Simple Lemmas Research Articles

Skeletal semantics and their interpretations

The development of mechanised language specification based on structured operational semantics, with applications to verified compilers and sound program analysis, requires huge effort. General theory and frameworks have been proposed to help with this effort. However, none of this work provides a systematic way of developing concrete and abstract semantics, connected together by a general consistency result. We introduce a skeletal semantics of a language, where each skeleton describes the complete semantic behaviour of a language construct. We define a general notion of interpretation , which provides a systematic and language-independent way of deriving semantic judgements from the skeletal semantics. We explore four generic interpretations: a simple well-formedness interpretation; a concrete interpretation; an abstract interpretation; and a constraint generator for flow-sensitive analysis. We prove general consistency results between interpretations, depending only on simple language-dependent lemmas. We illustrate our ideas using a simple While language.

A surjection problem leading to the Ax-Grothendieck theorem

ABSTRACTWe give a complete functional calculus for matrices of order two. The key tools are two simple Lemmas (Lemma 2.1 and Lemma 2.2, below), which are elementary but are somewhat new. As an application, we find the set of all polynomials P for which the matrix map on is surjective. The -version of the famous Ax-Grothendieck theorem is obtained as an immediate Corollary.

Optimal Slater-determinant approximation of fermionic wave functions

We study the optimal Slater-determinant approximation of an $N$-fermion wave function analytically. That is, we seek the Slater-determinant (constructed out of $N$ orthonormal single-particle orbitals) wave function having largest overlap with a given $N$-fermion wave function. Some simple lemmas have been established and their usefulness is demonstrated on some structured states, such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case of three fermions in six orbitals, which the celebrated Borland-Dennis discovery is about, the optimal Slater approximation wave function is proven to be built out of the natural orbitals in an interesting way. We also show that the Hadamard inequality is useful for finding the optimal Slater approximation of some special target wave functions.

Universality of random graphs and rainbow embedding

AbstractIn this paper we show how to use simple partitioning lemmas in order to embed spanning graphs in a typical member of . Let the maximum density of a graph H be the maximum average degree of all the subgraphs of H. First, we show that for , a graph w.h.p. contains copies of all spanning graphs H with maximum degree at most Δ and maximum density at most d. For , this improves a result of Dellamonica, Kohayakawa, Rödl and Rucińcki. Next, we show that if we additionally restrict the spanning graphs to have girth at least 7 then the random graph contains w.h.p. all such graphs for . In particular, if , the random graph therefore contains w.h.p. every spanning tree with maximum degree bounded by Δ. This improves a result of Johannsen, Krivelevich and Samotij.Finally, in the same spirit, we show that for any spanning graph H with constant maximum degree, and for suitable p, if we randomly color the edges of a graph with colors, then w.h.p. there exists a rainbow copy of H in G (that is, a copy of H with all edges colored with distinct colors). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 546–564, 2016

Hidden Variable Theory Supports Variability in Decay Rates of Nuclides

PROBLEM- The orthodox expectation is for decay rates to be strictly constant for all types of decay (β+, β-, EC, α). However empirical results show strong evidence for nuclides having variable decay rates, typically evident as periodicity. The volume of data available suggests this is a real phenomenon, not merely a spurious outcome of measurement errors. However the problem is complex because the data are conflicted for different decays. Consequently it is a significant challenge to explain how the variability might arise, what factors could be involved, and how the underlying mechanisms of causality might operate. PURPOSE- This paper develops a theoretical explanation of the variability of nuclide decay rates. APPROACH- The non-local hidden-variable solution provided by the Cordus theory was used, specifically its mechanics for neutrino-species interactions with nucleons. FINDINGS- It is predicted that the β-, β+ and electron capture processes are induced by pre-supply of neutrino-species, and that the effects are asymmetrical for those species. Also predicted is that different input energies are required, i.e. that a threshold effect exists. Four simple non-contentious lemmas are proposed with which it is straightforward to explain why β- and EC would be enhanced and correlate to solar neutrino flux (proximity & activity), and α emission unaffected. It is shown that the concept of a neutrino-species asymmetry makes sense of the broad patterns evident in the empirical data. IMPLICATIONS- The results support the variability of decay rates, on theoretical grounds. The type of decay (β+, β-, EC, α) is found to be a key variable in this theory, as is the type of neutrino species and its energy. ORIGINALITY- The novel contribution is the provision of a theoretical explanation for why decay rates would be variable. A detailed mechanism is presented for neutrino-species induced decay. Also novel is the prediction that the interaction is asymmetrical, and that the energy requirements are different for the various types of decay.

