8,073 research outputs found
Matrix product states and the quantum max-flow/min-cut conjectures
In this note we discuss the geometry of matrix product states with periodic
boundary conditions and provide three infinite sequences of examples where the
quantum max-flow is strictly less than the quantum min-cut. In the first we fix
the underlying graph to be a 4-cycle and verify a prediction of Hastings that
inequality occurs for infinitely many bond dimensions. In the second we
generalize this result to a 2d-cycle. In the third we show that the 2d-cycle
with periodic boundary conditions gives inequality for all d when all bond
dimensions equal two, namely a gap of at least 2^{d-2} between the quantum
max-flow and the quantum min-cut.Comment: 12 pages, 3 figures - Final version accepted for publication on J.
Math. Phy
A Fresh Look at Entropy and the Second Law of Thermodynamics
This paper is a non-technical, informal presentation of our theory of the
second law of thermodynamics as a law that is independent of statistical
mechanics and that is derivable solely from certain simple assumptions about
adiabatic processes for macroscopic systems. It is not necessary to assume
a-priori concepts such as "heat", "hot and cold", "temperature". These are
derivable from entropy, whose existence we derive from the basic assumptions.
See cond-mat/9708200 and math-ph/9805005.Comment: LaTex file. To appear in the April 2000 issue of PHYSICS TODA
Secants of Lagrangian Grassmannians
We study the dimensions of secant varieties of the Grassmannian of Lagrangian
subspaces in a symplectic vector space. We calculate these dimensions for third
and fourth secant varieties. Our result is obtained by providing a normal form
for four general points on such a Grassmannian and by explicitly calculating
the tangent spaces at these four points
General pseudoadditivity of composable entropy prescribed by existence of equilibrium
The concept of composability states that entropy of the total system composed
of independent subsystems is a function of entropies of the subsystems. Here,
the most general pseudoadditivity rule for composable entropy is derived based
only on existence of equilibrium.Comment: 12 page
Quasi-Homogeneous Thermodynamics and Black Holes
We propose a generalized thermodynamics in which quasi-homogeneity of the
thermodynamic potentials plays a fundamental role. This thermodynamic formalism
arises from a generalization of the approach presented in paper [1], and it is
based on the requirement that quasi-homogeneity is a non-trivial symmetry for
the Pfaffian form . It is shown that quasi-homogeneous
thermodynamics fits the thermodynamic features of at least some
self-gravitating systems. We analyze how quasi-homogeneous thermodynamics is
suggested by black hole thermodynamics. Then, some existing results involving
self-gravitating systems are also shortly discussed in the light of this
thermodynamic framework. The consequences of the lack of extensivity are also
recalled. We show that generalized Gibbs-Duhem equations arise as a consequence
of quasi-homogeneity of the thermodynamic potentials. An heuristic link between
this generalized thermodynamic formalism and the thermodynamic limit is also
discussed.Comment: 39 pages, uses RevteX. Published version (minor changes w.r.t. the
original one
Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator
The uniformity, for the family of exceptional Lie algebras g, of the
decompositions of the powers of their adjoint representations is well-known now
for powers up to the fourth. The paper describes an extension of this
uniformity for the totally antisymmetrised n-th powers up to n=9, identifying
(see Tables 3 and 6) families of representations with integer eigenvalues
5,...,9 for the quadratic Casimir operator, in each case providing a formula
(see eq. (11) to (15)) for the dimensions of the representations in the family
as a function of D=dim g. This generalises previous results for powers j and
Casimir eigenvalues j, j<=4. Many intriguing, perhaps puzzling, features of the
dimension formulas are discussed and the possibility that they may be valid for
a wider class of not necessarily simple Lie algebras is considered.Comment: 16 pages, LaTeX, 1 figure, 9 tables; v2: presentation improved, typos
correcte
Ultraperipheral photoproduction of vector mesons in the nuclear Coulomb field and the size of neutral vector mesons
We point out a significance of ultraperipheral photoproduction of vector
mesons in the Coulomb field of nuclei as a means of measuring the radius of the
neutral vector meson. This new contribution to the production amplitude is very
small compared to the conventional diffractive amplitude, but because of large
impact parameters inherent to the ultraperipheral Coulomb mechanism its impact
on the diffraction slope is substantial. We predict appreciable and strongly
energy dependent increase of the diffraction slope towards very small momentum
transfer.The magnitude of the effect is proportional to the mean radius squared
of the vector meson and is within the reach of high precision photoproduction
experiments, which gives a unique experimental handle on the size of vector
mesons
Notes on the Third Law of Thermodynamics.I
We analyze some aspects of the third law of thermodynamics. We first review
both the entropic version (N) and the unattainability version (U) and the
relation occurring between them. Then, we heuristically interpret (N) as a
continuity boundary condition for thermodynamics at the boundary T=0 of the
thermodynamic domain. On a rigorous mathematical footing, we discuss the third
law both in Carath\'eodory's approach and in Gibbs' one. Carath\'eodory's
approach is fundamental in order to understand the nature of the surface T=0.
In fact, in this approach, under suitable mathematical conditions, T=0 appears
as a leaf of the foliation of the thermodynamic manifold associated with the
non-singular integrable Pfaffian form . Being a leaf, it cannot
intersect any other leaf const. of the foliation. We show that (N) is
equivalent to the requirement that T=0 is a leaf. In Gibbs' approach, the
peculiar nature of T=0 appears to be less evident because the existence of the
entropy is a postulate; nevertheless, it is still possible to conclude that the
lowest value of the entropy has to belong to the boundary of the convex set
where the function is defined.Comment: 29 pages, 2 figures; RevTex fil
Chemical Potential and the Nature of the Dark Energy: The case of phantom
The influence of a possible non zero chemical potential on the nature
of dark energy is investigated by assuming that the dark energy is a
relativistic perfect simple fluid obeying the equation of state (EoS),
(). The entropy condition, ,
implies that the possible values of are heavily dependent on the
magnitude, as well as on the sign of the chemical potential. For , the
-parameter must be greater than -1 (vacuum is forbidden) while for not only the vacuum but even a phantomlike behavior () is
allowed. In any case, the ratio between the chemical potential and temperature
remains constant, that is, . Assuming that the dark energy
constituents have either a bosonic or fermionic nature, the general form of the
spectrum is also proposed. For bosons is always negative and the extended
Wien's law allows only a dark component with which includes
vacuum and the phantomlike cases. The same happens in the fermionic branch for
are permmited only if . The thermodynamics and statistical arguments constrain the
EoS parameter to be , a result surprisingly close to the maximal
value required to accelerate a FRW type universe dominated by matter and dark
energy ().Comment: 7 pages, 5 figure
Four lectures on secant varieties
This paper is based on the first author's lectures at the 2012 University of
Regina Workshop "Connections Between Algebra and Geometry". Its aim is to
provide an introduction to the theory of higher secant varieties and their
applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in
Mathematics & Statistics), Springer/Birkhause
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