436 research outputs found
Discrete Symmetry and Stability in Hamiltonian Dynamics
In this tutorial we address the existence and stability of periodic and
quasiperiodic orbits in N degree of freedom Hamiltonian systems and their
connection with discrete symmetries. Of primary importance in our study are the
nonlinear normal modes (NNMs), i.e periodic solutions which represent
continuations of the system's linear normal modes in the nonlinear regime. We
examine the existence of such solutions and discuss different methods for
constructing them and studying their stability under fixed and periodic
boundary conditions. In the periodic case, we employ group theoretical concepts
to identify a special type of NNMs called one-dimensional "bushes". We describe
how to use linear combinations such NNMs to construct s(>1)-dimensional bushes
of quasiperiodic orbits, for a wide variety of Hamiltonian systems and exploit
the symmetries of the linearized equations to simplify the study of their
destabilization. Applying this theory to the Fermi Pasta Ulam (FPU) chain, we
review a number of interesting results, which have appeared in the recent
literature. We then turn to an analytical and numerical construction of
quasiperiodic orbits, which does not depend on the symmetries or boundary
conditions. We demonstrate that the well-known "paradox" of FPU recurrences may
be explained in terms of the exponential localization of the energies Eq of
NNM's excited at the low part of the frequency spectrum, i.e. q=1,2,3,....
Thus, we show that the stability of these low-dimensional manifolds called
q-tori is related to the persistence or FPU recurrences at low energies.
Finally, we discuss a novel approach to the stability of orbits of conservative
systems, the GALIk, k=2,...,2N, by means of which one can determine accurately
and efficiently the destabilization of q-tori, leading to the breakdown of
recurrences and the equipartition of energy, at high values of the total energy
E.Comment: 50 pages, 13 figure
Almost Prime Coordinates for Anisotropic and Thin Pythagorean Orbits
We make an observation which doubles the exponent of distribution in certain
Affine Sieve problems, such as those considered by Liu-Sarnak, Kontorovich, and
Kontorovich-Oh. As a consequence, we decrease the known bounds on the
saturation numbers in these problems.Comment: 24 page
Bounds on the exponent of primitivity which depend on the spectrum and the minimal polynomial
AbstractSuppose A is an n × n nonnegative primitive matrix whose minimal polynomial has degree m. We conjecture that the well-known bound on the exponent of primitivity (n − 1)2 + 1, due to Wielandt, can be replaced by (m − 1)2 + 1. The only case for which we cannot prove the conjecture is when m ⩾ 5, the number of distinct eigenvalues of A is m − 1 or m, and the directed graph of A has no circuits of length shorter than m − 1, but at least one of its vertices lies on a circuit of length not shorter than m. We show that m(m − 1) is always a bound on the exponent, this being an improvement on Wielandt's bound when m < n. For the case in which A is also symmetric, the bound which we obtain is 2(m − 1). To obtain our results we prove a lemma which shows that for a (general) nonnegative matrix, the number of its distinct eigenvalues is an upper bound on the length of the shortest circuit in its directed graph
The cyclic sieving phenomenon: a survey
The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a
2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and
f(q) be a polynomial in q with nonnegative integer coefficients. Then the
triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we
have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g,
and w is a root of unity chosen to have the same order as g. It might seem
improbable that substituting a root of unity into a polynomial with integer
coefficients would have an enumerative meaning. But many instances of the
cyclic sieving phenomenon have now been found. Furthermore, the proofs that
this phenomenon hold often involve interesting and sometimes deep results from
representation theory. We will survey the current literature on cyclic sieving,
providing the necessary background about representations, Coxeter groups, and
other algebraic aspects as needed.Comment: 48 pages, 3 figures, the sedcond version contains numerous changes
suggested by colleagues and the referee. To appear in the London Mathematical
Society Lecture Note Series. The third version has a few smaller change
A Novel Provably Secure Key Agreement Protocol Based On Binary Matrices
In this paper, a new key agreement protocol is presented. The protocol uses
exponentiation of matrices over GF(2) to establish the key agreement. Security
analysis of the protocol shows that the shared secret key is indistinguishable
from the random under Decisional Diffie-Hellman (DDH) assumption for subgroup
of matrices over GF(2) with prime order, and furthermore, the analysis shows
that, unlike many other exponentiation based protocols, security of the
protocol goes beyond the level of security provided by (DDH) assumption and
intractability of Discrete Logarithm Problem (DLP). Actually, security of the
protocol completely transcends the reliance on the DLP in the sense that
breaking the DLP does not mean breaking the protocol. Complexity of brute force
attack on the protocol is equivalent to exhaustive search for the secret key
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
We analyze relationships between quantum computation and a family of
generalizations of the Jones polynomial. Extending recent work by Aharonov et
al., we give efficient quantum circuits for implementing the unitary
Jones-Wenzl representations of the braid group. We use these to provide new
quantum algorithms for approximately evaluating a family of specializations of
the HOMFLYPT two-variable polynomial of trace closures of braids. We also give
algorithms for approximating the Jones polynomial of a general class of
closures of braids at roots of unity. Next we provide a self-contained proof of
a result of Freedman et al. that any quantum computation can be replaced by an
additive approximation of the Jones polynomial, evaluated at almost any
primitive root of unity. Our proof encodes two-qubit unitaries into the
rectangular representation of the eight-strand braid group. We then give
QCMA-complete and PSPACE-complete problems which are based on braids. We
conclude with direct proofs that evaluating the Jones polynomial of the plat
closure at most primitive roots of unity is a #P-hard problem, while learning
its most significant bit is PP-hard, circumventing the usual route through the
Tutte polynomial and graph coloring.Comment: 34 pages. Substantial revision. Increased emphasis on HOMFLYPT,
greatly simplified arguments and improved organizatio
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