4,326 research outputs found
Slices, slabs, and sections of the unit hypercube
Using combinatorial methods, we derive several formulas for the volume of
convex bodies obtained by intersecting a unit hypercube with a halfspace, or
with a hyperplane of codimension 1, or with a flat defined by two parallel
hyperplanes. We also describe some of the history of these problems, dating to
Polya's Ph.D. thesis, and we discuss several applications of these formulas.Comment: 11 pages; minor corrections to reference
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
Variational data assimilation using targetted random walks
The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis (offline hindcasting). In either of these scenarios it can be important to assess uncertainties in the assimilated state. Ideally it would be desirable to have complete information concerning the Bayesian posterior distribution for unknown state, given data. The purpose of this paper is to show that complete computational probing of this posterior distribution is now within reach in the offline situation. In this paper we will introduce an MCMC method which enables us to directly sample from the Bayesian\ud
posterior distribution on the unknown functions of interest, given observations. Since we are aware that these\ud
methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however more sophisticated MCMC methods are available\ud
which do exploit derivative information. For simplicity of exposition we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number (Stokes flow) scenario in a two dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces
Carries, shuffling, and symmetric functions
The "carries" when n random numbers are added base b form a Markov chain with
an "amazing" transition matrix determined by Holte. This same Markov chain
occurs in following the number of descents or rising sequences when n cards are
repeatedly riffle shuffled. We give generating and symmetric function proofs
and determine the rate of convergence of this Markov chain to stationarity.
Similar results are given for type B shuffles. We also develop connections with
Gaussian autoregressive processes and the Veronese mapping of commutative
algebra.Comment: 23 page
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres
Evaluation of Harmonic Sums with Integrals
We consider the sums and
with being a positive
integer. We evaluate these sums with multiple integration, a modern technique.
First, we start with three different double integrals that have been previously
used in the literature to show which implies Euler's identity
Then, we generalize each integral in order to find the
considered sums. The dimensional analogue of the first integral is the
density function of the quotient of independent, nonnegative Cauchy random
variables. In seeking this function, we encounter a special logarithmic
integral that we can directly relate to The dimensional analogue of
the second integral, upon a change of variables, is the volume of a convex
polytope, which can be expressed as a probability involving certain pairwise
sums of independent uniform random variables. We use combinatorial
arguments to find the volume, which in turn gives new closed formulas for
and The dimensional analogue of the last integral, upon
another change of variables, is an integral of the joint density function of
Cauchy random variables over a hyperbolic polytope. This integral can be
expressed as a probability involving certain pairwise products of these random
variables, and it is equal to the probability from the second generalization.
Thus, we specifically highlight the similarities in the combinatorial arguments
between the second and third generalizations.Comment: Fixed Typos. To Appear in AMS Quarterly of Applied Mathematics
September 201
Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow
Fermat, Leibniz, Euler, and Cauchy all used one or another form of
approximate equality, or the idea of discarding "negligible" terms, so as to
obtain a correct analytic answer. Their inferential moves find suitable proxies
in the context of modern theories of infinitesimals, and specifically the
concept of shadow. We give an application to decreasing rearrangements of real
functions.Comment: 35 pages, 2 figures, to appear in Notices of the American
Mathematical Society 61 (2014), no.
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