Search results for "Mathematica"
showing 10 items of 7971 documents
Exact quantum algorithms have advantage for almost all Boolean functions
2014
It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.
Quantum lower bound for inverting a permutation with advice
2014
Given a random permutation $f: [N] \to [N]$ as a black box and $y \in [N]$, we want to output $x = f^{-1}(y)$. Supplementary to our input, we are given classical advice in the form of a pre-computed data structure; this advice can depend on the permutation but \emph{not} on the input $y$. Classically, there is a data structure of size $\tilde{O}(S)$ and an algorithm that with the help of the data structure, given $f(x)$, can invert $f$ in time $\tilde{O}(T)$, for every choice of parameters $S$, $T$, such that $S\cdot T \ge N$. We prove a quantum lower bound of $T^2\cdot S \ge \tilde{\Omega}(\epsilon N)$ for quantum algorithms that invert a random permutation $f$ on an $\epsilon$ fraction of…
An Empirical Study of the Relation Between Community Structure and Transitivity
2012
One of the most prominent properties in real-world networks is the presence of a community structure, i.e. dense and loosely interconnected groups of nodes called communities. In an attempt to better understand this concept, we study the relationship between the strength of the community structure and the network transitivity (or clustering coefficient). Although intuitively appealing, this analysis was not performed before. We adopt an approach based on random models to empirically study how one property varies depending on the other. It turns out the transitivity increases with the community structure strength, and is also affected by the distribution of the community sizes. Furthermore, …
Structural bias in population-based algorithms
2014
Abstract Challenging optimisation problems are abundant in all areas of science and industry. Since the 1950s, scientists have responded to this by developing ever-diversifying families of ‘black box’ optimisation algorithms. The latter are designed to be able to address any optimisation problem, requiring only that the quality of any candidate solution can be calculated via a ‘fitness function’ specific to the problem. For such algorithms to be successful, at least three properties are required: (i) an effective informed sampling strategy, that guides the generation of new candidates on the basis of the fitnesses and locations of previously visited candidates; (ii) mechanisms to ensure eff…
Mahonian STAT on words
2016
In 2000, Babson and Steingrimsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006.In 2010, Burstein refined some of these results by giving a bijection between permutations with a fixed value for the major index and those with the same value for STAT , where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingrimsson's paper. Several other statistics are preserved as well by Burstein's bijection.At…
Quantum finite multitape automata
1999
Quantum finite automata were introduced by C.Moore, J.P. Crutchfield, and by A.Kondacs and J.Watrous. This notion is not a generalization of the deterministic finite automata. Moreover, it was proved that not all regular languages can be recognized by quantum finite automata. A.Ambainis and R.Freivalds proved that for some languages quantum finite automata may be exponentially more concise rather than both deterministic and probabilistic finite automata. In this paper we introduce the notion of quantum finite multitape automata and prove that there is a language recognized by a quantum finite automaton but not by a deterministic or probabilistic finite automata. This is the first result on …
The minimal probabilistic and quantum finite automata recognizing uncountably many languages with fixed cutpoints
2019
Discrete Mathematics & Theoretical Computer Science ; vol. 22 no. 1 ; Automata, Logic and Semantics ; 1365-8050
Automata and Quantum Computing
2015
Quantum computing is a new model of computation, based on quantum physics. Quantum computers can be exponentially faster than conventional computers for problems such as factoring. Besides full-scale quantum computers, more restricted models such as quantum versions of finite automata have been studied. In this paper, we survey various models of quantum finite automata and their properties. We also provide some open questions and new directions for researchers. Keywords: quantum finite automata, probabilistic finite automata, nondeterminism, bounded error, unbounded error, state complexity, decidability and undecidability, computational complexity
Random Interruptions in Cooperation for Spectrum Sensing in Cognitive Radio Networks
2015
In this paper, a new cooperation structure for spectrum sensing in cognitive radio networks is proposed which outperforms the existing commonly-used ones in terms of energy efficiency. The efficiency is achieved in the proposed design by introducing random interruptions in the cooperation process between the sensing nodes and the fusion center, along with a compensation process at the fusion center. Regarding the hypothesis testing problem concerned, first, the proposed system behavior is thoroughly analyzed and its associated likelihood-ratio test (LRT) is provided. Next, based on a general linear fusion rule, statistics of the global test summary are derived and the sensing quality is cha…
A novel exact representation of stationary colored Gaussian processes (fractional differential approach)
2010
A novel representation of functions, called generalized Taylor form, is applied to the filtering of white noise processes. It is shown that every Gaussian colored noise can be expressed as the output of a set of linear fractional stochastic differential equations whose solution is a weighted sum of fractional Brownian motions. The exact form of the weighting coefficients is given and it is shown that it is related to the fractional moments of the target spectral density of the colored noise.