Matching Theorem Research Articles

New matching theorems for admissible type maps in topological vector spaces

New matching theorems for admissible type maps in topological vector spaces

A generalization of Petersen's matching theorem

One of the earliest results in graph theory is Petersen's matching theorem from 1891 which states that every bridgeless cubic graph contains a perfect matching. Since the vertex-connectivity and edge-connectivity in a connected cubic graph are equal, Petersen's theorem can be stated as follows: If G is a 2-connected 3-regular graph of order n, then α′(G)=12n, where α′(G) denotes the matching number of G. We generalize Petersen's theorem and prove that for k≥3 an odd integer, if G is a 2-connected k-regular graph of order n, then α′(G)≥(k2+k+6k2+2k+3)×n2. The case when k is even behaves differently. In this case, for k≥4 even, if G is a 2-connected k-regular graph of order n, then α′(G)≥(k2+4k2+k+2)×n2. For all k≥3, if G is a 2-connected graph of order n and maximum degree k that is not necessarily regular, then we show that α′(G)≥2nk+2. In all the above bounds, the extremal graphs are characterized.

Leray numbers of complexes of graphs with bounded matching number

Given a graph G on the vertex set V, the non-matching complex of G, denoted by NMk(G), is the family of subgraphs G′⊂G whose matching number ν(G′) is strictly less than k. As an attempt to extend the result by Linusson, Shareshian and Welker on the homotopy types of NMk(Kn) and NMk(Kr,s) to arbitrary graphs G, we show that (i) NMk(G) is (3k−3)-Leray, and (ii) if G is bipartite, then NMk(G) is (2k−2)-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex NMk(G), which vanishes in all dimensions d≥3k−4, and all dimensions d≥2k−3 when G is bipartite. As a corollary, we have the following rainbow matching theorem which generalizes a result by Aharoni, Berger, Chudnovsky, Howard and Seymour: Let E1,…,E3k−2 be non-empty edge subsets of a graph and suppose that ν(Ei∪Ej)≥k for every i≠j. Then E=⋃Ei has a rainbow matching of size k. Furthermore, the number of edge sets Ei can be reduced to 2k−1 when E is the edge set of a bipartite graph.

A simple Fourier analytic proof of the AKT optimal matching theorem

We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.

Filling random cycles

We compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert surface for a model of random knots first studied by Millett. This is a generalization of the classical Ajtai–Komlós–Tusnády optimal matching theorem from combinatorial probability. The author hopes for applications to the topology of random links, random maps between spheres, and other models of random geometric objects.

Art Gallery Problem with Rook and Queen Vision

How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that lfloor {frac{n}{2}} rfloor rooks or lfloor {frac{n}{3}} rfloor queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

Alternating signed bipartite graphs and difference-1 colourings

We investigate a class of 2-edge coloured bipartite graphs known as alternating signed bipartite graphs (ASBGs) that encode the information in alternating sign matrices. The central question is when a given bipartite graph admits an ASBG-colouring; a 2-edge colouring such that the resulting graph is an ASBG. We introduce the concept of a difference-1 colouring, a relaxation of the concept of an ASBG-colouring, and present a set of necessary and sufficient conditions for when a graph admits a difference-1 colouring. The relationship between distinct difference-1 colourings of a particular graph is characterised, and some classes of graphs for which all difference-1 colourings are ASBG-colourings are identified. One key step is Theorem 3.4.6, which generalises Hall's Matching Theorem by describing a necessary and sufficient condition for the existence of a subgraph H of a bipartite graph in which each vertex v of H has some prescribed degree r(v).

A comparison framework for interleaved persistence modules.

We present a generalization of the induced matching theorem of as reported by Bauer and Lesnick (in: Proceedings of the thirtieth annual symposium computational geometry 2014) and use it to prove a generalization of the algebraic stability theorem for -indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.

