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In this paper the fully nonlinear theory of finite deformations of an elastic solid is used to study the elastostatic field near the tip of a crack. The special elastic materials considered are such that the differential equations governing the equilibrium fields may lose ellipticity in the presence of sufficiently severe strains. The first problem considered involves finite anti-plane shear (Mode III) deformations of a cracked incompressible solid. The analysis is based on a direct asymptotic method, in contrast to earlier approaches which have depended on hodograph procedures. The second problem treated is that of plane strain of a compressible solid containing a crack under tensile (Mode I) loading conditions. The material is characterized by the so-called Blatz-Ko elastic potential. Again, the analysis involves only direct local considerations. For both the Mode III and Mode I problems, the loss of equilibrium ellipticity results in the appearance of curves ('elastostatic shocks') issuing from the crack-tip across which displacement gradients and stresses are discontinuous. (Author)

The method of coupled azimuthal potentials (CAP) was applied to a waveguide model of an axially symmetric engine inlet to analyze the fields in the region where the front face of the engine terminates the waveguide. Appropriate boundary conditions were derived and the finite element method was used to solve for the potentials. The Lagrangian of the CAP equations does not provide for the enforcement of Neumann boundary conditions. This prevents exact implementation of the correct boundary conditions for the azimuthal magnetic field. Dirichlet boundary conditions for the azimuthal electric and magnetic fields were enforced for standing wave condition in the inlet model with a conducting flat plate termination. The computed values for the interior field components were compared by evaluating the standard deviation. Three trials were performed with varying finite element mesh densities. It was found that as the mesh density increased, the standard deviation for the computed field components decreased. (Author)

This paper reconsiders the problem of determining the elastostatic field near the tip of a crack in an all-around infinite body deformed by a 'Mode III' loading at infinity to a state of anti-plane shear. The problem is treated for a class of incompressible, homogeneous, isotropic elastic materials whose constitutive laws permit a loss of ellipticity in the governing displacement equation of equilibrium at sufficiently severe shearing strains. The analysis represents a generalization of that reported in an earlier study and, as before, is carried out for the 'small-scale nonlinear crack problem', in which a crack of finite length is replaced by a semi-infinite one, and the nonlinear field far from the crack-tip is matched to the near field predicted by the linearized theory. The methods employed in the present paper are necessarily largely qualitative, since they apply to all materials in the class considered. The principal feature of the resulting elastic field is the presence of two symmetrically located curves issuing from the crack-tip and bearing discontinuities in displacement gradient and stress. (Author)

By providing highly realistic simulations of sound propagation through complex atmospheric and terrain environments, finite-difference time-domain (FDTD) techniques can potentially reduce development time and improve the battlefield performance of acoustic sensors. In this paper, we summarize recent progress in improving two key aspects of acoustic FDTD calculations for the atmosphere: (1) development of a rigorous implementation of sound propagation in a moving, inhomogeneous fluid, and (2) formulation and numerical implementation of time domain methods for handling sound interactions with partially reflecting ground surfaces. The new techniques are illustrated with highly detailed calculations of sound propagation through simulated, dynamic atmospheric turbulence fields and over a porous ground surface with viscous and thermal relaxation mechanisms.

Finite fields (also called Galois Fields) have been studied since their introduction by Evariste Galois in 1832 and the publication of his work in 1846. In the last few decades, finite fields have become important to information theory, coding theory, and cryptography. This report presents a simple method for representing a finite field in terms of powers of a single matrix over the integers modulo the characteristic of the field. The addition and multiplication in the field are immediately obtained as the results of ordinary matrix addition and multiplication. This representation called the canonical cyclic representation, makes it easy to understand the field structure and to carry out computations in the field.

Let P 2 denote the projective plane over a finite field F q . A pair of nonsingular conics ( A , B ) is said to satisfy the Poncelet triangle condition if, considered as conics in P 2 ( F q ) , they intersect transverally and there exists a triangle inscribed in A and circumscribed around B . It is shown in this article that a randomly chosen pair of conics satisfies the triangle condition with asymptotic probability 1/ q . We also make a conjecture based upon computer experimentation which predicts this probability for tetragons, pentagons and so on up to enneagons.

