Search

Let F be the finite field of order q and M (n , r , F) be the set of n × n matrices of rank r over the field F. For α ∈ F and A ∈ M (n , F) , let Z A , r α = { X ∈ M (n , r , F) | Tr (A X) = α }. In this article, we solve the problem of determining the cardinality of Z A , r α. We also solve the generalization of the problem to rectangular matrices. [ABSTRACT FROM AUTHOR]

Chronometric counting is a method to approximate the duration of a time interval by keeping track of the accumulation of its one-second subintervals. The ordinality of the number words is instrumental to this method, but whether also the magnitudes of these words affect the approximations remains unclear. The participants performed self-initiated and silent chronometric counting in different directions to produce target intervals prospectively. Two experiments were conducted. In Experiment 1, counting from 1- to 6-s target intervals started or stopped at zero. In Experiment 2, 1- or 2-s target intervals were counted with low-magnitude (1–3) or high-magnitude (4–6) number words. The participants were found to overproduce target intervals towards their shorter durations (Experiments 1 and 2) and, at a trend level, with downward rather than upward counting (Experiment 1 but not Experiment 2). They also produced target intervals as longer in duration with high- than low-magnitude number words (Experiment 2). The main findings suggest an involuntary magnitude effect of endogenously activated number words on subjective time. [ABSTRACT FROM AUTHOR]

This study aimed to estimate the number of distinct tooth colors using a large dataset of in-vivo CIELAB measurements. It further assessed the coverage error (CE) and coverage error percentage (CEP) of commonly used shade guides and determined the number of shades needed for an ideal guide, using the Euclidean distance (ΔEab) and thresholds for clinical perceptibility (PT) and acceptability (AT) as evaluation criteria. A total of 8153 untreated maxillary and mandibular anterior teeth were measured in vivo using calibrated dental photography. Cardinality was applied to determine the number of unique natural tooth colors. The CE and CEP were calculated for the Vita Classical and Vita 3D-Master shade guides, while the cardinality method was also used to estimate the number of shades required to adequately cover the estimated gamut of natural tooth colors. The cardinality analysis revealed 1173 unique natural tooth colors. The CE for the Vita Classical shade guide was 4.1 ΔEab, with a CEP of 75 % beyond AT, while the 3D-Master shade guide had a CE of 3.3 ΔEab and a CEP of 70 % beyond AT. Based on cardinality computation, 92 discrete shades are required to adequately cover the estimated gamut of natural tooth colors with a CE of 1.2 ΔEab and CEP of 0.3 % beyond AT. Cardinality computations estimated 1173 unique tooth colors while 92 discrete shades are estimated for full coverage. Such a number is impractical for physical shade guides, but new digital tools and 3D printing may offer future solutions. Both, the Vita Classical and 3D-Master shade guides do not fully represent the range of natural tooth colors. This study highlights the limitations of existing shade guides and underscores the potential for new developments. [ABSTRACT FROM AUTHOR]

State Early Learning Standards (ELS) are multi-function tools that inform early childhood instruction and practices. Using an established framework of early numerical development, this study assessed the prevalence of number, number relations, and number operations indicators in ELS, specifically indicators of counting, numeral knowledge, cardinality, relations among quantity and number, and operations. The type of quantity representation and the set sizes, or upper limits, assigned to counting, subitizing, and cardinality were also summarized. State ELS were organized into two groups: Group 1 (n = 23) included states with one set of indicators for preschool (e.g., ages three to five years) and Group 2 (n = 27) included states with at least two sets of indicators for preschool (e.g., 36 to 48 months and 48 to 60 months). Key findings include: (1) how states organize their ELS is associated with level of consensus for the upper limits and set sizes associated with counting and cardinality; (2) notable variability in the prevalence of early number indicators with gaps for indicators of advanced counting, cardinal principle knowledge, symbolic number relations, and ordering; and, (3) differences in how ELS address the type of quantity representation. Results are intended to highlight strengths and shortcomings of state ELS for number, number relations, and number operations and to offer considerations for future revisions. [ABSTRACT FROM AUTHOR]

In this paper we provide some general results about topological spaces X which satisfy starn- P , weakly starn- P and almost starn- P, for P∈ { κ ᴄᴄ , W κ L , ᴅ κ ᴄᴄ } , where κ is an infinite cardinal number. The particular cases when κ = ω , P ∈ { ᴄᴄᴄ , weakly Lindelöf , ᴅ ᴄᴄᴄ } are obtained. Furthermore, for the same classes of spaces defined by such P, by applying Erdős-Radó's theorem and using the rank l-diagonal notion, we establish some cardinal inequalities.

