Key Theorem Research Articles

On cycle-free lattices with high rate label codes

Etzion et al. have shown that high rate codes based on cycle-free Tanner graphs have minimum distance at most $2$. This result was extended by Sadeghi et al. to a small class of lattices based on Construction $D'$ only. In this paper, we prove a key theorem which relates the minimum distance of every lattice to the minimum distance of its label code. Then, using this powerful tool along with some new bounds on minimum distance of cycle-free group codes, we generalize those results to a large class of lattices here called RPS and PFP lattices. More importantly, we show that this class of cycle-free lattices are not so good in the view of coding gain.

On the Topology of Discrete Strategies

In this paper we explore a topological perspective of planning in the presence of uncertainty, focusing on tasks specified by goal states in discrete spaces. We introduce strategy complexes. A strategy complex is the collection of all plans for attaining all goals in a given space. Plans are like jigsaw pieces. Understanding how the pieces fit together in a strategy complex reveals structure. That structure characterizes the inherent capabilities of an uncertain system. By adjusting the jigsaw pieces in a design loop, one can build systems with desired competencies. The paper draws on representations from combinatorial topology, Markov chains, and polyhedral cones. Triangulating between these three perspectives produces a topological language for describing concisely the capabilities of uncertain systems, analogous to the concepts of reachability and controllability in other disciplines. The major nouns in this language are topological spaces. Three key theorems illustrate the sentences in this language. (a) Goal attain-ability: There exists a strategy for attaining a particular goal from anywhere in a system if and only if the strategy complex of a slightly modified system is homotopic to a sphere. (b) Full controllability : A system can move between any two states despite control uncertainty precisely when its strategy complex is homotopic to a sphere of dimension two less than the number of states. (c) General structure: Any system’s strategy complex is homotopic to the product of a spherical part, modeling full controllability on subspaces, and a general part, modeling adversarial capabilities. This paper contains a number of additional results required as stepping stones, along with many examples. We provide algorithms for computing the key structures described. Finally, we show that some interesting questions are hard. For instance, it is NP-complete to determine the most precisely attainable goal of a system with perfect sensing, but uncertain control.

The Key Theorem and the Bounds on the Rate of Uniform Convergence of Statistical Learning Theory on Quasi-Probability Spaces

Some properties of quasi-probability are further discussed.The definitions and properties of quasi-random variable and its distribution function,expected value and variance are then presented.Markov inequality,Chebyshev's inequality and the Khinchine's law of large numbers on quasi-probability spaces are also proved.Then the key theorem of learning theory on quasi-probability spaces is proved,and the bounds on the rate of uniform convergence of learning process on quasi-probability spaces are constructed.The investigations will help lay essential theoretical foundations for the systematic and comprehensive development of the quasi-statistical learning theory.

Some theoretical results of learning theory based on random sets in set‐valued probability space

PurposeThe purpose of this paper is to introduce some basic knowledge of statistical learning theory (SLT) based on random set samples in set‐valued probability space for the first time and generalize the key theorem and bounds on the rate of uniform convergence of learning theory in Vapnik, to the key theorem and bounds on the rate of uniform convergence for random sets in set‐valued probability space. SLT based on random samples formed in probability space is considered, at present, as one of the fundamental theories about small samples statistical learning. It has become a novel and important field of machine learning, along with other concepts and architectures such as neural networks. However, the theory hardly handles statistical learning problems for samples that involve random set samples.Design/methodology/approachBeing motivated by some applications, in this paper a SLT is developed based on random set samples. First, a certain law of large numbers for random sets is proved. Second, the definitions of the distribution function and the expectation of random sets are introduced, and the concepts of the expected risk functional and the empirical risk functional are discussed. A notion of the strict consistency of the principle of empirical risk minimization is presented.FindingsThe paper formulates and proves the key theorem and presents the bounds on the rate of uniform convergence of learning theory based on random sets in set‐valued probability space, which become cornerstones of the theoretical fundamentals of the SLT for random set samples.Originality/valueThe paper provides a studied analysis of some theoretical results of learning theory.

