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Linear equations are commonly used in various fields of science such as engineering, physics, computer science, technology, business, and economics. A recent and promising extension of fuzzy set theory is represented by hesitant fuzzy sets (HFSs). The promising characteristic of HFSs is that they permit to model a type of imprecise human knowledge that cannot be captured by classical fuzzy sets. As no research has been performed in the field of hesitant fuzzy equations, in this paper, hesitant parametric fuzzy set (HPFS) is introduced and operations on this set are investigated. Also, fully hesitant parametric fuzzy equation (FHPFE) in form of a x = b is solved for the first time in which a and b are known HPFS and x is an unknown HPFS. Then, the existence of different approximate solutions of FHPFE is investigated and proved. Finally, some numerical examples are presented that verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Recently, wide applications of fuzzy set theory have attracted the attention of both researchers and practitioners. Fuzzy portfolio develops as a new area in the research field of investment portfolio. This paper investigates the major research hotspots, development trend, and evolution of fuzzy portfolio, which provides a systematic review of the current fuzzy portfolio literature. CiteSpace, the most commonly used bibliometrics software, is used in this article. According to the 602 articles with 15132 references, several conclusions can be summarized as follows: (1) Fuzzy portfolio becomes increasingly interdisciplinary with the connections among "Computer Science" and so on. (2) Most contributive authors are Markowitz and Zadeh. (3) South China University of Technology makes excellent performance in this research area and China is the most influential country. (4) European Journal of Operations Research is the cradle of plenty of crucial fuzzy portfolio investigations. (5) We find some research hotspots helpful to make scientific predictions of future trends by analyzing the keywords. By utilizing the effective bibliometric methods, we provide a comprehensive analysis and deep insights into the fuzzy portfolio research, enabling the individuals, especially the new beginners who are interested in this area to learn fuzzy portfolio, which will be of great help for their future explorations. [ABSTRACT FROM AUTHOR]

With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]

In general, ordered algebraic structures, particularly ordered semigroups, play an important role in fuzzification in many applied areas, such as computer science, formal languages, coding theory, error correction, etc. Nowadays, the concept of ambiguity is important in dealing with a variety of issues related to engineering modeling problems, network theory, decision-making problems in real-life situations, and so on. Several theories have been developed by various researchers to overcome the difficulties that arise from uncertainty, including fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and many more. In this paper, we focus solely on neutrosophic set theory. In ordered semigroups, we define and investigate the properties of neutrosophic ℵ-ideals and neutrosophic ℵ-interior ideals. We also use neutrosophic K-ideals and neutrosophic ℵ-interior ideals to characterize ordered semigroups. [ABSTRACT FROM AUTHOR]

Recently, linear codes constructed from defining sets have been studied widely and they have many applications. For an odd prime p, let $$q=p^{m}$$ for a positive integer m and $$\mathrm {Tr}_{m}$$ the trace function from $$\mathbb {F}_{q}$$ onto $$\mathbb {F}_{p}$$ . In this paper, for a positive integer t, let $$D\subset \mathbb {F}^{t}_{q}$$ and $$D=\{(x_{1},x_{2}) \in (\mathbb {F}_{q}^{*})^{2} : \mathrm {Tr}_{m}(x_{1}+x_{2})=0\}$$ , we define a p-ary linear code $$\mathcal {C}_{D}$$ by where We compute the weight enumerators of the punctured codes $$\mathcal {C}_{D}$$ . [ABSTRACT FROM AUTHOR]

This paper deals with the specification problem in classical realizability (such as introduced by Krivine (2009 Panoramas et synthéses27)), which is to characterize the universal realizers of a given formula by their computational behaviour. After recalling the framework of classical realizability, we present the problem in the general case and illustrate it with some examples. In the rest of the paper, we focus on Peirce's law, and present two game-theoretic characterizations of its universal realizers. First, we consider the particular case where the language of realizers contains no extra instruction such as ‘quote’ (Krivine 2003 Theoretical Computer Science308 259–276). We present a first game $\mathds{G}$0 and show that the universal realizers of Peirce's law can be characterized as the uniform winning strategies for $\mathds{G}$0, using the technique of interaction constants. Then we show that in the presence of extra instructions such as ‘quote’, winning strategies for the game $\mathds{G}$0 are still adequate but no more complete. For that, we exhibit an example of a wild realizer of Peirce's law, that introduces a purely game-theoretic form of backtrack that is not captured by $\mathds{G}$0. We finally propose a more sophisticated game $\mathds{G}$1, and show that winning strategies for the game $\mathds{G}$1 are both adequate and complete in the general case, without any further assumption about the instruction set used by the language of classical realizers. [ABSTRACT FROM AUTHOR]

