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Folding and unfolding problems have been implicit since Albrecht Durer in the early 1500’s [Dur77], but have not been studied extensively until recently. Over the past few years, there has been a surge of interest in these problems in discrete and computationsl geometry. This paper gives a brief survey of some of the recent work in this area, subdivided into three sections based on the type of object being folded: linkages, paper, or polyhedra. See also [O’R98] for a related survey from this conference two years ago.

When decoding a threshold phenomenon is often observed: decoding deteriorates very suddenly around some critical value of the channel parameter. Threshold behaviour has been studied in many situations outside coding theory and a number of tools have been developped. One of those turns out to be particularly relevant to coding, namely the derivation of isoperimetric inequalities for product measures on Hamming spaces. we discuss this approach and derive consequences.

The core of the Wymore theory, or the tricotyledon theory of systems design, is a conceptual framework and the corresponding mathematical structure under which both continuous and discrete systems (or hybrid continuous/discrete systems) can be described. The mathematical structure was derived (in mid 1960’s) by integrating the theory of finite-state automata with the theory of continous systems described by differential equations. As such, it is essentially based on a generalized notion of the state transition structure.

In this short paper one overviews the two years development of kernel P systems (kP systems for short), a basic class of P systems combining features of different variants of such systems. The definition of kP systems is given, some examples illustrate various features of the model and the most significant results are presented.

To be an efficient tool for knowledge discovery, formal concept analysis has been paid more attention to and applied to many fields in recent years. Through studying the n-scale relation defined in this paper based on the formal context, we obtain some knowledge: left neighborhood sets and right neighborhood sets, which also belong to the powerset of partition on the object set, like the set of extents. Especially, when n is a special value, the corresponding left neighborhood approximate concepts and right neighborhood approximate concepts are the join-dense subsets of the property oriented concept lattice and the concept lattice of a formal context, respectively. And then, the whole lattices can be obtained.

Let be a complex algebraic hypersurface in not passing through the point . The generators of the fundamental group and the relations among them are described in terms of the real cone over with apex at . This description is a generalization to the algebraic case of Wirtinger's corepresentation of the fundamental group of a knot in . A new proof of Zariski's conjecture about commutativity of the fundamental group for a projective nodal curve is given in the second part of the paper based on the description of the generators and the relations in the group obtained in the first part.

This short paper is a summary of a survey talk given on the interplay between enumerative Combinatorics and Computer Science.

This paper formalises the followership in networks that agents following or unfollowing each other dynamically. The semantics is based on the basic hybrid logic and we extend the logic with a propositional action modality \([{a}\uparrow {\theta }]\) for the changes of followership. The main contribution of this paper is the completeness result. Moreover, we have proved that all pure axiomatic extensions have completeness and discussed some possible future works and extensions, in particular, some features of the extended action modality \([{a}\uparrow {\varphi }]\), like repetition regrets are discussed.

The paper expands upon the work by Wang [16], who proposes a single-agent modal logic framework for reasoning about “knowing how”. This paper proposes a more flexible semantics to the knowing-how operator. According to this semantics, an agent knows how to achieve \(\varphi \) given \(\psi \) if there exists a finite linear plan such that it will end up with some \(\varphi \)-state from any \(\psi \)-state by executing the plan, either fully or skipping some non-executable steps. We give a sound and complete axiomatization of this logic. Finally we introduce a suitable notion of bisimulation for this logic.