Computational Analysis of Logic Gates and Circuits Derived from Gene Systems Using Quantified Boolean Formula Methodology in CNF Format

Bio-inspired computing devices are playing a major role in spearheading the revolution of future electronic circuits. In this paper, we focus and describe computational analysis of gene logic circuits based on the quantified Boolean formulae approach using a QBF software tool. We base our current research on an established basic theorem and two simple lemmas for ‘NOT’ and ‘AND’ logic gates derived from bio-molecular systems and demonstrate the effectiveness of circuit analysis using a QBF Solver. Electronic circuits are established, proven and readily compatible, with real-world problems. Hence, we undertook research in this promising domain using biology as a source of inspiration for developing revolutionary computing platforms of the future.

Constructing Highly Incident Configurations

A geometric (q,k)-configuration is a collection of points and straight lines in the Euclidean plane in which each point lies on q lines and each line passes through k points. We say a (q,k)-configuration is highly incident when one (or both) of q or k is strictly greater than 4. In this paper, two simple lemmas are used to construct infinite classes of (2q,2k)-configurations for any q,k≥2; the resulting configurations have non-trivial dihedral symmetry. In particular, this construction produces the only known infinite class of symmetric 6-configurations.

On quantum scattering by δ'(x)} and quasi δ'(x) distributions

The zero-range interaction U(x) occurring in the one-dimensional, time-independent Schrödinger equation is regarded as a smoothed distribution characterized by a tiny length scale b such that the origin becomes an ordinary point. A neighbourhood around the origin is scanned by defining inner demarcation points a±≡ ±b/N and outer demarcation points b±≡ ±Nb with N >> 1. Then a sequence of simple Lemmas permits (i) construction of a systematic procedure for simultaneously solving the scattering wave function ψ(0) at the origin, its derivative ψ'(0) there, the transmission amplitude B, as well as the reflection amplitude D; and (ii) unambiguous application to scattering by the previously known δ'(x) and newly proposed quasi δ'(x) potentials in the Cauchy representation of various distributions.PACS No.: 03.65.Nk

A Transaction Costs Approach to Financial Assets Rates of Return

This paper expands on the extent of the gap between how academic teaching view market transactions and how the market professionals actually work, offering a sensible way of closing as much as possible such a gap. Firstly, transaction costs will be analysed from both the demand and supply sides of securities trading, featuring the essential role intermediaries usually performs. This analysis lead us to a careful survey of the structure of the transaction costs function, bearing in mind that different users meet different cost items. Next, the concept and usage of differential rates will be set forth, distinguishing the stock from the flow differential rates, in the framework of an innovative approach inclusive of information sets. Afterwards, it is showed that transaction costs can be measured by rates which are differential ones by themselves. This addresses to an explicit transaction costs function, within a multiplicative model of differential rates. Besides, three simple lemmas are proved to give foundations to some statements included in the paper. Finally, some numerical examples are provided to illustrate algorithms and concepts.

Shooting for π: the bowstring lemma

Some remarkable new series for π, in which the convergence is exponentially rapid, have been discovered by Bailey et al. [1, 2] using machine subroutines; see also Wagon [3]. An example is equation (17) below. Although the authors also provide algebraic proofs, these seem at first sight to be very special. The purpose of this note is to provide a more general demonstration and to deduce other instances of such series. It will be shown that all the results follow naturally from one or other of two simple lemmas.

Transition equations for isotropic flag manifolds

In analogy with transition equations for type A Schubert polynomials given by Lascoux and Schützenberger (1982), we give recursive formulas for computing representatives of the Schubert classes for the isotropic flag manifolds. These representatives are exactly the Schubert polynomials found in Billey and Haiman (1995). This new approach to finding Schubert polynomials is very closely related to the geometry of the flag manifold and has the advantage that it does not require explicit computations with divided difference operators. The generalized transition equations also lead to a recursion for Stanley symmetric functions and a new proof of Chevalley's intersection formula for Schubert varieties. The proofs involve a careful study of the Bruhat order for the Weyl groups and two simple lemmas for applying divided difference operators.

Adventures in sequent calculus modulo equations

We apply the notion of an oriented rewrite theory and the associated coherence techniques in order to construct a framework for theorem proving modulo equations. This is achieved using existing rewriting techniques and a few simple lemmas, and is intended to serve as a case study of the use of an oriented rewrite theory for building-in equality.

On the consistent histories approach to quantum mechanics

We review the consistent histories formulations of quantum mechanics developed by Griffiths, Omn\`es and Gell-Mann and Hartle, and describe the classification of consistent sets. We illustrate some general features of consistent sets by a few simple lemmas and examples. We consider various interpretations of the formalism, and examine the new problems which arise in reconstructing the past and predicting the future. It is shown that Omn\`es' characterisation of true statements --- statements which can be deduced unconditionally in his interpretation --- is incorrect. We examine critically Gell-Mann and Hartle's interpretation of the formalism, and in particular their discussions of communication, prediction and retrodiction, and conclude that their explanation of the apparent persistence of quasiclassicality relies on assumptions about an as yet unknown theory of experience. Our overall conclusion is that the consistent histories approach illustrates the need to supplement quantum mechanics by some selection principle in order to produce a fundamental theory capable of unconditional predictions.