In the paper we first construct a new cotorsion pair, in the category of chain complexes, from two given cotorsion pairs in the category of modules, and then we consider completeness of such pairs under certain conditions. [ABSTRACT FROM AUTHOR]

The k nearest-neighbour (kNN) algorithm has enjoyed much attention since its inception as an intuitive and effective classification method. Many further developments of kNN have been reported such as those integrated with fuzzy sets, rough sets, and evolutionary computation. In particular, the fuzzy and rough modifications of kNN have shown significant enhancement in performance. This paper presents another significant improvement, leading to a multi-functional nearest-neighbour (MFNN) approach which is conceptually simple to understand. It employs an aggregation of fuzzy similarity relations and class memberships in playing the critical role of decision qualifier to perform the task of classification. The new method offers important adaptivity in dealing with different classification problems by nearest-neighbour classifiers, due to the large and variable choice of available aggregation methods and similarity metrics. This flexibility allows the proposed approach to be implemented in a variety of forms. Both theoretical analysis and empirical evaluation demonstrate that conventional kNN and fuzzy nearest neighbour, as well as two recently developed fuzzy-rough nearest-neighbour algorithms can be considered as special cases of MFNN. Experimental results also confirm that the proposed approach works effectively and generally outperforms many state-of-the-art techniques. [ABSTRACT FROM AUTHOR]

The study of various decision problems for logic fragments has a long history in computer science. This paper is on the membership problem for a fragment of first-order logic over infinite words; the membership problem asks for a given language whether it is definable in some fixed fragment. The alphabetic topology was introduced as part of an effective characterization of the fragment Σ2 over infinite words. Here, Σ2 consists of the first-order formulas with two blocks of quantifiers, starting with an existential quantifier. Its Boolean closure is 픹Σ2. Our first main result is an effective characterization of the Boolean closure of the alphabetic topology, that is, given an ω-regular language L, it is decidable whether L is a Boolean combination of open sets in the alphabetic topology. This is then used for transferring Place and Zeitoun’s recent decidability result for 픹Σ2 from finite to infinite words. [ABSTRACT FROM AUTHOR]

A variety of methods for ranking fuzzy sets has been suggested. Generally, these methods fall under two main categories: a fuzzy-real sets mapping, a dominance relation of one fuzzy set over another. The original approach proposed in this paper belongs to the second category, as the ranking is based on the degree of inclusion in the MIN of two fuzzy numbers. The novelty lies mainly in the intuitive connection between the topological relationship of fuzzy shapes (triangles, trapezoids, etc.) and the measure of inclusion or dominance referred as inclusion index. This connection led to the classification of different topological relationships into classes identified by a binary pattern. This operation is referred to as Bitset Encoding. Consequently, the outcome of a ranking is already decided for most cases by merely identifying its pattern. Ultimately, the method is validated by the axiomatic system of Wang and Kerre and proven to be a reliable, efficient and strong potential alternative to the other prominent methods. [ABSTRACT FROM AUTHOR]

Decision problems in physics have been an active field of research for quite a few decades resulting in some interesting findings in recent years. However, such research investigations are based on a priori knowledge of theoretical computer science and the technical jargon of set theory. Here, I discuss a particular, but a significant, instance of how decision problems in physics can be realised without such specific prerequisites. I expose a hitherto unnoticed contradiction, that can be posed as as decision problem, concerning the oil drop experiment, thereby resolve it by refining the notion of 'existence' in physics. This consequently leads to the undecidability of the charge spectral gap through the notion of 'undecidable charges' which is in tandem with the completeness condition of a theory as was stated by Einstein, Podolsky and Rosen in their seminal work. Decision problems can now be realised in connection with basic physics, in general, rather than quantum physics, in particular, as per some recent claims. [ABSTRACT FROM AUTHOR]

