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Without revealing data in the plain, public key encryption with equality test can achieve the basic cryptographic functionalities required. However, the extended version, i.e., public encryption with filtered equality test (PKE-FET) can be used to only test the ciphertext only for some data. In this paper, we revisit the notion of PKE-FET on various security models. We provide a formal definition for the security of PKE-FET and present a new scheme to achieve a stronger level of security than the existing scheme.

The concept of ambiguity designates those situations where the information available to the decision maker is insufficient to form a probabilistic view of the world. Thus, it has provided the motivation for departing from the subjective expected utility (SEU) paradigm. Yet, the formalization of the concept is missing. This is a grave omission as it leaves nonexpected utility models hanging on shaky ground. In particular, it leaves unanswered basic questions such as the following: (1) Does ambiguity exist? (2) If so, which situations should be labeled as “ambiguous”? (3) Why should one depart from SEU in the presence of ambiguity? (4) If so, what kind of behavior should emerge in the presence of ambiguity? The present paper fills these gaps. Specifically, it identifies those information structures that are incompatible with SEU theory, and shows that their mathematical properties are the formal counterpart of the intuitive idea of insufficient information. These are used to give a formal definition of ambiguity and, consequently, to distinguish between ambiguous and unambiguous situations. Finally, the paper shows that behavior not conforming to SEU theory must emerge in correspondence of insufficient information and identifies the class of non-EU models that emerge in the face of ambiguity. The paper also proposes a new comparative definition of ambiguity, and discusses its relation with some of the existing literature.

This study attempts to reveal pre-service teachers’ conceptions, definitions, and understanding of quadrilaterals and their internal relationships in terms of personal and formal figural concepts via case of the parallelograms. To collect data, an open-ended question was addressed to 27 pre-service mathematics teachers, and clinical interviews were conducted with them. The factors influential on pre-service teachers’ definitions of parallelograms and conceptions regarding internal relationships between quadrilaterals were analyzed. The strongest result involved definitions based on prototype figures and partially seeing internal relationships between quadrilaterals via these definitions. As a different result from what is reported in the literature, it was found that the fact that rectangle remains as a special case of parallelogram in pre-service teachers’ figural concepts leads them not to adopt the hierarchical relationship. The findings suggested that learners were likely to recognize quadrilaterals by a special case of them and prototypical figures, even though they knew the formal definition in general. This led learners to have difficulty in understanding the inclusion relations of quadrilaterals.

This chapter introduces the basic concepts on copulas and Markov models. We review the formal definition of copulas with its fundamental properties. We then introduce Kendall’s tau as a measure of dependence structure for a pair of random variables, and its relationship with a copula. Examples of copulas are reviewed, such as the Clayton copula, the Gaussian copula, the Frank copula, and the Joe copula. Finally, we introduce the copula-based Markov chain time series models and their fundamental properties.

This chapter is a mathematical exposition of the algebraic approach to manifolds, which was sketched in intuitive terms in Chapter 1. A detailed answer to the following fundamental question is presented: given an abstract \(\mathbb R\)-algebra \(\mathcal F\), find a set (smooth manifold) M whose \(\mathbb R\)-algebra of (smooth) functions can be identified with \(\mathcal F\). This involves giving the formal definition of the notion of point (which turns out to have a nontrivial structure!) and of the classes of \(\mathbb R\)-algebras that need to be considered, in particular the so-called geometrical and complete ones.

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous spectrum. We review important previous results of Sunada and Aomoto and present several illuminating examples. We present many open problems and conjectures that we hope will stimulate further work., appeared as Adv. Math. 379 (2020), 107241; this version includes an addendum

In 1959, Duncan Luce published the famous paper entitled “On the possible psychophysical laws.” The results presented here were inspired by that paper and by the ensuing controversy centered on the invariance concept of “meaningfulness.” We give a formal definition of this concept. As we define it, the condition of meaningfulness is quite potent. In its context, relatively weak additional conditions may suffice for the derivation of precise scientific or geometric laws. We give several examples of such derivations, including the Pythagorean Theorem and Beer’s Law.

