Nonzero Constant Term Research Articles

Decomposition of matrices into products of commutators of quadratic matrices

Let F be a field, p ( x ) be a quadratic polynomial in F [ x ] with a non-zero constant term, and n ≥ 2 be a positive integer. We say that T ∈ M n ( F ) is p ( x ) -quadratic if p ( T ) = 0 . In this paper, given A ∈ SL n ( F ) , we show that A can be decomposed into a product of at most two commutators of p ( x ) -quadratic matrices if either p ( x ) has two distinct roots in F and A is non-scalar, or if F is algebraically closed and has characteristic different from 2. A similar result for matrices over real quaternion division rings is also provided.

Coalescence sum rule and the electric charge- and strangeness-dependences of directed flow in heavy ion collisions

The rapidity-odd directed flows (v1) of identified hadrons are expected to follow the coalescence sum rule when the created matter is initially in parton degrees of freedom and then hadronizes through quark coalescence. A recent study has considered the v1 of produced hadrons that do not contain u or d constituent quarks. It has constructed multiple hadron sets with a small mass difference but given difference in electric charge Δq and strangeness ΔS between the two sides, where a nonzero and increasing Δv1 with Δq has been proposed to be a consequence of electromagnetic fields. In this study, we examine the consequence of coalescence sum rule on the Δv1 of the hadron sets in the absence of electromagnetic fields. We find that in general Δv1≠0 for a hadron set with nonzero Δq and/or ΔS due to potential v1 differences between u¯ and d¯ and between s and s¯ quarks. We further propose methods to extract the coefficients for the Δq- and ΔS-dependences of the direct flow difference, where a nonzero constant term would indicate the breaking of the coalescence sum rule. The extraction methods are then demonstrated with transport model results.

Twist polynomials of delta-matroids

Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an infinite family of counterexamples for the conjecture and showed that essentially these are the only counterexamples. This is also obtained independently by Chumutov and Vignes-Tourneret and they posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sense for general delta-matroids. In this paper, we show that partial duality polynomials have delta-matroid analogues. We introduce the twist polynomials of delta-matroids and discuss its basic properties for delta-matroids. We give a characterization of even normal binary delta-matroids whose twist polynomials have only one term and then prove that the twist polynomial of a normal binary delta-matroid contains non-zero constant term if and only if its intersection graph is bipartite.

Global dynamics and evolution for the Szekeres system with nonzero cosmological constant term

The Szekeres system with cosmological constant term describes the evolution of the kinematic quantities for Einstein field equations in \({\mathbb {R}}^4\). In this study, we investigate the behavior of trajectories in the presence of cosmological constant. It has been shown that the Szekeres system is a Hamiltonian dynamical system. It admits at least two conservation laws, h and \(I_{0}\) which indicate the integrability of the Hamiltonian system. We solve the Hamilton–Jacobi equation, and we reduce the Szekeres system from \({\mathbb {R}}^4\) to an equivalent system defined in \({\mathbb {R}}^2\). Global dynamics are studied where we find that there exists an attractor in the finite regime only for positive valued cosmological constant and \(I_0<-2 .08\). Otherwise, trajectories reach infinity. For \(I_ {0}>0\) the origin of trajectories in \({\mathbb {R}}^2\) is also at infinity. Finally, we investigate the evolution of physical properties by using dimensionless variables different from that of Hubble-normalization conducing to a dynamical system in \({\mathbb {R}}^5\). We see that the attractor at the finite regime in \({\mathbb {R}}^5\) is related with the de Sitter universe for a positive cosmological constant.

Polynomials of degree 4 over finite fields representing quadratic residues

It is proved that in a finite field F of prime order p, where p is not one of finitely many exceptions, for every polynomial f(x) ∈ F[x] of degree 4 that has a nonzero constant term and is not of the form αg(x)2 there exists a primitive root β ∈ F such that f(β) is a quadratic residue in F. This refines a result of Madden and Vélez from 1982 about polynomials that represent quadratic residues at primitive roots.