Since the critical graphs of order n ≤ 14 were studied so far [1–4,6,11] , in this paper, we are concerned with 5-critical graphs of order 15. This paper gives the complete possible list of degree sequences of 5-critical graphs of order 15. [ABSTRACT FROM AUTHOR]

Many problems in Computer Science can be abstracted to the following question: given a set of objects and rules respectively, which new objects can be produced? In the paper, we consider a succinct version of the question: given a set of binary strings and several operations like conjunction and disjunction , which new binary strings can be generated? Although it is a fundamental problem, to the best of our knowledge, the problem hasn't been studied yet. In this paper, an O ( m 2 n ) algorithm is presented to determine whether a string s is representable by a set W , where n is the number of strings in W and each string has the same length m . However, looking for the minimum subset to represent a given string is shown to be NP -hard. Also, finding the smallest subset to represent each string in the original set is NP -hard. We establish tight inapproximability results and approximation algorithms for these NP -hard problems. In addition, we prove that counting the number of representable strings is # P -complete. We then explore how the problems change when the operator negation is available. For example, if the operator negation can be used, the counting problem is as simple as some power of 2. This difference may help us to better understand the problem. [ABSTRACT FROM AUTHOR]

This paper reviews the state of the art in the theory of multisets, i.e., mathematical models of sets with repetitions (duplicates or copies of their elements). The corresponding bibliography is categorized as follows: the general theory of multisets, reviews, and application of multisets, in particular, in computer science. [ABSTRACT FROM AUTHOR]

The Dual Post Correspondence Problem asks whether, for a given word α, there exists a pair of distinct morphisms σ,τ, one of which needs to be non-periodic, such that σ(α) = τ(α) is satisfied. This problem is important for the research on equality sets, which are a vital concept in the theory of computation, as it helps to identify words that are in trivial equality sets only. Little is known about the Dual PCP for words α over larger than binary alphabets, especially for so-called ratio-primitive examples. In the present paper, we address this question in a way that simplifies the usual method, which means that we can reduce the intricacy of the word equations involved in dealing with the Dual PCP. Our approach yields large sets of words for which there exists a solution to the Dual PCP as well as examples of words over arbitrary alphabets for which such a solution does not exist. [ABSTRACT FROM AUTHOR]

The problem of automated theorem finding is one of the 33 basic research problems in automated reasoning which was originally proposed by Wos in 1988, and it is still an open problem. To solve the problem, an approach of forward deduction based on the strong relevant logics was proposed. Following the approach, this paper presents a systematic methodology for automated theorem finding. To show the effectiveness of our methodology, the paper presents two case studies, one is automated theorem finding in NBG set theory and the other is automated theorem finding in Peano's arithmetic. Some known theorems have been found in our case studies. [ABSTRACT FROM AUTHOR]

Abstract: The influence of the kth coordinate on a Boolean function f : { 0 , 1 } n → { 0 , 1 } is the probability (with respect to the uniform measure on { 0 , 1 } n) that flipping x k changes the value f ( x ). The total influence I ( f ) is the sum of influences of the coordinates. The well‐known ‘Junta theorem’ of Friedgut [Combinatorica 18 (1998) 27–35] asserts that if I ( f ) ⩽ M, then f can be ε‐approximated by a function that depends on 2 O ( M / ε ) coordinates. Friedgut's theorem has a wide variety of applications in mathematics and theoretical computer science. For a function f with Pr [ f ( x ) = 1 ] = t ⩽ 1 / 2, the edge isoperimetric inequality on the cube implies that I ( f ) ⩾ 2 t log 2 ( 1 / t ). Kahn and Kalai (2006) asked, in the spirit of the Junta theorem, whether any f such that I ( f ) is within a constant factor of the minimum, can be ( ε t )‐approximated by a disjunctive normal form (DNF) of a ‘small’ size (that is, a union of a small number of subcubes). We answer the question by proving the following structure theorem: If I ( f ) ⩽ 2 t ( log 2 ( 1 / t ) + M ), then f can be ( ε t )‐approximated by a DNF of size 2 2 O ( M / ε ). The dependence on M is sharp up to the constant factor in the double exponent. [ABSTRACT FROM AUTHOR]

The notion of multigroups via multisets is of interest in non- classical groups. In this paper, we propose the notion of homomorphism of cuts of multigroups and obtain some related results. Some homomorphic prop- erties of upper cut of multigroups are also discussed. [ABSTRACT FROM AUTHOR]

