4,852 research outputs found
Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling
We consider the multivariate point process determined by the crossing times
of the components of a multivariate jump process through a multivariate
boundary, assuming to reset each component to an initial value after its
boundary crossing. We prove that this point process converges weakly to the
point process determined by the crossing times of the limit process. This holds
for both diffusion and deterministic limit processes. The almost sure
convergence of the first passage times under the almost sure convergence of the
processes is also proved. The particular case of a multivariate Stein process
converging to a multivariate Ornstein-Uhlenbeck process is discussed as a
guideline for applying diffusion limits for jump processes. We apply our
theoretical findings to neural network modeling. The proposed model gives a
mathematical foundation to the generalization of the class of Leaky
Integrate-and-Fire models for single neural dynamics to the case of a firing
network of neurons. This will help future study of dependent spike trains.Comment: 20 pages, 1 figur
Exact valence bond entanglement entropy and probability distribution in the XXX spin chain and the Potts model
By relating the ground state of Temperley-Lieb hamiltonians to partition
functions of 2D statistical mechanics systems on a half plane, and using a
boundary Coulomb gas formalism, we obtain in closed form the valence bond
entanglement entropy as well as the valence bond probability distribution in
these ground states. We find in particular that for the XXX spin chain, the
number N_c of valence bonds connecting a subsystem of size L to the outside
goes, in the thermodynamic limit, as = (4/pi^2) ln L, disproving a recent
conjecture that this should be related with the von Neumann entropy, and thus
equal to 1/(3 ln 2) ln L. Our results generalize to the Q-state Potts model.Comment: 4 pages, 2 figure
The antiferromagnetic transition for the square-lattice Potts model
We solve the antiferromagnetic transition for the Q-state Potts model
(defined geometrically for Q generic) on the square lattice. The solution is
based on a detailed analysis of the Bethe ansatz equations (which involve
staggered source terms) and on extensive numerical diagonalization of transfer
matrices. It involves subtle distinctions between the loop/cluster version of
the model, and the associated RSOS and (twisted) vertex models. The latter's
continuum limit involves two bosons, one which is compact and twisted, and the
other which is not, with a total central charge c=2-6/t, for
sqrt(Q)=2cos(pi/t). The non-compact boson contributes a continuum component to
the spectrum of critical exponents. For Q generic, these properties are shared
by the Potts model. For Q a Beraha number [Q = 4 cos^2(pi/n) with n integer]
the two-boson theory is truncated and becomes essentially Z\_{n-2}
parafermions. Moreover, the vertex model, and, for Q generic, the Potts model,
exhibit a first-order critical point on the transition line, i.e., the critical
point is also the locus of level crossings where the derivatives of the free
energy are discontinuous. In that sense, the thermal exponent of the Potts
model is generically nu=1/2. Things are profoundly different for Q a Beraha
number, where the transition is second order, with nu=(t-2)/2 determined by the
psi\_1 parafermion. As one enters the adjacant Berker-Kadanoff phase, the model
flows, for t odd, to a minimal model of CFT with c=1-6/t(t-1), while for t even
it becomes massive. This provides a physical realization of a flow conjectured
by Fateev and Zamolodchikov in the context of Z\_N integrable perturbations.
Finally, we argue that the antiferromagnetic transition occurs as well on other
two-dimensional lattices
Conformal boundary loop models
We study a model of densely packed self-avoiding loops on the annulus,
related to the Temperley Lieb algebra with an extra idempotent boundary
generator. Four different weights are given to the loops, depending on their
homotopy class and whether they touch the outer rim of the annulus. When the
weight of a contractible bulk loop x = q + 1/q satisfies -2 < x <= 2, this
model is conformally invariant for any real weight of the remaining three
parameters. We classify the conformal boundary conditions and give exact
expressions for the corresponding boundary scaling dimensions. The amplitudes
with which the sectors with any prescribed number and types of non contractible
loops appear in the full partition function Z are computed rigorously. Based on
this, we write a number of identities involving Z which hold true for any
finite size. When the weight of a contractible boundary loop y takes certain
discrete values, y_r = [r+1]_q / [r]_q with r integer, other identities
involving the standard characters K_{r,s} of the Virasoro algebra are
established. The connection with Dirichlet and Neumann boundary conditions in
the O(n) model is discussed in detail, and new scaling dimensions are derived.
When q is a root of unity and y = y_r, exact connections with the A_m type RSOS
model are made. These involve precise relations between the spectra of the loop
and RSOS model transfer matrices, valid in finite size. Finally, the results
where y=y_r are related to the theory of Temperley Lieb cabling.Comment: 28 pages, 19 figures, 2 tables. v2: added new section 3.2, amended
figures 17-18, updated reference
Detektion organischer Moleküle mit Hilfe von heizbaren Elektroden im negativen Potentialbereich
Es wurden direkt heizbare Bismutelektroden entwickelt, welche für den Einsatz im negativen Potentialbereichen geeignet sind. Mit diesen Elektroden konnten kleinste Konzentrationen organischer Moleküle detektieret werden. Weiterhin ist es gelungen, Hybridisierungen von Nukleinsäuren an geheizten Goldelektroden durchzuführen. Es ist dabei gelungen, sowohl DNA- als auch RNA-Stränge mit dem Komplex [OsO4(bipy)] zu markieren und zu detektieren. Durch den Einsatz erhöhter Temperaturen konnten außerdem Fehlpaarungen in DNA-Strängen zuverlässig nachgewiesen werden.For a bigger commensurable potential range we used directly heatable bismuth electrodes. With these electrodes it was possible to measure in a more negative potential. We could detect a very small amount of organic molecules. Furthermore we could detect hybridization reactions on heated gold electrodes. We used targed strands labeled with the complex [OsO4(bipy] and detect them on immobilized probes. It was possible to label DNA and RNA-strands. The use of higher temperatures enabled the separation of complementary DNA-targets and targets containing mismatches
Is the five-flow conjecture almost false?
The number of nowhere zero Z_Q flows on a graph G can be shown to be a
polynomial in Q, defining the flow polynomial \Phi_G(Q). According to Tutte's
five-flow conjecture, \Phi_G(5) > 0 for any bridgeless G.A conjecture by Welsh
that \Phi_G(Q) has no real roots for Q \in (4,\infty) was recently disproved by
Haggard, Pearce and Royle. These authors conjectured the absence of roots for Q
\in [5,\infty). We study the real roots of \Phi_G(Q) for a family of non-planar
cubic graphs known as generalised Petersen graphs G(m,k). We show that the
modified conjecture on real flow roots is also false, by exhibiting infinitely
many real flow roots Q>5 within the class G(nk,k). In particular, we compute
explicitly the flow polynomial of G(119,7), showing that it has real roots at
Q\approx 5.0000197675 and Q\approx 5.1653424423. We moreover prove that the
graph families G(6n,6) and G(7n,7) possess real flow roots that accumulate at
Q=5 as n\to\infty (in the latter case from above and below); and that
Q_c(7)\approx 5.2352605291 is an accumulation point of real zeros of the flow
polynomials for G(7n,7) as n\to\infty.Comment: 44 pages (LaTeX2e). Includes tex file, three sty files, and a
mathematica script polyG119_7.m. Many improvements from version 3, in
particular Sections 3 and 4 have been mostly re-writen, and Sections 7 and 8
have been eliminated. (This material can now be found in arXiv:1303.5210.)
Final version published in J. Combin. Theory
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