The improved QV signature scheme based on conic curves over ℤ n

The classical RSA is vulnerable to low private exponent attacks (LPEA) and has homomorphism. KMOV based on elliptic curve E n (a,b) over ℤ n can resist LPEA but still has homomorphism. QV over E n (a,b) not only can resist LPEA but also has no homomorphism. However, QV over E n (a,b) requires the existence of points whose order is M n = lcm{♯E p (a,b), ♯E q (a,b)}. This requirement is impractical for all general elliptic curves. Besides, the computation over En(a,b) is quite complicated. In this paper, we further study conic curve C n (a,b) over ℤ n and its corresponding properties, and advance several key theorems and corollaries for designing digital signature schemes, and point out that C n (a,b) always has some points whose order is M n = lcm{♯E p (a,b), ♯E q (a,b)}. Thereby we present an improved QV signature over C n (a,b), which inherits the property of non-homomorphism and can resist the Wiener attack. Furthermore, under the same security requirements, the improved QV scheme is easier than that over E n (a,b), with respect plaintext embedding, inverse elements computation, points computation and points’ order calculation. Especially, it is applicable to general conic curves, which is of great significance to the application of QV schemes.

Oscillation criteria for two-dimensional systems of first-order linear dynamic equations on time scales

(Communicated by John V. Baxley)Oscillation criteria for two-dimensional difference systems of first-order lineardifference equations are generalized and extended to arbitrary dynamic equa-tions on time scales. This unifies under one theory corresponding results fromdifferential systems, and includes second-order self-adjoint differential, differ-ence, and q-difference equations within its scope. Examples are given illustrat-ing a key theorem.

The theoretical fundamentals of learning theory based on fuzzy complex random samples

Statistical learning theory based on real-valued random samples has been regarded as one of the influential developments for small samples statistical estimation and learning. The key theorem of learning theory and the bounds on the rate of convergence of learning process are the most important theoretical fundamentals of the statistical learning theory. In this paper, we discuss a statistical learning theory based on fuzzy complex random samples. Firstly, the definition of fuzzy complex numbers is introduced and the fuzzy complex random variables along with their numeric characteristic are investigated. Secondly, we carry out further research focused on a special type of fuzzy complex number, namely rectangular fuzzy complex number and establish some properties and develop important theorems. We also prove the strong law of large numbers based on fuzzy complex random variables. Thirdly, the definitions of the fuzzy complex expected risk functional, the fuzzy complex empirical risk functional, the fuzzy complex empirical risk minimization principle and the consistency are provided and discussed. Finally, the key theorem of learning theory and the bounds on the rate of convergence of learning process based on fuzzy complex random samples are discussed.

Formal Languages and Groups as Memory

We present an exposition of the theory of M-automata and G-automata, or finite automata augmented with a multiply-only register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a group-theoretic interpretation and proof of a key theorem of Chomsky and Schützenberger from formal language theory.

Random fuzzy delayed renewal processes

In renewal processes, fuzziness and randomness often coexist intrinsically. Based on the random fuzzy theory, a delayed renewal process with random fuzzy interarrival times is proposed in this paper. Relations between the renewal number and interarrival times in such a process are investigated. Useful theorems such as the elementary renewal theorem, the Blackwell renewal theorem and the Smith key renewal theorem in a conventional delayed renewal process are extended to their counterparts for random fuzzy delayed renewal processes.

The theoretical foundations of statistical learning theory based on fuzzy number samples

Statistical learning theory based on real-valued random samples has been regarded as a better theory on statistical learning with small sample. The key theorem of learning theory and bounds on the rate of convergence of learning processes are important theoretical foundations of statistical learning theory. In this paper, the theoretical foundations of the statistical learning theory based on fuzzy number samples are discussed. The concepts of fuzzy expected risk functional, fuzzy empirical risk functional and fuzzy empirical risk minimization principle are redefined. The key theorem of learning theory based on fuzzy number samples is proved. Furthermore, the bounds on the rate of convergence of learning processes based on fuzzy number samples are discussed.

