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We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.

We show that smooth cubic hypersurfaces of dimension $n$ defined over a finite field ${\bf F}_q$ contain a line defined over ${\bf F}_q$ in each of the following cases: - $n=3$ and $q\ge 11$; - $n=4$, and $q=2$ or $q\ge 5$; - $n\ge 5$. For a smooth cubic threefold $X$, the variety of lines contained in $X$ is a smooth projective surface $F(X)$ for which the Tate conjecture holds, and we obtain information about the Picard number of $F(X)$ and its 5-dimensional principally polarized Albanese variety $A(F(X))$.

The aim of this paper is to study asymptotic geometric properties almost surely or/and in probability of extreme order statistics of an i.i.d. random field (potential) indexed by sites of multidimensional lattice cube, the volume of which unboundedly increases. We discuss the following topics: (I) high level exceedances, in particular, clustering of exceedances; (II) decay rate of spacings in comparison with increasing rate of extreme order statistics; (III) minimum of spacings of successive order statistics; (IV) asymptotic behavior of values neighboring to extremes and so on. The conditions of the results are formulated in terms of regular variation (RV) of the cumulative hazard function and its inverse. A relationship between RV classes of the present paper as well as their links to the well-known RV classes (including domains of attraction of max-stable distributions) are discussed. The asymptotic behavior of functionals (I)--(IV) determines the asymptotic structure of the top eigenvalues and the corresponding eigenfunctions of the large-volume discrete Schr\" odinger operators with an i.i.d. potential (Anderson Hamiltonian). Thus, another aim of the present paper is to review and comment a recent progress on the extreme value theory for eigenvalues of random Schr\" odinger operators as well as to provide a clear and rigorous understanding of the relationship between the top eigenvalues and extreme values of i.i.d. random potentials. We also discuss their links to the long-time intermittent behavior of the parabolic problems associated with the Anderson Hamiltonian via spectral representation of solutions.

In this paper we give a classification of complete intersection vanishing ideals on parameterized sets of clutter type over finite fields.

For an integer $r$, a prime power $q$, and a polynomial $f$ over a finite field ${\mathbb F}_{q^r}$ of $q^r$ elements, we obtain an upper bound on the frequency of elements in an orbit generated by iterations of $f$ which fall in a proper subfield of ${\mathbb F}_{q^r}$. We also obtain similar results for elements in affine subspaces of ${\mathbb F}_{q^r}$, considered as a linear space over ${\mathbb F}_q$.

We improve upper bounds of F. R. K. Chung and of M. Lu, D. Wan, L.-P. Wang, X.-D. Zhang on the diameter of some Cayley graphs constructed from polynomials over finite fields.

Let $q>2$ be a prime power and $f=-{\tt x}+t{\tt x}^q+{\tt x}^{2q-1}$, where $t\in\Bbb F_q^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q$ is even and $\text{Tr}_{q/2}(\frac 1t)=0$; (ii) $q\equiv 1\pmod 8$ and $t^2=-2$.

Let $k$ be a positive integer and $S_{2k}={\tt x}+{\tt x}^4+...+{\tt x}^{4^{2k-1}}\in\Bbb F_2[{\tt x}]$. It was recently conjectured that ${\tt x}+S_{2k}^{4^{2k}}+S_{2k}^{4^k+3}$ is a permutation polynomial of $\Bbb F_{4^{3k}}$. In this note, the conjecture is confirmed and a generalization is obtained.

A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently [Y] introduced a novel technique for constructing locally decodable codes and vastly improved the upper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work of [Y] and argue that further progress via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m=2^t-1 that has a prime factor p>m^\gamma yields a family of k(\gamma)-query locally decodable codes of length Exp(n^{1/t}). Conversely, if for some fixed k and all \epsilon > 0 one can use the technique of [Y] to obtain a family of k-query LDCs of length Exp(n^\epsilon); then infinitely many Mersenne numbers have prime factors arger than known currently.

Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical metrology and optics design of nanostructured components. As has been shown in previous benchmarks some of the presently used methods suffer from low convergence rates and/or low accuracy of the results and exhibit very long computation times which makes application to extended 2D layout patterns impractical. We address 3D simulation tasks by using a finite-element solver which has been shown to be superior to competing methods by several orders of magnitude in accuracy and computational time for typical microlithography simulations. We report on the current status of the solver, incorporating higher order edge elements, adaptive refinement methods, and fast solution algorithms. Further, we investigate the performance of the solver in the 3D simulation project of light diffraction off an alternating phase-shift contact-hole mask.

