Search results for "General Mathematics"

Error-Free Affine, Unitary, and Probabilistic OBDDs

2018

We introduce the affine OBDD model and show that zero-error affine OBDDs can be exponentially narrower than bounded-error unitary and probabilistic OBDDs on certain problems. Moreover, we show that Las Vegas unitary and probabilistic OBDDs can be quadratically narrower than deterministic OBDDs. We also obtain the same results for the automata versions of these models.

Quantum Strategies Are Better Than Classical in Almost Any XOR Game

2012

We initiate a study of random instances of nonlocal games. We show that quantum strategies are better than classical for almost any 2-player XOR game. More precisely, for large n, the entangled value of a random 2-player XOR game with n questions to every player is at least 1.21... times the classical value, for 1−o(1) fraction of all 2-player XOR games.

Any AND-OR Formula of Size N Can Be Evaluated in Time $N^{1/2+o(1)}$ on a Quantum Computer

2007

Consider the problem of evaluating an AND-OR formula on an $N$-bit black-box input. We present a bounded-error quantum algorithm that solves this problem in time $N^{1/2+o(1)}$. In particular, approximately balanced formulas can be evaluated in $O(\sqrt{N})$ queries, which is optimal. The idea of the algorithm is to apply phase estimation to a discrete-time quantum walk on a weighted tree whose spectrum encodes the value of the formula.

On a fundamental variational lemma for extremal quasiconformal mappings

1986

On the conical density properties of measures on $\mathbb{R}^n$

2005

We compare conical density properties and spherical density properties for general Borel measures on $\mathbb{R}^n$ . As a consequence, we obtain results for packing and Hausdorff measures $\mathcal{P}_h$ and $\mathcal{H}_h$ provided that the gauge function $h$ satisfies certain conditions. One consequence of our general results is the following: let $m, n\,{\in}\,\mathbb{N}, 0\,{\lt}\,s\,{\lt}\,m\,{\leq}\,n$ , $0\,{\lt}\,\eta\,{\lt}\,1$ , and suppose that $V$ is an $m$ -dimensional linear subspace of $\mathbb{R}^n$ . Let $\mu$ be either the $s$ -dimensional Hausdorff measure or the $s$ -dimensional packing measure restricted to a set $A$ with $\mu(A)\,{\lt}\,\infty$ . Then for $\mu$ -almos…

Stochastic factorizations, sandwiched simplices and the topology of the space of explanations

2003

We study the space of stochastic factorizations of a stochastic matrix V, motivated by the statistical problem of hidden random variables. We show that this space is homeomorphic to the space of simplices sandwiched between two nested convex polyhedra, and use this geometrical model to gain some insight into its structure and topology. We prove theorems describing its homotopy type, and, in the case where the rank of V is 2, we give a complete description, including bounds on the number of connected components, and examples in which these bounds are attained. We attempt to make the notions of topology accessible and relevant to statisticians.

Restriction of odd degree characters and natural correspondences

2016

Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a natural correspondence of characters, both for $GL_n(q)$ and $SL_n(q)$. A similar result is established for certain finite groups with self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of $S_n$ and those of $M$, where $M$ is any maximal subgroup of $S_n$ of odd index; as well as between the odd-degree irreducible characters of $G = GL_n(q)$ or $GU_n(q)$ with $q$ odd and those of $N_{G}…

Associative rings whose adjoint semigroup is locally nilpotent

2001

The set of all elements of an associative ring R, not necessarily with a unit element, forms a semigroup R ad under the circle operation \({r\circ s}={r+s+rs}\) on R. The ring R is called radical if R ad is a group. It is proved that the semigroup R ad is nilpotent of class n (in sense of A. Mal'cev or B. H. Neumann and T. Taylor) if and only if the ring R is Lie-nilpotent of class n. This yields a positive answer to a question posed by A. Krasil'nikov and independently considered by D. Riley and V. Tasic. It is also shown that the adjoint group of a radical ring R is locally nilpotent if and only if R is locally Lie-nilpotent.

On a representation theorem for finitely exchangeable random vectors

2016

A random vector $X=(X_1,\ldots,X_n)$ with the $X_i$ taking values in an arbitrary measurable space $(S, \mathscr{S})$ is exchangeable if its law is the same as that of $(X_{\sigma(1)}, \ldots, X_{\sigma(n)})$ for any permutation $\sigma$. We give an alternative and shorter proof of the representation result (Jaynes \cite{Jay86} and Kerns and Sz\'ekely \cite{KS06}) stating that the law of $X$ is a mixture of product probability measures with respect to a signed mixing measure. The result is "finitistic" in nature meaning that it is a matter of linear algebra for finite $S$. The passing from finite $S$ to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our p…

Una generalizzazione del birapporto sopra un anello

1990

We generalize to the case of the projective line over a (not necessarily commutative) ring the well-know theorem on the bijective maps preserving a given cross-ratio.