5,398 research outputs found
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure
Molecular Motors Interacting with Their Own Tracks
Dynamics of molecular motors that move along linear lattices and interact
with them via reversible destruction of specific lattice bonds is investigated
theoretically by analyzing exactly solvable discrete-state ``burnt-bridge''
models. Molecular motors are viewed as diffusing particles that can
asymmetrically break or rebuild periodically distributed weak links when
passing over them. Our explicit calculations of dynamic properties show that
coupling the transport of the unbiased molecular motor with the bridge-burning
mechanism leads to a directed motion that lowers fluctuations and produces a
dynamic transition in the limit of low concentration of weak links. Interaction
between the backward biased molecular motor and the bridge-burning mechanism
yields a complex dynamic behavior. For the reversible dissociation the backward
motion of the molecular motor is slowed down. There is a change in the
direction of the molecular motor's motion for some range of parameters. The
molecular motor also experiences non-monotonic fluctuations due to the action
of two opposing mechanisms: the reduced activity after the burned sites and
locking of large fluctuations. Large spatial fluctuations are observed when two
mechanisms are comparable. The properties of the molecular motor are different
for the irreversible burning of bridges where the velocity and fluctuations are
suppressed for some concentration range, and the dynamic transition is also
observed. Dynamics of the system is discussed in terms of the effective driving
forces and transitions between different diffusional regimes
More Evidence for the WDVV Equations in N=2 SUSY Yang-Mills Theories
We consider 4d and 5d N=2 supersymmetric theories and demonstrate that in
general their Seiberg-Witten prepotentials satisfy the
Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. General proof for the
Yang-Mills models (with matter in the first fundamental representation) makes
use of the hyperelliptic curves and underlying integrable systems. A wide class
of examples is discussed, it contains few understandable exceptions. In
particular, in perturbative regime of 5d theories in addition to naive field
theory expectations some extra terms appear, like it happens in heterotic
string models. We consider also the example of the Yang-Mills theory with
matter hypermultiplet in the adjoint representation (related to the elliptic
Calogero-Moser system) when the standard WDVV equations do not hold.Comment: LaTeX, 40 pages, no figure
Cut-and-Join operator representation for Kontsevich-Witten tau-function
In this short note we construct a simple cut-and-join operator representation
for Kontsevich-Witten tau-function that is the partition function of the
two-dimensional topological gravity. Our derivation is based on the Virasoro
constraints. Possible applications of the obtained expression are discussed.Comment: 5 pages, minor correction
Heuristic parameter-choice rules for convex variational regularization based on error estimates
In this paper, we are interested in heuristic parameter choice rules for
general convex variational regularization which are based on error estimates.
Two such rules are derived and generalize those from quadratic regularization,
namely the Hanke-Raus rule and quasi-optimality criterion. A posteriori error
estimates are shown for the Hanke-Raus rule, and convergence for both rules is
also discussed. Numerical results for both rules are presented to illustrate
their applicability
SDiff(2) and uniqueness of the Pleba\'{n}ski equation
The group of area preserving diffeomorphisms showed importance in the
problems of self-dual gravity and integrability theory. We discuss how
representations of this infinite-dimensional Lie group can arise in
mathematical physics from pure local considerations. Then using Lie algebra
extensions and cohomology we derive the second Pleba\'{n}ski equation and its
geometry. We do not use K\"ahler or other additional structures but obtain the
equation solely from the geometry of area preserving transformations group. We
conclude that the Pleba\'{n}ski equation is Lie remarkable
Lattice Gauge Fields Topology Uncovered by Quaternionic sigma-model Embedding
We investigate SU(2) gauge fields topology using new approach, which exploits
the well known connection between SU(2) gauge theory and quaternionic
projective sigma-models and allows to formulate the topological charge density
entirely in terms of sigma-model fields. The method is studied in details and
for thermalized vacuum configurations is shown to be compatible with
overlap-based definition. We confirm that the topological charge is distributed
in localized four dimensional regions which, however, are not compatible with
instantons. Topological density bulk distribution is investigated at different
lattice spacings and is shown to possess some universal properties.Comment: revtex4, 19 pages (24 ps figures included); replaced to match the
published version, to appear in PRD; minor changes, references adde
BGWM as Second Constituent of Complex Matrix Model
Earlier we explained that partition functions of various matrix models can be
constructed from that of the cubic Kontsevich model, which, therefore, becomes
a basic elementary building block in "M-theory" of matrix models. However, the
less topical complex matrix model appeared to be an exception: its
decomposition involved not only the Kontsevich tau-function but also another
constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition
function. The BGW tau-function can be represented either as a generating
function of all unitary-matrix integrals or as a Kontsevich-Penner model with
potential 1/X (instead of X^3 in the cubic Kontsevich model).Comment: 42 page
Exact 2-point function in Hermitian matrix model
J. Harer and D. Zagier have found a strikingly simple generating function for
exact (all-genera) 1-point correlators in the Gaussian Hermitian matrix model.
In this paper we generalize their result to 2-point correlators, using Toda
integrability of the model. Remarkably, this exact 2-point correlation function
turns out to be an elementary function - arctangent. Relation to the standard
2-point resolvents is pointed out. Some attempts of generalization to 3-point
and higher functions are described.Comment: 31 pages, 1 figur
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