Distributed Function Computation over Fields and Rings via Linear Compression of Sources

The setting considered in this paper is one of distributed function computation. More specifically, there is a collection of N sources possessing correlated information and a destination that would like to acquire a specific linear combination of the N sources. We address both the case when the common alphabet of the sources is a finite field and the case when it is a finite, commutative principal ideal ring with identity. The goal is to minimize the total amount of information needed to be transmitted by the N sources while enabling reliable recovery at the destination of the linear combination sought. One means of achieving this goal is for each of the sources to compress all the information it possesses and transmit this to the receiver. The Slepian-Wolf theorem of information theory governs the minimum rate at which each source must transmit while enabling all data to be reliably recovered at the receiver. However, recovering all the data at the destination is often wasteful of resources since the destination is only interested in computing a specific linear combination. An alternative explored here is one in which each source is compressed using a common linear mapping and then transmitted to the destination which then proceeds to use linearity to directly recover the needed linear combination. The article is part review and presents in part, new results. The portion of the paper that deals with finite fields is previously known material, while that dealing with rings is mostly new.Attempting to find the best linear map that will enable function computation forces us to consider the linear compression of source. While in the finite field case, it is known that a source can be linearly compressed down to its entropy, it turns out that the same does not hold in the case of rings. An explanation for this curious interplay between algebra and information theory is also provided in this paper.

Minimal Triangulations of Manifolds

Finding vertex-minimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertex-minimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present a brief survey on the works done in last 30 years on the following:(i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers n and d, construction of n-vertex triangulations of different d-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices.In Section 1, we have given all the definitions which are required for the remaining part of this article. A reader can start from Section 2 and come back to Section 1 as and when required. In Section 2, we have presented a very brief history of triangulations of manifolds. In Section 3,we have presented examples of several vertex-minimal triangulations. In Section 4, we have presented some interesting results on triangulations of manifolds. In particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem. In Section 5, we have stated several results on minimal triangulations without proofs. Proofs are available in the references mentioned there. We have also presented some open problems/conjectures in Sections 3 and 5.

Polynomial Approximation, Local Polynomial convexity, and Degenerate CR Singularities- II

We provide some conditions for the graph of a Holder-continuous function on (D) over bar, where (D) over bar is a closed disk in C, to be polynomially convex. Almost all sufficient conditions known to date - provided the function (say F) is smooth - arise from versions of the Weierstrass Approximation Theorem on (D) over bar. These conditions often fail to yield any conclusion if rank(R)DF is not maximal on a sufficiently large subset of (D) over bar. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in C(2) at an isolated complex tangency.

An embedded grid adaptation strategy for unstructured data based finite volume computations

A grid adaptation strategy for unstructured data based codes, employing a combination of hexahedral and prismatic elements, generalizable to tetrahedral and pyramidal elements has been developed.

Numerical analysis of transpiration cooling in carbon dioxide atmosphere at hypersonic Mach numbers

The numerical solutions are obtained for skin friction, heat transfer to the wall and growth of boundary layer along the flat plate by employing two dimensional Navier-Stokes equations governing the hypersonic flow coupled with species continuity equations. Flow fields have been computed along the flat plate in CO2 atmosphere in the presence of transpiration cooling using air and carbon dioxide.

On porous wave maker problems

The problem of generation of surface water waves at tile interface of two immiscible liquids by a onesided porous wave maker is studied in both the cases of water of infinite as well as finite depth by suitable application of the generalisation of Havelock's expansion theorem. The solution of the the problem of reflection of water waves due to a fixed porous wall is derived as a particular case.

Some Aspects of the Kobayashi and Carath,odory Metrics on Pseudoconvex Domains

The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Carath,odory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C (2), and on convex finite type domains in C (n) using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C (2) and convex finite type domains in C (n) in terms of Euclidean parameters. Second, a version of Vitushkin's theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C (1)-isometries of the Kobayashi and Carath,odory metrics on a smoothly bounded strongly pseudoconvex domain.

Vertical Uplift Resistance of a Group of Two Coaxial Anchors in Clay

The vertical uplift resistance for a group of two horizontal coaxial rigid strip anchors embedded in clay under undrained condition has been determined by using the upper bound theorem of limit analysis in combination with finite elements. An increase of undrained shear strength of soil mass with depth has been incorporated. The uplift factor F-c gamma has been computed. As compared to a single isolated anchor, a group of two anchors provides greater magnitude of the uplift resistance. For a given embedment ratio, the group of two anchors generates almost the maximum uplift resistance when the upper anchor is located midway between ground surface and the lower anchor. For a given embedment ratio, F-c gamma increases linearly with an increase in the normalized unit weight of soil mass up to a certain value before attaining a certain maximum magnitude; the maximum value of F-c gamma increases with an increase in embedment ratio. DOI: 10.1061/(ASCE)GT.19435606.0000599. (C) 2012 American Society of Civil Engineers.

Holomorphic Mappings between Domains in C-2

An extension theorem for holomorphic mappings between two domains in C-2 is proved under purely local hypotheses.

Spacelike pion form factor from analytic continuation and the onset of perturbative QCD

The factorization theorem for exclusive processes in perturbative QCD predicts the behavior of the pion electromagnetic form factor F(t) at asymptotic spacelike momenta t(= -Q(2)) < 0. We address the question of the onset energy using a suitable mathematical framework of analytic continuation, which uses as input the phase of the form factor below the first inelastic threshold, known with great precision through the Fermi-Watson theorem from pi pi elastic scattering, and the modulus measured from threshold up to 3 GeV by the BABAR Collaboration. The method leads to almost model-independent upper and lower bounds on the spacelike form factor. Further inclusion of the value of the charge radius and the experimental value at -2.45 GeV2 measured at JLab considerably increases the strength of the bounds in the region Q(2) less than or similar to 10 GeV2, excluding the onset of the asymptotic perturbative QCD regime for Q(2) < 7 GeV2. We also compare the bounds with available experimental data and with several theoretical models proposed for the low and intermediate spacelike region.

The force distribution function of an oscillator model of polymer stretching at constant velocity

Recent simulations of the stretching of tethered biopolymers at a constant speed v (Ponmurugan and Vemparala, 2011 Phys. Rev. E 84 060101(R)) have suggested that for any time t, the distribution of the fluctuating forces f responsible for chain deformation is governed by a relation of the form P(+ f)/ P(- f) = expgamma f], gamma being a coefficient that is solely a function of v and the temperature T. This result, which is reminiscent of the fluctuation theorems applicable to stochastic trajectories involving thermodynamic variables, is derived in this paper from an analytical calculation based on a generalization of Mazonka and Jarzynski's classic model of dragged particle dynamics Mazonka and Jarzynski, 1999 arXiv:cond-\textbackslashmat/9912121v1]. However, the analytical calculations suggest that the result holds only if t >> 1 and the force fluctuations are driven by white rather than colored noise; they further suggest that the coefficient gamma in the purported theorem varies not as v(0.15)T-(0.7), as indicated by the simulations, but as vT(-1).

On the structure of analytic vectors for the Schrodinger representation

This article deals with the structure of analytic and entire vectors for the Schrodinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.

The Dπ form factors from analyticity and unitarity

We study the shape parameters of the Dπ scalar and vector form factors using as input dispersion relations and unitarity for the moments of suitable heavy-light correlators evaluated with Operator Product Expansions, including O(α 2 s) terms in perturbative QCD. For the scalar form factor, a low energy theorem and phase information on the unitarity cut are implemented to further constrain the shape parameters. We finally determine points on the real axis and isolate regions in the complex energy plane where zeros of the form factors are excluded.