Uncertainty appears not only in real quantities but also in complex quantities. Complex uncertain process is essentially a sequence of complex uncertain variables indexed by time. In order to describe complex uncertain process, a formal definition of complex uncertain distribution is given in this paper, as well as the concepts of independence and variance. In addition, some properties of complex uncertain integral are presented.

Although we do not intend to give a general, formal definition, the stochastic mean-field dynamics we present in these notes can be conceived as the random evolution of a system comprised by N interacting components which is: (a) invariant in law for permutation of the components; (b) such that the contribution of each component to the evolution of any other is of order \(\frac{1}{N}\). The permutation invariance clearly does not allow any freedom in the choice of the geometry of the interaction; however, this is exactly the feature that makes these models analytically treatable, and therefore attractive for a wide scientific community.

According to the standard realistic interpretation of classical electrodynamics, the electromagnetic field is conceived as a real physical entity existing in space and time. The problem we address in this paper is how to understand this spatiotemporal existence, that is, how to describe the persistence of a field-like physical entity like electromagnetic field. First, we provide a formal description of the notion of persistence: we derive an “equation of persistence” constituting a necessary condition that the spatiotemporal distributions of the fundamental attributes of a persisting physical entity must satisfy. We then prove a theorem according to which the vast majority of the solutions of Maxwell’s equations, describing possible spatiotemporal distributions of the fundamental attributes of the electromagnetic field, violate the equation of persistence. Finally, we discuss the consequences of this result for the ontology of the electromagnetic field.

Strings, sequences and, ultimately, motifs play a key role in bioinformatics. Here, a motif is a biologically relevant pattern occurring in one or more input sequences. In this article, first we provide a formal definition of strings, sequences, matches, repetitions and motifs. Then, we present a detailed description of approaches to motif search, based on three possible taxonomies, namely: (1) the underlying algorithms; (2) the kinds of derived motif; (3) the kinds of considered match. After this, we illustrate the problem of assessing motif relevance, along with the current solutions. Finally, we draw some closing remarks.

International audience; The literature on inconsistency measures has ignored a distinction, that is, differentiating absolute measures and relative measures. An absolute measure gives the total amount of inconsistency in the knowledge base but a relative measure computes, by some criteria, the proportion of the base that is inconsistent. To compare the inconsistency measures, researchers have proposed postulates for such measures. We split these postulates into three groups: ones (including two new postulates) that relative measures should satisfy, ones inappropriate for relative measures, and ones that relative measures may satisfy. We obtain some new results upon the relationships between these groups of postulates. On these grounds, we introduce a formal definition for relative inconsistency measures. We consider some relative measures previously proposed and define several new ones that serve as examples. We show that all of these measures satisfy the new formal definition.

Some types of nouns in Bulgarian in view of their formal description This article presents a part of the formal description of nouns in Bulgarian made for the purposes of the Normative grammar of modern Bulgarian . It reviews the descriptive approach that has been adopted and some issues that can be solved through its implementation. In particular, cases of variation of forms within the paradigm of the nouns as well as types of nouns with “defective” paradigm are discussed. Niektore typy rzeczownikow w jezyku bulgarskim ze wzgledu na ich opis formalny W artykule przedstawiono cześc formalnego opisu rzeczownikow w jezyku bulgarskim. Opis zostal sporządzony na potrzeby Gramatyki normatywnej wspolczesnego jezyka bulgarskiego . Zaprezentowano przyjete kryteria opisu oraz ich zastosowanie na wybranych przykladach. W szczegolności omowiono przypadki wariancji form wewnątrz paradygmatu rzeczownikowego i przeanalizowano typy rzeczownikow o „wadliwym” paradygmacie.