Voltage instability prediction based on reactive power reserve of generating units and zone selection

In this study, a voltage instability prediction algorithm is proposed based on Thevenin impedance matching theorem. This algorithm consists of three important parts: an adaptive network zoning algorithm, the Thevenin-based voltage instability risk predictor (VIRP) and a third-order extrapolation technique. The zoning algorithm, based on the concept of electrical distance, is proposed to divide the main network into several zones after the occurrence of any large event. Each zone consists of some load buses and generating units have the most influences on their voltage stability. The proposed VIRP is defined similarly to the Thevenin indicator multiplied with a contribution factor. This factor is calculated for each zone, based on the lack of the reactive power reserve in its generating units, to increase the predictor value, especially in the proximity of the instability. So, VIRP could reduce the instability detection time with respect to the Thevenin indicator. The further reduction in the instability detection time is obtained by the well known cubic spline extrapolation technique. This function is applied to the values of VIRP to predict the instant of the instability. The proposed algorithm is tested on the modified New England 39-bus test system supplied with the voltage-source converter-high-voltage dc link.

Scaling and non-standard matching theorems

Consider the standard Gaussian measure μ on R2. Consider independent r.v.s (Xi)i≤N distributed according to μ, and an independent copy (Yi)i≤N of these r.v.s. We prove that, for some number C and N large, we have(1)(log⁡N)2C≤Einfπ⁡∑i≤Nd(Xi,Yπ(i))2≤C(log⁡N)2, where the infimum is over all permutations π of {1,…,N}. The striking point of this result is the factor (log⁡N)2. Indeed, if instead of μ we consider the uniform distribution on the unit square, it is well known that the proper factor is log⁡N. The upper bound was proved by Michel Ledoux (2017) [3].

Edge Computing Framework for Cooperative Video Processing in Multimedia IoT Systems

Multimedia Internet-of-Things (IoT) systems have been widely used in surveillance, automatic behavior analysis and event recognition, which integrate image processing, computer vision, and networking capabilities. In conventional multimedia IoT systems, videos captured by surveillance cameras are required to be delivered to remote IoT servers for video analysis. However, the long-distance transmission of a large volume of video chunks may cause congestions and delays due to limited network bandwidth. Nowadays, mobile devices, e.g., smart phones and tablets, are resource-abundant in computation and communication capabilities. Thus, these devices have the potential to extract features from videos for the remote IoT servers. By sending back only a few video features to the remote servers, the bandwidth starvation of delivering original video chunks can be avoided. In this paper, we propose an edge computing framework to enable cooperative processing on resource-abundant mobile devices for delay-sensitive multimedia IoT tasks. We identify that the key challenges in the proposed edge computing framework are to optimally form mobile devices into video processing groups and to dispatch video chunks to proper video processing groups. Based on the derived optimal matching theorem, we put forward a cooperative video processing scheme formed by two efficient algorithms to tackle above challenges, which achieves suboptimal performance on the human detection accuracy. The proposed scheme has been evaluated under diverse parameter settings. Extensive simulation confirms the superiority of the proposed scheme over other two baseline schemes.

Two-Sided, Unbiased Version of Hall's Marriage Theorem

The standard conditions in Hall's perfect matching theorem for a bipartite graph G require that all subsets from one side of G are expanding. The unbiased extension identifies mixtures of subsets from both sides such that their expansions imply the standard conditions—hence a perfect matching.

The solutions for the flow of micropolar fluid through an expanding or contracting channel with porous walls

The unsteady, two-dimensional laminar flow of an incompressible micropolar fluid in a channel with expanding or contracting porous walls is investigated. The governing equations are transformed into a coupled nonlinear two-points boundary value problem by a suitable similarity transformation. Unlike the classic Berman problem (Berman in J. Appl. Phys. 24:1232-1235, 1953), three new solutions (totally six solutions) and no-solution interval, which is one of important characteristics for the laminar flow through porous pipe with stationary wall (Terrill and Thomas in Appl. Sci. Res. 21:37-67, 1969), are found numerically for the first time. The multiplicity of the solutions is strictly dependent on the expansion ratio. Furthermore, the asymptotic solutions are constructed by the Lighthill method, which eliminates the singularity of the similarity solution, for large injection and by the matching theorem for the suction Reynolds number, respectively. The analytical solutions also are compared with the numerical ones and the results agree well.