The aim of this work is to describe subsets of Banach limits in terms of a certain functional characteristic. We compute radii and cardinalities for some of these subsets. [ABSTRACT FROM AUTHOR]

Expressions like two novels are traditionally taken to convey information about cardinality and are analyzed using a cardinality function. Salmon (Philosop Perspect 11:1–15, 1997), Liebesman (Australasian J Philos 93:21–42, 2015; Philos' Impr 16:1–25, 2016; In D. W. Zimmerman (ed.), Oxford studies in metaphysics, Oxford University Press, Oxford, forthcoming), and Haida and Trinh (in: Dočekal, Wagiel (eds) Formal approaches to number in Slavic and beyond, Language Science Press, Berlin, 2001) argue against this traditional account, claiming that it can't explain our use of expressions like two and a half novels. According to them, the proper analysis of such expressions requires a partiality measure, which maps entities to rational numbers such as 2.5. In this paper, we set out to defend the traditional account. To do so, we demonstrate that an analysis based on a partiality measure is inconsistent with the truth conditions of plural comparatives and equatives. We also show that such an account doesn't provide an adequate analysis for expressions like two novels and a half, nor for their French counterparts (e.g., deux romans et demi). Critically, such expressions contain no constituent that could refer to the number 2.5. We provide an alternative analysis for all these expressions, based on cardinality and an operation of non-overlapping summation. [ABSTRACT FROM AUTHOR]

Mapping skills between different codes to represent numerical information, such as number symbols (i.e., verbal number words and written digits) and non‐symbolic quantities, are important in the development of the concept of number. The aim of the current study is to investigate children's mapping skills by incorporating another numerical code that emerges at early stages in development, finger patterns. Specifically, the study investigates (i) the order in which mapping skills develop and the association with young children's understanding of cardinality; and (ii) whether finger patterns are processed similarly to symbolic codes or rather as non‐symbolic quantities. Preschool children (3‐year‐olds, N = 113, Mage = 40.8 months, SDage = 3.6 months; 4‐year‐olds, N = 103, Mage = 52.9 months, SDage = 3.4 months) both cardinality knowers and subset‐knowers, were presented with twelve tasks that assessed the mappings between number words, Arabic digits, finger patterns, and quantities. The results showed that children's ability to map symbolic numbers precedes the understanding that such symbols reflect quantities, and that children recognize finger patterns above their cardinality knowledge, suggesting that finger patterns are symbolic in essence. Research Highlights: Children are more accurate in mapping between finger patterns and symbols (number words and Arabic digits) than in mapping finger patterns and quantities, indicating that fingers are processed holistically as symbolic codes.Children can map finger patterns to symbols above their corresponding cardinality level even in subset‐knowers.Finger patterns may play a role in the process by which children learn to map symbols to quantities.Fingers patterns' use in the classroom context may be an adequate instructional and diagnostic tool. [ABSTRACT FROM AUTHOR]

In this paper we provide some general results about topological spaces X satisfying any of the following properties: starn-P, weakly starn-P or almost starn-P, for P C {kCC,WkL, DkCC}, where k is an infinite cardinal number. The particular cases when k = u, P C {ccc, weakly Lindelof, Dccc} are obtained. Furthermore, for the same classes of spaces defined by such P, by applying Erdos-Rado's theorem and using the rank l-diagonal notion, we establish some cardinal inequalities. [ABSTRACT FROM AUTHOR]

This paper continues the study of second-order signature properties—the characterization of the extreme 3-uniform hypergraph. Previously, bases were used to count extreme 3-uniform hypergraphs. However, the algorithm using this mechanism is extremely labor-intensive. The structure of the signature allows us to use it as a more efficient basis for the same problem. Here, we establish the nature of the mutual correspondence between the kind of second-order signature and extreme hypergraphs, and we present a new algorithm to find the power of the set of extreme 3-uniform hypergraphs through the set of their characteristic-signatures. New results obtained with the proposed tool are also presented. [ABSTRACT FROM AUTHOR]

The notion of Fuzzy Graphs (FGs) and Intuitionistic Fuzzy Graphs (IFGs) is generalized in the Picture Fuzzy Graph (PFG), which is a more common platform for expressing the degree of positive, negative, and neutral membership functions. Picture Fuzzy Soft Graphs (PFSGs) are powerful mathematical tools for modeling real-world vagueness. The concept of vulnerability parameters of PFSG is provided in this research work by introducing the novel cardinality, domination number, integrity, and Domination Integrity (ÐÏ) of PFSG. Furthermore, a decision-making method for the PFSG has been presented using ÐÏ to suggest an algorithm for determining the ideal location for establishing a city diagnosis center