We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.

In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the later. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism algebra, but also this fails over infinite fields. Still over finite fields we study the endomorphism rings of pure t-motives and criteria for the existence of isogenies. We obtain answers which are similar to Tate's famous results for abelian varieties.

Let $\mathbb{F}_{q}$ be a finite field, $\mathbb{F}_{q^s}$ be an extension of $\mathbb{F}_q$, let $f(x)\in \mathbb{F}_q[x]$ be a polynomial of degree $n$ with $\gcd(n,q)=1$. We present a recursive formula for evaluating the exponential sum $\sum_{c\in \mathbb{F}_{q^s}}\chi^{(s)}(f(x))$. Let $a$ and $b$ be two elements in $\mathbb{F}_q$ with $a\neq 0$, $u$ be a positive integer. We obtain an estimate for the exponential sum $\sum_{c\in \mathbb{F}^*_{q^s}}\chi^{(s)}(ac^u+bc^{-1})$, where $\chi^{(s)}$ is the lifting of an additive character $\chi$ of $\mathbb{F}_q$. Some properties of the sequences constructed from these exponential sums are provided also.

In this paper we study the generalized Erdos-Falconer distance problems in the finite field setting. The generalized distances are defined in terms of polynomials, and various formulas for sizes of distance sets are obtained. In particular, we develop a simple formula for estimating the cardinality of distance sets determined by diagonal polynomials. As a result, we generalize the spherical distance problems due to Iosevich and Rudnev and the cubic distance problems due to Iosevich and Koh. Moreover, our results are of higher dimensional version for Vu's work on two dimension. In addition, we set up and study the generalized pinned distance problems in finite fields. We give a nice generalization of some recent work in which the pinned distance problems related to spherical distances were investigated. Discrete Fourier analysis and exponential sum estimates play an important role in our proof.

We study chiral symmetry restoration with increasing temperature and density in gauge theories subject to mutually perpendicular electric and magnetic fields using holography. We determine the chiral symmetry breaking phase structure of the D3/D7 and D3/D5 systems in the temperature-density-electric field directions. A magnetic field may break the chiral symmetry and an additional electric field induces Ohm and Hall currents as well as restoring the chiral symmetry. At zero temperature the D3/D5 system displays a line of holographic BKT phase transitions in the density-electric field plane, while the D3/D7 system shows a mean-field phase transition. At intermediate temperatures, the transitions in the density-electric field plane are of first order at low density, transforming to second order at critical points as density rises. At high temperature the transition is only ever first order.

Motivated by communication through a network employing linear network coding, capacities of linear operator channels (LOCs) with arbitrarily distributed transfer matrices over finite fields are studied. Both the Shannon capacity C and the subspace coding capacity CSS are analyzed. By establishing and comparing a lower bound on C and an upper bound on CSS, it is demonstrated that CSS is strictly less than C for a broad class of LOCs. In general, evaluating CSS is difficult because it requires to solve the maximization of a non-concave function. However, it is shown that if a LOC has a unique subspace degradation, then CSS can be obtained by solving a convex optimization problem over rank distribution. A class of LOCs such that C = CSS, which includes the LOCs with uniform-given-rank transfer matrices as special cases, is shown to have a unique subspace degradation.

Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on $V$ are obtained, and the distribution of visible points into different congruence classes is investigated.

Root extraction is a classical problem in computers algebra. It plays an essential role in cryptosystems based on elliptic curves. In 2006, Barreto and Voloch proposed an algorithm to compute $r$th roots in ${F}_{q^m} $ for certain choices of $m$ and $q$. If $r\,||\,q-1$ and $ (m, r)=1, $ they proved that the complexity of their method is $\widetilde{\mathcal {O}}(r(\log m+\log\log q)m\log q) $. In this paper, we extend the Barreto-Voloch algorithm to the general case that $r\,||\,q^m-1$, without the restrictions $r\,||\,q-1$ and $(m, r)=1 $. We also specify the conditions that the Barreto-Voloch algorithm can be preferably applied.

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to numerical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.