In recent years, monotonic ordinal classification has increased the focus of attention for machine learning community. Real life problems frequently have monotonicity constraints. Many of the monotonic classifiers require that the input data sets satisfy the monotonicity relationships between its samples. To address this, a conventional strategy consists of relabeling the input data to achieve complete monotonicity. As an alternative, we explore the use of preprocessing algorithms without modifying the class label of the input data. In this paper we propose the use of training set selection to choose the most effective instances which lead the monotonic classifiers to obtain more accurate and efficient models, fulfilling the monotonic constraints. To show the benefits of our proposed training set selection algorithm, called MonTSS, we carry out an experimentation over 30 data sets related to ordinal classification problems. [ABSTRACT FROM AUTHOR]

In 1974 Lawson and Hanson produced a seminal active set strategy to solve least-squares problems with non-negativity constraints that remains popular today. In this paper we present TNT-NN, a new active set method for solving non-negative least squares (NNLS) problems. TNT-NN uses a different strategy not only for the construction of the active set but also for the solution of the unconstrained least squares sub-problem. This results in dramatically improved performance over traditional active set NNLS solvers, including the Lawson and Hanson NNLS algorithm and the Fast NNLS (FNNLS) algorithm, allowing for computational investigations of new types of scientific and engineering problems. For the small systems tested (5000 × 5000 or smaller), it is shown that TNT-NN is up to 95 × faster than FNNLS. Recent studies in rock magnetism have revealed a need for fast NNLS algorithms to address large problems (on the order of 10 5 × 10 5 or larger). We apply the TNT-NN algorithm to a representative rock magnetism inversion problem where it is 60× faster than FNNLS. We also show that TNT-NN is capable of solving large (45000 × 45000) problems more than 150 × faster than FNNLS. These large test problems were previously considered to be unsolvable, due to the excessive execution time required by traditional methods. [ABSTRACT FROM AUTHOR]

In the last decades, one issue that has received a lot of attention in classification problems is how to obtain better classifications. This problem becomes even more complicated when the number of classes is high. In this multiclass scenario, it is assumed that the class labels are independent of each other, and thus, most techniques and methods proposed to improve the performance of the classifiers rely on it. An alternative way to address the multiclass problem is to hierarchically distribute the classes in a collection of multiclass subproblems by reducing the number of classes involved in each local subproblem. In this paper, we propose a new method for inducing a class hierarchy from the confusion matrix of a multiclass classifier. Then, we use the class hierarchy to learn a tree-like hierarchy of classifiers for solving the original multiclass problem in a similar way as the top-down hierarchical classification approach does for working with hierarchical domains. We experimentally evaluate the proposal on a collection of multiclass datasets showing that, in general, the generated hierarchies not only outperforms the original (flat) classification but also hierarchical approaches based on other alternative ways of constructing the class hierarchy. [ABSTRACT FROM AUTHOR]

Let F be a set system on [ n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdős, Ko and Rado from 1961 says that, provided n ≥ 2 k, any such system has size at most $$(_{k - 1}^{n - 1})$$ . A natural question, which was asked by Ahlswede in 1980, is how many disjoint pairs must appear in a set system of larger size. Except for the case k = 2, solved by Ahlswede and Katona, this problem has remained open for the last three decades. In this paper, we determine the minimum number of disjoint pairs in small k-uniform families, thus confirming a conjecture of Bollobás and Leader in these cases. Moreover, we obtain similar results for two well-known extensions of the Erdős-Ko-Rado Theorem, determining the minimum number of matchings of size q and the minimum number of t-disjoint pairs that appear in set systems larger than the corresponding extremal bounds. In the latter case, this provides a partial solution to a problem of Kleitman and West. [ABSTRACT FROM AUTHOR]

