Computing Zeros of Analytic Functions in the Complex Plane without using Derivatives

A new approach to evaluating all multiple complex roots of analytical function f(z) confined to the specified rectangular domain of complex plane has been developed and implemented in Fortran code. Generally f (z), despite being holomorphic function, does not have a closed analytical form thereby inhibiting explicit evaluation of its derivatives. The latter constraint poses a major challenge to implementation of the robust numerical algorithm. This work is at the instrumental level and provides an enabling tool for solving a broad class of eigenvalue problems and polynomial approximations.

Algorithmic computation of knot polynomials of secondary structure elements of proteins

The classification of protein structures is an important and still outstanding problem. The purpose of this paper is threefold. First, we utilize a relation between the Tutte and homfly polynomial to show that the Alexander-Conway polynomial can be algorithmically computed for a given planar graph. Second, as special cases of planar graphs, we use polymer graphs of protein structures. More precisely, we use three building blocks of the three-dimensional protein structure-alpha-helix, antiparallel beta-sheet, and parallel beta-sheet-and calculate, for their corresponding polymer graphs, the Tutte polynomials analytically by providing recurrence equations for all three secondary structure elements. Third, we present numerical results comparing the results from our analytical calculations with the numerical results of our algorithm-not only to test consistency, but also to demonstrate that all assigned polynomials are unique labels of the secondary structure elements. This paves the way for an automatic classification of protein structures.

Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove that every symmetric tensor of unit norm in HoH is an infinite absolute convex combination of points of the form xox with x in the unit sphere of the Hilbert space. We use this to obtain explicit characterizations of the smooth points of the unit ball of HoH .

Polynomials on Banach spaces with unconditional bases

We study the classes of homogeneous polynomials on a Banach space with unconditional Schauder basis that have unconditionally convergent monomial expansions relative to this basis. We extend some results of Matos, and we show that the homogeneous polynomials with unconditionally convergent expansions coincide with the polynomials that are regular with respect to the Banach lattices structure of the domain.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

Escape times for rigid Brownian rotators in a bistable potential from the time evolution of the Green function and the characteristic time of the probability evolution

The greatest relaxation time for an assembly of three- dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions (averages of the Legendre polynomials) of the system. This is accomplished by studying the entire time evolution of the Green function (transition probability) by expanding the time dependent distribution as a Fourier series and proceeding to the zero frequency limit of the Laplace transform of that distribution. The procedure is entirely analogous to the calculation of the characteristic time of the probability evolution (the integral of the configuration space probability density function with respect to the position co-ordinate) for a particle undergoing translational diffusion in a potential; a concept originally used by Malakhov and Pankratov (Physica A 229 (1996) 109). This procedure allowed them to obtain exact solutions of the Kramers one-dimensional translational escape rate problem for piecewise parabolic potentials. The solution was accomplished by posing the problem in terms of the appropriate Sturm-Liouville equation which could be solved in terms of the parabolic cylinder functions. The method (as applied to rotational problems and posed in terms of recurrence relations for the decay functions, i.e., the Brinkman approach c.f. Blomberg, Physica A 86 (1977) 49, as opposed to the Sturm-Liouville one) demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function (albeit coupled to all the others in non-linear fashion via the underlying recurrence relation) is governed by a sum of decay functions. The method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non-axially symmetric potentials, where the distribution function is governed by two state variables. (C) 2001 Elsevier Science B.V. All rights reserved.

Factorial Design of Cement Slurries Containing Limestone Powder for Self-Consolidating Slurry-Infiltrated Fiber Concrete

Slurries with high penetrability for production of Self-consolidating Slurry Infiltrated Fiber Concrete (SIFCON) were investigated in this study. Factorial experimental design was adopted in this investigation to assess the combined effects of five independent variables on mini-slump test, plate cohesion meter, induced bleeding test, J-fiber penetration test and compressive strength at 7 and 28 days. The independent variables investigated were the proportions of limestone powder (LSP) and sand, the dosages of superplasticiser (SP) and viscosity agent (VA), and water-to-binder ratio (w/b). A two-level fractional factorial statistical method was used to model the influence of key parameters on properties affecting the behaviour of fresh cement slurry and compressive strength. The models are valid for mixes with 10 to 50% LSP as replacement of cement, 0.02 to 0.06% VA by mass of cement, 0.6 to 1.2% SP and 50 to 150% sand (% mass of binder) and 0.42 to 0.48 w/b. The influences of LSP, SP, VA, sand and W/B were characterised and analysed using polynomial regression which identifies the primary factors and their interactions on the measured properties. Mathematical polynomials were developed for mini-slump, plate cohesion meter, J-fiber penetration test, induced bleeding and compressive strength as functions of LSP, SP, VA, sand and w/b. The estimated results of mini-slump, induced bleeding test and compressive strength from the derived models are compared with results obtained from previously proposed models that were developed for cement paste. The proposed response models of the self-consolidating SIFCON offer useful information regarding the mix optimization to secure a highly penetration of slurry with low compressive strength

Influence of mix proprtions on rheolgy of cement grouts containing limestone powder