Rules of a formal description of Bulgarian inflected lexemes as a component of normative grammar A formal description of the set of grammatical means in Bulgarian as a component of its normative grammar should form the basis for adequate distinction and diagnosis of normative problems. It means that linguistic facts should be described precisely and exhaustively and without any presupposed normative interpretation. Accomplishment of such an objective may befeasible if the description is treated as the task of building the universal classification of forms in the Bulgarian language understood as the classification in which each of the classifying features finds its fulfilment through one of its values in each of the objects classified. The study is an attempt to determine the rules of building the classification of Bulgarian lexical items relevant to their formal morphological description as a transition from dichotomous to universal classification. Zasady formalnego opisu bulgarskich leksemow odmiennych jako komponentu gramatyki normatywnej Formalny opis inwentarza środkow gramatycznych jezyka bulgarskiego jako komponent jego gramatyki normatywnej powinien stanowic podstawe do adekwatnego wyodrebnienia i zdiagnozowania problemow normatywnych. Oznacza to, ze fakty jezykowe powinny byc opisane ściśle i wyczerpująco bez jakiejkolwiek z gory zalozonej interpretacji normatywnej. Osiągniecie tego celu wydaje sie mozliwe, jeśli do opisu podejśc jak do zadania zbudowania uniwersalnej klasyfikacji form jezyka bulgarskiego rozumianej jako klasyfikacji, w ktorej kazda z cech klasyfikacyjnych realizuje sie poprzez ktorąś ze swoich wartości w kazdym klasyfikowanym obiekcie. Praca ta stanowi probe ustalenia zasad budowania klasyfikacji jednostek leksykalnych jezyka bulgarskiego relewantnej dla ich formalnego opisu morfologicznego jako przejścia od klasyfikacji dychotomicznej do uniwersalnej.

A formal description of processes in cooperative full-truckload networks is developed. Based on this a matching concept between transport routes is introduced to prevent empty runs. As an application of this framework an optimization problem is formulated to find matchings, which guarantee a maximal number of matched transports. A solution of this optimization problem and strategies and algorithms for data reduction and augmentation are presented as well.

We are concerned with the stochastic process for which the likelihood of the data, although it does exist, is possibly unknown to us. In order to estimate parameter, we are led to appropriate estimating functions (EF, for short) including the (quasi) maximum likelihood score. A formal definition of the general non-ergodic estimating function is made in this paper. This can be viewed as a generalization of the non-ergodic maximum likelihood score (due to Basawa and Koul (1979), and Basawa and Scott (1983)) toward the theory of EFs. In addition, some characterizations on the non-ergodic EFs are made. It is interesting to note non-standard cases where non-stationary process may yield an ergodic EF while stationary process can produce a non-ergodic EF. Various examples are presented to illustrate the main results.

In this study, we formally investigate how to determine fuzzy/linguistic IF-THEN rules that are redundant in linguistic descriptions (systems of IF-THEN rules). We present a formal definition of redundancy and show that seemingly redundant rules can in fact be indispensable. These results apply to IF-THEN rules that use evaluative expressions (e.g., small and very big) and the inference method called perception-based logical deduction. However, they are also valid for inference systems with compatible design choices. We also describe an algorithm for the automatic detection and removal of redundant rules. Finally, we present an example of a linguistic description that is learned automatically from data and reduced using our algorithm.

This mixed-methods study examines the conceptual understanding of limit among 22 undergraduate engineering students from two different sections of the same introductory differential equations course. The participants’ concepts of limit (concept images and personal concept definitions) were examined using written tasks followed by one-on-one investigative interviews. The findings suggest that, at the end of their mathematics coursework, many engineering students do not consider employing a formal definition of limit to solve limit-based mathematics problems. Furthermore, the findings suggest that most participants’ personal concept definitions of limit are both inoperable for solving limit problems and inconsistent with the formal definition of limit.

This article proposes a comprehensive approach to computing a navigation graph for an indoor space. It focuses on a single floor, but the work is easily extensible to multi-level spaces. The approach proceeds by using a formal model, based on the combinatorial map but enhanced with geometric and semantic information. The process is almost fully automatic, taking as input the building plans providing the geometric structure of the floors and semantics of the building, such as functions of interior spaces, portals, etc. One of the novel aspects in this work was the use of combinatorial maps and their duals to provide a compact formal description of the topology and connectivity of the indoor structure represented by a connected, embedded graph. While making use of existing libraries for the more routine computational geometry involved, the research develops several new algorithms, including one for computing the local kernel of a region. The process is evaluated by means of a case study using part of a university building.