Mutually orthogonal latin squares based on cellular automata

We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter $d$ over an alphabet of $q$ elements generates a Latin square of order $q^{d-1}$, we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field $\mathbb{F}_q$ are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree $n$ over $\mathbb{F}_q$. Finally, we present a construction of MOLS generated by LBCA with irreducible polynomials and prove the maximality of the resulting sets, as well as a lower bound which is asymptotically close to their actual number.

Stability for hyperplane complements of type B/C and statistics on squarefree polynomials over finite fields

AbstractIn this paper, we explore a relationship between the topology of the complex hyperplane complements ℳBCn(ℂ) in type B/C and the combinatorics of certain spaces of degree-n polynomials over a finite field Fq. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras H*(ℳBCn(ℂ);ℂ), and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over Fq with non-zero constant term. This result is the type B/C analogue of a theorem due to Church, Ellenberg, and Farb in type A, and we include a new proof of their theorem. To establish these convergence results, we realize the sequences of cohomology algebras of the hyperplane complements as FIW-algebras finitely generated inFIW-degree 2, and we investigate the asymptotic behaviour of general families of algebras with this structure. We prove a negative result implying that this structure alone is not sufficient to prove the necessary convergence conditions. Our proof of convergence for the cohomology algebras involves the combinatorics of their relators.

Analytical Solutions for an Escape Problem in a Disc with an Arbitrary Distribution of Exit Holes Along Its Boundary

We analytically construct solutions for the mean first-passage time and splitting probabilities for the escape problem of a particle moving with continuous Brownian motion in a confining planar disc with an arbitrary distribution (i.e., of any number, size and spacing) of exit holes/absorbing sections along its boundary. The governing equations for these quantities are Poisson’s equation with a (non-zero) constant forcing term and Laplace’s equation, respectively, and both are subject to a mixture of homogeneous Neumann and Dirichlet boundary conditions. Our solutions are expressed as explicit closed formulae written in terms of a parameterising variable via a conformal map, using special transcendental functions that are defined in terms of an associated Schottky group. They are derived by exploiting recent results for a related problem of fluid mechanics that describes a unidirectional flow over “no-slip/no-shear” surfaces, as well as results from potential theory, all of which were themselves derived using the same theory of Schottky groups. They are exact up to the determination of a finite set of mapping parameters, which is performed numerically. Their evaluation also requires the numerical inversion of the parameterising conformal map. Computations for a series of illustrative examples are also presented.

On the universal central extension of hyperelliptic current algebras

Let p ( t ) ∈ C [ t ] p(t)\in \mathbb C[t] be a polynomial with distinct roots and nonzero constant term. We describe, using Faà de Bruno’s formula and Bell polynomials, the universal central extension in terms of generators and relations for the hyperelliptic current Lie algebras g ⊗ R \mathfrak g\otimes R whose coordinate ring is of the form R = C [ t , t − 1 , u | u 2 = p ( t ) ] R=\mathbb C[t,t^{-1},u\,|\, u^2=p(t)] .

A combinatorial equivalence relation for formal power series

In this paper we examine an equivalence relation on the set of formal power series with nonzero constant term. This is done both in terms of functional equations and also by interlacing two concepts from Riordan group theory, the A-sequence and the Bell subgroup. The best known example gives an equivalence class{1+z,1/(1−z),C(z),T(z),Q(z),…} where C(z),T(z) and Q(z) are generating functions of the Catalan numbers, ternary numbers and quaternary numbers, respectively. A power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given.

How many Boolean polynomials are irreducible?

We present a number of new results on the multiplicative structure of univariate polynomials over the Boolean and tropical semirings. We answer the question asked by Kim and Roush in 2005 by proving that almost all Boolean polynomials with nonzero constant term are irreducible. We also give a lower bound for the number of reducible polynomials, and we discuss the related issues for polynomials over tropical semiring.

Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: extracting information from transients.