The framework of Categorical Compositional Distributional models of meaning [3], inspired by category theory, allows one to compute the meaning of natural language phrases, given basic meaning entities assigned to words. Composing word meanings is the result of a functorial passage from syntax to semantics. To keep one from drowning in technical details, diagrammatic reasoning is used to represent the information flow of sentences that exists independently of the concrete instantiation of the model. Not only does this serve the purpose of clarification, it moreover offers computational benefits as complex diagrams can be transformed into simpler ones, which under coherence can simplify computation on the semantic side. Until now, diagrams for compact closed categories and monoidal closed categories have been used (see [2, 3]). These correspond to the use of pregroup grammar [12] and the Lambek calculus [9] for syntactic structure, respectively. Unfortunately, the diagrammatic language of Baez and Stay [1] has not been proven coherent. In this paper, we develop a graphical language for the (categorical formulation of) the nonassociative Lambek calculus [10]. This has the benefit of modularity where extension of the system are easily incorporated in the graphical language. Moreover, we show the language is coherent with monoidal closed categories without associativity, in the style of Selinger’s survey paper [17].

Rough Sets over generalized transitive relations like proto-transitive ones have been initiated recently by the present author. In a recent paper, approximation of proto-transitive relations by other relations was investigated and the relation with rough approximations was developed towards constructing semantics that can handle fragments of structure. It was also proved that difference of approximations induced by some approximate relations need not induce rough structures. In this research, the structure of rough objects is characterized and a theory of dependence for general rough sets is developed and used to internalize the Nelson-algebra based approximate semantics developed earlier by the present author. This is part of the different semantics of PRAX developed in this paper by her. The theory of rough dependence initiated in earlier papers is extended in the process. This paper is reasonably self-contained and includes proofs and extensions of representation of objects that have not been published earlier.

This paper proposes an analysis in a Feature-based Lexicalized Tree-Adjoining Grammar F-LTAG [9, 18] for deriving the semantic representations of various narrow focus constructions. Te paper presents an extension of the F-LTAG analysis by Balogh [3] based on the syntax-semantics approach by Kallmeyer & Romero [11] and the semantic-pragmatic analysis of focus by Balogh [2].

The codebook cardinality spectrum (CCS) of distributed arithmetic codes (DAC) provides an effective tool for analyzing the decoding complexity of the full-search DAC decoder. However, in our previous work, the study on CCS is limited to equiprobable binary sources. In this paper, by defining the CCS for nonuniform binary sources, we try to provide a more general theoretical analysis on this complex problem. At first, a general explicit form of the initial CCS is obtained by utilizing the Fourier transform. Then, we get a recursive equation for deducing the CCS forwardly. For the convenience of computation, the paper also provides a numerical method for calculating the CCS. Finally, we define the expansion factor for nonuniform binary sources to measure the complexity of the full-search DAC decoder.

In this paper, we construct an infinitary variant of the relational model of linear logic, where the exponential modality is interpreted as the set of finite or countable multisets. We explain how to interpret in this model the fixpoint operator Y as a Conway operator alternatively defined in an inductive or a coinductive way. We then extend the relational semantics with a notion of color or priority in the sense of parity games. This extension enables us to define a new fixpoint operator Y combining both inductive and coinductive policies. We conclude the paper by mentionning a connection between the resulting model of λ-calculus with recursion and higher-order model-checking.

In the paper Justification logics and hybrid logics, Melvin Fitting initiated the idea of combining justification logics and hybrid logics into a single system. However, for some technical reason, he presented a basic-hybrid justification logic \(\mathsf {JT}\) instead of \(\mathsf {J}\) with proving completeness theorem and realization theorem. The task of weakening this logic to \(\mathsf {J}\) was left as an open problem in conclusion. This paper aims at solving this problem to further facilitate the proof of the properties of this logic as well as the derivation of other members of this logic family.

The probing security model is very popular to prove the side-channel security of cryptographic implementations protected by masking. A common approach to secure nonlinear functions in this model is to represent them as polynomials over a binary field and to secure their nonlinear multiplications thanks to a method introduced by Ishai, Sahai and Wagner at Crypto 2003. Several schemes based on this approach have been published, leading to the recent proposal of Coron, Roy and Vivek which is currently the best known method when no particular assumption is made on the algebraic structure of the function. In the present paper, we revisit this idea by trading nonlinear multiplications for low-degree functions. Specifically, we introduce an algebraic decomposition approach in which a nonlinear function is represented as a sequence of functions with low algebraic degrees. We therefore focus on the probing-secure evaluation of such low-degree functions and we introduce three novel methods to tackle this particular issue. The paper concludes with a comparative analysis of the proposals, which shows that our algebraic decomposition method outperforms the method of Coron, Roy and Vivek in several realistic contexts.