The extraction of logically-independent fragments out of an ontology ABox can be useful for solving the tractability problem of querying ontologies with large ABoxes. In this paper, we propose a formal definition of an ABox module, such that it guarantees complete preservation of facts about a given set of individuals, and thus can be reasoned independently w.r.t. the ontology TBox. With ABox modules of this type, isolated or distributed (parallel) ABox reasoning becomes feasible, and more efficient data retrieval from ontology ABoxes can be attained. To compute such an ABox module, we present a theoretical approach and also an approximation for SHIQ ontologies. Evaluation of the module approximation on different types of ontologies shows that, on average, extracted ABox modules are significantly smaller than the entire ABox, and the time for ontology reasoning based on ABox modules can be improved significantly.

In this chapter we describe the diastasis function, a basic tool introduced by Calabi (Ann Math 58:1–23, 1953) which is fundamental to study Kahler immersions of Kahler manifolds into complex space forms . In Sect. 1.1 we define the diastasis function and summarize its basic properties, while in Sect. 1.2 we describe the diastasis functions of complex space forms, which represent the basic examples of Kahler manifolds. Finally, in Sect. 1.3 we give the formal definition of what a Kahler immersion is and prove that the indefinite Hilbert space constitutes a universal Kahler manifold, in the sense that it is a space in which every real analytic Kahler manifold can be locally Kahler immersed.

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of tornadoes, in the phenomenon of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we introduce new theoretical concepts which enhance topological surgery with the observed forces and dynamics. To do this, we first extend the formal definition to a continuous process caused by local forces. Next, for modeling phenomena which do not happen on arcs or surfaces but are 2-dimensional or 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further introduce the notion of embedded surgery in $S^3$ for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effect of the process lies beyond the initial manifold, such as the formation of black holes. Finally, we connect these new theoretical concepts with a dynamical system and we present it as a model for both 2-dimensional 0-surgery and natural phenomena exhibiting a `hole drilling' behavior. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood., Comment: 54 pages, 34 figures

The intuition of causation is so fundamental that almost every research study in life sciences refers to this concept. However, a widely accepted formal definition of causal influence between observables is still missing. In the framework of linear Langevin networks without feedback (linear response models) we propose a measure of causal influence based on a new decomposition of information flows over time. We discuss its main properties and we compare it with other information measures like the transfer entropy. We are currently unable to extend the definition of causal influence to systems with a general feedback structure and nonlinearities.

In a conventional threshold k out of n visual cryptographic scheme (( k,n)-VCS, for short), one secret image P can be encoded into n seemingly random transparencies (called shares) such that the superimposed result of any group of k or more transparencies can reveal P to our eyes, while that of less than k ones cannot. To ease the management and identification, the shares may look meaningful, instead of seemingly random, pictures. Given secret image P shared by n participants and cover image C, we study ( k, n)-VCS with meaningful shares (denoted as ( k, n)-VCS-MS) in this letter where the n shares could be recognized as the meaningful cover C and their superimpositions follow the threshold requirements of ( k, n)-VCS. We present a formal definition to ( k, n)-VCS-MS and develop an efficient construction by way of integer linear programming. Experimental results demonstrate the effectiveness of our construction.

Strong designated verifier proxy signature (SDVPS) enables a proxy signer to sign messages on behalf of an original signer, and only the designated verifier can be convinced of the validity of the proxy signature, and the verifier cannot prove it to any third party. It is a combination of proxy signature and strong designated verifier signature. In this paper, we define a new formal definition and a new security model of existential unforgery of SDVPS, and then analyze a recently proposed SDVPS scheme, and give four attacks to the scheme, and further we give out an improved SDVPS scheme, and prove it in the new security model. Performance analysis shows the improved scheme is efficient and suitable for electronic commerce.

Association rule mining often produces massive rules, many of which do not meet users’ interest. To address the problem of evaluating and sorting useful association rules, a variety of interestingness measures has been proposed during the past two decades. Nevertheless, current interestingness measures are inadequate to reveal the essential characters hidden in the data. Causality indicates not only that the variables are related, but also how varying a variable is likely to induce a change of another, therefore it is more useful in prediction and reasoning. In this paper, we introduce causality to measure the interestingness of association rules and give a formal definition of casual criterion. Furthermore, we propose a new framework for generating causal association rules using feature selection, and implement an algorithm to efficiently mine the causal association rules. Experiment result on both real and synthetic datasets show that the algorithm performs better than traditional algorithms, and can discover causal association rules from databases efficiently.

