Nontrivial Root Research Articles

Temperature measurement from perturbations

The notion of configuration temperature is extended to discontinuous systems by identifying the temperature as the nontrivial root of several integral equations regarding the distribution of the energy change upon configuration perturbations. The relations are generalized to pressure and a distribution mean force.

Degree of roots of disk twists on 3-dimensional handlebodies

Margalit and Schleimer (Geom Topol 13(3):1495–1497, 2009) discovered a nontrivial root of the Dehn twist about a nonseparating curve on a closed oriented connected surface. McCullough and Rajeevsarathy (Geom Dedicata 151(1):397–409, 2011) and Monden (Rocky Mt J Math, to appear) obtained the evaluation of the degrees of roots of Dehn twists. In this paper, we discuss existence and degrees of homeomorphisms whose power is equal to disk twist about a nonseparating disk in the mapping class group of the 3-dimensional handlebody.

On square roots of the Haar state on compact quantum groups

The paper is concerned with the extension of the classical study of probability measures on a compact group which are square roots of the Haar measure, due to Diaconis and Shahshahani, to the context of compact quantum groups. We provide a simple characterisation for compact quantum groups which admit no non-trivial square roots of the Haar state in terms of their corepresentation theory. In particular it is shown that such compact quantum groups are necessarily of Kac type and their subalgebras generated by the coefficients of a fixed two-dimensional irreducible corepresentation are isomorphic (as finite quantum groups) to the algebra of functions on the group of unit quaternions. An example of a quantum group whose Haar state admits no nontrivial square root and which is neither commutative nor cocommutative is given.

Complex Hadamard matrices and equiangular tight frames

In this paper, we give a new construction of parametric families of complex Hadamard matrices of square orders, and connect them to equiangular tight frames. The results presented here generalize some of the recent ideas of Bodmann et al. [3] and extend the list of known equiangular tight frames. In particular, a (36,21)-frame coming from a nontrivial cube root signature matrix is obtained for the first time.

Symmetric invariant cocycles on the duals of q-deformations

We prove that for q ∈ C ⁎ not a nontrivial root of unity any symmetric invariant 2-cocycle for a completion of U q g is the coboundary of a central element. Equivalently, a Drinfeld twist relating the coproducts on completions of U q g and U g is unique up to coboundary of a central element. As an application we show that the spectral triple we defined in an earlier paper for the q-deformation of a simply connected semisimple compact Lie group G does not depend on any choices up to unitary equivalence.

The rationality of the moduli space of genus-4 curves endowed with an order-3 subgroup of their Jacobian

Let C be a smooth, irreducible complex projective curve of genus g and let η ∈ Pic0(C) be a nontrivial nth root of the trivial bundle OC. For several different reasons, special attention has been paid, now and in the recent past, to the moduli spaces Rg,n of pairs (C, η) as above and to its possible compactifications (see e.g. [CapCasC]). For instance, they are generalizations of the case n = 2, the so-called Prym moduli spaces, usually denoted by Rg. Since they are related to the theory of Prym varieties, the interest in this case occupies a prominent position. In particular, many results on the Kodaira dimension of Rg are now available, while classical geometric descriptions of Rg exist for g ≤ 7. More precisely, let us mention that Farkas and Ludwig [FLu] proved that Rg is of general type for g ≥ 14 and g = 15. On the other hand, unirational parameterizations of Rg are known for g ≤ 7 [Cat, D, Do, ILoS, Ve1, Ve2]. One can also consider the moduli spaces Rg,〈n〉 of pairs (C,Z/nZ), where C is a smooth, irreducible complex projective curve of genus g and Z/nZ is a cyclic subgroup of order n of Pic0(C). As Rg,〈2〉 = Rg,2, these mouli spaces are generalizing the Prym moduli spaces in a (slightly) different way. In contrast to the case n = 2, not very much is known about Rg,n and Rg,〈n〉 for n > 2. In particular, the (probably short) list of all pairs (g, n) such that Rg,n and Rg,〈n〉 have negative Kodaira dimension is not known. The rationality of Rg,n and Rg,〈n〉 has been proved in some cases of very low genus: the case of R4 is a result of Catanese [Cat]. The rationality of R3 was proved by Katsylo in [Ka]. Independent proofs are also due to Catanese and to Dolgachev; see [D] (also for R2). Recently, the rationality of R3,3 and of R3,〈3〉 has been proven by Catanese and the first author [BCat]. To complete the picture, we recall that R1,n is an irreducible curve for every prime n and that its geometric genus is well known.

