87 research outputs found
Diagonal approximation of the form factor of the unitary group
The form factor of the unitary group U(N) endowed with the Haar measure
characterizes the correlations within the spectrum of a typical unitary matrix.
It can be decomposed into a sum over pairs of ``periodic orbits'', where by
periodic orbit we understand any sequence of matrix indices. From here the
diagonal approximation can be defined in the usual fashion as a sum only over
pairs of identical orbits. We prove that as we take the dimension to
infinity, the diagonal approximation becomes ``exact'', that is converges to
the full form factor.Comment: 9 page
How to Quantize Outputs of a Binary Symmetric Channel to Bits?
Suppose that is obtained by observing a uniform Bernoulli random vector
through a binary symmetric channel with crossover probability .
The "most informative Boolean function" conjecture postulates that the maximal
mutual information between and any Boolean function is
attained by a dictator function. In this paper, we consider the "complementary"
case in which the Boolean function is replaced by
, namely, an bit
quantizer, and show that
for any such . Thus, in this case, the optimal function is of the form
.Comment: 5 pages, accepted ISIT 201
Strongly intersecting integer partitions
We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe
Candidate One-Way Functions and One-Way Permutations Based on Quasigroup String Transformations
In this paper we propose a definition and construction of a new family of
one-way candidate functions , where
is an alphabet with elements. Special instances of these functions can have
the additional property to be permutations (i.e. one-way permutations). These
one-way functions have the property that for achieving the security level of
computations in order to invert them, only bits of input are needed.
The construction is based on quasigroup string transformations. Since
quasigroups in general do not have algebraic properties such as associativity,
commutativity, neutral elements, inverting these functions seems to require
exponentially many readings from the lookup table that defines them (a Latin
Square) in order to check the satisfiability for the initial conditions, thus
making them natural candidates for one-way functions.Comment: Submitetd to conferenc
A discrete isodiametric result: the Erd\H{o}s-Ko-Rado theorem for multisets
There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new
results (and problems) concerning families of -intersecting -element
multisets of an -set and point out connections to coding theory and
classical geometry. We establish the conjecture that for such
a family can have at most members
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