This paper updates my talk on Cache Blocking for Dense Linear Algorithms since 1985 given at PPAM 11; see [11]. We again apply Dimension Theory to matrices in the Fortran and C programming languages. New Data Structures (NDS) for matrices are given. We use the GCD algorithm to transpose a \(n\) by \(m\) matrix \(A\) in CMO order, standard layout, in-place. Algebra and Geometry are used to make this idea concrete and practical; it is the reason for title of our paper: make a picture of any matrix by the GCD algorithm to convert it into direct sum of square submatrices. The picture is Geometry and the GCD algorithm is Algebra. Also, the in-place transposition of the GKK and TT algorithms will be compared. Finally, the importance of using negative integers will be used to give new results about subtraction and finding primitive roots which also make a priori in-place transpose more efficient.

This is the first part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. This theory generalizes the tensor category theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a “conformal vertex algebra” or even more generally, for a “Mobius vertex algebra.” We do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. As in the earlier series of papers, our tensor product functors depend on a complex variable, but in the present generality, the logarithm of the complex variable is required; the general representation theory of vertex operator algebras requires logarithmic structure. This work includes the complete proofs in the present generality and can be read independently of the earlier series of papers. Since this is a new theory, we present it in detail, including the necessary new foundational material. In addition, with a view toward anticipated applications, we develop and present the various stages of the theory in the natural, general settings in which the proofs hold, settings that are sometimes more general than what we need for the main conclusions. In this paper (Part I), we give a detailed overview of our theory, state our main results and introduce the basic objects that we shall study in this work. We include a brief discussion of some of the recent applications of this theory, and also a discussion of some recent literature.

In the first part of the paper we recall the coalgebraic approach to handling the so-called invisible transitions that appear in different state-based systems semantics. We claim that these transitions are always part of the unit of a certain monad. Hence, coalgebras with internal moves are exactly coalgebras over a monadic type. The rest of the paper is devoted to supporting our claim by studying two important behavioural equivalences for state-based systems with internal moves, namely: weak bisimulation and trace semantics. We continue our research on weak bisimulations for coalgebras over order enriched monads. The key notions used in this paper and proposed by us in our previous work are the notions of an order saturation monad and a saturator. A saturator operator can be intuitively understood as a reflexive, transitive closure operator. There are two approaches towards defining saturators for coalgebras with internal moves. Here, we give necessary conditions for them to yield the same notion of weak bisimulation. Finally, we propose a definition of trace semantics for coalgebras with silent moves via a uniform fixed point operator. We compare strong and weak bisimilation together with trace semantics for coalgebras with internal steps.

We propose a modular method for verifying the correctness of a Grobner basis candidate. For an inhomogeneous ideal I, we propose to check that a Grobner basis candidate G is a subset of I by computing an exact generating relation for each g in G by the given generating set of I via a modular method. The whole procedure is implemented in Risa/Asir, which is an open source general computer algebra system. By applying this method we succeeded in verifying the correctness of a Grobner basis candidate computed in Romanovski et al (2007). In their paper the candidate was computed by a black-box software system and it has been necessary to verify the candidate for ensuring the mathematical correctness of the paper.

The purpose of this paper is to introduce and study Ishikawa iterative algorithms for solving the SFP in the setting of infinite-dimensional Hilbert spaces. The main results presented in this paper improve and extend some recent results done by Xu [Iterative methods for the split feasibility problem in infinite-dimensional Hilbert space, Inverse Problems 26 (2010) 105018]. At the end we prove that the accumulation of errors in Ishikawa iterative CQ algorithm is bounded in certain range.

