Fractional Matching Research Articles

Sharp lower bounds on the fractional matching number

A fractional matching of a graph G is a function f from E(G) to the interval [0,1] such that ∑e∈Γ(v)f(e)≤1 for each v∈V(G), where Γ(v) is the set of edges incident to v. The fractional matching number of G, written α∗′(G), is the maximum of ∑e∈E(G)f(e) over all fractional matchings f. For G with n vertices, m edges, positive minimum degree d, and maximum degree D, we prove α∗′(G)≥max{mD,n−md,dD+dn}. For the first two bounds, equality holds if and only if each component of G is r-regular or is bipartite with all vertices in one part having degree r, where r=D for the first bound and r=d for the second. Equality holds in the third bound if and only if G is regular or is (d,D)-biregular.

Fractional matchings in hypergraphs

We find an exact formula for the minimum number of edges in a hypergraph which guarantees a fractional matching of cardinality s in the case where sn is an integer.

Minimum number of edges in a hypergraph guaranteeing a perfect fractional matching and the MMS conjecture

In this paper we prove the Ahlswede-Khachatrian conjecture [1] up to a finite number of cases, which can be checked using modern computers. This conjecture implies the conjecture from [2] and the Manickam-Miklos-Singhi conjecture.

A NOTE ON PERFECT FUZZY MATCHING

The notion of matching in a fuzzy graph could be defined using the concept of effective edges (8) or by fractional matching (4). In this paper, we derive a necessary condition for a fuzzy graph on a cycle or a complete graph or a stargraph to have a perfect fuzzy matching. Also we discuss perfect fuzzy matching on strong regular fuzzy graphs.

Fractional and integer matchings in uniform hypergraphs

Our main result improves bounds of Markström and Ruciński on the minimum d-degree which forces a perfect matching in a k-uniform hypergraph on n vertices. We also extend bounds of Bollobás, Daykin and Erdős by asymptotically determining the minimum vertex degree which forces a matching of size t<n/2(k−1) in a k-uniform hypergraph on n vertices. Further asymptotically tight results on d-degrees which force large matchings are also obtained. Our approach is to prove fractional versions of the above results and then translate these into integer versions.

In-phase motion of Josephson vortices in stacked SNS Josephson junctions: effect of ordered pinning

The dynamics of Josephson vortices (fluxons) in artificial stacks of superconducting–normal–superconducting Josephson junctions is investigated using the anisotropic time-dependent Ginzburg–Landau theory in the presence of a square/rectangular array of pinning centers (holes). For small values of the applied drive, fluxons in different junctions move out of phase, forming a periodic triangular lattice. A rectangular lattice of moving fluxons is observed at larger currents, which is in agreement with previous theoretical predictions (Koshelev and Aranson 2000 Phys. Rev. Lett. 85 3938). This ‘superradiant’ flux-flow state is found to be stable in a wide region of applied current. The stability range of this ordered state is considerably larger than the one obtained for the pinning-free sample. Clear commensurability features are observed in the current–voltage characteristics of the system with pronounced peaks in the critical current at (fractional) matching fields. The effect of density and strength of the pinning centers on the stability of the rectangular fluxon lattice is discussed. Predicted synchronized motion of fluxons in the presence of ordered pinning can be detected experimentally using the rf response of the system, where enhancement of the Shapiro-like steps is expected due to the synchronization.

Envy-free two-player [formula omitted]-cake and three-player two-cake divisions

Cloutier, Nyman, and Su (2005) initiated the study of envy-free cake-cutting problems involving several cakes. They showed that when there are two players and two or three cakes it is possible to find envy-free cake-divisions requiring few cuts, under natural assumptions.We prove that such a result also exists when there are two players and any number of cakes and when there are three players and two cakes. The proof relies on the fractional matching number in m-partite hypergraphs.

Realization of Artificial Ice Systems for Magnetic Vortices in a Superconducting MoGe Thin Film with Patterned Nanostructures

We report an anomalous matching effect in MoGe thin films containing pairs of circular holes arranged in such a way that four of those pairs meet at each vertex point of a square lattice. A remarkably pronounced fractional matching was observed in the magnetic field dependences of both the resistance and the critical current. At the half matching field the critical current can be even higher than that at zero field. This has never been observed before for vortices in superconductors with pinning arrays. Numerical simulations within the nonlinear Ginzburg-Landau theory reveal a square vortex ice configuration in the ground state at the half matching field and demonstrate similar characteristic features in the field dependence of the critical current, confirming the experimental realization of an artificial ice system for vortices for the first time.

