Finite Analog Research Articles

Representing polynomials as characteristic polynomials via the Stieltjes transform

The Stieltjes transform SA of an infinite lower triangular matrix A with nonzero diagonal entries is defined by SA=A−1A¯ where A¯ is the matrix obtained from A by deleting its initial row. In this paper, we express a sequence of polynomials as the characteristic polynomials of the Stieltjes transforms using a highly structured infinite lower triangular matrix called a Riordan matrix. As a result, computation of the zeros of such polynomials becomes amenable to iterative methods for computing eigenvalues, or to eigenvalue location theorems such as the Geršgorin theorem. We also describe a finite analog of the polynomial correspondence and its relationship to eigenvalue regions. As an application, the recurrence relations for several polynomial sequences are obtained using the Stieltjes transform.

Small Kakeya sets in non-prime order planes

The finite field analog of the classical Kakeya problem asks the smallest possible size for a set of points in the Desarguesian affine plane which contains a line in every direction. This problem has been definitively solved by Blokhuis and Mazzocca (2008), who show that in AG(2,s), s a prime power, the smallest possible size of such a set is 12s(s+1). In this paper we examine a new construction of an infinite family of sets in AG(2,s) containing a line in every direction, where s=qe with q a prime power and e an integer with e>1. These sets have size 12q2e+O(q2e−1), which is small in the sense that the lower bound is also 12q2e plus smaller order terms. In addition, we discuss the minimality of our sets, showing that they contain no proper subset containing a line in every direction.

Is bimodality a sufficient condition for a first-order phase transition existence?

Here we present two explicit counterexamples to the widely spread beliefs about an exclusive role of bimodality as the first-order phase transition signal. On the basis of an exactly solvable statistical model generalizing the statistical multifragmentation model of nuclei, we demonstrate that the bimodal distributions can naturally appear both in infinite and in finite systems without a phase transition. In the first counterexample a bimodal distribution appears in an infinite system at the supercritical temperatures due to the negative values of the surface tension coefficient. In the second counterexample we explicitly demonstrate that a bimodal fragment distribution appears in a finite volume analog of a gaseous phase. In contrast to the statistical multifragmentation model, the developed statistical model corresponds to the compressible nuclear liquid with the tricritical endpoint located at one third of the normal nuclear density. The suggested parameterization of the liquid phase equation of state is consistent with the L. Van Hove axioms of statistical mechanics and it does not lead to an appearance of the nonmonotonic isotherms in the macroscopic mixed phase region which are typical for the classical models of the Van der Waals type. Peculiarly, such a way to account for the nuclear liquid compressibility automatically leads to an appearance of an additional state that in many respects resembles the physical antinuclear matter.

Minimal Kakeya Sets

AbstractBlokhuis and Mazzocca (A. Blokhuis and F. Mazzocca, The finite field Kakeya problem (English summary). Building bridges. Bolyai Soc Math Stud 19 (2008) 205–218) provide a strong answer to the finite field analog of the classical Kakeya problem, which asks for the minimum size of a point set in an affine plane π that contains a line in every direction. In this article, we consider the related problem of minimal Kakeya sets, namely Kakeya sets containing no smaller Kakeya sets, and provide an interesting infinite family of minimal Kakeya sets that are not of extremal size.

On the sum of the squared multiplicities of the distances in a point set over finite spaces

We study a finite analog of a conjecture of Erd?s on the sum of the squared multiplicities of the distances determined by an n-element point set. Our result is based on an estimate of the number of hinges in spectral graphs.

The Number of Occurrences of a Fixed Spread among n Directions in Vector Spaces over Finite Fields

We study a finite analog of a problem of Erdős, Hickerson and Pach on the maximum number of occurrences of a fixed angle among n directions in three-dimensional spaces.

Residual distribution finite element method for convection-dominated problems

In this paper, we present a new stabilized finite element method for the simulation of nonlinear convection-dominated systems of PDEs. We seek to combine the robustness of upwind finite-volume methods for the discretization of strongly shocked flows with the ease and accuracy of C0 finite element reconstruction. This is achieved by exploiting the similarities between traditional SUPG method and finite volume residual distribution schemes. By constructing the finite element analog of finite-volume fluctuation-splitting schemes, we combine the ability to simulate PDEs with strongly discontinuous solutions of upwind finite-volume methods with the reliable accuracy of finite element reconstruction, which traditional finite-volume methods often lack on unstructured grids. Finally, we test the proposed algorithm with challenging problems drawn from hypersonic compressible flow and semiconductor device simulation.