This paper examines the syntax of additive compound numerals in Modern Standard Arabic (MSA), uncovering their unique properties related to number morphology, definiteness, and Case assignment within numeral–noun constructions. These properties necessitate a constituency analysis which reveals that compound numerals have the structure of copulative compounds in MSA, and they are phrases, not functional heads. Drawing on the distinction between inherent, lexical, and structural Cases, this paper posits that the accusative Case on the numerals is an inherent Case, inaccessible to syntactic transformations. Furthermore, the analysis of numeral–noun constructions as numerically quantified phrases (NQPs) explains the assignment of a structural accusative Case or the inherent genitive Case on the quantified noun, based on the overtness of NQ0. Finally, the paper addresses the intriguing question of how NQPs in MSA, despite lacking a nominative Case, can assume the subject position and govern agreement in both verbal and verbless sentences. [ABSTRACT FROM AUTHOR]

In this paper,  (⊙ , ∨) -multiderivations on an MV-algebra A are introduced, the relations between  (⊙ , ∨) -multiderivations and  (⊙ , ∨) -derivations are discussed. The set  MD (A)  of  (⊙ , ∨) -multiderivations on A can be equipped with a preorder, and  (MD (A) / ∼ , ≼)  can be made into a partially ordered set with respect to some equivalence relation ∼. In particular, for any finite MV-chain  L n ,  (MD (L n) / ∼ , ≼)  becomes a complete lattice. Finally, a counting principle is built to obtain the enumeration of  MD (L n) . [ABSTRACT FROM AUTHOR]

This research aims to show the effects of adding cardinality constraints to limit the number of different cross-sections used in simultaneous sizing and shape optimization of truss structures. The optimal solutions for sizing and shape optimized trusses result in a generally high, and impractical, number of different cross-sections being used. This paper presents the influence of constraining the number of different cross-sections used on the optimal results to bring the scientific results closer to the applicable results. The savings achieved using the cardinality constraint are expected to manifest in more than just the minimization of weight but in all the other aspects of truss construction, such as labor, assembly time, total weld length, surface area to be treated, transport, logistics, and so on. It is expected that the optimal weight of the structures would be greater than when not using this constraint; however, it would still be below conventionally sized structures and have the added benefits derived from the simplicity and elegance of the solution. The results of standard test examples for each different cardinality constraint value are shown and compared to the same examples using only a single cross-section on all bars and the overall optimal solution, which does not have the cardinality constraint. An additional comparison is made with results of just the sizing optimization from previously published research where authors first used the same cardinality constraint. [ABSTRACT FROM AUTHOR]

I propose an account of possible worlds. According to the account, possible worlds are pluralities of sentences in an extremely large language. This account avoids a problem, relating to the total number of possible worlds, that other accounts face. And it has several additional benefits. [ABSTRACT FROM AUTHOR]

The notion of Fuzzy Graphs (FGs) and Intuitionistic Fuzzy Graphs (IFGs) is generalized in the Picture Fuzzy Graph (PFG), which is a more common platform for expressing the degree of positive, negative, and neutral membership functions. Picture Fuzzy Soft Graphs (PFSGs) are powerful mathematical tools for modeling real-world vagueness. The concept of vulnerability parameters of PFSG is provided in this research work by introducing the novel cardinality, domination number, integrity, and Domination Integrity (ÐÏ) of PFSG. Furthermore, a decision-making method for the PFSG has been presented using ÐÏ to suggest an algorithm for determining the ideal location for establishing a city diagnosis center. [ABSTRACT FROM AUTHOR]

This research presents a novel portfolio optimization model that incorporates asset preselection. This model aims to demonstrate how using Long and Short-Term Memory and Sharpe Ratio Maximization can enhance the efficiency of portfolios.The suggested approach consists of three stages, each with practical applications. During the initial phase, the data is gathered. In the second phase, the LSTM network, a commonly employed tool in predicting stock price movements, is utilized to anticipate the time series of stock closing prices. The third stage of the process focuses on stock selection and determining the appropriate weighting for each stock in the portfolio. The proposed approach is tested and carefully validated using the daily closing prices of ten equities from the FTSE 100. The results demonstrate the model’s resilience and efficacy, as the portfolios generated using the anticipated and validation data exhibit high similarity. Given the importance of selecting the right stocks for portfolio optimization, this study will combine asset preselection with portfolio weighting. Furthermore, this study utilized a Long Short-Term Memory network to forecast the optimization model’s parameters accurately and introduced a novel portfolio optimization model.