A computerised system for deriving symbolic unreliability expression

The failure event of a network can be expressed as the simultaneous failure of all the paths from source to terminal. This event can be expanded into disjoint events by using three simple lemmas to obtain the symbolic unreliability expression [2]. This paper first describes an algorithm, illustrated with a manual example, to derive the minimal paths between source and terminal. This is followed by the description of a computerised system which first derives all the minimal paths and then the unreliability expression using Ahmad-Jamil lemmas [2]. The results of the program execution on three different networks along with a discussion on the results are also presented.

Calculations of systems unreliability by algebraic manipulation of failure event

Computations of reliability of a given system is an important field of research in reliability engineering. There are many techniques available in the literature. Some of these are based on Boolean algebra's relations. In these papers unions of events have been reduced to give the reliability of a system. Reduction of union of events involve a lot of work and give rise to a large number of terms. In this paper we have adopted a different approach. We first set up some simple algebraic lemmas on intersections and give their proofs. Next we represent the system failure event as the intersection of paths failure events. By applying our lemmas on this intersection of events step by step we are able to expand the system failure event in the form of unions of mutually exclusive events. Simple substitution of probabilities of success or failure, as the case may be, of the events involved in the union of expanded form gives the unreliability of the system. The algorithm is explained by an example.

Antidirected Subtrees of Directed Graphs

The purpose of this paper is to prove the following result:Theorem. Let T be a directed tree with k arcs and with no directed path of length 2. Then if G is any directed graph with n points and at least 4kn arcs, T is a subgraph of G.It would be appropriate to call T an antidirected tree or a source-sink tree, since every point either has all its arcs directed outward or all inward. As N. G. de Bruijn has noted (personal communication), such a linear bound in n cannot hold if T is replaced by any directed graph other than a union of such trees. The above theorem strengthens one of Graham [1], where an implicit bound of c(k)n is obtained, where c(k) is exponentially large. The proof we give here is also shorter. We first give two simple lemmas. Both are essentially due to ErdÄs, but it is not clear where either first appeared. Their proofs are easy, so we give them here for completeness; in neither case do we state quite the best possible result.

Projective modules over rings with many units

Projective modules over rings with many units

On the Periodicity of Compositions of Entire Functions. II

In (1) the author suggested the following research problem. Does there exist a non-periodic entire function ƒ such that ƒƒ is periodic? My aim in this note is to give a partial answer to this question and, more generally, to give a partial solution to the following problem: if ƒ and g are entire functions and ƒ(g) is periodic, what can one say about g? These results also extend a previous result of mine; for details, see (2, Theorem 4). We begin with some simple lemmas.

Application of the Mayer Method to the Melting Problem

The antiferromagnetic Ising model can also describe a gas with purely repulsive intermolecular forces, and a comparison of the two is made, the transition of such a `gas' to a solid-like structure being the analogue of the appearance of an antiferromagnetic structure. Three simple lemmas enable the asymptotic behaviour of the Mayer cluster sums to be related to other combinatorial problems on the lattice, such as that studied by Hammersley and Broadbent (1957). Some information on these problems is already available, but a detailed discussion is reserved to another paper. The Mayer formalism can be generalized to deal with the antiferromagnetic problem, with the conclusion that the transition is probably first-order and associated with a singularity of the Mayer b-series on the positive real axis of the z variable, but that there is a closer one on the negative real axis. The Mayer β-series can probably be extended beyond the transition and corresponds to a metastable liquid. Its divergence probably represents the absolute limit of the metastable state. The symmetry of the particular model used implies a second transition at a still higher density which may be likened to a change of crystal structure. (The gas of completely rigid molecules is a limiting case, and has only one transition, the transition curve then being a linear relation between P and T.) These considerations agree well with other work on the antiferromagnetic Ising model. A `solid-gas' critical temperature seems most unlikely for any realistic interaction, through it could occur for a sufficiently `soft' one.

Theory of Resonance Topology of Fully Aromatic Hydrocarbons. I

The computation of resonance forms of any specified number of unpaired electrons (``radicality'') of aromatic hydrocarbons is reduced to combinatorial problems, many of which can be solved on the basis of five simple lemmas. In this first of two papers, only unexcited (Kekulé) forms are computed. Algorithms are deduced for deriving the number N of such forms for classes covering practically all nonreticulate aromatic hydrocarbons and a number of singly, doubly, or triply infinite series of reticulate ones. One lemma leads to a simplification of the combinatorial problems from the hexagonal to the square lattice (``dot diagrams''). Points of purely mathematical interest arise and may merit further study.