A Langevin process describing diffusion in a periodic potential landscape has a time-dependent diffusion constant, which means that its average mean-squared displacement (MSD) only becomes linear at late times. The long-time, or effective diffusion, constant can be estimated from the slope of a linear fit of the MSD at late times. Due to the crossover between a short time microscopic diffusion constant, which is independent of the potential, to the effective late-time diffusion constant, a linear fit of the MSD will not in general pass through the origin and will have a nonzero constant term. Here we address how to compute the constant term and provide explicit results for Brownian particles in one dimension in periodic potentials. We show that the constant is always positive and that at low temperatures it depends on the curvature of the minimum of the potential. For comparison we also consider the same question for the simpler problem of a symmetric continuous time random walk in discrete space. Here the constant can be positive or negative and can be used to determine the variance of the hopping time distribution.

FIRST GAP SIZE OF COEFFICIENTS OF A MODULAR FORM

We find an upper bound on the least positive exponent of a nonzero Fourier coefficient for a modular form with a nonzero constant term for under certain conditions where N is a prime number or .

Sums of two square-zero matrices over an arbitrary field

The problem to express an n × n matrix A as the sum of two square-zero matrices was first investigated by Wang and Wu [2] for matrices over the complex field. This paper investigates the problem over an arbitrary field F. It is shown that, if char ( F ) ≠ 2 , then A ∈ M n ( F ) is the sum of two square-zero matrices if and only if A is similar to a matrix of the form N ⊕ X ⊕ ( - X ) ⊕ ⊕ i = 1 m C ( g i ( x 2 ) ) , where N is nilpotent, X is nonsingular, and each C ( g i ( x 2 ) ) is a companion matrix associated with an even-power poly nomial with nonzero constant term. If F is of characteristic two, the term X ⊕ ( - X ) falls away. If F is of characteristic zero and algebraically closed, the term ⊕ i = 1 m C ( g i ( x 2 ) ) falls away and the result of Wang and Wu is obtained.

Asymptotic Rational Approximation To Pi: Solution of an "Unsolved Problem'' Posed By Herbert Wilf

The webpage of Herbert Wilf describes eight Unsolved Problems. Here, we completely resolve the third of these eight problems. The task seems innocent: find the first term of the asymptotic behavior of the coefficients of an ordinary generating function, whose coefficients naturally yield rational approximations to $\pi$. Upon closer examination, however, the analysis is fraught with difficulties. For instance, the function is the composition of three functions, but the innermost function has a non-zero constant term, so many standard techniques for analyzing function compositions will completely fail. Additionally, the signs of the coefficients are neither all positive, nor alternating in a regular manner. The generating function involves both a square root and an arctangent. The complex-valued square root and arctangent functions each rely on complex logarithms, which are multivalued and fundamentally depend on branch cuts. These multiple values and branch cuts make the function extremely tedious to visualize using Maple. We provide a complete asymptotic analysis of the coefficients of Wilf's generating function. The asymptotic expansion is naturally additive (not multiplicative); each term of the expansion contains oscillations, which we precisely characterize. The proofs rely on complex analysis, in particular, singularity analysis (which, in turn, rely on a Hankel contour and transfer theorems).

The number of irreducible polynomials of degree [formula omitted] over [formula omitted] with given trace and constant terms

We study the number N γ ( n , c , q ) of irreducible polynomials of degree n over F q where the trace γ and the constant term c are given. Under certain conditions on n and q , we obtain bounds on the maximum of N γ ( n , c , q ) varying c and γ . We show with concrete examples how our results improve the previously known bounds. In addition, we improve upper and lower bounds of any N γ ( n , c , q ) when n = a ( q − 1 ) for a nonzero constant term c and a nonzero trace γ . As a byproduct, we give a simple and explicit formula for the number N ( n , c , q ) of irreducible polynomials over F q of degree n = q − 1 with a prescribed primitive constant term c .