We consider generalised root identities for zeta functions of curves over finite fields, \zeta_{k}, and compare with the corresponding analysis for the Riemann zeta function. We verify numerically that, as for \zeta, the \zeta_{k} do satisfy the generalised root identities and we investigate these in detail for the special cases of \mu=0,-1\:\&\:-2. Unlike for \zeta, however, we show that in the setting of zeta functions of curves over finite fields the \mu=-2 root identity is consistent with the Riemann hypothesis (RH) proved by Weil. Comparison of this analysis with the corresponding calculations for \zeta illuminates the fact that, even though both \zeta and \zeta_{k} have both Euler and Hadamard product representations, it is the detailed structure of the counting function, N(T), which drives the Cesaro computations on the root side of these identities and thereby determines the implications of the root identities for RH in each setting.

The calculation of the outer inviscid flow about a rectangular wing moving at supersonic speeds is reported. The inviscid equations of motion governing the flow generated by the wing form a set of hyperbolic differential equations. The flow field about the rectangular wing is separated into three regions consisting of the forebody, the afterbody, and the wing wake. Solutions for the forebody are obtained using conical flow techniques while the afterbody and the wing wake regions are treated as initial value problems. The numerical solutions are compared in the two dimensional regions with known exact solutions.

We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many points a curve over GF(2) of genus 17 can have, by finding one with 18 points. The search is aided by computer; in some cases it is exhaustive for this type of curve of genus up to 50.

We give the distribution of points on smooth superelliptic curves over a fixed finite field, as their degree goes to infinity. We also give the distribution of points on smooth m-fold cyclic covers of the line, for any m, as the degree of their superelliptic model goes to infinity. This builds on previous work of Kurlberg, Rudnick, Bucur, David, Feigon, and Lalin for p-fold cyclic covers, but the limits taken differ slightly and the resulting distributions are interestingly different.

Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-\kappa}$ and $r$ an arbitrarily fixed positive integer. Assuming $\kappa=\kappa(r)>0$ and $p$ large enough, it is shown that any proportional subset $A\subset G$ contains non-trivial arithmetic progressions of length $r$. The main ingredient is the Szemer\'{e}di-Green-Tao theorem.

The distribution of the number of points on abelian covers of $\mathbb{P}1(\mathbb{F}_q)$ ranging over an irreducible moduli space has been answered in recent work by the author. Bucur, et al. determined the distribution over the whole moduli space for curves with Gal$(K(C)/K)$ a prime cyclic. In this paper, we prove a result towards determining the distribution over the whole moduli space of curves with Gal$(K(C)/K)$ any abelian group. We successfully determine the distribution in the case Gal$(K(C)/K)$ is a power of a prime cyclic.

We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many nonsingular polynomial matrices are mutually left coprime. These results are used to estimate the number of reachable and observable linear systems as well as the number of non-catastrophic convolutional codes. Moreover, we are able to achieve an asymptotic formula for the probability that a parallel connected linear system is reachable.

In this short note, we study the distribution of spreads in a point set $\mathcal{P} \subseteq \mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $\varepsilon > 0$, if $|\mathcal{P}| \geq (1+\varepsilon) q^{\lceil d/2 \rceil}$, then $\mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $\mathcal{P} \subset \mathbb{F}_q^d$ of size $|\mathcal{P}| = q^{\lceil d/2 \rceil}$ that determine at most one spread.

We study the model theory of $(\mathbb{F};

In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a question in arithmetic mirror symmetry.

Let $P, Q\in \mathbb{F}_q[X]\setminus\{0\}$ be two coprime polynomials over the finite field $\mathbb{F}_q$ with $\operatorname{deg}{P} > \operatorname{deg}{Q}$. We represent each polynomial $w$ over $\mathbb{F}_q$ by \[w=\sum_{i=0}^k\frac{s_i}{Q}{\left(\frac{P}{Q}\right)}^i\] using a rational base $P/Q$ and digits $s_i\in\mathbb{F}_q[X]$ satisfying $\operatorname{deg}{s_i} < \operatorname{deg}{P}$. Digit expansions of this type are also defined for formal Laurent series over $\mathbb{F}_q$. We prove uniqueness and automatic properties of these expansions. Although the $\omega$-language of the possible digit strings is not regular, we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal Laurent series is automatic if and only if the series is algebraic over $\mathbb{F}_q[X]$. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's $3/2$-problem.