A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory with relations and applications to many other areas of mathematics as well as to theoretical computer science. A central problem in the area due to Erdős and Rado from 1960 asks for the minimum number of sets of size r needed to guarantee the existence of a sunflower of a given size. Despite a lot of recent attention including a polymath project and some amazing breakthroughs, even the asymptotic answer remains unknown. We study a related problem first posed by Duke and Erdős in 1977 which requires that in addition the intersection size of the desired sunflower be fixed. This question is perhaps even more natural from a graph theoretic perspective since it asks for the Turán number of a hypergraph made by the sunflower consisting of k edges, each of size r and with common intersection of size t. For a fixed size of the sunflower k, the order of magnitude of the answer has been determined by Frankl and Füredi. In the 1980's, with certain applications in mind, Chung, Erdős and Graham considered what happens if one allows k, the size of the desired sunflower, to grow with the size of the ground set. In the three uniform case, r=3, the correct dependence on the size of the sunflower has been determined by Duke and Erdős and independently by Frankl and in the four uniform case by Bucić, Draganić, Sudakov and Tran. We resolve this problem for any uniformity, by determining up to a constant factor the n-vertex Turán number of a sunflower of arbitrary uniformity r, common intersection size t and with the size of the sunflower k allowed to grow with n. [ABSTRACT FROM AUTHOR]

Answer Set Programming, or ASP for short, has become a popular and sophisticated approach to declarative problem solving. Its popularity is due to its attractive modeling-grounding-solving workflow that provides an easy approach to problem solving, even for laypersons outside computer science. However, in contrast to ASP's ease of use, the high degree of sophistication of the underlying technology makes it even hard for ASP experts to put ideas into practice whenever this involves modifying ASP's machinery. For addressing this issue, this tutorial aims at enabling users to build their own ASP-based systems. More precisely, we show how the ASP system clingo can be used for extending ASP and for implementing customized special-purpose systems. To this end, we propose two alternatives. We begin with a traditional AI technique and show how metaprogramming can be used for extending ASP. This is a rather light approach that relies on clingo 's reification feature to use ASP itself for expressing new functionalities. The second part of this tutorial uses traditional programming (in Python) for manipulating clingo via its application programming interface. This approach allows for changing and controlling the entire model-ground-solve workflow of ASP. Central to this is clingo 's new Application class that allows us to draw on clingo 's infrastructure by customizing processes similar to the one in clingo. For instance, we may apply manipulations to programs' abstract syntax trees, control various forms of multi-shot solving, and set up theory propagators for foreign inferences. A cross-sectional structure, spanning meta as well as application programming, is clingo 's intermediate format, aspif , that specifies the interface among the underlying grounder and solver. We illustrate the aforementioned concepts and techniques throughout this tutorial by means of examples and several nontrivial case studies. In particular, we show how clingo can be extended by difference constraints and how guess-and-check programming can be implemented with both meta and application programming. [ABSTRACT FROM AUTHOR]

Parameterized Approximation is a topic of considerable interest in the field of Parameterized Complexity. In the past decade, new color coding-related techniques, including the breakthrough representative sets technique, have been proven extremely powerful in the design of fast parameterized algorithms. Furthermore, packing problems, which are often solved via color coding-related techniques, have enjoyed a race towards obtaining the fastest parameterized algorithms that solve them. Therefore, it is natural to ask whether packing problems admit efficient parameterized approximation algorithms. In this paper, we answer this question affirmatively. We present tradeoffs that improve the running times of algorithms for well-known special cases of the 3 -Set k -Packing problem at the cost of their accuracy. Consider a packing problem for which there is no known algorithm with approximation ratio α , and a parameter k . If the value of an optimal solution is at least k , we seek a solution of value at least αk ; otherwise, we seek an arbitrary solution. Clearly, if the best known parameterized algorithm that finds a solution of value t runs in time O ⁎ ( f ( t ) ) for some function f , we are interested in running times better than O ⁎ ( f ( α k ) ) . Our main contribution lies in the adaptation of notions fundamental to the representative sets technique to the design of approximation algorithms: We introduce the definition of “approximate lopsided universal sets”, combine partial executions of representative sets-based algorithms with approximation algorithms, and adapt the iterative expansion framework (in the context of representative sets) to the design of parameterized approximation algorithms. Our ideas are intuitive, and may be relevant to the design of other parameterized approximation algorithms. [ABSTRACT FROM AUTHOR]