Sparse optimization–fitting data with a low-cardinality linear model–is addressed through the minimization of a cardinality-penalized least-squares function, for which dedicated branch-and-bound algorithms clearly outperform generic mixed-integer-programming solvers. Three acceleration techniques are proposed for such algorithms. Convex relaxation problems at each node are addressed with dual approaches, which can early prune suboptimal nodes. Screening methods are implemented, which fix variables to their optimal value during the node evaluation, reducing the subproblem size. Numerical experiments show that the efficiency of such techniques depends on the node cardinality and on the structure of the problem matrix. Last, different exploration strategies are proposed to schedule the nodes. Best-first search is shown to outperform the standard depth-first search used in the related literature. A new strategy is proposed which first explores the nodes with the lowest least-squares value, which is shown to be the best at finding the optimal solution–without proving its optimality. A C++ solver with compiling and usage instructions is made available. [ABSTRACT FROM AUTHOR]

• Children perform worse on give-N compared to how many? • Subitizing, give-N , and how many? performance is best represented by separate factors. • Common measures of cardinal number knowledge should not be used interchangeably. In the current study we investigated performance on the how many (how many?) and give-N tasks. We first investigated the relative performance of three-to-five-year-old children (N = 393; M = 4.75 years, SD =.75 years) on these tasks. Replicating prior work, we found that children performed worse on give-N compared to how many? and this performance gap increased as the set size increased. This performance gap remained even after controlling for children's counting skill. We next conducted a series of confirmatory factor analyses to identify the best fitting factor structure of three tasks that measure children's understanding of set sizes across both small and large sets: a subitizing measure, how many? , and give-N. We found that the best fitting factor structure contained separate factors for each task. This suggests that each measure provides task-specific variance not captured by the other cardinality measures. These results highlight the different aspects of children's understanding of cardinality. [ABSTRACT FROM AUTHOR]

Some studies have found that genetic algorithms using a single population for portfolio optimization with cardinality constraints often converge slowly and do not achieve the best results. One way to improve the performance of these algorithms has been to incorporate an additional population that acts as a seeker of local maxima and minima; this increases the likelihood of finding the global optimum of the solution in a shorter time. This document seeks to identify the in-sample and out-of-sample performance of an equity portfolio with cardinality constraints using genetic algorithms with a single population and with a double population, using the Dow Jones index as the universe. The results show that performance can be affected by the parameters selected for optimization, so it is important to consider the error in estimating the mean and variance of the portfolio. [ABSTRACT FROM AUTHOR]

Copyright of DIS - Dislessia, Discalculia & Disturbi di Attenzione is the property of Edizioni Centro Studi Erickson SpA and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The concept of infinity refers to either an unending process or a limitless quantity. Aristotle introduced two types of infinity: potential infinity and actual infinity. Potential infinity refers to a never-ending process, and actual infinity refers to a collection containing infinitely many elements. This paper presents a descriptive study of the concept of infinity and discusses its properties through set theory and calculus. Infinity plays a central role in the formation and development of mathematics, specifically in limit, derivative, and integral. Moreover, the similarities and differences between potential infinity and actual infinity are explained with the help of set theory and integral differential calculus. The relationship between mathematics and infinity is a vital one. Since infinity is an elusive and contradictory idea without mathematical tools, it is hard to understand it, and there is no other knowledge to explain and make it understandable. By the way, in the absence of infinity, mathematics will never survive. This paper provided some examples to show that without employing mathematics, solving problems involving infinity based on human intuitions or weak induction may provide inaccurate results or lead to contradictions. Therefore, this paper suggested that using mathematical tools is essential for solving problems involving infinity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the compactness and cardinality of the space C(X, Y ) of continuous functions from a topological space X to Y equipped with the regular topology. We prove that different forms of compactness, such as sequential compactness, countable compactness, and pseudocompactness, coincide on a subset of C(X, Y ) with regular topology. Moreover, we prove the comparison and coincidence of regular topology with the graph topology on the space C(X, Y ). Furthermore, we examine various cardinal invariants, such as density, character, pseudocharacter, etc., on the space C(X, Y ) equipped with the regular topology. In addition, we define a type of equivalence between X and Y in terms of C(X) and C(Y ) endowed with the regular topology and investigate certain cardinal invariants preserved by this equivalence. [ABSTRACT FROM AUTHOR]