Hamming distance from irreducible polynomials over $\mathbb {F}_2$

We study the Hamming distance from polynomials to classes of polynomials that share certain properties of irreducible polynomials. The results give insight into whether or not irreducible polynomials can be effectively modeled by these more general classes of polynomials. For example, we prove that the number of degree $n$ polynomials of Hamming distance one from a randomly chosen set of $\lfloor 2^n/n \rfloor$ odd density polynomials, each of degree $n$ and each with non-zero constant term, is asymptotically $(1-e^{-4}) 2^{n-2}$, and this appears to be inconsistent with the numbers for irreducible polynomials. We also conjecture that there is a constant $c$ such that every polynomial has Hamming distance at most $c$ from an irreducible polynomial. Using exhaustive lists of irreducible polynomials over $\mathbb{F}_2$ for degrees $1 ≤ n ≤ 32$, we count the number of polynomials with a given Hamming distance to some irreducible polynomial of the same degree. Our work is based on this "empirical" study.

Conditioning-Annealing Studies of Natural Organic Matter Solids Linking Irreversible Sorption to Irreversible Structural Expansion

The assumption of reversibility underpins the sorption term in current models dealing with the fate and impact of organic compounds in the environment, yet experimentally sorption of organic compounds in soils and sediments often shows "irreversible" behaviors such as hysteresis and the conditioning effect (enhanced repeat sorption). The objective of this study was to test whether a glassy polymer irreversibility model applies to natural organic matter (NOM) solids. Irreversible sorption in polymers is believed to be caused by irreversible expansion and creation of internal micropores by penetrating molecules, leading to enhanced affinity during desorption or subsequent resorption. Using chlorobenzene as a conditioning agent and polychlorinated benzenes as test compounds in a second sorption step, we observed conditioning effects for a peat soil, a soil humic acid, and a model glassy polymer, poly- (vinyl chloride), but not for a model rubbery polymer, poly- (ethylene). The conditioning effect for the two natural solids, probed bythe enhancement in the sorption distribution coefficient of 1,2,4-trichlorobenzene, relaxed upon sample annealing between 45 and 91 degrees C in a manner similar to the relaxation of free volume and enthalpy of glassy polymers. Relaxation of the conditioning effect in the NOM solids depended on annealing temperature and, at a given temperature, followed a double additive exponential rate law with a nonzero constant term descriptive of the final state that depends inversely on temperature. At environmentally relevant temperatures, the conditioning effect may "never" completely relax. The results provide compelling evidence for the glassy, nonequilibrium nature of natural organic matter solids and for irreversible structural expansion as a cause of irreversible sorption.

Undecidability in matrices over Laurent polynomials

We show that it is undecidable for finite sets S of upper triangular ( 4 × 4 ) -matrices over Z [ x , x − 1 ] whether or not all elements in the semigroup generated by S have a nonzero constant term in some of the Laurent polynomials of the first row. This result follows from a representations of the integer weighted finite automata by matrices over Laurent polynomials.

Even permutations and oriented sets: their shifted asymmetry index series

The present authors introduced recently (see [Adv. in Appl. Math. 32 (2004) 576]) the notion of shifted asymmetry index series denoted Γ F , ξ , for an arbitrary species of structures, F, and a series of weights, ξ. This series generalizes the classical asymmetry index series, Γ F , of G. Labelle [Discrete Math. 99 (1992) 141] to take into account the substitution of species with non-zero constant term [Combinatoire Énumérative, Lecture Notes in Math., vol. 1234, Springer-Verlag, 1986, pp. 126–159]. In [Adv. in Appl. Math. 32 (2004) 576], we can find explicit formulas for the shifted asymmetry index series of usual species (sets, cycles, permutation, trees, …). The goal of this paper is to complement [Adv. in Appl. Math. 32 (2004) 576] by computing closed formulas for the series Γ E ± , ξ and Γ ALT , ξ , of the species E ± of oriented sets and ALT of even permutation of their elements. Oriented sets are, by definition, totally ordered sets modulo an even permutation of their elements. Such structures were used, for example, by Pólya and Read [Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987] in the context of combinatorial chemistry problems (vertex-sets of oriented simplices).