On extensional waves in a poroelastic cylinder within the framework of viscosity-extended Biot theory: the case of traction-free open-pore cylindrical surface

SUMMARY For a fluid saturated porous cylinder, in which the saturating fluid is allowed to move freely across the traction-free cylindrical boundary surface, the dispersion relation for extensional deformation, established over four decades ago by Gardner within the framework of Biot's theory, yield a trivial and two non-trivial roots. One of the non-trivial roots corresponds to the coupled Biot fast-compressional and shear waves and it is the poroelastic analogue of the extensional wave of the elasticity theory. The nature of the second non-trivial root is not fully understood yet. Also there is the suggestion that the scope of applicability of Gardner's first non-trivial root might be limited. Apart from these matters, since the Biot theory is also incapable of accounting for the viscous bulk- and shear-relaxations that occur within the saturating Newtonian fluid because its fluid stress tensor is devoid of the viscous stress tensor part, therefore, we re-examined this poroelastic extensional problem with viscosity-extended Biot constitutive relations. The erstwhile trivial third root of the Biot–Gardner framework is not non-vanishing in this viscosity-extended Biot framework. In the frequency regime below the Biot relaxation frequency, the other two non-trivial roots in this framework are the same as those obtained by Gardner within the classical Biot framework. We find there is nothing dubious about the formula of Gardner's first root. The scenario in which it is not expected to hold true, in fact the Biot theory itself breaks down. We find the Gardner's second root to be essentially the Biot slow-compressional wave, the out-of-phase/differential compressional motion of the two phases. The new non-vanishing third root is akin to the slow-shear process, the out-of-phase/differential shear motion of the two phases, and a process analogue to the Biot slow-compressional wave.

Surmounting the Cartesian Cut Through Philosophy, Physics, Logic, Cybernetics, and Geometry: Self-reference, Torsion, the Klein Bottle, the Time Operator, Multivalued Logics and Quantum Mechanics

In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology—after Merleau-Ponty, Heidegger and Rosen—and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further related to the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown’s primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Muses which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological in-formation for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classical-quantum transformer (Hadamard’s gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succinctly described above.

Noncoercive sums of squares in [formula omitted

Positive definite forms f ∈ R [ x 1 , … , x n ] which are sums of squares of forms of R [ x 1 , … , x n ] are constructed to have the additional property that the members of any collection of forms whose squares sum to f must share a nontrivial complex root in C n .

Dehn twists have roots

We construct nontrivial roots of Dehn twists about nonseparating curves.

Improved Analytical Bounds for Gambler?s Ruin Probabilities

Given integer-valued wagers Feller (1968) has established upper and lower bounds on the probability of ruin, which often turn out to be very close to each other. However, the exact calculation of these bounds depends on the unique non-trivial positive root of the equation Φ(ρ) = 1, where Φ is the probability generating function for the wager. In the situation of incomplete information about the distribution of the wager, one is interested in bounds depending only on the first few moments of the wager. Ethier and Khoshnevisan (2002) derive bounds depending explicitly on the first four moments. However, these bounds do not make the best possible use of the available information. Based on the theory of s-convex extremal random variables among arithmetic and real random variables, a substantial improvement can be given. By fixed first four moments of the wager, the obtained new bounds are nearly perfect analytical approximations to the exact bounds of Feller.

Subgroups of the Spinor Group that Contain a Split Maximal Torus. II

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin(n,R) over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial root element. In the present paper, we complete the proof of a result announced by the author in 1990 and prove the standardness of all intermediate subgroups, provided that n=2l and \(\left| K \right| \geqslant 9\). For an algebraically closed K, this follows from a classical result of Borel and Tits, and for a finite K this was proved by Seitz. Similar results for subgroups of the orthogonal groups SO(n,R) were previously obtained by the author not only for fields, but for any commutative semilocal rings R with residue fields large enough. Bibliography: 52 titles.

A contraction of the Lucas polygon

The celebrated Gauss-Lucas theorem states that all the roots of the derivative of a complex non-constant polynomialpplie in the convex hull of the roots ofpp, called the Lucas polygon ofpp. We improve the Gauss-Lucas theorem by proving that all the nontrivial roots ofp′p’lie in a smaller convex polygon which is obtained by a strict contraction of the Lucas polygon ofpp.

Scaling, multiscaling, and nontrivial exponents in inelastic collision processes.

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large-velocity tail, P(v,t) approximately v(-sigma). The exponent sigma(d,epsilon), a nontrivial root of an integral equation, varies continuously with the spatial dimension d and the dissipation coefficient epsilon. Although the velocity distribution follows a scaling form, its moments exhibit multiscaling asymptotic behavior. Furthermore, the velocity autocorrelation function decays algebraically with time, A(t)=<v(0).v(t)> approximately t(-alpha), with a nonuniversal dissipation-dependent exponent alpha=1/epsilon. In the forced case, the steady state Fourier transform is obtained via a cumulant expansion. Even in this case, velocity correlations develop and the velocity distribution is non-Maxwellian.