Let $P, Q\in \mathbb{F}_q[X]\setminus\{0\}$ be two coprime polynomials over the finite field $\mathbb{F}_q$ with $\operatorname{deg}{P} > \operatorname{deg}{Q}$. We represent each polynomial $w$ over $\mathbb{F}_q$ by \[w=\sum_{i=0}^k\frac{s_i}{Q}{\left(\frac{P}{Q}\right)}^i\] using a rational base $P/Q$ and digits $s_i\in\mathbb{F}_q[X]$ satisfying $\operatorname{deg}{s_i} < \operatorname{deg}{P}$. Digit expansions of this type are also defined for formal Laurent series over $\mathbb{F}_q$. We prove uniqueness and automatic properties of these expansions. Although the $\omega$-language of the possible digit strings is not regular, we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal Laurent series is automatic if and only if the series is algebraic over $\mathbb{F}_q[X]$. Finally, we study relations between digit expansions of formal Laurent series and a finite fields version of Mahler's $3/2$-problem.

Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $\pi_X:X\to X/G$. We characterize rational tensor fields $\tau$ on $X/G$ such that the {\it pull back} of $\tau $ is regular on $X$: these are precisely all $\tau$ such that $\operatorname{div}_{R_{X/G}}(\tau)\ge 0$ where $R_{X/G}$ is the {\it reflection divisor} of $X/G$ and $\operatorname{div}_{R_{X/G}}(\tau)$ is the {\it $R_{X/G}$-divisor} of $\tau$. We give some applications, in particular to the generalization of Solomon's theorem. In the last section we show that if $V$ is a finite dimensional vector space and $G$ a finite subgroup of $\operatorname{GL}(V)$, then each automorphism $\psi$ of $V/G$ admits a biregular lift $\phi: V\to V$ provided that $\psi$ maps the regular stratum to itself and $\psi_*(R_{X/G})=R_{X/G}$.

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non hyperelliptic locus M_3^{nh} of the moduli space M_3 of curves of genus 3. By adding to these computations the knowed results on the hyperelliptic locus we obtain a complete picture of these strata for M_3.

We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| > K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain a similar result, albeit with the slightly weaker condition |V| >> K^{-O(K)}|A|.

We prove that a sufficiently large subset of the $d$-dimensional vector space over a finite field with $q$ elements, $ {\Bbb F}_q^d$, contains a copy of every $k$-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.

Reported here are the results of a systematic study of a model of the direct-current electromagnetic pump. Of particular interest is the motion imparted to the electrically conducting fluid in the rectangular duct by the body forces that result from applied electric and magnetic fields. The purpose of the investigation is to associate the observed fluid motion with the characteristics of the electric and magnetic fields which cause them. The experiments were carried out with electromagnetic fields that moved a stream of copper sulphate solution through a clear plastic channel. Ink filaments injected into the stream ahead of the region where the fields were applied identify the motion of the fluid elements as they passed through the test channel. Several magnetic field configurations were employed with a two-dimensional electric current distribution in order to study and identify the magnitude of some of the effects on the fluid motion brought about by nonuniformities in the electromagnetic fields. A theoretical analysis was used to guide and evaluate the identification of the several fluid motions observed. The agreement of the experimental data with the theoretical predictions is satisfactory. It is found that sizable variations in the velocity profile and pressure head of the output stream are produced by the shape of the electric and magnetic fields.

The various formulations of Maxwell's equations are reviewed with emphasis on those formulations which most readily form analogies with Navier's equations. Analogies involving scalar and vector potentials and electric and magnetic field components are presented. Formulations allowing for media with dielectric and conducting properties are emphasized. It is demonstrated that many problems in electromagnetism can be solved using the NASTRAN finite element code. Several fundamental problems involving time harmonic solutions of Maxwell's equations with known analytic solutions are solved using NASTRAN to demonstrate convergence and mesh requirements. Mesh requirements are studied as a function of frequency, conductivity, and dielectric properties. Applications in both low frequency and high frequency are highlighted. The low frequency problems demonstrate the ability to solve problems involving media inhomogeneity and unbounded domains. The high frequency applications demonstrate the ability to handle problems with large boundary to wavelength ratios.

Interim report for period 1 October 1983-September 1984--Cover

We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3 isogeny classes. We also show how things become more intricate from H^4 onwards, for more that 3 isogeny classes.

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.

In this paper, we first find the distribution of nth power residues modulo a prime $p$ by analyzing sums involving Dirichlet characters. We then extend this method to characterize the distribution of powers in arbitrary finite fields.

