12,347 research outputs found
Constructing Nonabelian (1,0) Hypermultiplet Theories in Six Dimensions
We construct a class of nonabelian superconformal (1,0) hypermultiplet
theories in six dimensions by introducing an abelian auxiliary field. The gauge
fields of this class of theories are non-dynamical, and this class of theories
can be viewed as Chern-Simons-matter theories in 6D.Comment: 5 pages, minor changes, reference adde
Spectral gap and logarithmic Sobolev constant for continuous spin systems
The aim of this paper is to study the spectral gap and the logarithmic
Sobolev constant for continuous spin systems. A simple but general result for
estimating the spectral gap of finite dimensional systems is given by Theorem
1.1, in terms of the spectral gap for one-dimensional marginals. The study of
the topic provides us a chance, and it is indeed another aim of the paper, to
justify the power of the results obtained previously. The exact order in
dimension one (Proposition 1.4), and then the precise leading order and the
explicit positive regions of the spectral gap and the logarithmic Sobolev
constant for two typical infinite-dimensional models are presented (Theorems
6.2 and 6.3). Since we are interested in explicit estimates, the computations
become quite involved. A long section (Section 4) is devoted to the study of
the spectral gap in dimension one.Comment: 40 pages, 2 figure
Progress on Hardy-type Inequalities
This paper surveys some of our recent progress on Hardy-type inequa\-lities
which consist of a well-known topic in Harmonic Analysis. In the first section,
we recall the original probabilistic motivation dealing with the stability
speed in terms of the -theory. A crucial application of a result by
Fukushima and Uemura (2003) is included. In the second section, the non-linear
case (a general Hardy-type inequality) is handled with a direct and analytic
proof. In the last section, it is illustrated that the basic estimates
presented in the first two sections can still be improved considerably.Comment: 13 pages, 5 figures, Festschrift Masatoshi Fukushima: In Honor of
Masatoshi Fukushima's Sanju. World Scientific, 201
Eigenvalues, inequalities and ergodic theory
This paper surveys the main results obtained during the period 1992-1999 on
three aspects mentioned at the title. The first result is a new and general
variational formula for the lower bound of spectral gap (i.e., the first
non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on
Riemannian manifolds or Markov chains (\S 1). Here, a probabilistic
method-coupling method is adopted. The new formula is a dual of the classical
variational formula. The last formula is actually equivalent to Poincar\'e
inequality. To which, there are closely related logarithmic Sobolev inequality,
Nash inequality, Liggett inequality and so on. These inequalities are treated
in a unified way by using Cheeger's method which comes from Riemannian
geometry. This consists of \S 2. The results on these two aspects are mainly
completed by the author joint with F. Y. Wang. Furthermore, a diagram of the
inequalities and the traditional three types of ergodicity is presented (\S 3).
The diagram extends the ergodic theory of Markov processes. The details of the
methods used in the paper will be explained in a subsequent paper under the
same title.Comment: 6 page
Practical Criterion for Uniqueness of -processes
The note begins with a short story on seeking for a practical sufficiency
theorem for the uniqueness of time-continuous Markov jump processes, starting
around 1977. The general result was obtained in 1985 for the processes with
general state spaces. To see the sufficient conditions are sharp, a dual
criterion for non-uniqueness was obtained in 1991. This note is restricted
however to the discrete state space (then the processes are called
-processes or Markov chains), for which the sufficient conditions just
mentioned are showing at the end of the note to be necessary. Some examples are
included to illustrate that the sufficient conditions either for uniqueness or
for non-uniqueness are not only powerful but also sharp.Comment: 11 page
OSp(4|4) superconformal currents in three-dimensional N=4 Chern-Simons quiver gauge theories
We prove explicitly that the general D=3, N=4 Chern-Simons-matter (CSM)
theory has a complete OSp(4|4) superconformal symmetry, and construct the
corresponding conserved currents. We re-derive the OSp(5|4) superconformal
currents in the general N=5 theory as special cases of the OSp(4|4) currents by
enhancing the supersymmetry from N=4 to N=5. The closure of the full OSp(4|4)
superconformal algebra is verified explicitly.Comment: 23 pages, published in PR
Symplectic Three-Algebra Unifying N=5,6 Superconformal Chern-Simons-Matter Theories
We define a 3-algebra with structure constants being symmetric in the first
two indices. We also introduce an invariant anti-symmetric tensor into this
3-algebra and call it a symplectic 3-algebra. The general N=5 superconformal
Chern-Simons-matter (CSM) theory with SO(5) R-symmetry in three dimensions is
constructed by using this algebraic structure. We demonstrate that the
supersymmetry can be enhanced to N=6 if the sympelctic 3-algebra and the fields
are decomposed in a proper fashion. By specifying the 3-brackets, some
presently known N=5, 6 superconformal theories are described in terms of this
unified 3-algebraic framework. These include the N=5, Sp(2N) X O(M) CSM theory
with SO(5) R-symmetry , the N=6, Sp(2N) X U(1) CSM theory with SU(4)
R-symmetry, as well as the ABJM theory as a special case of U(M) X U(N) theory
with SU(4) R-symmetry.Comment: 31 pages, minor changes, final results remain the sam
Criteria for Discrete Spectrum of 1D Operators
For discrete spectrum of 1D second-order differential/difference operators
(with or without potential (killing), with the maximal/minimal domain), a pair
of unified dual criteria are presented in terms of two explicit measures and
the harmonic function of the operators. Interes\-tingly, these criteria can be
read out from the ones for the exponential convergence of four types of
stability studied earlier, simply replacing the `finite supremum' by `vanishing
at infinity'. Except a dual technique, the main tool used here is a transform
in terms of the harmonic function, to which two new practical algorithms are
introduced in the discrete context and two successive approximation schemes are
reviewed in the continuous context. All of them are illustrated by examples.
The main body of the paper is devoted to the hard part of the story, the easier
part but powerful one is delayed to the end of the paper.Comment: 31 pages, 1 figure in Commun. Math. Stat. (2015
OSp(5|4) Superconformal Symmetry of N=5 Chern-Simons Theory
We demonstrate that the general D=3, N=5 Chern-Simons matter theory possesses
a full OSp(5|4) superconformal symmetry, and construct the corresponding
superconformal currents. The closure of the superconformal algebra is verified
in detail. We also show that the conserved OSp(6|4) superconformal currents in
the general N=6 theory can be obtained as special cases of the OSp(5|4)
currents by enhancing the R-symmetry of the N=5 theory from USp(4) to SU(4).Comment: 24 pages, minor changes, version published in Nucl.Phys.
Global algorithms for maximal eigenpair
This paper is a continuation of \ct{cmf16} where an efficient algorithm for
computing the maximal eigenpair was introduced first for tridiagonal matrices
and then extended to the irreducible matrices with nonnegative off-diagonal
elements. This paper introduces two global algorithms for computing the maximal
eigenpair in a rather general setup, including even a class of real (with some
negative off-diagonal elements) or complex matrices.Comment: 20 page
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