Let P 2 denote the projective plane over a finite field F q . A pair of nonsingular conics ( A , B ) is said to satisfy the Poncelet triangle condition if, considered as conics in P 2 ( F q ) , they intersect transverally and there exists a triangle inscribed in A and circumscribed around B . It is shown in this article that a randomly chosen pair of conics satisfies the triangle condition with asymptotic probability 1/ q . We also make a conjecture based upon computer experimentation which predicts this probability for tetragons, pentagons and so on up to enneagons.

Duursma, I. M., & Park, S. (2010). Coset bounds for algebraic geometric codes.  Finite Fields and Their Applications ,  16 (1), 36-55.

Let $C$ be a smooth projective curve of genus $g \ge 1$ over a finite field $\F$ of cardinality $q$. In this paper, we first study $\#\J_C$, the size of the Jacobian of $C$ over $\F$ in case that $\F(C)/\F(X)$ is a geometric Galois extension. This improves results of Shparlinski \cite{shp}. Then we study fluctuations of the quantity $\log \#\J_C-g \log q$ as the curve $C$ varies over a large family of hyperelliptic curves of genus $g$. For fixed genus and growing $q$, Katz and Sarnak showed that $\sqrt{q}\left(\log \# \J_C-g \log q\right)$ is distributed as the trace of a random $2g \times 2g$ unitary symplectic matrix. When the finite field is fixed and the genus grows, we find the limiting distribution of $\log \#\J_C-g \log q$ in terms of the characteristic function. When both the genus and the finite field grow, we find that $\sqrt{q}\left(\log \# \J_C-g \log q\right)$ has a standard Gaussian distribution.

I give a formula for the zeta function of a projective toric hypersurface over a finite field and estimate its Newton polygon. As an application this formula allows us to compute the exact number of rational points on the families of Calabi-Yau manifolds in Mirror Symmetry.

We present an elementary proof that the Schur polynomial corresponding to an increasing sequence of exponents (c_0,..., c_{n-1}) with c_0 = 0 is irreducible over every field of characteristic p whenever the numbers d_i = c_{i+1} - c_i are all greater than 1, not divisible by p, and satisfy gcd(d_i, d_{i+1}) = 1 for every i.

We present a deterministic 2^O(t)q^{(t-2)(t-1)+o(1)} algorithm to decide whether a univariate polynomial f, with exactly t monomial terms and degree

Cheng and Wan have related the decoding of Reed-Solomon codes to the computation of discrete logarithms over finite fields, with the aim of proving the hardness of their decoding. In this work, we experiment with solving the discrete logarithm over GF(q^h) using Reed-Solomon decoding. For fixed h and q going to infinity, we introduce an algorithm (RSDL) needing O (h! q^2) operations over GF(q), operating on a q x q matrix with (h+2) q non-zero coefficients. We give faster variants including an incremental version and another one that uses auxiliary finite fields that need not be subfields of GF(q^h); this variant is very practical for moderate values of q and h. We include some numerical results of our first implementations.

We show that, as in de Rham cohomology over the complex numbers, the value of the entropy of an automorphism of the surface over a finite field $\F_q $ is taken on the span of the N\'eron-Severi group inside of $\ell$-adic cohomology. v2: (some) typos removed, exposition (partly) improved.

We consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method. The solution's two-point correlation can efficiently be approximated by means of an $\mathcal{H}$-matrix, in particular if the correlation length is rather short or the correlation kernel is non-smooth. Since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the $\mathcal{H}$-matrix format, we can solve the correspondent $\mathcal{H}$-matrix equation in essentially linear time by using the $\mathcal{H}$-matrix arithmetic. Numerical experiments for three-dimensional finite element discretizations for several correlation lengths and different smoothness are provided. They validate the presented method and demonstrate that the computation times do not increase for non-smooth or shortly correlated data.