This paper treats 6j symbols or their orthonormal forms as a function of two variables spanning a square manifold which we call the “screen”. We show that this approach gives important and interesting insight. This two dimensional perspective provides the most natural extension to exhibit the role of these discrete functions as matrix elements that appear at the very foundation of the modern theory of classical discrete orthogonal polynomials. Here we present 2D and 1D recursion relations that are useful for the direct computation of the orthonormal 6j, which we name U. We present a convention for the order of the arguments of the 6j that is based on their classical and Regge symmetries, and a detailed investigation of new geometrical aspects of the 6j symbols. Specifically we compare the geometric recursion analysis of Schulten and Gordon with the methods of this paper. The 1D recursion relation, written as a matrix diagonalization problem, permits an interpretation as a discrete Shrodinger-like equations and an asymptotic analysis illustrates semiclassical and classical limits in terms of Hamiltonian evolution.

The present paper explores the interaction between two recursion-theoretic notions: program self-reference and learning partial recursive functions in the limit. Kleene’s Recursion Theorem formalises the notion of program self-reference: It says that given a partial-recursive function ψ p there is an index e such that the e-th function ψ e is equal to the e-th slice of ψ p . The paper studies constructive forms of Kleene’s recursion theorem which are inspired by learning criteria from inductive inference and also relates these constructive forms to notions of learnability. For example, it is shown that a numbering can fail to satisfy Kleene’s Recursion Theorem, yet that numbering can still be used as a hypothesis space when learning explanatorily an arbitrary learnable class. The paper provides a detailed picture of numberings separating various versions of Kleene’s Recursion Theorem and learnability.

This paper combines semantics and syntax to study the displaced separable form of verb+object separable words AB in modern Chinese in a comprehensive perspective, namely the "N's+B+Ax" form. The paper discusses and analyzes the characteristics of such a form and selection and restriction within the quantitative verb+object separable words AB. It further analyzes the internal semantic features and rules of use for separable words falling into this category. The outcome of the study can serve as a reference for computer linguistics and foreign language teaching.

In this paper we present product-form solutions from the point of view of stochastic process algebra. In previous work [16] we have shown how to derive product-form solutions for a formalism called Labelled Markov Automata (LMA). LMA are very useful as their relation with the Continuous Time Markov Chains is very direct. The disadvantage of using LMA is that the proofs of properties are cumbersome. In fact, in LMA it is not possible to use the inductive structure of the language in a proof. In this paper we consider a simple stochastic process algebra that has the great advantage of simplifying the proofs. This simple language has been inspired by PEPA [10], however, detailed analysis of the semantics of cooperation will show the differences between the two formalisms. It will also be shown that the semantics of the cooperation in process algebra influences the correctness of the derivation of the product-form solutions.

In this paper, the author introduces a structure called M-type bag structure which can be defined on a non-empty set associated with an indiscernibilty relation. It can be observed that an M-type bag structure represents a bag if the indiscernibility relation be defined in such a way that any two elementsof the set are indiscernible under a given set of criteria that considers the values of some predefined attribute set. This paper further studies some algebraic properties of M-type bag structures.

Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0,1}, to fuzzy logic, where the truth values belong to the unit interval [0,1]. These functions are not only fundamental for fuzzy logic systems, fuzzy control, approximate reasoning and expert systems, but they also play a significant role in mathematical fuzzy logic, in fuzzy mathematical morphology and image processing, in defining fuzzy subsethood measures and in solving fuzzy relational equations. This volume collects 8 research papers on fuzzy implication functions. Three articles focus on the construction methods, on different ways of generating new classes and on the common properties of implications and their dependencies. Two articles discuss implications defined on lattices, in particular implication functions in interval-valued fuzzy set theories. One paper summarizes the sufficient and necessary conditions of solutions for one distributivity equation of implication. The following paper analyzes compositions based on a binary operation * and discusses the dependencies between the algebraic properties of this operation and the induced sup-* composition. The last article discusses some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. These papers aim to present todays state-of-the-art in this area.