391,324 research outputs found
A characterization of covering equivalence
Let A={a_s(mod n_s)}_{s=1}^k and B={b_t(mod m_t)}_{t=1}^l be two systems of
residue classes. If |{1\le s\le k: x=a_s (mod n_s)}| and |{1\le t\le l: x=b_t
(mod m_t)}| are equal for all integers x, then A and B are said to be covering
equivalent. In this paper we characterize the covering equivalence in a simple
and new way. Using the characterization we partially confirm a conjecture of R.
L. Graham and K. O'Bryant
Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs
A cyclic proof system, called CLKID-omega, gives us another way of
representing inductive definitions and efficient proof search. The 2005 paper
by Brotherston showed that the provability of CLKID-omega includes the
provability of LKID, first order classical logic with inductive definitions in
Martin-L\"of's style, and conjectured the equivalence. The equivalence has been
left an open question since 2011. This paper shows that CLKID-omega and LKID
are indeed not equivalent. This paper considers a statement called 2-Hydra in
these two systems with the first-order language formed by 0, the successor, the
natural number predicate, and a binary predicate symbol used to express
2-Hydra. This paper shows that the 2-Hydra statement is provable in
CLKID-omega, but the statement is not provable in LKID, by constructing some
Henkin model where the statement is false
Statistical solutions of hyperbolic conservation laws I: Foundations
We seek to define statistical solutions of hyperbolic systems of conservation
laws as time-parametrized probability measures on -integrable functions. To
do so, we prove the equivalence between probability measures on spaces
and infinite families of \textit{correlation measures}. Each member of this
family, termed a \textit{correlation marginal}, is a Young measure on a
finite-dimensional tensor product domain and provides information about
multi-point correlations of the underlying integrable functions. We also prove
that any probability measure on a space is uniquely determined by certain
moments (correlation functions) of the equivalent correlation measure.
We utilize this equivalence to define statistical solutions of
multi-dimensional conservation laws in terms of an infinite set of equations,
each evolving a moment of the correlation marginal. These evolution equations
can be interpreted as augmenting entropy measure-valued solutions, with
additional information about the evolution of all possible multi-point
correlation functions. Our concept of statistical solutions can accommodate
uncertain initial data as well as possibly non-atomic solutions even for atomic
initial data.
For multi-dimensional scalar conservation laws we impose additional entropy
conditions and prove that the resulting \textit{entropy statistical solutions}
exist, are unique and are stable with respect to the -Wasserstein metric on
probability measures on
H\"older equivalence of the value function for control-affine systems
We prove the continuity and the H\"older equivalence w.r.t.\ an Euclidean
distance of the value function associated with the cost of the
control-affine system \dot q = \drift(q)+\sum_{j=1}^m u_j f_j(q), satisfying
the strong H\"ormander condition. This is done by proving a result in the same
spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The
techniques used are based on a reduction of the control-affine system to a
linear but time-dependent one, for which we are able to define a generalization
of the nilpotent approximation and through which we derive estimates for the
shape of the reachable sets. Finally, we also prove the continuity of the value
function associated with the cost of time-dependent systems of the form
.Comment: 25 pages, some correction
Quantum Calogero-Moser Models: Integrability for all Root Systems
The issues related to the integrability of quantum Calogero-Moser models
based on any root systems are addressed. For the models with degenerate
potentials, i.e. the rational with/without the harmonic confining force, the
hyperbolic and the trigonometric, we demonstrate the following for all the root
systems: (i) Construction of a complete set of quantum conserved quantities in
terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal
R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the
Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of
the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack
polynomials are defined for all root systems as unique eigenfunctions of the
Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v)
Algebraic construction of all excited states in terms of creation operators.
These are mainly generalisations of the results known for the models based on
the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure
Scaling and Crossover Functions for the Conductance in the Directed Network Model of Edge States
We consider the directed network (DN) of edge states on the surface of a
cylinder of length L and circumference C. By mapping it to a ferromagnetic
superspin chain, and using a scaling analysis, we show its equivalence to a
one-dimensional supersymmetric nonlinear sigma model in the scaling limit, for
any value of the ratio L/C, except for short systems where L is less than of
order C^{1/2}. For the sigma model, the universal crossover functions for the
conductance and its variance have been determined previously. We also show that
the DN model can be mapped directly onto the random matrix (Fokker-Planck)
approach to disordered quasi-one-dimensional wires, which implies that the
entire distribution of the conductance is the same as in the latter system, for
any value of L/C in the same scaling limit. The results of Chalker and Dohmen
are explained quantitatively.Comment: 10 pages, REVTeX, 2 eps figure
On the Equivalence of Kirkman-Steiner Triple Systems and Sets of Mutually Orthogonal Latin Squares
5 pages, 1 article*On the Equivalence of Kirkman-Steiner Triple Systems and Sets of Mutually Orthogonal Latin Squares* (Hedayat, A.; Raktoe, B. L.) 5 page
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