We use the uniform probability distribution as well as the natural density to calculate the probability that finitely many polynomials are pairwise coprime. It will turn out that the formulas for the two considered probability measures asymptotically coincide but differ in the exact values. Moreover, we compute the natural density of mutually left coprime polynomial matrices and compare the result with the formula one gets using the uniform probability distribution. The achieved estimations are not as precise as in the scalar case but again we can show asymptotic coincidence.

We use the uniform probability distribution as well as the natural density to calculate the probability that finitely many polynomials are pairwise coprime. It will turn out that the formulas for the two considered probability measures asymptotically coincide but differ in the exact values. Moreover, we compute the natural density of mutually left coprime polynomial matrices and compare the result with the formula one gets using the uniform probability distribution. The achieved estimations are not as precise as in the scalar case but again we can show asymptotic coincidence.

In this paper we determine the number of isomorphism classes of superspecial abelian varieties $A$ over the prime field $\Fp$ such that the relative Frobenius morphism $\pi_A$ satisfying $\pi_A^2=-p$.

In 2005 Coates, Fukaya, Kato, Sujatha, and Venjakob formulated a noncommutative Iwasawa main conjecture for l-adic Lie extensions of number fields. To provide evidence for this main conjecture we formulate and prove an analogous statement for l-adic Lie extensions of separated schemes of finite type over a finite field of characteristic prime to l.

Let p>2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fractional field of the Witt ring of k. Let G and H be finite flat commutative group schemes killed by p over O_K and k[[u]], respectively. In this paper, we show the upper and the lower ramification subgroups of G and H in the sense of Abbes-Saito are naturally isomorphic to each other when they are associated to the same Kisin module.

In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in \cite{HI07} and \cite{HIKR07}. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions. In particular, our machinery enables us to prove that if $E \subset {\Bbb F}_q^d$, $d \ge 4$, the $d$-dimensional vector space over a finite field ${\Bbb F}_q$, of size much greater than $q^{\frac{d}{2}}$, and if $E$ is a product set, then the set of volumes of $d$-dimensional parallelepipeds determined by $E$ covers ${\Bbb F}_q$. This result is sharp as can be seen by taking $E$ to equal to $A \times A \times ... \times A$, where $A$ is a sub-field of ${\Bbb F}_q$ of size $\sqrt{q}$. In three dimensions we establish the same result if $|E| \gtrsim q^{{15/8}}$. We prove in three dimensions that the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^{{3/2}}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows.

In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced by a modified hyperscaling relation. As a result, standard formulations of finite size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free energy cost \Delta F of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, \Delta F proportional to $L^\theta$, with $\theta$ the violation of hyperscaling critical exponent, and L the linear extension of the system. This modified behavior facilitates a number of new numerical approaches that can be used to locate critical points in random field systems from finite size simulation data. We test and confirm the new approaches on two random field systems in three dimensions, namely the random field Ising model, and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.

The first construction of strongly secure quantum ramp secret sharing by Zhang and Matsumoto had an undesirable feature that the dimension of quantum shares must be larger than the number of shares. By using algebraic curves over finite fields, we propose a new construction in which the number of shares can become arbitrarily large for fixed dimension of shares.

We study analogues for the Tate-Shafarevich group for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields.

The finite-element method is a preferred numerical method when electromagnetic fields at high accuracy are to be computed in nano-optics design. Here, we demonstrate a finite-element method using hp-adaptivity on tetrahedral meshes for computation of electromagnetic fields in a device with rough textures. The method allows for efficient computations on meshes with strong variations in element sizes. This enables to use precise geometry resolution of the rough textures. Convergence to highly accurate results is observed.

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations. Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the polynomial equations is constant. As a consequence we design polynomial time quantum algorithms for two algebraic hidden structure problems: for the hidden subgroup problem in certain semidirect product p-groups of constant nilpotency class, and for the multi-dimensional univariate hidden polynomial graph problem when the degree of the polynomials is constant.

