Kilohertz laser frequency sensing with Brillouin mutually modulated cross-gain modulation

Using mutually modulated cross-gain modulation, Stokes optical frequency changes are converted into modulation phase changes with high sensitivity. In the slow-light transition regime, we demonstrate kilohertz sensitivity to the Stokes optical carrier frequency. The sensitivity is inversely proportional to the modulation frequency of the pump and Stokes beams.     

Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes

A uniform random vector over a simplex is generated. An explicit expression for the first moment of its largest spacing is derived. The result is used in a proposed diagnostic tool which examines the validity of random number generators. It is then shown that the first moment of the largest uniform spacing is related to the dependence measure of random vectors following any extreme value distribution. The main result is proved by a geometric proof as well as by a probabilistic one.     

Optimization over Young diagrams

Journal of Algebraic Combinatorics - We consider the problem of finding a Young diagram minimizing the sum of evaluations of a given pair of functions on the parts of the associated pair of...     

Some mappings of pfriodic groups

For the rational integera and the rational primeb, letP(a,b) be the set of primesp such that . A natural integerq satisfies (i) iffq is a power product fromP(a,b). In the (additively written) Abelian groupG, g→ag permutes the elements ofG ^# in cycles whose lengths are multiples ofb, but not ofb ², iffG is aπ-group withπ⊂P (a,b). The casea=−2,b=2 has combinational applications.     

2-Adic valuations of certain ratios of products of factorials and applications

We prove the conjecture of Falikman-Friedland-Loewy on the parity of the degrees of projective varieties of n×n complex symmetric matrices of rank at most k. We also characterize the parity of the degrees of projective varieties of n×n complex skew symmetric matrices of rank at most 2p. We give recursive relations which determine the parity of the degrees of projective varieties of m×n complex matrices of rank at most k. In the case the degrees of these varieties are odd, we characterize the minimal dimensions of subspaces of n×n skew symmetric real matrices and of m×n real matrices containing a nonzero matrix of rank at most k. The parity questions studied here are also of combinatorial interest since they concern the parity of the number of plane partitions contained in a given box, on the one hand, and the parity of the number of symplectic tableaux of rectangular shape, on the other hand.     

N-fold integer programming and nonlinear multi-transshipment

The multi-transshipment problem is NP-hard already for two commodities over bipartite networks. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the polynomial time solvability of the problem in two broad situations. First, for any fixed number of commodities and number of suppliers, we solve the problem over bipartite networks with variable number of consumers in polynomial time. This is very natural in operations research applications where few facilities serve many customers. Second, for every fixed network, we solve the problem with variable number of commodities in polynomial time.     

Modular Symbols and the Topological Nonrigidity of Arithmetic Manifolds

In this paper we address the existence of smooth manifolds proper homotopy equivalent to nonuniform arithmetic manifolds M = Γ\G/K that are not homeomorphic to it. While the manifolds M are properly rigid if rank_ℚ(Γ) ≤ 2, we show that the so‐called virtual structure group has infinite rank as a ℚ‐vector space if rank_ℚ(Γ) ≥ 4.© 2015 Wiley Periodicals, Inc.     

The Novikov Conjecture for Linear Groups

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C^*-algebra theory.     

The signature operator at 2

It is well known that the signature operator on a manifold defines a K-homology class which is an orientation after inverting 2. Here we address the following puzzle: What is this class localized at 2, and what special properties does it have? Our answers include the following:

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the K-homology class Δ_M of the signature operator is a bordism invariant;

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the reduction mod 8 of the K-homology class of the signature operator is an oriented homotopy invariant;

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the reduction mod 16 of the K-homology class of the signature operator is not an oriented homotopy invariant.