Abstract: In this paper one generalizes the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Many examples are presented. Distinctions between NS and IFS are underlined.

Abstract: We present a unified approach to the method of Nehari manifold for functionals which have a local minimum at 0 and we give several examples where this method is applied to the problem of finding ground states and multiple solutions for nonlinear elliptic boundary value problems. We also consider a recent generalization of this method to problems where 0 is a saddle point of the functional.

Abstract: Acquisition of new sensorimotor knowledge by imitation is a promising paradigm for robot learning. To be effective, action learning should not be limited to direct replication of movements obtained during training but must also enable the generation of actions in situations a robot has never encountered before. This paper describes a methodology that enables the generalization of the available sensorimotor knowledge. New actions are synthesized by the application of statistical methods, where the goal and other characteristics of an action are utilized as queries to create a suitable control policy, taking into account the current state of the world. Nonlinear dynamic systems are employed as a motor representation. The proposed approach enables the generation of a wide range of policies without requiring an expert to modify the underlying representations to account for different task-specific features and perceptual feedback. The paper also demonstrates that the proposed methodology can be integrated with an active vision system of a humanoid robot. 3-D vision data are used to provide query points for statistical generalization. While 3-D vision on humanoid robots with complex oculomotor systems is often difficult due to the modeling uncertainties, we show that these uncertainties can be accounted for by the proposed approach.

Abstract: We propose a generalization of Paillier’s probabilistic public-key system, in which the expansion factor is reduced and which allows to adjust the block length of the scheme even after the public key has been fixed, without losing the homomorphic property. We show that the generalization is as secure as Paillier’s original system and propose several ways to optimize implementations of both the generalized and the original scheme. We construct a threshold variant of the generalized scheme as well as zero-knowledge protocols to show that a given ciphertext encrypts one of a set of given plaintexts, and protocols to verify multiplicative relations on plaintexts. We then show how these building blocks can be used for applying the scheme to efficient electronic voting. This reduces dramatically the work needed to compute the final result of an election, compared to the previously best known schemes. We show how the basic scheme for a yes/no vote can be easily adapted to casting a vote for up to t out of L candidates. The same basic building blocks can also be adapted to provide receipt-free elections, under appropriate physical assumptions. The scheme for 1 out of L elections can be optimized such that for a certain range of the other parameter values, the ballot size is logarithmic in L.

Abstract: In this paper one generalizes the intuitionistic fuzzy set (IFS), paraconsistent set, and intuitionistic set to the neutrosophic set (NS). Many examples are presented. Distinctions between NS and IFS are underlined.

Abstract: When clustering produces more than one candidate to partition a finite set of objects O, there are two approaches to validation (i.e., selection of a “best” partition, and implicitly, a best value for c , which is the number of clusters in O). First, we may use an internal index, which evaluates each partition separately. Second, we may compare pairs of candidates with each other, or with a reference partition that purports to represent the “true” cluster structure in the objects. This paper generalizes many of the classical indices that have been used with outputs of crisp clustering algorithms so that they are applicable for candidate partitions of any type (i.e., crisp or soft, with soft comprising the fuzzy, probabilistic, and possibilistic cases). Space prevents inclusion of all of the possible generalizations that can be realized this way. Here, we concentrate on the Rand index and its modifications. We compare our fuzzy-Rand index with those of Campello, Hullermeier and Rifqi, and Brouwer, and show that our extension of the Rand index is O(n), while the other three are all O(n2). Numerical examples are given to illustrate various facets of the new indices. In particular, we show that our indices can be used, even when the partitions are probabilistic or possibilistic, and that our method of generalization is valid for any index that depends only on the entries of the classical (i.e., four-pair types) contingency table for this problem.

Abstract: We propose a notion of weak (n+k,n)–category, which we call (n+k,n)–Θ–spaces. The (n+k,n)–Θ–spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal’s category Θn. This notion is a generalization of that of complete Segal spaces (which are precisely the (∞,1)–Θ–spaces). Our main result is that the above model category is cartesian.

Abstract: In this note we give a simple proof and a new generalization of the Hermite-Hadamard inequality.