As evaluation methods for logic programs have become more sophisticated, the classes of programs for which termination can be guaranteed have expanded. From the perspective of ar set programs that include function symbols, recent work has identified classes for which grounding routines can terminate either on the entire program [Calimeri et al. 2008] or on suitable queries [Baselice et al. 2009]. From the perspective of tabling, it has long been known that a tabling technique called subgoal abstraction provides good termination properties for definite programs [Tamaki and Sato 1986], and this result was recently extended to stratified programs via the class of bounded term-size programs [Riguzzi and Swift 2013]. In this article, we provide a formal definition of tabling with subgoal abstraction resulting in the SLG SA algorithm. Moreover, we discuss a declarative characterization of the queries and programs for which SLG SA terminates. We call this class strongly bounded term-size programs and show its equivalence to programs with finite well-founded models. For normal programs, strongly bounded term-size programs strictly includes the finitely ground programs of Calimeri et al. [2008]. SLG SA has an asymptotic complexity on strongly bounded term-size programs equal to the best known and produces a residual program that can be sent to an answer set programming system. Finally, we describe the implementation of subgoal abstraction within the SLG-WAM of XSB and provide performance results.

This paper introduces two virtual knot theory “analogues” of a well-known family of invariants for knots in thickened surfaces: the Grishanov–Vassiliev finite-type invariants of order two. The first, called the three loop isotopy invariant, is an invariant of virtual knots while the second, called the three loop framed isotopy invariant, is a regular isotopy invariant of framed virtual knots. The properties of these invariants are investigated at length. In addition, we make precise the informal notion of “analogue”. Using this formal definition, it is proved that a generalized three loop invariant is a virtual knot theory analogue of a generalization of the Grishanov–Vassiliev invariants of order two.

In this study, we consider the informal and formal definition of limit on the basis of middle and high school curriculum, and then analyze the reason of difficulties experienced when sophomores learn the formal definition($\epsilon $-$\delta$ procedure) of limit. We conducted teaching of the formal definition of limit with sophomores and analyzed their errors which were appeared when they applied to limits problems. In addition, we try to improve the understanding of $\epsilon $-$\delta$ procedure of the limit taught in analysis.

Length and girth are central to measure the size of tube shaped objects. This paper extends circular helical curves to general tubular shapes and proposes a novel method for measuring their length and girth. We call the extended circular helixes quasi-helical curves. A formal definition, as well as a set of practical algorithms for quasi-helixes, is presented in this paper. Experimental results demonstrate that our method is fast, intrinsic, insensitive to noises, invariant to triangulation and resolution. Furthermore, quasi-helical curves can also be used in classifying 3D shapes and designing vector fields on surfaces of revolution.

A formal definition of fractional integrals in the complex matrix variate case is given here. This definition will encompass all the various fractional integral operators introduced by various authors in the real scalar and matrix cases. The new definition is introduced in terms of M-convolutions of products and ratios of matrices in the complex domain. Their connections to statistical distribution theory, Mellin convolutions, M-transforms and Mellin transform are pointed out. Some basic properties are given and a pathway extension of the new definition is also given. The pathway extension will provide a switching mechanism to move among three different families of functions.

A theory for quantile based hydrologic model selection and model structure deficiency assessment is presented. The paper demonstrates that the degree to which a model selection problem is constrained by the model structure (measured by the Lagrange multipliers of the constraints) quantifies structural deficiency. This leads to a formal definition of model structure deficiency (or rigidity). Model structure deficiency introduces a bias in the prediction of an observed quantile, which is often not equal across quantiles. Structure deficiency is therefore diagnosed when any two quantile predictions for a given model structure cross since unequal bias across quantiles result in quantile predictions crossing. The analysis further suggests that the optimal value of quantile specific loss functions order different model structures by its structure deficiencies over a range of quantiles. In addition to such novelties, quantile hydrologic model selection is a frequentist approach that seeks to complement existing Bayesian approaches to hydrological model uncertainty.

We give a formal definition of a representative sample, but roughly speaking, it is a scaled-down version of the population, capturing its characteristics. New methods for selecting representative ...

