1,257 research outputs found
The Mixed Vector Current Correlator <0|T(V^3_\mu V^8_\nu )|0> To Two Loops in Chiral Perturbation Theory
The isospin-breaking correlator of the product of flavor octet vector
currents, and , is computed to
next-to-next- to-leading (two-loop) order in Chiral Perturbation Theory. Large
corrections to both the magnitude and -dependence of the one-loop result
are found, and the reasons for the slow convergence of the chiral series for
the correlator given. The two-loop expression involves a single
counterterm, present also in the two-loop expressions for
and , which counterterm
contributes a constant to the scalar correlator . The
feasibility of extracting the value of this counterterm from other sources is
discussed. Analysis of the slope of the correlator with respect to using
QCD sum rules is shown to suggest that, even to two-loop order, the chiral
series for the correlator may not yet be well-converged.Comment: 32 pages, uses REVTEX and epsfig.sty with 7 uuencoded figures. Entire
manuscript available as a ps file at
http://www.physics.adelaide.edu.au/theory/home.html Also available via
anonymous ftp at ftp://adelphi.adelaide.edu.au/pub/theory/ADP-95-27.T181.p
Lucas' theorem: its generalizations, extensions and applications (1878--2014)
In 1878 \'E. Lucas proved a remarkable result which provides a simple way to
compute the binomial coefficient modulo a prime in terms of
the binomial coefficients of the base- digits of and : {\it If is
a prime, and are the
-adic expansions of nonnegative integers and , then
\begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}.
\end{equation*}}
The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of
Lucas}), plays an important role in Number Theory and Combinatorics. In this
article, consisting of six sections, we provide a historical survey of Lucas
type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas
like theorems for some generalized binomial coefficients, and some their
applications.
In Section 1 we present the fundamental congruences modulo a prime including
the famous Lucas' theorem. In Section 2 we mention several known proofs and
some consequences of Lucas' theorem. In Section 3 we present a number of
extensions and variations of Lucas' theorem modulo prime powers. In Section 4
we consider the notions of the Lucas property and the double Lucas property,
where we also present numerous integer sequences satisfying one of these
properties or a certain Lucas type congruence. In Section 5 we collect several
known Lucas type congruences for some generalized binomial coefficients. In
particular, this concerns the Fibonomial coefficients, the Lucas -nomial
coefficients, the Gaussian -nomial coefficients and their generalizations.
Finally, some applications of Lucas' theorem in Number Theory and Combinatorics
are given in Section 6.Comment: 51 pages; survey article on Lucas type congruences closely related to
Lucas' theore
Hori-mological projective duality
Kuznetsov has conjectured that Pfaffian varieties should admit
non-commutative crepant resolutions which satisfy his Homological Projective
Duality. We prove half the cases of this conjecture, by interpreting and
proving a duality of non-abelian gauged linear sigma models proposed by Hori.Comment: 55 pages. V2: slightly rewritten to take advantage of the
`non-commutative Bertini theorem' recently proved by the authors and Van den
Bergh. V3: lots of changes in exposition following referees' comments.
Section 5 has been mostly cut because it was boring. To appear in Duke Math.
J. V3: added funder acknowledgemen
Effects of disorder on two strongly correlated coupled chains
We study the effects of disorder on a system of two coupled chain of strongly
correlated fermions (ladder system), using renormalization group. The stability
of the phases of the pure system is investigated as a function of interactions
both for fermions with spin and spinless fermions. For spinless fermions the
repulsive side is strongly localized whereas the system with attractive
interactions is stable with respect to disorder, at variance with the single
chain case. For fermions with spins, the repulsive side is also localized, and
in particular the d-wave superconducting phase found for the pure system is
totally destroyed by an arbitrarily small amount of disorder. On the other hand
the attractive side is again remarkably stable with respect to localization. We
have also computed the charge stiffness, the localization length and the
temperature dependence of the conductivity for the various phases. In the range
of parameter where d-wave superconductivity would occur for the pure system the
conductivity is found to decrease monotonically with temperature, even at high
temperature, and we discuss this surprising result. For a model with one site
repulsion and nearest neighbor attraction, the most stable phase is an orbital
antiferromagnet . Although this phase has no divergent superconducting
fluctuation it can have a divergent conductivity at low temperature. We argue
based on our results that the superconductivity observed in some two chain
compounds cannot be a simple stabilization of the d-wave phase found for a pure
single ladder. Applications to quantum wires are discussed.Comment: 47 pages, ReVTeX , 8 eps figures submitted to PR
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