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory, some are conditional under certain widely believed conjectures, and others are purely heuristic in nature.

If K/F is a finite abelian Galois extension of global fields whose Galois group has exponent t, we prove that there exists a short exact sequence that has as a consequence that if t is square free, then Dec(K/F)=Br_{t}(K/F) which we use to show that prime exponent division algebras over Henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras. Finally, we construct a counterexample to the result for higher exponent division algebras.

In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant -19. We prove that any such curve can be given by an explicit equation of certain form. Using these equations we obtain a table of maximal and minimal curves over finite fields with discriminant -19 of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.

Let S be a smooth cubic surface defined over a field K. As observed by Segre and Manin, there is a secant and tangent process on S that generates new K-rational points from old. It is natural to ask for the size of a minimal generating set for S(K). In a recent paper, for fields K with at least 13 elements, Siksek showed that if S contains a skew pair of K-lines then S(K) can be generated from one point. In this paper we prove the corresponding version of this result for fields K having at least 4 elements, and slightly milder results for #K=2 or 3.

We determine upper bounds on the number of rational points of an affine or projective algebraic set defined over an extension of a finite field by a system of polynomial equations, including the case where the algebraic set is not defined over the finite field by itself. A special attention is given to irreducible but not absolutely irreducible algebraic sets, which satisfy better bounds. We study the case of complete intersections, for which we give a decomposition, coarser than the decomposition in irreducible components, but more directly related to the polynomials defining the algebraic set. We describe families of algebraic sets having the maximum number of rational points in the affine case, and a large number of points in the projective case. Nous d\'eterminons des majorations du nombre de points d'un ensemble alg\'ebrique affine ou projectif, d\'efini sur une extension d'un corps fini par un syst\`eme d'\'equations polynomiales, y compris dans le cas o\`u l'ensemble alg\'ebrique n'est pas d\'efini sur le corps fini lui-m\^eme. Une attention particuli\`ere est port\'ee aux ensemble alg\'ebriques irr\'eductibles mais non absolument irr\'eductibles, pour lesquels nous obtenons de meilleures bornes. Nous \'etudions le cas des intersections compl\`etes, pour lesquelles nous construisons une d\'ecomposition moins fine que la d\'ecomposition en composantes irr\'eductibles, mais plus directement li\'ee aux polyn\^omes qui d\'efinissent l'ensemble alg\'ebrique. Enfin, nous construisons des familles d'ensembles alg\'ebriques atteignant le nombre maximum de points rationnels dans le cas affine, et comportant de nombreux points dans le cas projectifs.

We study special values of zeta functions of singular varieties over finite fields. We give a new formula of special values by constructing a morphism of homology theories, which we call regulator, from higher Chow group to weight homology. Our regulator is defined by using the notion of weight complex for varieties over a perfect field, which was introduced by Gillet and Soule. The main idea of the proof of our formula of special values is to use weight spectral sequence of homology theories, whose E1 terms are homology groups for smooth projective schemes. Also, to calculate special values, we prove that the weight complex for any variety over a perfect field is bounded. This boundedness result was known by Gillet and Soule in the case that the base field admits resolution of singularities, but not in general.

In this paper we improve our previous results on classification of groups of points on abelian varieties over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $k$-isogeny class.

Permutation polynomials have been a subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are described in the literature so far. In this paper we present a number of permutation trinomials over finite fields, which are of different forms.

