2,073 research outputs found
An note on the maximization of matrix valued Hankel determinants with application
In this note we consider the problem of maximizing the determinant of moment matrices of matrix measures. The maximizing matrix measure can be characterized explicitly by having equal (matrix valued) weights at the zeros of classical (one dimensional) orthogonal polynomials. The results generalize classical work of Schoenberg (1959) to the case of matrix measures. As a statistical application we consider several optimal design problems in linear models, which generalize the classical weighing design problems. --Matrix measures,Hankel matrix,orthogonal polynomials,approximate optimal designs,spring balance weighing designs
Matrix measures and random walks
In this paper we study the connection between matrix measures and random walks with a tridiagonal block transition matrix. We derive sufficient conditions such that the blocks of the n-step transition matrix of the Markov chain can be represented as integrals with respect to a matrix valued spectral measure. Several stochastic properties of the processes are characterized by means of this matrix measure. In many cases this measure is supported in the interval [-1, 1]. The results are illustrated by several examples including random walks on a grid and the embedded chain of a queuing system. --Markov chain,block tridiagonal transition matrix,spectral measure,matrix measure,quasi birth and death processes,canonical moments
D-optimal designs via a cocktail algorithm
A fast new algorithm is proposed for numerical computation of (approximate)
D-optimal designs. This "cocktail algorithm" extends the well-known vertex
direction method (VDM; Fedorov 1972) and the multiplicative algorithm (Silvey,
Titterington and Torsney, 1978), and shares their simplicity and monotonic
convergence properties. Numerical examples show that the cocktail algorithm can
lead to dramatically improved speed, sometimes by orders of magnitude, relative
to either the multiplicative algorithm or the vertex exchange method (a variant
of VDM). Key to the improved speed is a new nearest neighbor exchange strategy,
which acts locally and complements the global effect of the multiplicative
algorithm. Possible extensions to related problems such as nonparametric
maximum likelihood estimation are mentioned.Comment: A number of changes after accounting for the referees' comments
including new examples in Section 4 and more detailed explanations throughou
Solution of a Generalized Stieltjes Problem
We present the exact solution for a set of nonlinear algebraic equations
. These
were encountered by us in a recent study of the low energy spectrum of the
Heisenberg ferromagnetic chain \cite{dhar}. These equations are low
(density) ``degenerations'' of more complicated transcendental equation of
Bethe's Ansatz for a ferromagnet, but are interesting in themselves. They
generalize, through a single parameter, the equations of Stieltjes,
, familiar from Random Matrix theory.
It is shown that the solutions of these set of equations is given by the
zeros of generalized associated Laguerre polynomials. These zeros are
interesting, since they provide one of the few known cases where the location
is along a nontrivial curve in the complex plane that is determined in this
work.
Using a ``Green's function'' and a saddle point technique we determine the
asymptotic distribution of zeros.Comment: 19 pages, 4 figure
Physical state of 2-methylbutane-1,2,3,4-tetraol in pure and internally mixed aerosols
2-Methylbutane-1,2,3,4-tetraol (hereafter named tetraol) is an important
oxidation product of isoprene and can be considered as a marker compound for
isoprene-derived secondary organic aerosols (SOAs). Little is known about
this compound's physical phase state, although some field observations
indicate that isoprene-derived secondary organic aerosols in the tropics tend
to be in a liquid rather than a solid state. To gain more knowledge about the
possible phase states of tetraol and of tetraol-containing SOA particles, we
synthesized tetraol as racemates as well as enantiomerically enriched
materials. Subsequently the obtained highly viscous dry liquids were
investigated calorimetrically by differential scanning calorimetry revealing
subambient glass transition temperatures Tg. We also show that
only the diastereomeric isomers differ in their Tg values, albeit
only by a few kelvin. We derive the phase diagram of water–tetraol mixtures
over the whole tropospheric temperature and humidity range from determining
glass transition temperatures and ice melting temperatures of aqueous tetraol
mixtures. We also investigated how water diffuses into a sample of dry
tetraol. We show that upon water uptake two homogeneous liquid domains form
that are separated by a sharp, locally constrained concentration gradient.
Finally, we measured the glass transition temperatures of mixtures of tetraol
and an important oxidation product of α-pinene-derived SOA:
3-methylbutane-1,2,3-tricarboxylic acid (3-MBTCA). Overall, our results imply
a liquid-like state of isoprene-derived SOA particles in the lower troposphere
at moderate to high relative humidity (RH), but presumably a semisolid or even
glassy state at upper tropospheric conditions, particularly at low relative
humidity, thus providing experimental support for recent modeling calculations.</p
The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials
We present a general, asymptotical solution for the discretised harmonic
oscillator. The corresponding Schr\"odinger equation is canonically conjugate
to the Mathieu differential equation, the Schr\"odinger equation of the quantum
pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian
of an isolated Josephon junction or a superconducting single-electron
transistor (SSET), we obtain an asymptotical representation of Mathieu
functions. We solve the discretised harmonic oscillator by transforming the
infinite-dimensional matrix-eigenvalue problem into an infinite set of
algebraic equations which are later shown to be satisfied by the obtained
solution. The proposed ansatz defines a new class of generalised Hermite
polynomials which are explicit functions of the coupling parameter and tend to
ordinary Hermite polynomials in the limit of vanishing coupling constant. The
polynomials become orthogonal as parts of the eigenvectors of a Hermitian
matrix and, consequently, the exponential part of the solution can not be
excluded. We have conjectured the general structure of the solution, both with
respect to the quantum number and the order of the expansion. An explicit proof
is given for the three leading orders of the asymptotical solution and we
sketch a proof for the asymptotical convergence of eigenvectors with respect to
norm. From a more practical point of view, we can estimate the required effort
for improving the known solution and the accuracy of the eigenvectors. The
applied method can be generalised in order to accommodate several variables.Comment: 18 pages, ReVTeX, the final version with rather general expression
A note on the invariant distribution of a quasi-birth-and-death process
The aim of this paper is to give an explicit formula of the invariant
distribution of a quasi-birth-and-death process in terms of the block entries
of the transition probability matrix using a matrix-valued orthogonal
polynomials approach. We will show that the invariant distribution can be
computed using the squared norms of the corresponding matrix-valued orthogonal
polynomials, no matter if they are or not diagonal matrices. We will give an
example where the squared norms are not diagonal matrices, but nevertheless we
can compute its invariant distribution
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