High-level small-step operational semantics for transactions

Software transactions have received significant attention as a way to simplify shared-memory concurrent programming, but insufficient focus has been given to the precise meaning of software transactions or their interaction with other language features. This work begins to rectify that situation by presenting a family of formal languages that model a wide variety of behaviors for software transactions. These languages abstract away implementation details of transactional memory, providing high-level definitions suitable for programming languages. We use small-step semantics in order to represent explicitly the interleaved execution of threads that is necessary to investigate pertinent issues. We demonstrate the value of our core approach to modeling transactions by investigating two issues in depth. First, we consider parallel nesting, in which parallelism and transactions can nest arbitrarily. Second, we present multiple models for weak isolation, in which nontransactional code can violate the isolation of a transaction. For both, type-and-effect systems let us soundly and statically restrict what computation can occur inside or outside a transaction. We prove some key language-equivalence theorems to confirm that under sufficient static restrictions, in particular that each mutable memory location is used outside transactions or inside transactions (but not both), no program can determine whether the language implementation uses weak isolation or strong isolation.

The Key Renewal Theorem for a Transient Markov Chain

We consider a time-homogeneous real-valued Markov chain Xn, n≥0. Suppose that this chain is transient, that is, Xn generates a σ-finite renewal measure. We prove the key renewal theorem under the condition that this chain has jumps that are asymptotically homogeneous at infinity and asymptotically positive drift.

CLIFFORD AND EXTENSOR CALCULUS AND THE RIEMANN AND RICCI EXTENSOR FIELDS OF DEFORMED STRUCTURES (M, ∇′, η) AND (M, ∇, g)

Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of a triple (M, ∇, g) is investigated for each particular open set U ⊂ M through the introduction of a geometric structure on U, i.e. a triple (U, γ, g), where γ is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related to gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover, we study the important case of a class of deformed Levi–Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.

Operator Methods, Abelian Processes and Dynamic Conditioning

A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field. It also leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the mathematical framework self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can effectively use block-factorization algorithms. Finally, we discuss the method of dynamic conditioning that allows one to dynamically correlate over possibly even hundreds of processes in a numerically noiseless framework while preserving marginal distributions.

Processor sharing: A survey of the mathematical theory

During last few decades the Egalitarian Processor Sharing (EPS) has gained a prominent role in applied probability, especially, in queueing theory and its computer applications. While the EPS paradigm emerged in 1967 as an idealization of round-robin (RR) scheduling algorithm in time-sharing computer systems, it has recently capture renewed interest as a powerful concept for modeling WEB servers. This paper summarizes the most important results concerning the exact solutions for the M/GI/1 queue with egalitarian processor sharing. The material is drawn, mainly, from recent authors' papers which are supplemented, in small degree, by other related results. Many of the further results are established under the direct influence of our earlier papers. Our main purpose is to give a survey of state-of-the-art with regard to main achievements of the contemporary theory of the M/GI/1 queueing system with processor sharing. The focus is on the methods and techniques of exact and asymptotic analysis of this queueing system. In contrast to the standard surveys, the abridged proofs (or their ideas) of some key theorems and corollaries are included in the paper. We outline recent developments in exact analysis of the M/GI/1-EPS queue with further emphasis on time-dependent (transient) probability distributions of the main characteristics. In particular, the present paper includes the results on the joint time-dependent distribution of the sojourn time of a job arriving at time t with the service demand (length) u, and of the number of jobs at time t- in the M/GI/1 queue with egalitarian processor sharing, which obtained in form of multiple transforms. We also show how the non-stationary solutions can be used to obtain known and new results which allow to predict the behaviour of the EPS queue and to yield additional insights into its new unexpected properties. We also discuss a number of limit theorems arising under the study of the processor sharing queues.