To contain the competitive influence spread in social networks is to maximize the influence of one participant and contain the influence of its opponent. It is desirable to develop effective strategies for influence spread of the participants themselves instead of blocking the influence spread of their opponents. In this paper, we extend the linear threshold model to establish the diffusion-containment model, abbreviated as D-C model, by incorporating the realistic specialties and characteristics of the containment of competitive influence spread. Then, we discuss the influence spread mechanism for the D-C model, and give the algorithm for the propagation of the diffusion influence (D-influence) and containment influence (C-influence). Further, we define the sub-modular set function of the C-influence in the D-C model and consequently give a greedy algorithm for solving the problem of maximizing the competitive influence containment approximately. Experimental results show the feasibility of our method. [ABSTRACT FROM AUTHOR]

Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo- b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step. [ABSTRACT FROM AUTHOR]

In the committee coordination problem , a committee consists of a set of professors and committee meetings are synchronized, so that each professor participates in at most one committee meeting at a time. In this paper, we propose two snap-stabilizing distributed algorithms for the committee coordination. Snap-stabilization is a versatile property which requires a distributed algorithm to efficiently tolerate transient faults. Indeed, after a finite number of such faults, a snap-stabilizing algorithm immediately operates correctly, without any external intervention. We design snap-stabilizing committee coordination algorithms enriched with some desirable properties related to concurrency , (weak) fairness , and a stronger synchronization mechanism called 2- Phase Discussion . In our setting, all processes are identical and each process has a unique identifier. The existing work in the literature has shown that (1) in general, fairness cannot be achieved in committee coordination, and (2) it becomes feasible if each professor waits for meetings infinitely often. Nevertheless, we show that even under this latter assumption, it is impossible to implement a fair solution that allows maximal concurrency . Hence, we propose two orthogonal snap-stabilizing algorithms, each satisfying 2-phase discussion, and either maximal concurrency or fairness. The algorithm that implements fairness requires that every professor waits for meetings infinitely often. Moreover, for this algorithm, we introduce and evaluate a new efficiency criterion called the degree of fair concurrency . This criterion shows that even if it does not satisfy maximal concurrency, our snap-stabilizing fair algorithm still allows a high level of concurrency. [ABSTRACT FROM AUTHOR]

In 1998, Chartrand, E. Salehi and P. Zhang introduced the concept of graph partition dimension. This is a variant of graph metric dimension concept introduced independently by Slater in 1975 and Harary & Melter in 1976. In this paper, we determine the partition dimension of specific classes of trees, namely homogeneous caterpillars and homogeneous banana trees. In particular, we characterize all trees in these classes with partition dimension three. [ABSTRACT FROM AUTHOR]

The assumption that the training and testing samples are drawn from the same distribution is violated under covariate shift setting, and most algorithms for the covariate shift setting try to first estimate distributions and then reweight samples based on the distributions estimated. Due to the difficulty of estimating a correct distribution, previous methods can not get good classification performance. In this paper, we firstly present two types of covariate shift problems. Rather than estimating the distributions, we then desire an effective method to select a maximum subset following the target testing distribution based on feature space split from the auxiliary set or the target training set. Finally, we prove that our subset selection method can consistently deal with both scenarios of covariate shift. Experimental results demonstrate that training a classifier with the selected maximum subset exhibits good generalization ability and running efficiency over those of traditional methods under covariate shift setting. [ABSTRACT FROM AUTHOR]

A detailed and extensive empirical study of empty prediction of multi-label classification is conducted in this paper and to the best of our knowledge this work is the first empirical study of this problem. Total 8 state of the art multi-label classification methods, BR, CC, CLR, HOMER, RAkEL, ECC, MLkNN, and BRkNN, are compared on 11 datasets. The empirical results clearly answer the two research questions, (1) whether empty prediction problems happen in commonly used state of the art multi-label classification methods and what their empty prediction rates (EPR) on different test sets are and (2) what multi-label classification methods are with overall highest/lowest EPRs. Specifically, it is empirically shown that every method considered all made empty predictions on different datasets. In addition, several thresholding methods which in theory can solve empty prediction are compared. The clear answers to the two research questions and the experimental findings are the main contributions of this work to multi-label classification. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of smg-closed sets in biminimal structure space and a new notion of a pair wise smg-closed set is defined and studied some of its properties. [ABSTRACT FROM AUTHOR]

The minimum eccentric dominating graph MmD E (G) of a graph G is defined to be the intersection graph on the minimum eccentric dominating sets of G. In this paper minimum eccentric dominating graph of some families of graphs are studied. [ABSTRACT FROM AUTHOR]