Simple Proofs Of Known Results Research Articles

Unified Approach to Probability Problems and Estimation Algorithms Associated With Symmetric Functions

In this article we introduce a simple powerful methodology where we replace the independent variables λ1,...,λnin various symmetric functions as well as in Vieta’s formulas by the indication functions of the events Ai,i = 1,...,n, i.e., λi = 1(Ai),i = 1,...,n. Both the random variable K that counts the number of events that actually occurred and the proposed obvious identity Π_n ^i=1(z−1(Ai)) ≡ (z−1)^KZ^(n−K) that solely depends on K play a central role in this article. Just by choosing different values for z (real, complex, and random) and taking expectations of the various functions we provide other simple proofs of known results as well as obtain new results. The estimation algorithms for computing the expected elementary symmetric functions via least squares based on IFFT in the complex domain (z ∈ C) and least squares or linear programming in the real domain (z ∈ R) are noteworthy. Similarly, we we use Newton’s identities and some well known inequalities to obtain new results and inequalities. Then, we give an algorithm that exactly computes the distribution of K (i.e., q_k:= P(K = k), k = 0,1,...,n) for finite sample spaces. Finally, we give the conclusion and area for further research.

Transverse Surgery on Knots in Contact 3-Manifolds

We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. One main aim of this paper is to show that in many contexts, transverse surgery is a more natural tool than surgery on Legendrian knots. We reinterpret contact ( ± 1 ) (\pm 1) -surgery on Legendrian knots as transverse surgery on transverse push-offs, allowing us to give simpler proofs of known results. We give the first result on the tightness of inadmissible transverse surgery (cf. contact ( + 1 ) (+1) -surgery) for contact manifolds with vanishing Heegaard Floer contact invariant. In particular, inadmissible transverse r r -surgery on the connected binding of a genus g g open book that supports a tight contact structure preserves tightness if r > 2 g − 1 r > 2g-1 .

Hobson's formula for Dunkl operators and its applications

ABSTRACTWe generalize classical Hobson's formula concerning partial derivatives of radial functions on a Euclidean space to a formula in the Dunkl analysis. As applications we give new simple proofs of known results involving Maxwell's representation of harmonic polynomials, Bochner–Hecke identity, Pizzetti formula for spherical mean, and Rodrigues formula for generalized Hermite polynomials.

Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves

A new method is proposed for searching for trapped modes in several problems of linear water wave theory. In the case of submerged bodies, the method gives simple proofs of known results, and the condition obtained is completely new in the case of surface piercing bodies. Bibliography: 24 titles.

Some formulas related to Hurwitz–Lerch zeta functions

In this paper, we consider multiplication formulas and their inversion formulas for Hurwitz–Lerch zeta functions. Inversion formulas give simple proofs of known results, and also show generalizations of those results. Next, we give a generalization of digamma and gamma functions in terms of Hurwitz–Lerch zeta functions, and consider its properties. In all the sections, various results are always proved by multiplication and inversion formulas.

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.

A homological solution for the Gauss code problem in arbitrary surfaces

Let P ¯ be a sequence of length 2 n in which each element of { 1 , 2 , … , n } occurs twice. Let P ′ be a closed curve in a closed surface S having n points of simple self-intersections, inducing a 4-regular graph embedded in S which is 2-face colorable. If the sequence of auto-intersections along P ′ is given by P ¯ , we say that P ′ is a 2-face colorable solution for the Gauss code P ¯ on surface S or a lacet for P ¯ on S. In this paper we show (by using surface homology theory mod 2), that the set of lacets for P ¯ on S are in 1–1 correspondence with the tight solutions of a system of quadratic equations over the Galois field GF ( 2 ) . If S is the 2-sphere, the projective plane or the Klein bottle, the corresponding quadratic systems are equivalent to linear ones. In consequence, algorithmic characterizations for the existence of solutions on these surfaces are available. For the two first surfaces this produces simple proofs of known results. The algorithmic characterization for the existence of solutions on the Klein bottle is new. We provide a polynomial algorithm to resolve the issue.

A Polytope Approach to the Optimal Assembly Problem

The problem of assembling components into series modules to maximize the system reliability has been intensively studied in the literature. Invariably, the methods employed exploit special properties of the reliability function through standard analytical optimization techniques. We propose a geometric approach by exploiting the assembly polytope --- a polytope generated by the potential assembly configurations. The new approach yields simpler proofs of known results, as well as new results about systems where the number of components in a module is not fixed, but subject to lower and upper bounds.

Reduced Form Auctions Revisited

This note uses the Theorem of the Alternative to prove new results on the implementability of general, asymmetric auctions, and to provide simpler proofs of known results for symmetric auctions. The tradeoff is that type spaces are taken to be finite.

On best transitive approximations to simple graphs

In this paper, we investigate both combinatorial and complexity aspects of the problem of finding best transitive approximations to simple graphs. These problems are addressed in an interlocked way. We provide new and simple proofs of known results and in addition prove some new theorems.

On the Degree, Nonlinearity, Algebraic Thickness, and Nonnormality of Boolean Functions, With Developments on Symmetric Functions

The two main criteria evaluating, from cryptographic viewpoint, the complexity of Boolean functions are the nonlinearity and the algebraic degree. Two other criteria can also be considered: the algebraic thickness and the nonnormality. Simple proofs are given that, asymptotically, almost all Boolean functions have high algebraic thicknesses and are deeply nonnormal, as well as they have high algebraic degrees and high nonlinearities. We also study in detail the relationship between nonnormality and nonlinearity. We derive simple proofs of known results on symmetric Boolean functions and we prove several new and more general results on a class containing all symmetric functions.