CONFORMAL $\mathcal{s}$$\mathcal{l}$2 ENVELOPING ALGEBRAS AS AMBISKEW POLYNOMIAL RINGS

AbstractWe study a three parameter deformation $\mathcal{U}_{abc}$ of $\mathcal{U}(\mathfrak{sl}_2)$ introduced by Le Bruyn in 1995. Working over an arbitrary algebraically closed field of characteristic zero, we determine the centres, the finite-dimensional irreducible representations, and, when the parameter $a$ is not a non-trivial root of unity, the prime ideals of those $\mathcal{U}_{abc}$, with $ac\neq0$, which are conformal as ambiskew polynomial rings.AMS 2000 Mathematics subject classification: Primary 16W35; 17B37. Secondary 16S36; 16S80

Bounds on gambler's ruin probabilities in terms of moments

Consider a wager that is more complicated than simply winning or losing the amount of the bet. For example, a pass line bet with double odds is such a wager, as is a bet on video poker using a specified drawing strategy. We are concerned with the probability that, in an independent sequence of identical wagers of this type, the gambler loses L or more betting units (i.e., the gambler is “ruined”) before he wins W or more betting units. Using an idea of Markov, Feller established upper and lower bounds on the probability of ruin, bounds that are often very close to each other. However, his formulation depends on finding a positive nontrivial root of the equation φ (ρ )=1, where φ is the probability generating function for the wager in question. Here we give simpler bounds, which rely on the first few moments of the specified wager, thereby making such gambler's ruin probabilities more easily computable.

Deterministic exponential heteroskedasticity, a weakly stationary unit-root process and a useful diagnostic test

A specific form of deterministic exponential heteroskedasticity is examined. A non-trivial unit root process which has exponentially heteroskedastic innovation and as a consequence, a variance that vanishes asymptotically is detailed. Such a unit root stochastic process, with exponential heteroskedasticity, may be perceived as weakly stationary by the usual unit root tests. In view of the importance of deterministic exponential heteroskedasticity, a new general diagnostic test for detecting the presence of deterministic exponential heteroskedasticity is developed.

Fourier–Bessel analysis of patterns in a circular domain

This paper explores the use of the Fourier–Bessel analysis for characterizing patterns in a circular domain. A set of stable patterns is found to be well-characterized by the Fourier–Bessel functions. Most patterns are dominated by a principal Fourier–Bessel mode [ n, m] which has the largest Fourier–Bessel decomposition amplitude when the control parameter R is close to a corresponding non-trivial root ( ρ n, m ) of the Bessel function. Moreover, when the control parameter is chosen to be close to two or more roots of the Bessel function, the corresponding principal Fourier–Bessel modes compete to dominate the morphology of the patterns.

Closed-form displacement analysis of 10-link 1-DOF mechanisms: Part 2 – polynomial solutions

This paper presents closed-form polynomial solutions to the displacement analysis problem of planar 10-link mechanisms with 1 degree of freedom (DOF). Using the successive elimination procedure, the input–output (I/O) polynomials as well as the number of assembly configurations for five mechanisms resulting from two 10-link kinematic chains are presented. It is shown that the displacement analysis problems for all five mechanisms can be reduced to a univariate polynomial devoid of any extraneous roots. This univariate polynomial is the I/O polynomial of the mechanism. In addition, one of the examples also illustrates how trigonometric manipulations in conjunction with tangent half-angle substitutions can lead to non-trivial extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these extraneous roots are also presented.

A Framework for Closed-Form Displacement Analysis of Planar Mechanisms

This paper presents a closed-form approach, based on the theory of resultants, to the displacement analysis problem of planar n-link mechanisms. The successive elimination procedure presented herein generalizes the Sylvester’s dialytic eliminant for the case when p equations (p ≥ 3) are to be solved in p unknowns. Conditions under which the method of successive elimination can be used to reduce p equations (in p unknowns) into a univariate polynomial, devoid of extraneous roots, are presented. This univariate polynomial corresponds to the I/O polynomial of the mechanism. A comprehensive treatment is also presented on some of the problems associated with the conversion of transcendental loop-closure equations, into an algebraic form, using tangent half-angle substitutions. It is shown how trigonometric manipulations in conjunction with tangent half-angle substitutions can lead to non-trivial extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these extraneous roots are presented. The computational procedure is illustrated through the displacement analysis of a 10-link 1-DOF mechanism with 4 independent loops.