s rule are for and with the Cramer’s rule is . For an invertible matrix A, Cramer’s rule can be written as .

Microtopography effects on pedogenesis in the mudstone-derived soils of the hilly mountainous regions

Topography is a critical factor that determines the characteristics of regional soil formation. Small-scale topographic changes are referred to microtopographies. In hilly mountainous regions, the redistribution of water and soil materials caused by microtopography is the main factor affecting the spatial heterogeneity of soil and the utilization of land resources. In this study, the influence of microtopography on pedogenesis was investigated using soil samples formed from mudstones with lacustrine facies deposition in the middle of the Sichuan Basin. Soil profiles were sampled along the slopes at the summit, shoulder, backslope, footslope, and toeslope positions. The morphological, physicochemical, and geochemical attributes of profiles were analyzed. The results showed that from the summit to the toeslope, soil thickness increased significantly and profile configuration changed from A–C to A–B–C. The total contents of Ca and Na decreased at the summit, backslope, and footslope, while the total contents of Al, Fe and Mg showed an opposite trend. On the summit and shoulder of the hillslope, weathered materials were transported away by gravity and surface erosion, exposing new rocks. As a result, soil development in these areas was relatively weak. In flat areas such as the footslope and toeslope with sufficient water conditions, the addition of weathered components and the prolonged contact between water, soil, and sediment led to further chemical weathering, resulting in highly developed characteristics. Microtopography may influence physicochemical properties, chemical weathering, and redistribution of water and materials, causing variations in pedogenic characteristics at different slope positions.

PROVING THE CORRECTNESS OF THE EXTENDED SERIAL GRAPH-VALIDATION QUEUE SCHEME IN THE CLIENT-SERVER SYSTEM

Numerous studies have been conducted to develop concurrency control schemes that can be applied to client-server systems, such as the Extended Serial Graph-Validation Queue (SG-VQ) scheme. Extended SG-VQ is a control concurrency scheme in client-server system which implements object caching on the client side and locking strategy on the server side. This scheme employs validation algorithms based on queues on the client side and graphs on the server side. This research focuses on the mathematical analysis of the correctness of the Extended SG-VQ scheme using serializability as the criterion that needs to be achieved. Implementing a cycle-free transaction graph is a necessary and sufficient condition to achieve serializability. In this research, the serializability of the Extended SG-VQ scheme has been proven through the exposition of ten definitions, two propositions, three lemmas, and one theorem.

Trace Anomaly Redefined in a Convention for Pontryagin Equivalent to a Generalized Wick

The lack of Unitarity is sought after, and is first resolved by an extraction from a unite-scale diffeomorphic transformation. The same result can second be found independently and is based on an orbital-wise Pfaffian differential satisfying a Conformal geodesic. Such a fundamental reason is borne out in the multiple methods for the Pontryagin Chiral Fermions density anomalies resolutions being either zero or imaginary results, were, then fore, contradictory or randomly correct outcomes due to the eigenvalue non-separable sorting. Confirming then an equivalent (1st as necessary and 2nd as sufficient) condition for Unitarity is via a regularization for the zero component of the Dirac Matrix \(\gamma^{0}\), and a generalization of the Wick rotation, whilst both above hypotheses (may directly) be contouring around the Einstein Gravity.

(Almost) complete characterization of the stability of a discrete-time Hawkes process with inhibition and memory of length two

Abstract We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with parameters $a_1,\ldots,a_p \in \mathbb{R}$ , $p\in\mathbb{N}$ , and $\lambda \ge 0$ , such that, for all $n\ge p$ , conditioned on $X_0,\ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.