Pinning control for synchronization of nonlinear complex dynamical network with suboptimal SDRE controllers

Based on the state-dependent Riccati equation (SDRE) technique, in this paper, a suboptimal pinning control scheme is proposed to synchronize linearly coupled complex networks. The Lyapunov direct method is used to analyze the stability of the closed-loop control system, where it leads to a LMI criterion for pinning synchronization. It is shown that the time interval for synchronization of the proposed SDRE controllers is faster comparing with the results in the latest literatures. It is also shown that the minimum required coupling weights for the network synchronization in a finite desired time is decreased when some specified nodes in the network are pinned with the SDRE controllers. Based on the proposed criterion for pinned nodes selection, the network performances for different topological structures are investigated and the results are compared. The results indicate that the coupling weights for network synchronization in finite desired time in random Erdos Reiny networks are minimum when the pinned nodes are selected based on the minimum matching theorem. In small-world and scale-free networks, the minimum required coupling weight for network synchronization in finite desired time decreases when the highest degree nodes are pinned.

The Optimal Use of Rate-Limited Randomness in Broadcast Channels With Confidential Messages

In coding schemes for the wire-tap channel or for broadcast channels with confidential messages, it is well-known that the sender needs to use stochastic encoding to avoid information about the transmitted confidential message from being leaked to an eavesdropper. In this paper, we investigate the tradeoff between the rate of random numbers needed to realize the stochastic encoding and the rates of common, private, and confidential messages. For the direct theorem, we use the superposition coding scheme for the wire-tap channel, recently proposed by Chia and El Gamal, and its strong security is proved. The matching converse theorem is also established. Our result clarifies that a combination of ordinary stochastic encoding and channel prefixing by channel simulation is suboptimal.

The Equilibrium Existence Theorem for Systems of General Quasiequilibrium Problems in GFC-Spaces

In this paper, the GFC-KKM mapping is introduced and a GFC-KKM theorem is established in GFC-spaces. As applications, a matching theorem and a maximal element theorem are obtained. Our results unify, improve and generalize some known results in recent reference. Finally, an equilibrium existence theorem for systems of general quasiequilibrium problems is yielded in GFC-spaces.

Applications of Fan's matching theorem in MKKM spaces

Using some concepts in MKKM space, we prove the generalized forms of open and closed versions of Fan’s matching theorem. The S -weakly KKM multimap is applied to obtain a type of KKM theorem. As an application, a Fan type minimax theorem is proved. Moreover, a minimax inequality is given in these new settings. Mathematics subject classification (2010): 26A51, 26B25, 54H25, 55M20, 47H10, 54A05.

Ky Fan Matching Theorems for MFC Mappings in FC-Metric Spaces with the Application to Minimax Inequalities

In this paper, MFC mappings is introduced and Ky Fan matching theorems for MFC mappings with transfer compactly open values are established in noncompact complete FC-metric spaces. As applications, a Fan-Browder type coincidence theorem and a minimax inequality are obtained. Our results unify, improve and generalize some known results in recent literature.

A Matching Theorem in GFC-Spaces with Application to Saddle Points

In this paper, a matching theorem for weakly transfer compactly open valued mappings is established in GFC-spaces. As applications, a fixed point theorem, a minimax inequality and a saddle point theorem are obtained in GFC-spaces. Our results unify, improve and generalize some known results in recent reference.

FIXED POINTS AND ALTERNATIVE PRINCIPLES

In a recent paper, M. Balaj [B] established an alternative principle. The principle was applied to a matching theorem of Ky Fan type, an analytic alternative, a minimax inequality, and existence of solutions of a vector equilibrium theorem. Based on the first author's fixed point theorems, in the present paper, we obtain generalizations of the main result of Balaj [B] and their applications.