Abstract: Pattern generalization is considered one of the prominent routes for introducing students to algebra. However, not all generalizations are algebraic. In the use of pattern generalization as a route to algebra, we �teachers and educators� thus have to remain vigilant in order not to confound algebraic generalizations with other forms of dealing with the general. But how to distinguish between algebraic and non-algebraic generalizations? On epistemological and semiotic grounds, in this article I suggest a characterization of algebraic generalizations. This characterization helps to bring about a typology of algebraic and arithmetic generalizations. The typology is illustrated with classroom examples.

Abstract: Most generalization bounds in learning theory are based on some measure of the complexity of the hypothesis class used, independently of any algorithm. In contrast, the notion of algorithmic stability can be used to derive tight generalization bounds that are tailored to specific learning algorithms by exploiting their particular properties. However, as in much of learning theory, existing stability analyses and bounds apply only in the scenario where the samples are independently and identically distributed. In many machine learning applications, however, this assumption does not hold. The observations received by the learning algorithm often have some inherent temporal dependence. This paper studies the scenario where the observations are drawn from a stationary φ-mixing or β-mixing sequence, a widely adopted assumption in the study of non-i.i.d. processes that implies a dependence between observations weakening over time. We prove novel and distinct stability-based generalization bounds for stationary φ-mixing and β-mixing sequences. These bounds strictly generalize the bounds given in the i.i.d. case and apply to all stable learning algorithms, thereby extending the use of stability-bounds to non-i.i.d. scenarios. We also illustrate the application of our φ-mixing generalization bounds to general classes of learning algorithms, including Support Vector Regression, Kernel Ridge Regression, and Support Vector Machines, and many other kernel regularization-based and relative entropy-based regularization algorithms. These novel bounds can thus be viewed as the first theoretical basis for the use of these algorithms in non-i.i.d. scenarios.

Abstract: In this research article, I present evidence of the existence of visual templates in pattern generalization activity. Such templates initially emerged from a 3-week design-driven classroom teaching experiment on pattern generalization involving linear figural patterns and were assessed for existence in a clinical interview that was conducted four and a half months after the teaching experiment using three tasks (one ambiguous, two well defined). Drawing on the clinical interviews conducted with 11 seventh- and eighth-grade students, I discuss how their visual templates have spawned at least six types of algebraic generalizations. A visual template model is also presented that illustrates the distributed and a dynamically embedded nature of pattern generalization involving the following factors: pattern goodness effect; knowledge/action effects; and the triad of stage-driven grouping, structural unit, and analogy.

Abstract: We implement the dressing method for a novel integrable generalization of the nonlinear Schrodinger equation. As an application, explicit formulas for the N-soliton solutions are derived. As a by-product of the analysis, we find a simplification of the formulas for the N-solitons of the derivative nonlinear Schrodinger equation given by Huang and Chen.

Abstract: We introduce a natural generalization of sub-modular set cover and exact active learning with a finite hypothesis class (query learning). We call this new problem interactive submodular set cover. Applications include advertising in social networks with hidden information. We give an approximation guarantee for a novel greedy algorithm and give a hardness of approximation result which matches up to constant factors. We also discuss negative results for simpler approaches and present encouraging early experimental results.

Abstract: As an important method of terrain representation, a DEM usually needs to be generalized at multiple resolutions in order to adapt to different applications. The preservation of main landscape features is an important constraint in DEM generalization. The traditional generalization method based on signal processing by resampling or low-pass filtering is just a data compression operation rather than the abstraction of real information. This study develops a structured analysis method to generalize DEM data through the identification of minor valleys and filling the corresponding depression positions. The generalization process has two steps: geographic decision and geometric operation. According to their hydrological significance, the unimportant valley branches are detected and their corresponding coverage is filled by raising the terrain to make the terrain surface smoother. In contrast to the conventional algorithms based on image processing, this method is able to retain the main geographical characteristics more effectively in terrain representation.