Farkas' Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach

We briefly consider several formulations of Farkas' Lemma first. Then we assume the setting of two vector spaces, one of them being linearly ordered, over a linearly ordered field till the end of this article. In this setting, we state a generalized version of Farkas' Lemma and prove it in a purely linear-algebraic way. Afterwards, we present Theorems of Motzkin, Tucker, Carver, Dax, and some other theorems of the alternative that characterize consistency of a finite system of linear inequalities. We also mention the Key Theorem, which is a related result. Finally, we use Farkas' Lemma to prove the Duality Theorem for linear programming (with a finite number of linear constraints). The Duality Theorem that is proved here covers, among others, linear programming in a real vector space of finite or infinite dimension and lexicographic linear programming.

Slowly oscillating functions in semigroup compactifications and convolution algebras

The paper proposes a unified approach to many key theorems proved in the last twenty years in different areas of abstract harmonic analysis. This approach is based on the so-called slowly oscillating functions which were introduced in coarse geometry. In addition to this method being the most natural and simple, it also leads to the generalisation of some of the results and to the achievement of some new results. Several of these results concern the topological centres of convolution algebras and semigroup compactifications.

The Better Bubbling Lemma

BCD [Barendregt, Henk, Mario Coppo and Mariangiola Dezani-Ciancaglini, A filter lambda model and the completeness of type assignment, JSL 48 (1983), 931–940] relies for its modeling of λ calculus in intersection type filters on a key theorem which I call BL (for the Bubbling Lemma, following someone). This lemma has been extended in [Castagna, Giuseppe, and Alain Frisch. A gentle introduction to semantic subtyping. In PPDP05, ACM Press (full version) and ICALP05, LNCS volume 3580, Springer-Verlag (summary), 2005. Joint ICALP-PPDP keynote talk; Dezani-Ciancaglini, M., A. Frisch, E. Giovannetti and Y. Motohama, 2002. The Relevance of Semantic Subtyping, Electronic Notes in Theoretical Computer Science 70 No. 1, 15pp] to encompass Boolean structure, including specifically union types; this extended lemma I call BBL (the Better Bubbling Lemma). There are resonances, explored in [Dezani-Ciancaglini, M., R.K. Meyer and Y. Motohama, The semantics of entailment omega, NDJFL 43 (2002), 129–145] and [Dezani-Ciancaglini, M., A. Frisch, E. Giovannetti and Y. Motohama, 2002. The Relevance of Semantic Subtyping, Electronic Notes in Theoretical Computer Science 70 No. 1, 15pp], between intersection and union type theories and the already existing minimal positive relevant logic B+ of [Routley, Richard, and Robert K. Meyer, The semantics of entailment III, JPL 1 (1972), 192–208]. (Indeed [Pal, Koushik, and Robert K. Meyer. 2005. Basic relevant theories for combinators at levels I and II, AJL 3, http://www.philosophy.unimelb.edu.au/2005/2003_2.pdf, 19 pages] applies BL and BBL to get further results linking combinators to relevant theories and propositions.) On these resonances the filters of algebra become the theories of logic. The semantics of [Meyer, Robert K., Yoko Motohama and Viviana Bono, Truth translations of relevant logics, in B. Brown and F. Lepage, eds., “Truth and Probability: Essays in Honour of Hugues Leblanc,” pages 59–84, College Publications, King's College, London, 2005] yields here a new and short proof of BBL, which takes account of full Boolean structure by encompassing not only B+ but also its conservative Boolean extension CB [Routley, Richard, and Robert K. Meyer, The semantics of entailment III, JPL 1 (1972), 192–208; Meyer, Robert K., 1995. Types and the Boolean system CB+, cslab.anu.edu.au/ftp/techreports/SRS/1995/TR-SRS-5-95/meyer.ps.gz; Meyer, Robert K., Yoko Motohama and Viviana Bono, Truth translations of relevant logics, in B. Brown and F. Lepage, eds., “Truth and Probability: Essays in Honour of Hugues Leblanc,” pages 59–84, College Publications, King's College, London, 2005.].

A Diffusion Approximation for Markov Renewal Processes

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.

A Diffusion Approximation for Markov Renewal Processes

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.