An algorithm for weighted fractional matroid matching

Let M be a matroid on ground set E with rank function r. A subset l⊆E is called a line when r(l)∈{1,2}. Given a finite set L of lines in M, a vector x∈R+L is called a fractional matching when ∑l∈Lxla(F)l⩽r(F) for every flat F of M. Here a(F)l is equal to 0 when l∩F=∅, equal to 2 when l⊆F and equal to 1 otherwise. We refer to ∑l∈Lxl as the size of x.It was shown by Chang et al. [S. Chang, D. Llewellyn, J. Vande Vate, Matching 2-lattice polyhedra: finding a maximum vector, Discrete Math. 237 (2001) 29–61], that a maximum size fractional matching can be found in polynomial time. In this paper we give a polynomial time algorithm to find, for given weight function w:L→Q, a maximum weight fractional matching. A simple reference to the equivalence of separation and optimization does not lead to such an algorithm, since no direct method for polynomial time separation is known for this polytope.

On-site energy matching indices for buildings with energy conversion, storage and hybrid grid connections

In order to meet EU objective of nearly zero-energy building, mismatch problem is an inevitable side effect, which requires suitable criteria to quantitatively evaluate the problem. There are two commonly used basic indices for matching analysis: on-site energy fraction (OEF) and on-site energy matching (OEM). OEF indicates the proportion of the load covered by the on-site generated energy, while OEM indicates the proportion of the on-site generated energy that is used in the load rather than being dumped or exported. However, with rapid development of energy technology, matching problems become more complicated when a system involves different energy forms, energy conversions, storages, and hybrid grid connections. In order to meet these challenges, six matching indices are defined based on the extension of the two basic indices. Four examples at one time-step are presented for the usage of these extended indices’ equations. Furthermore, in order to evaluate the applicability of these extended indices in a dynamic simulation process, the hourly and daily matching situations of one Finnish single-family house with hybrid grid connections are simulated for one summer's day. It shows that the extended matching indices are powerful tools for assessing the matching situation of increasingly complicated on-site energy systems.

NAHR-mediated copy-number variants in a clinical population: mechanistic insights into both genomic disorders and Mendelizing traits.

We delineated and analyzed directly oriented paralogous low-copy repeats (DP-LCRs) in the most recent version of the human haploid reference genome. The computationally defined DP-LCRs were cross-referenced with our chromosomal microarray analysis (CMA) database of 25,144 patients subjected to genome-wide assays. This computationally guided approach to the empirically derived large data set allowed us to investigate genomic rearrangement relative frequencies and identify new loci for recurrent nonallelic homologous recombination (NAHR)-mediated copy-number variants (CNVs). The most commonly observed recurrent CNVs were NPHP1 duplications (233), CHRNA7 duplications (175), and 22q11.21 deletions (DiGeorge/velocardiofacial syndrome, 166). In the ∼25% of CMA cases for which parental studies were available, we identified 190 de novo recurrent CNVs. In this group, the most frequently observed events were deletions of 22q11.21 (48), 16p11.2 (autism, 34), and 7q11.23 (Williams-Beuren syndrome, 11). Several features of DP-LCRs, including length, distance between NAHR substrate elements, DNA sequence identity (fraction matching), GC content, and concentration of the homologous recombination (HR) hot spot motif 5′-CCNCCNTNNCCNC-3′, correlate with the frequencies of the recurrent CNVs events. Four novel adjacent DP-LCR-flanked and NAHR-prone regions, involving 2q12.2q13, were elucidated in association with novel genomic disorders. Our study quantitates genome architectural features responsible for NAHR-mediated genomic instability and further elucidates the role of NAHR in human disease.

The mental representations of fractions: adults' same-different judgments.

Two experiments examined whether the processing of the magnitude of fractions is global or componential. Previously, some authors concluded that adults process the numerators and denominators of fractions separately and do not access the global magnitude of fractions. Conversely, others reported evidence suggesting that the global magnitude of fractions is accessed. We hypothesized that in a fraction matching task, participants automatically extract the magnitude of the components but that the activation of the global magnitude of the whole fraction is only optional or strategic. Participants carried out same/different judgment tasks. Two different tasks were used: a physical matching task and a numerical matching task. Pairs of fractions were presented either simultaneously or sequentially. Results showed that participants only accessed the representation of the global magnitude of fractions in the numerical matching task. The mode of stimulus presentation did not affect the processing of fractions. The present study allows a deeper understanding of the conditions in which the magnitude of fractions is mentally represented by using matching tasks and two different modes of presentation.

Highly efficient broadband second harmonic generation in a tapered zinc telluride slab using total internal reflection fractional quasi phase matching

We analytically describe broadband second-harmonic generation in a tapered zinc telluride (ZnTe) slab using total internal reflection fractional quasi-phase matching. Refractive index of ZnTe shows minimum deviation over a wide wavelength range resulting in increased coherence length which leads to fractional quasi-phase matching scenario, ensuring highly efficient second harmonic output. The simulated results, considering absorption and reflection losses, indicate 3dB bandwidth of 192.5nm, centered at 8.583μm, with conversion efficiency as high as 19.6% in a 30mm ZnTe slab. The effects of variations in incident angle at air–material interface, length and tapering angle of the slab have also been analyzed.