In this paper the parameters of cement grout affecting rheological behaviour and compressive strength are investigated. Factorial experimental design was adopted in this investigation to assess the combined effects of the following factors on fluidity, rheological properties, induced bleeding and compressive strength: water/binder ratio (W/B), dosage of superplasticiser (SP), dosage of viscosity agent (VA), and proportion of limestone powder as replacement of cement (LSP). Mini-slump test, Marsh cone, Lombardi plate cohesion meter, induced bleeding test, coaxial rotating cylinder viscometer were used to evaluate the rheology of the cement grout and the compressive strengths at 7 and 28 days were measured. A two-level fractional factorial statistical model was used to model the influence of key parameters on properties affecting the fluidity, the rheology and compressive strength. The models are valid for mixes with 0.35-0.42 W/B, 0.3-1.2% SP, 0.02-0.7% VA (percentage of binder) and 12-45% LSP as replacement of cement. The influences of W/B, SP, VA and LSP were characterised and analysed using polynomial regression which can identify the primary factors and their interactions on the measured properties. Mathematical polynomials were developed for mini-slump, plate cohesion meter, inducing bleeding, yield value, plastic viscosity and compressive strength as function of W/B, SP, VA and proportion of LSP. The statistical approach used highlighted the limestone powder effect and the dosage of SP and VA on the various rheological characteristics of cement grout

On solvability of linear differential equations in ${\Bbb R}^{\Bbb N}$

We construct $x^0$ in ${\Bbb R}^{\Bbb N}$ and a row-finite matrix $T=\{T_{i,j}(t)\}_{i,j\in\N}$ of polynomials of one real variable $t$ such that the Cauchy problem $\dot x(t)=T_tx(t)$, $x(0)=x^0$ in the Fr\'echet space $\R^\N$ has no solutions. We also construct a row-finite matrix $A=\{A_{i,j}(t)\}_{i,j\in\N}$ of $C^\infty(\R)$ functions such that the Cauchy problem $\dot x(t)=A_tx(t)$, $x(0)=x^0$ in ${\Bbb R}^{\Bbb N}$ has no solutions for any $x^0\in{\Bbb R}^{\Bbb N}\setminus\{0\}$. We provide some sufficient condition of solvability and of unique solvability for linear ordinary differential equations $\dot x(t)=T_tx(t)$ with matrix elements $T_{i,j}(t)$ analytically dependent on $t$.

Efficient analysis, design, and filter applications of EBG waveguide with periodic resonant loads

An efficient analysis and design of an electromagnetic-bandgap (EBG) waveguide with resonant loads is presented. Equivalent-circuit analysis is employed to demonstrate the differences between EBG waveguides with resonant and nonresonant loadings. As a result of the resonance, transmission zeros at finite frequencies emerge. The concept is demonstrated in E-plane waveguides. A generic fast and efficient formulation is presented, which starts from the generalized scattering matrix of the unit cell and derives the dispersion properties of the infinite structure. Both real and imaginary parts of the propagation constant are derived and discussed. The Floquet wavelength and impedance are also presented. The theoretical results are validated by comparison with simulations of a finite structure and experimental results. The application of the proposed EBG waveguide in the suppression of the spurious passband of a conventional E-plane filter is presented by experiment.

The unit ball of the complex P(3H)

Let H be a two-dimensional complex Hilbert space and P(3H) the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, P(3H), from which we deduce that the unit sphere of P(3H) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of P(3H) remains extreme as considered as an element of L(3H). Finally we make a few remarks about the geometry of the unit ball of the predual of P(3H) and give a characterization of its smooth points.

Exoplanets or Dynamic Atmospheres? The Radial Velocity and Line Shape Variations of 51 Pegasi and tau Bootis

The stars 51 Pegasi and tau Bootis show radial velocity variations that have been interpreted as resulting from companions with roughly Jovian mass and orbital periods of a few days. Gray and Gray & Hatzes reported that the radial velocity signal of 51 Peg is synchronous with variations in the shape of the line lambda 6253 Fe I; thus, they argue that the velocity signal arises not from a companion of planetary mass but from dynamic processes in the atmosphere of the star, possibly nonradial pulsations. Here we seek confirming evidence for line shape or strength variations in both 51 Peg and tau Boo, using R = 50,000 observations taken with the Advanced Fiber Optic Echelle. Because of our relatively low spectral resolution, we compare our observations with Gray's line bisector data by fitting observed line profiles to an expansion in terms of orthogonal (Hermite) functions. To obtain an accurate comparison, we model the emergent line profiles from rotating and pulsating stars, taking the instrumental point-spread function into account. We describe this modeling process in detail. We find no evidence for line profile or strength variations at the radial velocity period in either 51 Peg or in tau Boo. For 51 Peg, our upper limit for line shape variations with 4.23 day periodicity is small enough to exclude with 10 sigma confidence the bisector curvature signal reported by Gray & Hatzes; the bisector span and relative line depth signals reported by Gray are also not seen, but in this case with marginal (2 sigma ) confidence. We cannot, however, exclude pulsations as the source of 51 Peg's radial velocity variation because our models imply that line shape variations associated with pulsations should be much smaller than those computed by Gray & Hatzes; these smaller signals are below the detection limits both for Gray & Hatzes's data and for our own. tau Boo's large radial velocity amplitude and v sin i make it easier to test for pulsations in this star. Again we find no evidence for periodic line shape changes, at a level that rules out pulsations as the source of the radial velocity variability. We conclude that the planet hypothesis remains the most likely explanation for the existing data.