Handsaw Quiver Varieties and Finite W-Algebras

Following Braverman-Finkelberg-Feigin-Rybnikov (arXiv:1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W-algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang-Mills theory with surface operators. Our new observation is that the C^*-fixed point set of a handsaw quiver variety is isomorphic to a graded quiver variety of type A, which was introduced by the author in connection with the representation theory of a quantum affine algebra. As an application, simple modules of the W-algebra are described in terms of IC sheaves of graded quiver varieties of type A, which were known to be related to Kazhdan-Lusztig polynomials. This gives a new proof of a conjecture by Brundan-Kleshchev on composition multiplicities on Verma modules, which was proved by Losev, in a wider context, by a different method.

Finite Element Simulations of Mechanical Properties of Solid Expandable Tubular Threaded Joint

In this paper, threaded joint dedicated for the solid expandable tubular is taken as the basic structure and specific parameters for the threaded joint is set. 16Mn improved material is selected as the material, and MSC/Marc software is adopted to conduct finite element analog study on technical performance of the threaded joint and to analyze the distribution and strain condition of microstress and macrostress for the thread after make-up as well as the tensile strength and sealing property for the joint after make-up. Multiple important data concerning the designed expandable tubular threaded joint have been obtained through the study. It is convinced that the solid expandable tubular threaded joint shall adopt be a flush one with both internal and external walls parallel and level with the tubular body, and the type of thread shall take a shape of reversed hook. Besides, the expandable tubular threaded joint shall adopt the sealing structure with the rubber seal as the main sealing type and metallic seal auxiliary.

A Finite Analog of the AGT Relation I: Finite W-Algebras and Quasimaps’ Spaces

Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on $${\mathbb{P}^2}$$ . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a “finite analog” of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from $${\mathbb{P}^1}$$ to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra $${U(\mathfrak{g},e)}$$ associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of [5] when P is the Borel subgroup. We prove our conjecture for G = GL(N), using the works of Brundan and Kleshchev interpreting the algebra $${U(\mathfrak{g},e)}$$ in terms of certain shifted Yangians.

Impact of topology on the dynamical organization of cooperation in the prisoner’s dilemma game

The way cooperation organizes dynamically strongly depends on the topology of the underlying interaction network. We study this dependence using heterogeneous scale-free networks with different levels of (a) degree-degree correlations and (b) enhanced clustering, where the number of neighbors of connected nodes are correlated and the number of closed triangles are enhanced, respectively. Using these networks, we analyze a finite population analog of the evolutionary replicator dynamics of the prisoner's dilemma, the latter being a two-player game with two strategies, defection and cooperation, whose payoff matrix favors defection. Both topological features significantly change the dynamics with respect to the one observed for fully randomized scale-free networks and can strongly facilitate cooperation even for a large temptation to defect, and should hence be considered as important factors in the evolution and sustainment of cooperation.

A 3-Tap FIR Filter With Cascaded Distributed Tap Amplifiers for Equalization Up to 40 Gb/s in 0.18-<tex>$mu$</tex>m CMOS

This paper describes a 3-tap finite impulse response programmable analog filter in 0.18-mum CMOS. The filter is intended for polarization-mode dispersion compensation in 40-Gb/s long-haul single-mode optical fiber links. A novel filter topology is employed, whereby each tap gain comprises a cascade of two distributed amplifiers with adjustable gain. The same lumped LC ladders that provide the 25-ps tap delays also serve as artificial transmission lines for the distributed tap amplifiers. The prototype is 1 mm times 1 mm (900 mumtimes600 mum active area) and consumes up to 70 mW from a 1.8-V supply depending on the tap gains. Linear equalization is demonstrated over a channel with 17 dB of loss at 20 GHz. A 50 mV per side eye amplitude with 14 dB signal-to-noise ratio is demonstrated at 30 Gb/s, and a modest 39 mV per side eye amplitude with 10 dB signal-to-noise ratio is achieved at 40 Gb/s

Why is quantum physics based on complex numbers?

The modern quantum theory is based on the assumption that quantum states are represented by elements of a complex Hilbert space. It is expected that in future quantum theory the number field will not be postulated but derived from more general principles. We consider the choice of the number field in a quantum theory based on a finite field. We assume that the symmetry algebra is the finite field analog of the de Sitter algebra so(1,4) and consider spinless irreducible representations of this algebra. It is shown that the finite field analog of complex numbers is the minimal extension of the residue field modulo p for which the representations are fully decomposable.

Average Run Length and the OC Curve of Sampling Plans

ABSTRACT This paper amplifies the distinction between Type A and Type B sampling plans for individual lots and processes, respectively. It introduces the f-binomial distribution as a Poisson type finite analog to the hypergeometric distribution for use in constructing appropriate Type A OC curves for defects. The use of average run length in characterizing Type B sampling plans is also presented as a missing measure in the evaluation of such plans.

