The minimum forcing number of perfect matchings in the hypercube

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim (1998) conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is $2^{n?2}$, for all $n\ge 2$. This was revised by Riddle (2002), who conjectured that it is at least $2^{n?2}$, and proved it for all even $n$. We show that the revised conjecture holds for all $n\ge 2$. The proof is based on simple linear algebra.     

Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts

In this paper, we propose to enhance Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations for polynomial programming problems by developing cutting plane strategies using concepts derived from semidefinite programming. Given an RLT relaxation, we impose positive semidefiniteness on suitable dyadic variable-product matrices, and correspondingly derive implied semidefinite cuts. In the case of polynomial programs, there are several possible variants for selecting such particular variable-product matrices on which positive semidefiniteness restrictions can be imposed in order to derive implied valid inequalities. This leads to a new class of cutting planes that we call v-semidefinite cuts. We explore various strategies for generating such cuts, and exhibit their relative effectiveness towards tightening the RLT relaxations and solving the underlying polynomial programming problems in conjunction with an RLT-based branch-and-cut scheme, using a test-bed of problems from the literature as well as randomly generated instances. Our results demonstrate that these cutting planes achieve a significant tightening of the lower bound in contrast with using RLT as a stand-alone approach, thereby enabling a more robust algorithm with an appreciable reduction in the overall computational effort, even in comparison with the commercial software BARON and the polynomial programming problem solver GloptiPoly.     

A note on ratio and product type estimators

Summary The use of ratio and product estimators for the mean of a finite population is well known. This paper proposes transformed estimators obtained through parametric linear combination of the ratio, or product, and the usual unbiased estimator of the mean for any sample design. To the first degree of approximation, the proposed estimators have smaller mean square error than that of the ratio, product and the usual unbiased estimator, for suitable choice of the parameter. The superiority of the proposed estimators over others for small samples has been studied empirically.     

Subsystem entropy for ? sofic shifts

We prove that any ?^d shift of finite type with positive topological entropy has a family of subsystems of finite type whose entropies are dense in the interval from zero to the entropy of the original shift. We show a similar result for ?^d sofic shifts, and also show every ?^d sofic shift can be covered by a ?^d shift of finite type arbitrarily close in entropy.     

Ramsey theory for hypergroups

Semigroup Forum - In this paper, Ramsey theory for discrete hypergroups is introduced with emphasis on polynomial hypergroups, discrete orbit hypergroups and hypergroup deformations of semigroups....     

Divisibility of certain $$\ell $$ -regular partitions by 2

The Ramanujan Journal - For a positive integer $$\ell $$ , let $$b_{\ell }(n)$$ denote the number of $$\ell $$ -regular partitions of a nonnegative integer n. Motivated by some recent conjectures...     

An approximate max-flow min-cut relation for undirected multicommodity flow,with applications

In this paper, we prove the first approximate max-flow min-cut theorem for undirected multicommodity flow. We show that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is the sum of demands. Moreover, our theorem yields an algorithm for finding a cut that is approximately minimumrelative to the flow that must cross it. We use this result to obtain an approximation algorithm for T. C. Hu's generalization of the multiway-cut problem. This algorithm can in turn be applied to obtain approximation algorithms for minimum deletion of clauses of a 2-CNF formula, via minimization, and other problems. We also generalize the theorem to hypergraph networks; using this generalization, we can handle CNF clauses with an arbitrary number of literals per clause.Most of the results in this paper were presented in preliminary form in Approximation through multicommodity flow,Proceedings, 31th Annual Symposium on Foundations of Computer Science (1990), pp. 726–737.Research supported by the National Science Foundation under NSF grant CDA 8722809, by the Office of Naval and the Defense Advanced Research Projects Agency under contract N00014-83-K-0146, and ARPA Order No. 6320, Amendament 1.Research supported by NSF grant CCR-9012357 and by an NSF Presidential Young Investigator Award.     

Circumference, chromatic number and online coloring

Erd?s conjectured that if G is a triangle free graph of chromatic number at least k≥3, then it contains an odd cycle of length at least k ^2?o(1) [13,15]. Nothing better than a linear bound ([3], Problem 5.1.55 in [16]) was so far known. We make progress on this conjecture by showing that G contains an odd cycle of length at least Ω(k log logk). Erd?s’ conjecture is known to hold for graphs with girth at least five. We show that if a graph with girth four is C ₅ free, then Erd?s’ conjecture holds. When the number of vertices is not too large we can prove better bounds on χ. We also give bounds on the chromatic number of graphs with at most r cycles of length 1 mod k, or at most s cycles of length 2 mod k, or no cycles of length 3 mod k. Our techniques essentially consist of using a depth first search tree to decompose the graph into ordered paths, which are then fed to an online coloring algorithm. Using this technique we give simple proofs of some old results, and also obtain several other results. We also obtain a lower bound on the number of colors which an online coloring algorithm needs to use to color triangle free graphs.     

Structural characterization of microwave-synthesized zinc-substituted cobalt ferrite nanoparticles

Microwave combustion technique modified by post treatment procedure is used to synthesize single-phase spinel ferrites of cobalt, zinc, and substituted magnetic nanoparticles of typical size 390 Å. The post treatment does not alter the crystal structure but increases the crystallinity. This is confirmed by powder x-ray diffraction and Fourier Transform Infrared (FTIR) studies. Citric acid is used as a fuel. The fresh synthesized sample shows an impurity phase in x-ray and in FTIR. This is due to the unreacted citrate molecule adsorbed on the particle surface. It is shown that by treating the sample with 0.1 M HCl, we can eliminate the impurity phase, and one can obtain a pure single phase. The magnetization at 8 kOe increases by nearly 8% after the removal of impurity. In order to remove surface adsorbed OH^? ion, samples are treated with NaCl and heated to 200°C for four hours. The XRD result indicates that after the treatment neither the crystallite size nor the distribution changes, but it removes OH^? ion. This is also confirmed by FTIR analysis. Thus, this modified technique can be used to synthesize pure nanocrystalline samples of spinel ferrites.     

Early Detection of Breast Cancer: Synthesis and Characterization of Novel Target Specific NIR-Fluorescent Estrogen Conjugate for Molecular Optical Imaging

Estrogen induced proliferation of existing mutant cells is widely understood to be the major risk determining factor in the development of breast cancer. Hence determination of the Estrogen Receptor[ER] status is of paramount importance. We have carried out the synthesis and characterization of a novel NIR fluorescent dye conjugate aimed at measuring ER+ve status in-vivo. The conjugate was synthesized by ester formation between 17-β estradiol and a cyanine dye namely: bis-1, 1-(4-sulfobutyl) indotricarbocyanine-5-carboxylic acid, sodium salt. The replacement of the sodium ion in the ester by a larger glucosammonium ion was found to enhance the hydrophilicity and reduce the toxic effect on cell lines. The excitation and emission peaks for the dye were recorded as 750 and 788 nm respectively; ideal for non-invasive optical imaging owing to minimal tissue attenuation and auto-fluorescence at these wavelengths. The dye (NIRDC1) has a significant drop in plasma-protein binding therefore leading to marked improvement in pharmacokinetic profile such as dye evacuation in comparison to ICG. In addition the dye showed enhanced fluorescence quantum yield, molar extinction coefficient and linearity in fluorescence relative to ICG. This dye can be potentially used as a target specific exogenous contrast agent in molecular optical imaging for early detection of breast cancer.