We consider the problem of existence and enumeration of primitive TSRs of order n over any finite field. Here we prove the existence of primitive TSRs of order two over finite fields of characteristic two and establish an equivalence between primitive TSRs and primitive polynomials of special form. A conjecture regarding the existence of these special type of primitive polynomials is submitted by us along with some experimental verification. Further we have attempted to enumerate primitive TSRs of order two over finite fields of characteristic two. Finally we give a general search algorithm for primitive TSRs of odd order over any finite field and in particular of order two over fields of characteristic two.

We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed.

In this article, we present an emerging field of quantum chemistry at finite temperature. We discuss its recent developments on both theoretical and experimental fronts.We describe and analyze several experimental investigations related to the temperature effects on the structure, electronic spectra,or bond rupture forces for molecules. This includes the study of the temperature impact on the pathway shifts for the protein unfolding by atomic force microscopy, the temperature dependence of the absorption spectra of electrons in solvents, and temperature influence over the intermolecular forces measured by the AFM. On the theoretical side, we review a recent advancement made by the author in the coming fields of quantum chemistry at finite temperature. Starting from Bloch equation, we have derived the sets of hierarchy equations for the reduced density operators in both canonical and grand canonical ensembles. They provide a law according to which the reduced density operators vary in temperature for the identical and interacting many-body particles. By taking the independent particle approximation, we have solved the equation in the case of a grand canonical ensemble, and obtained an eigenequation for the molecular orbitals at finite temperature. The explicit expression for the temperature-dependent Fock operator is also given. They will form a foundation for the study of the molecular electronic structures and their interplay with the finite temperature. Furthermore, we clarify the physics concerning the temperature effect on the electronic structure or processes of molecules which is crucial for both theoretical understanding and computational study.Finally,we summarize our discussion and point out the theoretical and computational issues for the future explorations in the fields of quantum chemistry at finite temperature.

Integral expressions are determined for the elastic displacement and stress fields due to stationary or moving dislocation loops in three dimensional, not necessarily isotropic, finite samples. A line integral representation is found for the stress field, thus satisfying the expectation that stresses should depend on the location of the dislocation loop, but not on the location of surfaces bounded by such loops that are devoid of physical significance. In the stationary case the line integral representation involves a ``vector potential'' that depends on the specific geometry of the sample, through its Green's function: a specific combination of derivatives of the elastic stress produced by the Green's function appropriate for the sample is divergenceless, so it is the curl of this ``vector potential''. This ``vector potential'' is explicitely determined for an isotropic half space and for a thin plate. Earlier specific results in these geometries are recovered as special cases. In the non stationary case a line integral representation can be obtained for the time derivative of the stress field. This, combined with the static result, assures a line integral representation for the time dependent stress field.

The article contents suggestions on how to perform the Fast Fourier Transform over Large Finite Fields. The technique is to use the fact that the multiplicative groups of specific prime fields are surprisingly composite.

Polarization phenomenon over any finite field $\mathbb{F}_{q}$ with size $q$ being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arikan for the binary field, as well as its extension to a prime field by Sasoglu, Telatar, and Arikan. In this paper, a necessary and sufficient condition of a matrix over a finite field $\mathbb{F}_q$ is shown under which any source and channel are polarized. Furthermore, the result of the speed of polarization for the binary alphabet obtained by Arikan and Telatar is generalized to arbitrary finite field. It is also shown that the asymptotic error probability of polar codes is improved by using the Reed-Solomon matrix, which can be regarded as a natural generalization of the $2\times 2$ binary matrix used in the original proposal by Arikan

A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of Chung.

A new algebraic Cayley graph is constructed using finite fields. Its connectedness and diameter bound are studied via Weil's estimate for character sums. These graphs provide a new source of expander graphs, extending classical results of Chung.

We present a mathematical model: dynamical systems over finite sets (DSF), and we show that Boolean and discrete genetic models are special cases of DFS. In this paper, we prove that a function defined over finite sets with different number of elements can be represented as a polynomial function over a finite field. Given the data of a function defined over different finite sets, we describe an algorithm to obtain all the polynomial functions associated to this data. As a consequence, all the functions defined in a regulatory network can be represented as a polynomial function in one variable or in several variables over a finite field. We apply these results to study the reverse engineering problem.