215 research outputs found
Quasi-exactly Solvable Lie Superalgebras of Differential Operators
In this paper, we study Lie superalgebras of matrix-valued
first-order differential operators on the complex line. We first completely
classify all such superalgebras of finite dimension. Among the
finite-dimensional superalgebras whose odd subspace is nontrivial, we find
those admitting a finite-dimensional invariant module of smooth vector-valued
functions, and classify all the resulting finite-dimensional modules. The
latter Lie superalgebras and their modules are the building blocks in the
construction of QES quantum mechanical models for spin 1/2 particles in one
dimension.Comment: LaTeX2e using the amstex and amssymb packages, 24 page
Distribution of Eigenvalues for the Modular Group
The two-point correlation function of energy levels for free motion on the
modular domain, both with periodic and Dirichlet boundary conditions, are
explicitly computed using a generalization of the Hardy-Littlewood method. It
is shown that ion the limit of small separations they show an uncorrelated
behaviour and agree with the Poisson distribution but they have prominent
number-theoretical oscillations at larger scale. The results agree well with
numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital,
but can be obtained upon request from the first Autho
Spectral evolution of the SU(4) Kondo effect from the single impurity to the two-dimensional lattice
We describe the evolution of the SU(4) Kondo effect as the number of magnetic
centers increases from one impurity to the two-dimensional (2D) lattice. We
derive a Hubbard-Anderson model which describes a 2D array of atoms or
molecules with two-fold orbital degeneracy, acting as magnetic impurities and
interacting with a metallic host. We calculate the differential conductance,
observed typically in experiments of scanning tunneling spectroscopy, for
different arrangements of impurities on a metallic surface: a single impurity,
a periodic square lattice, and several sites of a rectangular cluster. Our
results point towards the crucial importance of the orbital degeneracy and
agree well with recent experiments in different systems of iron(II)
phtalocyanine molecules deposited on top of Au(111) [N. Tsukahara et al., Phys.
Rev. Lett. 106, 187201 (2011)], indicating that this would be the first
experimental realization of an artificial 2D SU(4) Kondo-lattice system.Comment: 17 pages, 4 figures. New version contains an Appendix with details of
the derivation of the Hamiltonian Eq.(2), derivation of the slave-boson
mean-field equations, and an estimation of the upper bounds of the RKKY
interactio
Cryptography from Information Loss
© Marshall Ball, Elette Boyle, Akshay Degwekar, Apoorvaa Deshpande, Alon Rosen, Vinod. Reductions between problems, the mainstay of theoretical computer science, efficiently map an instance of one problem to an instance of another in such a way that solving the latter allows solving the former.1 The subject of this work is “lossy” reductions, where the reduction loses some information about the input instance. We show that such reductions, when they exist, have interesting and powerful consequences for lifting hardness into “useful” hardness, namely cryptography. Our first, conceptual, contribution is a definition of lossy reductions in the language of mutual information. Roughly speaking, our definition says that a reduction C is t-lossy if, for any distribution X over its inputs, the mutual information I(X; C(X)) ≤ t. Our treatment generalizes a variety of seemingly related but distinct notions such as worst-case to average-case reductions, randomized encodings (Ishai and Kushilevitz, FOCS 2000), homomorphic computations (Gentry, STOC 2009), and instance compression (Harnik and Naor, FOCS 2006). We then proceed to show several consequences of lossy reductions: 1. We say that a language L has an f-reduction to a language L0 for a Boolean function f if there is a (randomized) polynomial-time algorithm C that takes an m-tuple of strings X = (x1, . . ., xm), with each xi ∈ {0, 1}n, and outputs a string z such that with high probability, L0(z) = f(L(x1), L(x2), . . ., L(xm)) Suppose a language L has an f-reduction C to L0 that is t-lossy. Our first result is that one-way functions exist if L is worst-case hard and one of the following conditions holds: f is the OR function, t ≤ m/100, and L0 is the same as L f is the Majority function, and t ≤ m/100 f is the OR function, t ≤ O(m log n), and the reduction has no error This improves on the implications that follow from combining (Drucker, FOCS 2012) with (Ostrovsky and Wigderson, ISTCS 1993) that result in auxiliary-input one-way functions. 2. Our second result is about the stronger notion of t-compressing f-reductions – reductions that only output t bits. We show that if there is an average-case hard language L that has a t-compressing Majority reduction to some language for t = m/100, then there exist collision-resistant hash functions. This improves on the result of (Harnik and Naor, STOC 2006), whose starting point is a cryptographic primitive (namely, one-way functions) rather than average-case hardness, and whose assumption is a compressing OR-reduction of SAT (which is now known to be false unless the polynomial hierarchy collapses). Along the way, we define a non-standard one-sided notion of average-case hardness, which is the notion of hardness used in the second result above, that may be of independent interest
Linking Topological Quantum Field Theory and Nonperturbative Quantum Gravity
Quantum gravity is studied nonperturbatively in the case in which space has a
boundary with finite area. A natural set of boundary conditions is studied in
the Euclidean signature theory, in which the pullback of the curvature to the
boundary is self-dual (with a cosmological constant). A Hilbert space which
describes all the information accessible by measuring the metric and connection
induced in the boundary is constructed and is found to be the direct sum of the
state spaces of all Chern-Simon theories defined by all choices of
punctures and representations on the spatial boundary . The integer
level of Chern-Simons theory is found to be given by , where is the cosmological constant and is a
breaking phase. Using these results, expectation values of observables which
are functions of fields on the boundary may be evaluated in closed form. The
Beckenstein bound and 't Hooft-Susskind holographic hypothesis are confirmed,
(in the limit of large area and small cosmological constant) in the sense that
once the two metric of the boundary has been measured, the subspace of the
physical state space that describes the further information that the observer
on the boundary may obtain about the interior has finite dimension equal to the
exponent of the area of the boundary, in Planck units, times a fixed constant.
Finally,the construction of the state space for quantum gravity in a region
from that of all Chern-Simon theories defined on its boundary confirms the
categorical-theoretic ``ladder of dimensions picture" of Crane.Comment: TEX File, Minor Changes Made, 59 page
Strong Uniform Attractors for Non-Autonomous Dissipative PDEs with non translation-compact external forces
We give a comprehensive study of strong uniform attractors of non-autonomous
dissipative systems for the case where the external forces are not translation
compact. We introduce several new classes of external forces which are not
translation compact, but nevertheless allow to verify the attraction in a
strong topology of the phase space and discuss in a more detailed way the class
of so-called normal external forces introduced before. We also develop a
unified approach to verify the asymptotic compactness for such systems based on
the energy method and apply it to a number of equations of mathematical physics
including the Navier-Stokes equations, damped wave equations and
reaction-diffusing equations in unbounded domains
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