Return of the EMC effect: Finite nuclei

A light front formalism for deep inelastic lepton scattering from finite nuclei is developed. In particular, the nucleon plus momentum distribution and a finite system analog of the Hugenholtz-van Hove theorem are presented. Using a relativistic mean field model, numerical results for the plus momentum distribution and ratio of bound to free nucleon structure functions for Oxygen, Calcium and Lead are given. We show that we can incorporate light front physics with excellent accuracy while using easily computed equal time wavefunctions. Assuming nucleon structure is not modified in-medium we find that the calculations are not consistent with the binding effect apparent in the data not only in the magnitude of the effect, but in the dependence on the number of nucleons.

Dynamical approximation by recurrent neural networks

We examine the approximating power of recurrent networks for dynamical systems through an unbounded number of iterations. It is shown that the natural family of recurrent neural networks with saturated linear transfer functions and synaptic weight matrices of rank 1 are essentially equivalent to feedforward neural networks with recurrent layers. Therefore, they inherit the universal approximation property of real-valued functions in one variable in a stronger sense, namely through an unbounded number of iterations and approximation guaranteed to be within O(1/ n), with n neurons and possibly lateral synapses allowed in the hidden-layer. However, they are not as complex in their dynamical behavior as systems defined by Turing machines. It is further proved that every continuous dynamical system can be approximated through all iterations, by both finite analog and boolean networks, when one requires approximation of given arbitrary exact orbits of the (perhaps unknown) map. This result no longer holds when the orbits of the given map are only available as contaminated orbits of the approximant net due to the presence of random noise (e.g., due to digital truncations of analog activations). Neural nets can nonetheless approximate large families of continuous maps, including chaotic maps and maps sensitive to initial conditions. A precise characterization of what maps can be approximated fault-tolerantly by analog and discrete neural networks for unboundedly many iterations remains an open problem.

Discrete Dispersion Relations and Taylor-Galerkin FEM for Transport of Dispersed Fronts

A time and space accurate method for computing the movement of an adsorbing contaminant in a carrier fluid is presented. The method uses a finite element analog to a previously described finite difference approach which can accurately describe movement of highly convected, but dispersing fronts. An operator splitting scheme between convection and dispersion is employed which provides fourth order time accuracy on the convection step. The higher accuracy for the convection step is also accompanied with belter mass conservation properties than is common for most algorithms which model purely convective problems. The method is demonstrated on problems varying in one dimension by applications to pure convection-dispersion, the nonlinear Burger's equation, convection-reaction equations, and convection-dispersion through an adsorbing medium. The model predictions are found to provide accurate and stable results.

Tricriticality and the Blume-Capel model: A Monte Carlo study within the microcanonical ensemble

The microcanonical partition function ${\ensuremath{\Omega}}_{\ensuremath{\alpha},N}(E)$ of a three-dimensional ferromagnetic simple cubic Blume-Capel model is calculated for several coupling ratios $\ensuremath{\alpha}$ close to and on the first-order side of the tricritical point. To this end a single Monte Carlo simulation of a suitably extended partition function for systems of $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ spins with $L\ensuremath{\in}{8,10,12,14,18}$ was performed. A finite system analog of the latent heat is introduced in order to define a tricritical point in finite systems as well. An empirical scaling of its coordinates to the thermodynamic limit yields ${\ensuremath{\alpha}}^{(t)}=2.84479\ifmmode\pm\else\textpm\fi{}0.00030$ for the tricritical coupling ratio, and ${k}_{B}{T}^{(t)}/J=1.4182\ifmmode\pm\else\textpm\fi{}0.0055$ for the tricritical transition temperature. The results are compared with values obtained on fcc lattices.

Numerical evidence of an embryonic orientational phase transition in small nitrogen clusters.

Using molecular dynamics simulation we provide some evidence of an embryonic solid-solid phase transition in (${\mathrm{N}}_{2}$${)}_{13}$ cluster. This transition, as characterized in terms of the underlying potential energy surface, is accompanied by the release of the molecular orientational degree of freedom. This transition is also observed in (${\mathrm{N}}_{2}$${)}_{55}$ but not in (C${\mathrm{O}}_{2}$${)}_{13}$. It is presumably a finite size analog of the bulk rigid to plastic phase transition, which is known to occur in solid ${\mathrm{N}}_{2}$ but not in C${\mathrm{O}}_{2}$.

Power flow analysis of beam members with multiple wave types

A power flow finite element method (PFFEM) is developed by Nefske and Sung ( ASME J. Vib. Stress Reliability Des. 111, 1989) for the steady state high frequency vibration analysis of beams in which there is propagation of one wave type (bending). A finite element thermal analog presented herein for power flow analysis of beam members in which both bending and longitudinal waves are present is a further development of the PFFEM. Conducting bar elements model mechanical power flow in the beam and also power transmission, reflection and conversion at structural joints. Convection elements model internal power dissipation. Properties of joint elements are in general non-linear and depend upon the power level at the joint as well as transmission and reflection coefficients. The thermal analog is verified for particular linear cases wherein properties of the joint elements are dependent only upon the transmission and reflection coefficients.