Simulation of extracellular fields is one of the substantial methods used in the area of computational neuroscience. Its most common usage is validation of experimental methods as EEG and extracellular spike recordings or modeling of physiological phenomena which can not be easily determined empirically. Continuous experimental work has been re-raising the importance of polarization effects between neuronal structures to neuronal communication. As this effects relies on very small potential changes, better modeling methods are necessary to quantify the weak electrical fields in the microscopic scale in a more realistic way. An important factor of influence on local field effects in the hippocampal formation is the heterogeneous resistivity of extracellular tissue. The vast majority of modeling studies consider the extracellular space to be homogeneous while experimentally, it has been shown that the stratum pyramidale has two times higher resistivity then other hippocampal layers. Common simulation methods for extracellular electrical fields based on the point source approximation are bound to describe the resistance of the space with a single, linear factor. We propose that models should be based on the space- and time-dependent Maxwell equations in order to account for heterogeneous properties of the extracellular space and specific arrangements of neurons in dense hippocampal layers. To demonstrate the influence of heterogeneous extracellular resistivity and neuronal spatial orientation on modeling results, we combine solutions of classical compartment models with spatiotemporal PDEs solved by the FEM. With the help of these methods, we show that the inclusion of heterogeneous resistivity has a substantial impact on voltages in close proximity to emitting neurons, increasing the extracellular potentials substantially compared to the homogeneous variant.

We use elementary methods to prove an incidence theorem for points and spheres in $\mathbb{F}_q^n$. As an application, we show that any point set of $P\subset \mathbb{F}_q^2$ with $|P|\geq 5q$ determines a positive proportion of all circles. The latter result is an analogue of Beck's Theorem for circles which is optimal up to multiplicative constants.

We count invertible Schr\"odinger operators (perturbations by diagonal matrices of the adjacency matrix) over finite fieldsfor trees, cycles and complete graphs.This is achieved for trees through the definition and use of local invariants (algebraic constructions of perhapsindependent interest).Cycles and complete graphs are treated by ad hoc methods.

Massless and massive scalar fields and massless spinor fields are considered at arbitrary temperatures in four dimensional ultrastatic curved spacetime. Scalar models under consideration can be either conformal or nonconformal and include selfinteraction. The one-loop nonlocal effective action at finite temperature and free energy for these quantum fields are found up to the second order in background field strengths using the covariant perturbation theory. The resulting expressions are free of infrared divergences. Spectral representations for nonlocal terms of high temperature expansions are obtained.

First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful. Connections are made with symmetric function theory, Markov chains, potential theory, Rogers-Ramanujan type identities, quivers, and various measures on partitions.

We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.

Integral expressions are determined for the elastic displacement and stress fields due to stationary or moving dislocation loops in finite samples. These general expressions are valid for anisotropic media as well. Specifically for the stress fields, a line integral representation is found, thus showing rigorously the independence of the stress fields with respect to the choice of slip planes. In the stationary case the line integral representation involves calculating a "vector potential" dependent on the specific geometry of the sample. Two examples of geometries, isotropic half space and thin plate, are shown where the "vector potential" has been explicitly determined. With this general method one recovers some earlier specific results in these geometries.

We study thermodynamic behaviors of the antiferromagnetic zigzag spin chain in magnetic fields, using the density-matrix renormalization group method for the quantum transfer matrix. We focus on the thermodynamics of the system near the critical fields in the ground-state magnetization process($M$-$H$ curve): the saturation field, the lower critical field associated with excitation gap, and the field at the middle-field cusp singularity. We calculate magnetization, susceptibility and specific heat of the zigzag chain in magnetic fields at finite temperatures, and then discuss how the calculated quantities reflect the low-lying excitations of the system related with the critical behaviors in the $M$-$H$ curve.

We present a new algorithm for computing $m$-th roots over the finite field $\F_q$, where $q = p^n$, with $p$ a prime, and $m$ any positive integer. In the particular case $m=2$, the cost of the new algorithm is an expected $O(\M(n)\log (p) + \CC(n)\log(n))$ operations in $\F_p$, where $\M(n)$ and $\CC(n)$ are bounds for the cost of polynomial multiplication and modular polynomial composition. Known results give $\M(n) = O(n\log (n) \log\log (n))$ and $\CC(n) = O(n^{1.67})$, so our algorithm is subquadratic in $n$.

