Extending an edge-coloring

When can a k-edge-coloring of a subgraph K of a graph G be extended to a k-edge-coloring of G? One necessary condition is that for all X ? E(G) - E(K), where μ_i(X) is the maximum cardinality of a subset of X whose union with the set of edges of K colored i is a matching. This condition is not sufficient in general, but is sufficient for graphs of very simple structure. We try to locate the border where sufficiency ends.     

Infinite-order phase transition in a classical spin system

For an exactly soluble classical spin model with long-range inhomogeneous coupling it is proved that in the absence of external magnetic field the free energy is aC function of the temperature at the critical point.     

Parallel and superfast algorithms for Hankel systems of equations

Summary Utilizing kernel structure properties a unified construction of Hankel matrix inversion algorithms is presented. Three types of algorithms are obtained: 1)O(n ²) complexity Levinson type, 2)O (n) parallel complexity Schur-type, and 3)O(n log² n) complexity asymptotically fast ones. All algorithms work without additional assumption (like strong nonsingularity).     

Impulse systems with fixed moments of shocks of general position: The structure of the set of moments of the shocks

Ordinary differential equations with impulse action are considered at fixed moments of time when finite limits points are admitted to the set of moments of the shocks. The topological structure of this set is clarified: it must be resolvable, in particular nowhere dense in R.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 378–383, March, 1990.     

Tracking poles and representing Hankel operators directly from data

Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL ² fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL ² which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH , analytic, approximation toG relative to theL norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF     

Delaunay graphs are almost as good as complete graphs

LetS be any set ofN points in the plane and let DT(S) be the graph of the Delaunay triangulation ofS. For all pointsa andb ofS, letd(a, b) be the Euclidean distance froma tob and let DT(a, b) be the length of the shortest path in DT(S) froma tob. We show that there is a constantc (≤((1+√5)/2) π≈5.08) independent ofS andN such that $$\frac{{DT(a,b)}}{{d(a,b)}}< c.$$      

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

Effects of flanking band proximity, number, and modulation pattern on comodulation masking release

Comodulation masking release for a 700-Hz pure-tone signal was investigated as a function of the number and spectral positions of 20-Hz-wide comodulated flanking bands. In the first experiment, all stimuli were presented diotically. CMR was examined as a function of the number of flanking bands present, in conditions where the bands were arranged symmetrically around the signal frequency, were below the signal frequency, or were above the signal frequency. The number of flanking bands ranged from one to eight, and the magnitude of the diotic CMR ranged from approximately 5-16 dB. The results indicated: (1) bands closer to the signal resulted in larger masking release, and (2) more bands gave rise to larger CMR (but with diminishing returns above two flanking bands). Two additional sets of diotic conditions were examined and compared to the condition where all eight comodulated flanking bands were present: In one set of conditions, two of the eight flanking bands were removed; in the other set of conditions, two of the eight flanking bands were replaced with bands (termed "deviant" bands) that were not comodulated with respect to the other bands. There was very little effect of reducing eight bands to six, even when the removed bands were relatively near the signal frequency; however, CMR was substantially reduced when deviant bands were introduced, particularly when the deviant bands were placed relatively near the signal frequency. These reductions in CMR were slightly greater when each of the deviant bands had a unique modulation pattern (bideviant bands) than when the two deviant bands themselves shared the same modulation pattern (codeviant bands). In the second experiment, dichotic conditions were examined where the number and spectral positions of the flanking bands in the nonsignal ear were varied (the signal ear received only a 20-Hz-wide noise band centered on the signal frequency). The magnitude of the dichotic CMR ranged from approximately 2-10 dB, depending on condition. Effects of proximity and the number of flanking bands were similar to the effects obtained in diotic conditions. For both the diotic and the dichotic data, the effects of proximity were more consistent with an interpretation based upon across-channel processing than upon a within-channel interaction. The results obtained using deviant bands indicate that it is difficult for the auditory system to disregard the modulation pattern of flanking bands that differ from the modulation pattern of the on-signal band, particularly if such bands are proximal to the signal frequency.