Sur les types des propositions composées

Il s'agit d'un problème combinatoire de logique formelle, formulé par Jevons; il sera expliqué en détails dans ce qui suit (voir no. 1). Jevona luimême n'a traité le problème que dans les cas les plus simples (n = 1, 2, 3); un cas plus difficile (n = 4) a été traité par Clifford; le cas général (n quelconque) a été à peine abordé. Le but de ce travail est de faire remarquer que ce problème de Jevons et de Clifford est contenu comme cas particutier dans un problème combinatoire général que j'ai traité ailleurs. La méthode générale ramène le problème présent à l'étude d'un certain groupe de permutations d'ordre n!2 n , étroitement lié au groupe symétrique d'ordre n!. J'ai fait les calculs nécessaires pour n = 1, 2, 3, 4. Mes résultats numèriques sont complètement en accord avec les résultats de Jevons, mais ils ne s'accordent qu'en partie avec les résultats de Clifford. Une proposition peut être vraie ou fausse. On peut exprimer la même chose en disant que nous pouvons attribuer à une proposition l'une ou l'autre des deux "valeurs logiques” qui s'excluent mutuellement: la "vérité” et la "fausseté.

Multiplier Sequences for Simple Sets of Polynomials

In this paper we give a new characterization of simple sets of polynomials B with the property that the set of B-multiplier sequences contains all Q-multiplier sequence for every simple set Q. We characterize sequences of real numbers which are multiplier sequences for every simple set Q, and obtain some results toward the partitioning of the set of classical multiplier sequences

Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems

We consider a class of Ising spin systems on a set \Lambda of sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the |\Lambda| Lee-Yang zeros in the complex z = exp{2\beta h} plane, where \beta is the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of \beta. In some cases one obtains, in an appropriately taken \beta to infinity limit, a gas of hard objects on a set \Lambda'; the fugacity for the limiting system is a rescaling of z and the Lee-Yang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the negative real axis for all \beta. One zero-temperature limit of this type, for example, is a monomer-dimer system; our results thus generalize, to finite \beta, a well-known result of Heilmann and Lieb that the Lee-Yang zeros of monomer-dimer systems are real and negative.Comment: Plain TeX. Seventeen pages, five figures from .eps files. Version 2 corrects minor errors in version

The support of the limit distribution of optimal Riesz energy points on sets of revolution in R3\mathbb{R}^{3}

Let A be a compact set in the right-half plane and $\Gamma(A)$ the set in $\mathbb{R}^{3}$ obtained by rotating A about the vertical axis. We investigate the support of the limit distribution of minimal energy point charges on $\Gamma(A)$ that interact according to the Riesz potential 1/r^{s}, 0<s<1, where r is the Euclidean distance between points. Potential theory yields that this limit distribution coincides with the equilibrium measure on $\Gamma(A)$ which is supported on the outer boundary of $\Gamma(A)$. We show that there are sets of revolution $\Gamma(A)$ such that the support of the equilibrium measure on $\Gamma(A)$ is {\bf not} the complete outer boundary, in contrast to the Coulomb case s=1. However, the support of the limit distribution on the set of revolution $\Gamma(R+A)$ as R goes to infinity, is the full outer boundary for certain sets A, in contrast to the logarithmic case (s=0)

A New Class of Non-Linear Stability Preserving Operators

We extend Br\"and\'en's recent proof of a conjecture of Stanley and describe a new class of non-linear operators that preserve weak Hurwitz stability and the Laguerre-P\'olya class.Comment: Fixed typos, spelling, and updated links in reference

Coherent states in fermionic Fock-Krein spaces and their amplitudes

We generalize the fermionic coherent states to the case of Fock-Krein spaces, i.e., Fock spaces with an idefinite inner product of Krein type. This allows for their application in topological or functorial quantum field theory and more specifically in general boundary quantum field theory. In this context we derive a universal formula for the amplitude of a coherent state in linear field theory on an arbitrary manifold with boundary.Comment: 20 pages, LaTeX + AMS + svmult (included), contribution to the proceedings of the conference "Coherent States and their Applications: A Contemporary Panorama" (Marseille, 2016); v2: minor corrections and added axioms from arXiv:1208.503

Equivalence between Poly\'a-Szeg\H{o} and relative capacity inequalities under rearrangement

The transformations of functions acting on sublevel sets that satisfy a P\'olya-Szeg\H{o} inequality are characterized as those being induced by transformations of sets that do not increase the associated capacity.Comment: 9 page

On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at infinity. Even in the special case of Euclidean space, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.Comment: 18 page

Sums of magnetic eigenvalues are maximal on rotationally symmetric domains

The sum of the first n energy levels of the planar Laplacian with constant magnetic field of given total flux is shown to be maximal among triangles for the equilateral triangle, under normalization of the ratio (moment of inertia)/(area)^3 on the domain. The result holds for both Dirichlet and Neumann boundary conditions, with an analogue for Robin (or de Gennes) boundary conditions too. The square similarly maximizes the eigenvalue sum among parallelograms, and the disk maximizes among ellipses. More generally, a domain with rotational symmetry will maximize the magnetic eigenvalue sum among all linear images of that domain. These results are new even for the ground state energy (n=1).Comment: 19 pages, 1 figur

Active Brownian Motion in Threshold Distribution of a Coulomb Blockade Model

Randomly-distributed offset charges affect the nonlinear current-voltage property via the fluctuation of the threshold voltage of Coulomb blockade arrays. We analytically derive the distribution of the threshold voltage for a model of one-dimensional locally-coupled Coulomb blockade arrays, and propose a general relationship between conductance and the distribution. In addition, we show the distribution for a long array is equivalent to the distribution of the number of upward steps for aligned objects of different height. The distribution satisfies a novel Fokker-Planck equation corresponding to active Brownian motion. The feature of the distribution is clarified by comparing it with the Wigner and Ornstein-Uhlenbeck processes. It is not restricted to the Coulomb blockade model, but instructive in statistical physics generally.Comment: 4pages, 3figure