The concept of proxy re-encryption (PRE) was first introduced by Blaze et al. Since then several CCA secure PRE schemes have been presented. Weng et al. proposed a non-transitive bidirectional PRE scheme without pairings and proved that their scheme to be PRE-CCA secure in the random oracle model under the CDH assumption. However, we show that Weng et als non-transitive scheme is still transitive under the formal definition of non-transitivity presented in this paper. Moreover, the transitivity of Weng et al.s scheme also implies that it is not PRE-CCA secure.

Two different types of functional dependencies are compared: dependencies that are functional due to the laws of nature and dependencies that are functional if all involved agents behave rationally. The first type of dependencies was axiomatized by Armstrong. This article gives a formal definition of the second type of functional dependencies in terms of strategic games and describes a sound and complete axiomatization of their properties. The axiomatization is significantly different from the Armstrong’s axioms.

The growing complexity of systems increases the complexity of reliability modelling and implies the need to model both the dynamics of systems and the reparability of components. The qualitative reliability analyses aim at finding the minimal representation of all the scenarios of failures and repairs of the components, leading to the system failure. This minimal representation is the set of Minimal Cut Sequences. In order to provide a formal definition of these specific scenarios, whatever the risk analysis model used, this paper proposes coherence rules for dynamic and repairable systems whose dysfunctional scenarios are modelled by a finite automaton.

Equivalence relations and partitions are two interconnected ideas that play important roles in advanced mathematics. While students encounter the informal notion of equivalence in many courses, the formal definition of an equivalence relation is typically introduced in a junior level transition-to-proof course. This paper reports the results of a classroom teaching experiment comparing the effectiveness of traditional lecture versus in-class group activities in introducing equivalence relations. Student understanding was assessed using task-based interviews outside of class. Students involved in group activities made a stronger connection between equivalence relations and partitions. Furthermore, they were more flexible and successful in proving or disproving that a given relation formed an equivalence relation.

Shift registers with linear feedback are frequently used. They owe their popularity to very well developed theoretical base. Registers with feedback of prime polynomials are of particular practical importance. They are willingly applied as test sequence generators and test response compactors. The article presents an attempt to extend the class of registers with linear feedback. Basing on the formal description of the register, the algorithms of register transformation are proposed. It allows to obtain the registers with equivalent graphs., [1] I. Gosciniak, “Linear Registers with Mixed Feedback, in Polish; Rejestry liniowe z mieszanym sprzȩżeniem zwrotnym,” Pomiary Automatyka Kontrola, no. 1, pp. 4–6, 1996., [2] K. Iwasaki, “Analysis and proposal of signature circuits for LSI testing,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 7, no. 1, pp. 84–90, 1988., [3] L.-T. Wang, N. Touba, R. Brent, H. Xu, and H. Wang, “On Designing Transformed Linear Feedback Shift Registers with Minimum Hardware Cost – Technical Report,” Computer Engineering Research Center Department of Electrical & Computer Engineering The University of Texas at Austin, 2011., [4] J. Rajski, J. Tyszer, M. Kassab, and N. Mukherjee, “Method for Synthesizing Linear Finite State Machines,” U.S. Patent, No. 6,353,842, 2002., [5] I. Gosciniak, “Equivalent Form of Linear Feedback Shift Registers,” in XIXth National Conference Circuit Theory and Eletronic Networks, 1996, pp. 115–120., [6] L. Alaus, D. Noguet, and J. Palicot, “A Reconfigurable LFSR for Tristandard SDR Transceiver, Architecture and Complexity Analysis,” in Digital System Design Architectures, Methods and Tools, 2008. DSD ’08. 11th EUROMICRO Conference on. IEEE Computer Society, 2008, pp. 61–67., [7] R. Ash, Information Theory. John Wiley & Sons, 1967., [8] M. Kopec, “Can Nonlinear Compactors Be Better than Linear Ones?” IEEE Trans. Comput., no. 11, pp. 1275–1282, 1995., [9] A. Gucha and L. Kinney, “Relating the Cyclic Behaviour of Linear Intrainverted Feedback shift Registers,” IEEE Transactions on Computers, vol. 41, no. 9, pp. 1088–1100, 1992.