Abstract: The versatility of exponential families, along with their attendant convexity properties, make them a popular and effective statistical model. A central issue is learning these models in high-dimensions, such as when there is some sparsity pattern of the optimal parameter. This work characterizes a certain strong convexity property of general exponential families, which allow their generalization ability to be quantified. In particular, we show how this property can be used to analyze generic exponential families under L_1 regularization.

Abstract: PAC-Bayes bounds are among the most accurate generalization bounds for classifiers learned from independently and identically distributed (IID) data, and it is particularly so for margin classifier...

Abstract: A meta-analysis of the reliability of the scores from a specific test, also called reliability generalization, allows the quantitative synthesis of its properties from a set of studies. It is usually assumed that part of the variation in the reliability coefficients is due to some unknown and implicit mechanism that restricts and biases the selection of participants in the studies' samples. Sometimes this variation has been reduced by adjusting the coefficients by a formula associated with range restrictions. We propose a framework in which that variation is included (instead of adjusted) in the models intended to explain the variability and in which parallel analyses of the studies' means and variances are performed. Furthermore, the analysis of the residuals enables inferences to be made about the nature of the variability accounted for by moderator variables. The meta-analysis of the 3 studies' statistics-reliability coefficient, mean, and variance--allows psychometric inferences about the test scores. A numerical example illustrates the proposed framework.

Abstract: Blough (1975) proposed an elemental model of generalization and discrimination phenomena in which stimuli from physical dimensions, such as wavelength, are conceptualized as overlapping sets of ele

Abstract: Over the past twenty years, cartographers have become increasingly concerned with the nature and quality of cartographic data in digital format. One significant area of research has been in the generalization of these digital data. Specifically, algorithms have been developed to (1) weed out unnecessary detail, (2) smooth sharp angularity, (3) displace two features coalescing due to scale change, and (4) enhance certain characteristics of the data. This paper presents an analysis of nine algorithms developed for simplifying the superfluous detail in digital cartographic lines. A series of geometric measures were designed to evaluate the changes produced by simplification. These included both single attribute measures, which evaluate changes such as line length and angularity, and displacement measures, which evaluate geometric displacement such as area. The results of this analysis, using four digitized naturally occurring lines and their simplifications, indicates that four algorithms: DOUGLAS, OPHEIM, REUMANN, and LANG are mathematically superior. Although the Douglas routine was slightly better—in terms of area displacement—than the other three, it is the most computationally complex algorithm. Thus for certain mapping tasks, such as in thematic cartography, other routines such as LANG tolerancing appear more appropriate.

Abstract: A general definition and a criterion (a necessary and sufficient condition) are formulated for an arbitrary set of external factors to selectively influence a corresponding set of random entities (generalized random variables, with values in arbitrary observation spaces), jointly distributed at every treatment (a set of factor values containing precisely one value of each factor). The random entities are selectively influenced by the corresponding factors if and only if the following condition, called the joint distribution criterion, is satisfied : there is a jointly distributed set of random entities, one entity for every value of every factor, such that every subset of this set that corresponds to a treatment is distributed as the original variables at this treatment. The distance tests (necessary conditions) for selective influence previously formulated for two random variables in a two-by-two factorial design (Kujala & Dzhafarov, 2008, J. Math. Psychol., 52, 128–144) are extended to arbitrary sets of factors and random variables. The generalization turns out to be the simplest possible one: the distance tests should be applied to all two-by-two designs extractable from a given set of factors.

Abstract: The purpose of the current study was to investigate the effect of three procedures designed to promote the generalization of fluent responding of 12 addition facts to 12 related subtraction facts. The first procedure was completed to determine whether increases in the number of digits correct per minute computed on addition facts would generalize to related subtraction facts. If students were fluent in the targeted addition problems and no generalization was detected, a second procedure was implemented where students completed a conceptual lesson. If no generalization was observed during this phase, a third procedure was implemented where students completed addition problems presented in a cloze format. Results failed to demonstrate generalized fluency gains across any of the procedures. Discussion focuses on future directions of generalization research to further our understanding of the conditions needed for generalization to occur.

Abstract: In this note we give an extended version of Combinatorial Nullstellensatz, with weaker assumption on nonvanishing monomial We also present an application of our result in a situation where the original theorem does not seem to work