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We develop explicit formulas and algorithms for arithmetic in radical function fields K/k(x) over finite constant fields. First, we classify which places of k(x) whose local integral bases have an easy monogenic form, and give explicit…
An algorithm for the automatic Feynman diagram (FD) generation is presented in this paper. The algorithm starts directly from the definition formula of FD, and is simple in concept and easy for coding. The symmetry factor for each FD is…
The ultimate random number generators are those certified to be unpredictable -- including to an adversary. The use of simple quantum processes promises to provide numbers that no physical observer could predict but, in practice, unwanted…
This paper presents an expression to compute the exact period of a recursive random number generator based on d-sequences. Using the multi-recursive version of this generator we can produce large number of pseudorandom sequences.
We consider the general problem of modeling temporal data with long-range dependencies, wherein new observations are fully or partially predictable based on temporally-distant, past observations. A sufficiently powerful temporal model…
A characterization of finitely generated shift-invariant subspaces is given when generators are g-minimal. An algorithm is given for the determination of the coefficients in the well known representation of the Fourier transform of an…
Despite its long history, a canonical formulation of quantum ergodicity that applies to general classes of quantum dynamics, including driven systems, has not been fully established. Here we introduce and study a notion of quantum…
A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response…
We combine dependent types with linear type systems that soundly and completely capture polynomial time computation. We explore two systems for capturing polynomial time: one system that disallows construction of iterable data, and one,…
Random numbers are central to cryptography and various other tasks. The intrinsic probabilistic nature of quantum mechanics has allowed us to construct a large number of quantum random number generators (QRNGs) that are distinct from the…
Random numbers are a fundamental ingredient for many applications including simulation, modelling and cryptography. Sound random numbers should be independent and uniformly distributed. Moreover, for cryptographic applications they should…
We present a simple and robust construction of a real-time quantum random number generator (QRNG). Our minimalist approach ensures stable operation of the device as well as its simple and straightforward hardware implementation as a…
An integer $a$ is a quadratic nonresidue for a prime $p$ if $x^2 \equiv a \bmod p$ has no solution. Quadratic nonresidues may be found by probabilistic methods in polynomial time. However, without assuming the Generalized Riemann…
We present an algorithm that takes as input a graph $G$ with weights on the vertices, and computes a maximum weight independent set $S$ of $G$. If the input graph $G$ excludes a path $P_k$ on $k$ vertices as an induced subgraph, the…
Consider the power pseudorandom-number generator in a finite field ${\mathbb F}_q$. That is, for some integer $e\ge2$, one considers the sequence $u,u^e,u^{e^2},\dots$ in ${\mathbb F}_q$ for a given seed $u\in {\mathbb F}_q^\times$. This…
Non-deterministic random bits are needed in many scientific fields. Unfortunately today's computers are very limited in ability to produce them. We present here a method for extraction of non-deterministic random bits from random physics…
The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations…
Let k be a field, let R be a ring of polynomials in a finite number of variables over k, let D be the ring of k-linear differential operators of R and let f be a non-zero element of R. It is well-known that R_f, with its natural D-module…
We adopt the general formalism for analyzing evolution of gaussian states of quantized fields in time-dependent backgrounds in the Schrodinger picture (presented in detail in arXiv:0708.1233 and 0708.1237) to study the example of a…
Randomness is an essential resource and plays important roles in various applications ranging from cryptography to simulation of complex systems. Certified randomness from quantum process is ensured to have the element of privacy but…