We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field F_q and a finite extension E/F_0 such that the sequence \mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with the following: What can we say about the invariants of \mathcal{E}; i.e., the asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over F_q with finitely many prescribed invariants being positive, and towers of function fields over F_q, for q a square, with at least one positive invariant and certain prescribed invariants being zero. We show the existence of recursive towers attaining the Drinfeld-Vladut bound of order r, for any r\geq 1 with q^r a square. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.

We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field F_q and a finite extension E/F_0 such that the sequence \mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with the following: What can we say about the invariants of \mathcal{E}; i.e., the asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over F_q with finitely many prescribed invariants being positive, and towers of function fields over F_q, for q a square, with at least one positive invariant and certain prescribed invariants being zero. We show the existence of recursive towers attaining the Drinfeld-Vladut bound of order r, for any r\geq 1 with q^r a square. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.

We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field F_q and a finite extension E/F_0 such that the sequence \mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with the following: What can we say about the invariants of \mathcal{E}; i.e., the asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over F_q with finitely many prescribed invariants being positive, and towers of function fields over F_q, for q a square, with at least one positive invariant and certain prescribed invariants being zero. We show the existence of recursive towers attaining the Drinfeld-Vladut bound of order r, for any r\geq 1 with q^r a square. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.

We consider a tower of function fields F=(F_n)_{n\geq 0} over a finite field F_q and a finite extension E/F_0 such that the sequence \mathcal{E):=(EF_n)_{n\goq 0} is a tower over the field F_q. Then we deal with the following: What can we say about the invariants of \mathcal{E}; i.e., the asymptotic number of places of degree r for any r\geq 1 in \mathcal{E}, if those of F are known? We give a method based on explicit extensions for constructing towers of function fields over F_q with finitely many prescribed invariants being positive, and towers of function fields over F_q, for q a square, with at least one positive invariant and certain prescribed invariants being zero. We show the existence of recursive towers attaining the Drinfeld-Vladut bound of order r, for any r\geq 1 with q^r a square. Moreover, we give some examples of recursive towers with all but one invariants equal to zero.

We study the number of the vectors determined by two sets in d-dimensional vector spaces over finite fields. We observe that the lower bound of cardinality for the set of vectors can be given in view of an additive energy or the decay of the Fourier transform on given sets. As an application of our observation, we find sufficient conditions on sets where the Falconer distance conjecture for finite fields holds in two dimension. Moreover, we give an alternative proof of the theorem, due to Iosevich and Rudnev, that any Salem set satisfies the Falconer distance conjecture for finite fields.

In recent work, Benjamin Schumacher and Michael~D. Westmoreland investigate a version of quantum mechanics which they call "modal quantum theory" but which we prefer to call "discrete quantum theory". This theory is obtained by instantiating the mathematical framework of Hilbert spaces with a finite field instead of the field of complex numbers. This instantiation collapses much the structure of actual quantum mechanics but retains several of its distinguishing characteristics including the notions of superposition, interference, and entanglement. Furthermore, discrete quantum theory excludes local hidden variable models, has a no-cloning theorem, and can express natural counterparts of quantum information protocols such as superdense coding and teleportation. Our first result is to distill a model of discrete quantum computing from this quantum theory. The model is expressed using a monadic metalanguage built on top of a universal reversible language for finite computations, and hence is directly implementable in a language like Haskell. In addition to superpositions and invertible linear maps, the model includes conventional programming constructs including pairs, sums, higher-order functions, and recursion. Our second result is to relate this programming model to relational programming, e.g., a pure version of Prolog over finite relations. Surprisingly discrete quantum computing is identical to conventional logic programming except for a small twist that is responsible for all the ``quantum-ness.'' The twist occurs when merging sets of answers computed by several alternatives: the answers are combined using an "exclusive" version of logical disjunction. In other words, the two branches of a choice junction exhibit an "interference" effect: an answer is produced from the junction if it occurs in one or the other branch but not both.

Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.