1,431,613 research outputs found
SH Realization of Minimal Model CFT: Triality, Poset and Burge Condition
Recently an orthogonal basis of -algebra (AFLT basis) labeled
by -tuple Young diagrams was found in the context of 4D/2D duality.
Recursion relations among the basis are summarized in the form of an algebra
SH which is universal for any . We show that it has an
automorphism which is referred to as triality. We study the level-rank duality
between minimal models, which is a special example of the automorphism. It is
shown that the nonvanishing states in both systems are described by or
Young diagrams with the rows of boxes appropriately shuffled. The reshuffling
of rows implies there exists partial ordering of the set which labels them. For
the simplest example, one can compute the partition functions for the partially
ordered set (poset) explicitly, which reproduces the Rogers-Ramanujan
identities. We also study the description of minimal models by SH. Simple
analysis reproduces some known properties of minimal models, the structure of
singular vectors and the -Burge condition in the Hilbert space.Comment: 1+38 pages and 12 figures. v2: typos corrected + comments adde
Ramsey numbers of cubes versus cliques
The cube graph Q_n is the skeleton of the n-dimensional cube. It is an
n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N
such that every graph of order N contains the cube graph Q_n or an independent
set of order s. Burr and Erdos in 1983 asked whether the simple lower bound
r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large.
We make progress on this problem, obtaining the first upper bound which is
within a constant factor of the lower bound.Comment: 26 page
Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property
We find a sufficient condition that \H is not level based on a reduction
number. In particular, we prove that a graded Artinian algebra of codimension 3
with Hilbert function cannot be level
if , and that there exists a level O-sequence of codimension 3 of
type \H for for . Furthermore, we show that \H is
not level if , and also
prove that any codimension 3 Artinian graded algebra cannot be level if
\beta_{1,d+2}(\Gin(I))=\beta_{2,d+2}(\Gin(I)). In this case, the Hilbert
function of does not have to satisfy the condition .
Moreover, we show that every codimension graded Artinian level algebra
having the Weak-Lefschetz Property has the strictly unimodal Hilbert function
having a growth condition on for every
where
In particular, we find that if is of codimension 3, then for every and , and prove that
if is a codimension 3 Artinian algebra with an -vector
such that h_{d-1}-h_d=2(h_d-h_{d+1})>0 \quad \text{and}
\quad \soc(A)_{d-1}=0 for some , then is
-regular and \dim_k\soc(A)_d=h_d-h_{d+1}.Comment: 25 page
Modular Invariance, Finiteness, and Misaligned Supersymmetry: New Constraints on the Numbers of Physical String States
We investigate the generic distribution of bosonic and fermionic states at
all mass levels in non-supersymmetric string theories, and find that a hidden
``misaligned supersymmetry'' must always appear in the string spectrum. We show
that this misaligned supersymmetry is ultimately responsible for the finiteness
of string amplitudes in the absence of full spacetime supersymmetry, and
therefore the existence of misaligned supersymmetry provides a natural
constraint on the degree to which spacetime supersymmetry can be broken in
string theory without destroying the finiteness of string amplitudes.
Misaligned supersymmetry also explains how the requirements of modular
invariance and absence of physical tachyons generically affect the distribution
of states throughout the string spectrum, and implicitly furnishes a
two-variable generalization of some well-known results in the theory of modular
functions.Comment: standard LaTeX; 55 pages, 4 figures. (Note: This replaced version
matches the version which was published in Nuclear Physics B.
Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets
In this paper we describe a singly exponential algorithm for computing the
first Betti number of a given semi-algebraic set. Singly exponential algorithms
for computing the zero-th Betti number, and the Euler-Poincar\'e
characteristic, were known before. No singly exponential algorithm was known
for computing any of the individual Betti numbers other than the zero-th one.
We also give algorithms for obtaining semi-algebraic descriptions of the
semi-algebraically connected components of any given real algebraic or
semi-algebraic set in single-exponential time improving on previous results
A Simple Shell Model for Quantum Dots in a Tilted Magnetic Field
A model for quantum dots is proposed, in which the motion of a few electrons
in a three-dimensional harmonic oscillator potential under the influence of a
homogeneous magnetic field of arbitrary direction is studied. The spectrum and
the wave functions are obtained by solving the classical problem. The ground
state of the Fermi-system is obtained by minimizing the total energy with
regard to the confining frequencies. From this a dependence of the equilibrium
shape of the quantum dot on the electron number, the magnetic field parameters
and the slab thickness is found.Comment: 15 pages (Latex), 3 epsi figures, to appear in PhysRev B, 55 Nr. 20
(1997
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