Schur-positive sets of permutations via products and grid classes

Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur-positive is a long-standing problem in algebraic combinatorics. In this paper, we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.     

On Lipschitz extension from finite subsets

We prove that for every n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X, a Banach space (Z, \({\left\| \right\|_Z}\)) and a 1-Lipschitz function f: S → Z such that the Lipschitz constant of every function F: X → Z that extends f is at least a constant multiple of \(\sqrt {\log n} \). This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X and a function f: S → ?₂ that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ?₂ that extends f satisfies \({\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}\). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].     

An adjoint-based pressure correction scheme

Incompressible flows are solenoidal, i. e. the divergence of the flow field is zero. This is an algebraic constraint on the solution in time. The pressure has to be determined, so that the constraint is fulfilled. To calculate the pressure often a Poisson equation is derived, which is then solved by an iterative method. Instead of this the constraint is here formulated as an optimization problem. The objective functional is taken as the square of the norm of the divergence. A gradient based optimization is performed to calculate the pressure in every time step of the simulation. By this an alternative iterative scheme is derived. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enumeration and randomized constructions of hypertrees

Over 30 years ago, Kalai proved a beautiful d‐dimensional analog of Cayley's formula for the number of n‐vertex trees. He enumerated d‐dimensional hypertrees weighted by the squared size of their (d ? 1)‐dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of d‐hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of d‐hypertrees. In addition, we study a random 1‐out model of d‐complexes where every (d ? 1)‐dimensional face selects a random d‐face containing it, and show that it has a negligible d‐dimensional homology.     

Q-switching an all-fiber laser using acousto-optic null coupler

A new method for Q-switching an all-fiber laser is presented. It is based on induced acoustic long period grating operating on a null coupler, which acts as acoustically controlled tunable output coupler. Q-switching is achieved by switching on and off the acoustic wave in a burst mode, thereby generating laser pulses that are ~400 times shorter than the acoustically controlled coupler’s rise time. Output pulse energy of 22 μJ and temporal width of ~100 ns were measured at a wavelength of 1.54 μm.     

Statistics on wreath products and generalized binomial-stirling numbers

Various statistics on wreath products are defined via canonical words, “colored” right to left minima and “colored” descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to a MacMahon-type equi-distribution theorem over subsets with prescribed statistics. Partially supported by Minerva Grant No. 8441 and by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272. Partially supported by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272.     

No directed fractal percolation in zero area

We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {p _k}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN ¹⁻ k is partitioned intoN ² congruent subsquares, and each of these is retained with probabilityp _k. independent of all others. We show that if Π_k p _k =0 (i.e., if the area of the limiting set vanishes a.s.), then a.s. the limiting set contains no directed crossings of the unit square (a directed crossing is a path that crosses the unit square from left to right, and moves only up, down, and to the right).     

Application of the thermodynamics of curved boundary layers to the scaled particle theory of hard-sphere fluids

The thermodynamics of curved boundary layers, with the assumption that the distance between the surface of a fluid cavity and its surface of tension is a quadratic function of the cavity radius, is applied to the exact thermo-dynamic expression forG, the central function of scaled particle theory. The coefficients in the quadratic representation are determined so as to satisfyall five of the known exact conditions onG valid for cavity radii between one-half and one molecular diameter. The results of the calculation are displayed as the hard-sphere equation of state, the boundary tension associated with the surface of tension, and the distance between the cavity surface and the surface of tension. Although the hard-sphere equation of state obtained by this method using all five conditions onG is more accurate than in the case where only two or three conditions are used, the original scaled particle theory, in whichG itself was represented simply by a quadratic function of inverse powers of cavity radius, still yields the more accurate equation of state. Nevertheless, the present approach limits approximations to the distance between the cavity surface and the surface of tension, a small quantity in itself. The path to a still more improved theory remains well defined, contingent upon the discovery of additional exact conditions, and does not depend, as in the original scaled particle theory, on a form forG arrived at in a somewhat ad hoc manner.Research supported under NSF Grant #GP-12408.