РАЗРАБОТКА БАЗОВЫХ ПОДХОДОВ К ПРИВЛЕЧЕНИЮ И УДЕРЖАНИЮ КАДРОВ В ГОРОДАХ НАУКИ И ВЫСОКИХ ТЕХНОЛОГИЙ

In modern Russia, the problem of technological sovereignty and import substitution of high-tech products is acute. To solve the problems set by the Russian government, it is necessary to launch the production of a wide range of high-tech products as soon as possible. The implementation of the plans is impossible without solving the key tasks of introducing modern technological equipment and increasing human resources. At the moment, a prompt solution to these problems is required, and the previously developed and adopted territorial development strategies assume a long-term perspective. The main scientific organizations and high-tech businesses are located in large regional centers, as well as in small towns with high scientific and technological potential. In large regional centers, a scientific and technological ecosystem has historically been formed, allowing for the creation of conditions for the development of high technologies and scientific research. The situation is more complex in small core cities with pronounced scientific and technological specialization in centers of scientific and technological leadership. At the present stage, the main problem in the development of such cities is attracting and retaining highly qualified personnel. It is necessary to take into account the fact that attracting and retaining specialists are two related processes, but different in essence. The first is characterized by the migration activity of the population, the second - by satisfaction with living in the territory and working conditions. Most often, the factors influencing these processes have common directions, but differ from each other in their characteristics and content. When deciding on a place of work, a specialist evaluates the proposed conditions that are related to his expectations. When retaining specialists, realities are compared with the expectations that were when hired. If they do not comply, the specialist leaves the place of work. In this regard, it is important for the management of the territory and organizations to minimize the differences between expectations and the actual state of the labor and urban environment in order to retain high-level specialists. The work examines the main factors influencing and allowing to solve the problems of attracting and retaining talented youth and highly professional specialists in cities with high scientific and technological potential, taking into account modern realities and requirements. Research shows that the main factor ensuring the attraction of talented youth is the availability of high wages. It is a necessary but not sufficient condition for their retention. Additional factors play an important role in retaining specialists, including modern sociocultural and comfortable urban space, which includes various aspects of the lives of specialists and their families.

Properties of a Linear Operator Involving Lambert Series and Rabotnov Function

This work is an attempt to apply Lambert series in the theory of univalent functions. We first consider the Hadamard product of Rabotnov function and Lambert series with coefficients derived from the arithmetic function σn to introduce a normalized linear operator JRα,βz. We then acquire sufficient conditions for JRα,βz to be univalent, starlike and convex, respectively. Furthermore, we discuss the inclusion results in some special classes, namely, spiral-like and convex spiral-like subclasses. In addition, we extend the findings by incorporating two Robin’s inequalities, one of which is analogous to the Riemann hypothesis.

Dynamical Behaviors of Stochastic SIS Epidemic Model with Ornstein–Uhlenbeck Process

Controlling infectious diseases has become an increasingly complex issue, and vaccination has become a common preventive measure to reduce infection rates. It has been thought that vaccination protects the population. However, there is no fully effective vaccine. This is based on the fact that it has long been assumed that the immune system produces corresponding antibodies after vaccination, but usually does not achieve the level of complete protection for undergoing environmental fluctuations. In this paper, we investigate a stochastic SIS epidemic model with incomplete inoculation, which is perturbed by the Ornstein–Uhlenbeck process and Brownian motion. We determine the existence of a unique global solution for the stochastic SIS epidemic model and derive control conditions for the extinction. By constructing two suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. Furthermore, we obtain the exact expression of the probability density function near the pseudo-equilibrium point of the stochastic model while addressing the four-dimensional Fokker–Planck equation under the same conditions. Finally, we conduct several numerical simulations to validate the theoretical results.

Explosion Rates for Continuous-State Branching Processes in a Lévy Environment

AbstractHere, we study the long-term behaviour of the non-explosion probability for continuous-state branching processes in a Lévy environment when the branching mechanism is given by the negative of the Laplace exponent of a subordinator. In order to do so, we study the law of this family of processes in the infinite mean case and provide necessary and sufficient conditions for the process to be conservative, i.e. that the process does not explode in finite time a.s. In addition, we establish precise rates for the non-explosion probabilities in the subcritical and critical regimes, first found by Palau et al. (ALEA Lat Am J Probab Math Stat 13(2):1235–1258, 2016) in the case when the branching mechanism is given by the negative of the Laplace exponent of a stable subordinator.