41,223 research outputs found
Formulas for Generalized Two-Qubit Separability Probabilities
To begin, we find certain formulas , for . These yield that part of the total
separability probability, , for generalized (real, complex,
quaternionic,\ldots) two-qubit states endowed with random induced measure, for
which the determinantal inequality holds. Here
denotes a density matrix, obtained by tracing over the pure states
in -dimensions, and , its partial transpose.
Further, is a Dyson-index-like parameter with for the
standard (15-dimensional) convex set of (complex) two-qubit states. For ,
we obtain the previously reported Hilbert-Schmidt formulas, with (the real
case) , (the standard complex case)
, and (the quaternionic case) ---the three simply equalling . The factors
are sums of polynomial-weighted generalized hypergeometric
functions , , all with argument . We find number-theoretic-based formulas for the upper
() and lower () parameter sets of these functions and, then,
equivalently express in terms of first-order difference
equations. Applications of Zeilberger's algorithm yield "concise" forms,
parallel to the one obtained previously for . For
nonnegative half-integer and integer values of , has
descending roots starting at . Then, we (C. Dunkl and I) construct
a remarkably compact (hypergeometric) form for itself. The
possibility of an analogous "master" formula for is, then,
investigated, and a number of interesting results found.Comment: 78 pages, 5 figures, 15 appendices, to appear in Adv. Math.
Phys--verification in arXiv:1701.01973 of 8/33-two-qubit Hilbert-Schmidt
separability probability conjecture note
Weak dentability index of spaces
We compute the weak-dentability index of the spaces where is a
countable compact space. Namely , whenever . More generally,
if is a scattered compact whose height
satisfies with an
countable
Clustering in a hyperbolic model of complex networks
In this paper we consider the clustering coefficient and clustering function
in a random graph model proposed by Krioukov et al.~in 2010. In this model,
nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes
are connected if they are at most a certain hyperbolic distance from each
other. It has been shown that this model has various properties associated with
complex networks, e.g. power-law degree distribution, short distances and
non-vanishing clustering coefficient. Here we show that the clustering
coefficient tends in probability to a constant that we give explicitly
as a closed form expression in terms of and certain special
functions. This improves earlier work by Gugelmann et al., who proved that the
clustering coefficient remains bounded away from zero with high probability,
but left open the issue of convergence to a limiting constant. Similarly, we
are able to show that , the average clustering coefficient over all
vertices of degree exactly , tends in probability to a limit
which we give explicitly as a closed form expression in terms of
and certain special functions. We are able to extend this last result also to
sequences where grows as a function of . Our results show
that scales differently, as grows, for different ranges of
. More precisely, there exists constants depending on
and , such that as , if , if and
when . These
results contradict a claim of Krioukov et al., which stated that the limiting
values should always scale with as we let grow.Comment: 127 page
Convergence Radii for Eigenvalues of Tri--diagonal Matrices
Consider a family of infinite tri--diagonal matrices of the form
where the matrix is diagonal with entries and the matrix
is off--diagonal, with nonzero entries The spectrum of is discrete. For small the
-th eigenvalue is a well--defined analytic
function. Let be the convergence radius of its Taylor's series about It is proved that R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq
\alpha <11/6.$
Anomalous thermal expansion of SbTe topological insulator
We have investigated the temperature dependence of the linear thermal
expansion along the hexagonal c axis (), in-plane resistivity
(), and specific heat () of the topological insulator SbTe
single crystal. exhibits a clear anomaly in the temperature region
204-236 K. The coefficient of linear thermal expansion decreases
rapidly above 204 K, passes through a deep minimum at around 225 K and then
increases abruptly in the region 225-236 K. is negative in the
interval 221-228 K. The temperature dependence of both and can
be described well by the Debye model from 2 to 290 K, excluding the region
around the anomaly in
Computing Hypercircles by Moving Hyperplanes
Let K be a field of characteristic zero, alpha algebraic of degree n over K.
Given a proper parametrization psi of a rational curve C, we present a new
algorithm to compute the hypercircle associated to the parametrization psi. As
a consequence, we can decide if the curve C is defined over K and, if not, to
compute the minimum field of definition of C containing K. The algorithm
exploits the conjugate curves of C but avoids computation in the normal closure
of K(alpha) over K.Comment: 16 page
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