Palindromy and Ambiguous Ideals Revisited

The purpose of this article is to revisit the relationship between ambiguous ideals and palindromy in the simple continued fraction expansions of quadratic irrationals begun by the first author and A. J. van der Poorten (1995,Bull. Austral. Math. Soc.51, 215–233). We present simpler proofs of known results, new interrelationships, and correct some misinterpretations. We do this via the infrastructure of real quadratic fields. The conclusion is that palindromy is ambiguity, when properly viewed.

Contributions to a General Theory of View-Obstruction Problems, II

In view-obstruction problems, congruent copies of a closed, centrally symmetric, convex bodyC, centred at the points of the shifted lattice (12, 12, ..., 12)+Znin Rn, are expanded uniformly. The expansion factor required to touch a given subspaceLis denoted byν(C,L) and for each dimensiond, 1≤d≤n−1, the relevant expansion factors are used to determine a supremumν(C,d)=sup {ν(C,L): dimL=d,Lnot contained in a coordinate hyperplane}.Here a method for obtaining upper bounds onν(C,L) for “rational” subspacesLis given. This leads to many interesting results, e.g. it follows that the supremaν(C,d) are always attained and a general isolation result always holds. The method also applies to give simple proofs of known results for three dimensional spheres. These proofs are generalized to obtainν(B,n−2) and a Markoff type chain of related isolations for spheresBin Rnwithn≥4. In another part of the paper, the subspaces occurring in view-obstruction problems are generalized to arbitrary flats. This generalization is related to Schoenberg's problem of billiard ball motion. Several results analogous to those forν(C,L) andν(C,d) are obtained.

SPARSE Reduces Conjunctively to TALLY

Polynomials over finite fields are used to show that any sparse set can conjunctively reduce to a tally set. This leads to new results and to simple proofs of known results about various classes that lie between P and P/poly.

A representation theorem for higher level reduced Witt rings

In [6] Becker and Rosenberg defined the reduced Witt ring WAK) of a formally real field K, with respect to a higher level preorder Tin K. WAK) is a subring of the ring C(X,, @) of continuous c-valued functions on the space X, of signatures of K which are trivial on T, and is built up from higher level T-forms over K. This higher level reduced theory is a natural generalization of the reduced theory of quadratic forms over fields. Spaces of Signatures were introduced in [18] to provide an abstract framework for the higher level reduced theory over fields, along the lines of Marshall’s Spaces of Orderings [13-H] treatment of the classical reduced theory of quadratic forms. There are many advantages to this axiomatic approach. It allows for a unified study of reduced Witt rings over fields and skew fields (Powers has shown that Becker and Rosenberg’s study of higher level reduced Witt rings can also be carried out over skew fields [22]). The absence of technical considerations such as valuation theory not only yields simpler proofs of known results, but also new results about the theory over fields [ 183. The Space of Orderings version [ 151 of the result which we prove here has found application in real algebraic geometry, and we hope that our abstract higher level result might also prove useful in this context. The question which concerns us here is the so-called representation problem: given FE C(X, @), how can we tell if F=f for some f~ W(X)? This has been solved for the higher level reduced Witt ring of a field by Becker and Rosenberg [6] (see [S] also), and in the case of Spaces of Orderings by Marshall [15], both solutions being generalizations of the original representation theorems of Becker and Brlicker [4, 5.31 and Brown and Marshall [9, 4.21 (in the context of the reduced quadratic form theory over fields). Under a 2-power assumption we simultaneously extend all of these representation theorems to our abstract higher level setting. This entails studying fans and more importantly a generalization called quasifans. We not only obtain a common proof of the representation 105 0021-8693/88 $3.00

Fundamental Solutions and Asymptotic Behaviour for the $p$-Laplacian Equation

We establish the uniqueness of fundamental solutions to the $p$-Laplacian equation $$ \\mathrm {(PLE)} \\; u_t = \\mathrm {div} (|Du|^{p-2}Du), \\; p > 2, $$ defined for $x \\in \\mathbb R^N$, $0 < t < T$. We derive from this result the asymptotic behaviour of nonnegative solutions with finite mass, i.e. such that $u(\\cdotp, t) \\in L^1(\\mathbb R^N)$. Our methods also apply to the porous medium equation $$ \\mathrm {(PME)} \\; u_t = \\Delta (u^m), \\; m > 1, $$ giving new and simpler proofs of known results. We finally introduce yet another method of proving asymptotic results based on the idea of asymptotic radial symmetry. This method can be useful in dealing with more general equations.

Renormalization-group fixed points of generaln-vector models

We make a general study of symmetry and stability of the fixed points of the quartic Hamiltonian of an $n$-component field (or order parameter) for $n\ensuremath{\ne}4$. Simple proofs of known results are given. Among new results, we show that when it exists the stable fixed point is unique; we give some precision on its symmetry and on its attractor basin.

Characterizations and computational complexity of systolic trellis automata

Systolic trellis automata are simple models for VLSI. We characterize the computing power of these models in terms of Turing machines. The characterizations are useful in proving new results as well as giving simpler proofs of known results. We also derive lower and upper bounds on the computational complexity of the models.

Groups in which every Finitely Generated Subgroup is almost a Free Factor

In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

The numerically stable reconstruction of a Jacobi matrix from spectral data

We show how to construct, from certain spectral data, a discrete inner product for which the associated sequence of monic orthogonal polynomials coincides with the sequence of appropriately normalized characteristic polynomials of the left principal submatrices of the Jacobi matrix. The generation of these orthogonal polynomials via their three-term recurrence relation, as popularized by Forsythe, then provides a stable means of computing the entries of the Jacobi matrix. Our construction provides, incidentally, very simple proofs of known results concerning the existence and uniqueness of a Jacobi matrix satisfying given spectral data and its continuous dependence on those data.