By a well-known result of Green (Proc R Soc A 237:574–581, 1956) and the formal definition of Ellis and Wiegold (Bull Austral Math Soc 60:191–196, 1999), there is an integer t, say corank(G), such that \({|\mathcal{M}(G)| = p^{\frac{1}{2}n(n-1)-t}}\). In Niroomand (J Algebra 322:4479–4482, 2009), the author showed for a non-abelian group G, corank(G) ≥ logp(|G|)−2 and classified the structure of all non-abelian p-groups of corank logp(|G|)−2. In the present paper, we are interesting to characterize the structure of all p-groups of corank logp(|G|)−1.

The article is devoted to the study of an abstract function of choice and classes of such functions. We use the language of upper cuts for a formal description of binary relations. This approach makes is possible to consider different models of choice from unified theoretical positions.

We study Hilbert space operators $${A=\oplus_{i \in \mathbb{N} } A_i}$$ which are consistent in the sense that each A i+1 contains a copy of A i . The formal definition is reminiscent of the classical ordering on projections in a von Neumann algebra. It is shown that if the powers of A are simultaneously consistent, then A must be reflexive. This is applied to study reflexivity of power partial isometries.

In this paper, we propose a formal definition of a multi-argument function distance. The conventional definition of a distance is extended to apply to collections of more than two elements. Some general properties of multidistances are investigated and significant examples are exhibited. Finally, we introduce the concept of ball centered at a list in order to extend the one of the usual ball in a metric space.

This study investigates 3 preservice secondary mathematics teachers’ understandings of limits of sequences and their changing conceptions of limit during and after instruction involving interactive, dynamic sketches embodying the formal definition of the limit of a sequence. Manipulating a coherent visual representation of the formal definition in the contexts of various sequences, coupled with answering carefully chosen questions and completing interview tasks before, during, and after technologyenhanced instruction, gave the participants opportunities to investigate and reflect on their own concept image as they compared their understandings to the results of the actions they performed on the sketch. The participants modified their conceptions of limit and strengthened their understandings of formal limit ideas as they integrated the visual representation with the symbolic definition.

The purpose of this research is to introduce a new approach to the decomposition of the Gini measure in terms of concordance and discordance shares: a new kind of dependence, the Gini rank dependence (GRD), and its formal definition are provided.

Geographical space is a major concept in spatial analysis. Curiously enough, no attempt has so far been made to provide a general and rigorous presentation. In this paper, a formal definition of the basic geographical components is given. For that purpose, topology and measure theory are used. A general definition of geographical space is then proposed and some properties are examined. Finally, the relations between the proposed definition and the particular representations used in spatial analysis are discussed.

A (t,n) secret sharing divides a secret into n shares in such a way that any t or more than t shares can reconstruct the secret; but fewer than t shares cannot reconstruct the secret. In this paper, we extend the idea of a (t,n) secret sharing scheme and give a formal definition on the (n,t,n) secret sharing scheme based on Pedersen's (t,n) secret sharing scheme. We will show that the (t,n) verifiable secret sharing (VSS) scheme proposed by Benaloh can only ensure that all shares are t-consistent (i.e. any subset of t shares defines the same secret); but shares may not satisfy the security requirements of a (t,n) secret sharing scheme. Then, we introduce new notions of strong t-consistency and strong VSS. A strong VSS can ensure that (a) all shares are t-consistent, and (b) all shares satisfy the security requirements of a secret sharing scheme. We propose a strong (n,t,n) VSS based on Benaloh's VSS. We also prove that our proposed (n,t,n) VSS satisfies the definition of a strong VSS.

The standard method to retrieve information can be formally defined as follows. To ask a query, one gives the properties of the entities to be retrieved, and the answer is the set of all the entities that satisfy the query. Another method, is to ask the overall information about a given entity, and the answer is the corresponding information. An example of the first kind of query is: “what are the drugs that contain a given molecule?”, an example of the second kind is: “what are the properties of a given drug?”.This latter method has deserved very few researches though it has great potential practical applications. However, it raises many non trivial issues. The first one is to find a precise definition of the fact that a piece of information “is about” a given entity. We recall the formal definition that have been proposed in formal classical logic, and the main properties that follow from this definition. The second one is to adapt existing automated deduction methods to compute this new kind of answer, using either deduction or abduction techniques.Finally, we present potential extensions to our definition and guidelines for automated deduction strategies.