573 research outputs found
The CMV bispectral problem
A classical result due to Bochner classifies the orthogonal polynomials on
the real line which are common eigenfunctions of a second order linear
differential operator. We settle a natural version of the Bochner problem on
the unit circle which answers a similar question concerning orthogonal Laurent
polynomials and can be formulated as a bispectral problem involving CMV
matrices. We solve this CMV bispectral problem in great generality proving
that, except the Lebesgue measure, no other one on the unit circle yields a
sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear
differential operator of arbitrary order. Actually, we prove that this is the
case even if such an eigenfunction condition is imposed up to finitely many
orthogonal Laurent polynomials.Comment: 25 pages, final version, to appear in International Mathematics
Research Notice
Some taste substances are direct activators of G-proteins
Amphiphilic substances may stimulate cellular events through direct activation of G-proteins. The present experiments indicate that several amphiphilic sweeteners and the bitter tastant, quinine, activate transducin and Gi/Go-proteins. Concentrations of taste substances required to activate G-proteins in vitro correlated with those used to elicit taste. These data support the hypothesis that amphiphilic taste substances may elicit taste through direct activation of G-proteins
New remarks on the Cosmological Argument
We present a formal analysis of the Cosmological Argument in its two main
forms: that due to Aquinas, and the revised version of the Kalam Cosmological
Argument more recently advocated by William Lane Craig. We formulate these two
arguments in such a way that each conclusion follows in first-order logic from
the corresponding assumptions. Our analysis shows that the conclusion which
follows for Aquinas is considerably weaker than what his aims demand. With
formalizations that are logically valid in hand, we reinterpret the natural
language versions of the premises and conclusions in terms of concepts of
causality consistent with (and used in) recent work in cosmology done by
physicists. In brief: the Kalam argument commits the fallacy of equivocation in
a way that seems beyond repair; two of the premises adopted by Aquinas seem
dubious when the terms `cause' and `causality' are interpreted in the context
of contemporary empirical science. Thus, while there are no problems with
whether the conclusions follow logically from their assumptions, the Kalam
argument is not viable, and the Aquinas argument does not imply a caused
origination of the universe. The assumptions of the latter are at best less
than obvious relative to recent work in the sciences. We conclude with mention
of a new argument that makes some positive modifications to an alternative
variation on Aquinas by Le Poidevin, which nonetheless seems rather weak.Comment: 12 pages, accepted for publication in International Journal for
Philosophy of Religio
On the number of simple arrangements of five double pseudolines
We describe an incremental algorithm to enumerate the isomorphism classes of
double pseudoline arrangements. The correction of our algorithm is based on the
connectedness under mutations of the spaces of one-extensions of double
pseudoline arrangements, proved in this paper. Counting results derived from an
implementation of our algorithm are also reported.Comment: 24 pages, 16 figures, 6 table
Determining All Universal Tilers
A universal tiler is a convex polyhedron whose every cross-section tiles the
plane. In this paper, we introduce a certain slight-rotating operation for
cross-sections of pentahedra. Based on a selected initial cross-section and by
applying the slight-rotating operation suitably, we prove that a convex
polyhedron is a universal tiler if and only if it is a tetrahedron or a
triangular prism.Comment: 18 pages, 12 figure
Askey-Wilson Type Functions, With Bound States
The two linearly independent solutions of the three-term recurrence relation
of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22],
are slightly modified so as to make it transparent that these functions satisfy
a beautiful symmetry property. It essentially means that the geometric and the
spectral parameters are interchangeable in these functions. We call the
resulting functions the Askey-Wilson functions. Then, we show that by adding
bound states (with arbitrary weights) at specific points outside of the
continuous spectrum of some instances of the Askey-Wilson difference operator,
we can generate functions that satisfy a doubly infinite three-term recursion
relation and are also eigenfunctions of -difference operators of arbitrary
orders. Our result provides a discrete analogue of the solutions of the purely
differential version of the bispectral problem that were discovered in the
pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction
Occupation Time for Classical and Quantum Walks
This is a personal tribute to Lance Littlejohn on the occasion of his 70th birthday. It is meant as a present to him for many years of friendship. It is not written in the “Satz-Beweis” style of Edmund Landau or even in the format of a standard mathematics paper. It is rather an invitation to a fairly new, largely unexplored, topic in the hope that Lance will read it some afternoon and enjoy it. If he cares about complete proofs he will have to wait a bit longer; we almost have them but not in time for this volume. We hope that the figures will convince him and other readers that the phenomena displayed here are interesting enough
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