A remark on the problem of nonnegative k-subset sums

Given a set of n real numbers with a nonnegative sum, consider the family of all its k-element subsets with nonnegative sums. How small can the size of this family be? We show that this problem is closely related to a problem raised by Ahlswede and Khachatrian in [1]. The latter, in a special case, is nothing else but the problem of determining a minimal number c n (k) such that any k-uniform hypergraph on n vertices having c n (k) + 1 edges has a perfect fractional matching. We show that results obtained in [1] can be applied for the former problem. Moreover, we conjecture that these problems have in general the same solution.

On Randomized Approximation for Finding a Level Ideal of a Poset and the Generalized Median Stable Matchings

This study is concerned with finding a level ideal (LI) of a partially ordered set (poset). Given a finite poset P, the level of each element p ∈ P is defined as the number of ideals that do not include p, then the problem is to find the ith LI–the ideal consisting of elements whose levels are less than a given integer i. The concept of a level ideal is naturally derived from the generalized median stable matchings, introduced by Teo and Sethuraman [Teo, C. P., J. Sethuraman. 1998. The geometry of fractional stable matchings and its applications. Math. Oper. Res. 23(4) 874–891] in the context of “fairness” of matchings in a stable marriage problem. Cheng [Cheng, C. T. 2010. Understanding the generalized median stable matchings. Algorithmica 58(1) 34–51] showed that finding the ith LI is #P-hard when i = Θ(N), where N is the total number of ideals of P. This paper shows that finding the ith LI is #P-hard even if i = Θ(N1/c), where c is an arbitrary constant at least one. Meanwhile, we present a polynomial time exact algorithm when i = O((log N)c′), where c′ is an arbitrary positive constant. We also devise two randomized approximation schemes for the ideals of a poset, by using an oracle of an almost-uniform sampler.

Large matchings in uniform hypergraphs and the conjectures of Erdős and Samuels

In this paper we study degree conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erdős on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum d-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also discuss an application to a problem of finding an optimal data allocation in a distributed storage system.

Vortex states of Bi2Sr2CaCu2O8+y with antidot array probed by c-axis transport measurements

To study the vortex states in Bi2Sr2CaCu2O8+y (Bi2212) high- T c superconductor with periodic array of antidots, we have measured the c -axis critical current in a stack of the intrinsic Josephson junctions with a square lattice of antidots, which gives us information on the c -axis correlation of the pancake vortices. Without the antidot array, the c -axis critical current can probe the first-order vortex lattice melting transition. Additionally, a suppression of the critical current in the vicinity of zero magnetic field is found, which may be related to the instability of vortex lattice under the parallel current flow. With the square lattice of antidots, enhancements of the critical current due to the matching effect are observed in both the integer and fractional matching fields. By mapping the critical current on H - T diagram, the influence of the matching effect on the phase boundary of vortex solid and liquid is shown.

A Market Approach to Fractional Matching

I study a model where people can be matched in fractions. That is to say, whether a pair is matched is not a binary matter. They can be matched with some intensity. Each has a fixed availability: the total intensity with which he is matched to his partners. I propose a notion of competitive equilibrium for this model. Since consumption sets are compact, preferences are satiated. Furthermore, the natural notion of an endowment is that each person owns himself and hence is on the boundary of his consumption set. For these two reasons, exact equilibria may not exist for some economies. For each e > 0, I define an approximate equilibrium that I call “e–double-indexed price equilibrium” (eDIP). I show that eDIP matches are efficient. By picking e small enough, one can find an eDIP match arbitrarily close to the core.

Theory of Fractional Order Elements Based Impedance Matching Networks

Fractional order circuit elements (inductors and capacitors) based impedance matching networks are introduced for the first time. In comparison to the conventional integer based L-type matching networks, fractional matching networks are much simpler and versatile. Any complex load can be matched utilizing a single series fractional element, which generally requires two elements for matching in the conventional approach. It is shown that all the Smith chart circles (resistance and reactance) are actually pairs of completely identical circles. They appear to be single for the conventional integer order case, where the identical circles completely overlap each other. The concept is supported by design equations and impedance matching examples.

Resonance structure of dynamic fractional Stark ladders in laser-driven biased superlattices

The resonance structure of an electronic Floquet state in a dynamic fractional Stark ladder (DFSL) is examined based on the scattering theory applied to a dressed potential resulting from renormalization of a laser–electron interaction to an original potential. Here, the DFSL is realized in laser-driven biased superlattices with a fractional matching ratio of a Bloch frequency to a laser frequency. It is revealed that, in contrast to a conventional understanding, the DFSL resonance position and lifetime tend to redshift and shorten, respectively, with an increase in strength of the laser field, and further, these show irregular changes in a limited region of the strength. The underlying physics is discussed in detail.