We review and compare five ways of assigning totally ordered sizes to subsets of the natural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and a-numerosity. Generalised densities and a-numerosities lack uniqueness, which can be traced to intangibles: objects that can be proven to exist in ZFC while no explicit example of them can be given. As a sixth and final formalism, we consider a recent proposal by Trlifajová (2024), which we call c-numerosity. It is fully constructive and uniquely determined, but assigns merely partially ordered numerosity values. By relating all six formalisms to each other in terms of the underlying limit operations, we get a better sense of the intrinsic limitations in determining the sizes of subsets of N. [ABSTRACT FROM AUTHOR]

Finger gnosia and fine motor skills (FMS) are assumed to play a key role in the development of arithmetic abilities, but their contribution to early numerical skills (i.e., enumeration skills and cardinality) has received little attention so far. The purpose of this study was to investigate the predictive value of finger gnosia and FMS to enumeration, cardinal, and arithmetical abilities and how these different dimensions contribute to arithmetic development. Overall, 3- to 5-year-old preschoolers were asked to perform tasks assessing enumeration, cardinality, and early arithmetic, as well as finger gnosia and FMS. FMS, involving either static or dynamic fine finger movement, were examined using 3D motion analyses. Using hierarchical regression, FMS were found to be the best predictor of both cardinality and early arithmetic skills, while finger gnosia did not predict the additional variance of arithmetic performance when FMS and age were considered in the regression model. Moreover, neither finger gnosia nor FMS were significant predictors of enumeration. Mediation analyses indicated that knowledge of the cardinal principle fully mediates the relationship between FMS and arithmetic skills, suggesting that FMS contribute to cardinal principle knowledge development, which would be a gateway to more complex arithmetical processing. Public Significance Statement: This study suggests that fine motor skills contribute to the development of cardinality, which would be a gateway to more complex arithmetic processing. [ABSTRACT FROM AUTHOR]

Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus 'grassroots mathematics'. We begin with a brief review of $\mathsf {FO}(\#)$ , first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, $\mathsf {MFO}(\#)$. We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced. Next, we investigate a series of strengthenings of $\mathsf {MFO}(\#)$ , again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system $\mathsf {ML}(\#)$ that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of $\mathsf {ML}(\#)$ , and provide a new kind of bisimulation matching the expressive power of the language. As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of $\mathsf {FO}(\#)$ by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of 'mass' or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables. Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite 'semantic automata'. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the idea of how to construct transfinite numbers and study transfinite arithmetic. The research method used is a literature review, which involves collecting various sources such as scientific papers and books related to Cantorian set theory, infinity, ordinal or transfinite arithmetic, as well as the connection between infinity and theology. The study also involves constructing the objects of study, namely ordinal numbers such as finite ordinals and transfinite ordinals, and examining their arithmetic properties. The results of this research include the methods of constructing both finite and transfinite ordinal numbers using two generation principles. Both finite and transfinite ordinal numbers are defined as well-orderings that are also transitive sets. Arithmetic of finite ordinal numbers is well-known, but the arithmetic of transfinite ordinal numbers will be introduced in this paper, including addition, multiplication, and exponentiation.

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existence of the number sense is still pending the investigation of a further issue, namely, whether the mechanisms the brain uses to assess numerosity qualify as numerical or non-numerical.

Development of the cardinality principle, an understanding that the last number-word recited in counting a collection of items specifies the number of items in that collection, is a critical milestone in developing a concept of number. Researchers in early number development have endeavored to theorize its development. Here we critique two widely respected hypotheses that explain cardinality-principle development as building on young learners' ability to subitize small numbers. These hypotheses consider subitizing to be the basis of cardinality-principle development. We argue that there is a qualitative and significant difference between subitizing and the cardinality principle and that the explanation provided are insufficient to account for a change of this magnitude. We then propose a conjecture intended to better explain this change. The conjecture describes counting as the medium for a series of reflective abstractions leading to the cardinality principle. [ABSTRACT FROM AUTHOR]

Both convex and non-convex penalties have been widely proposed to tackle the sparse index tracking problem. Owing to their good property of generating sparse solutions, penalties based on the least absolute shrinkage and selection operator (LASSO) and its variations are often suggested in the stream of convex penalties. However, the LASSO-type penalty is often shown to have poor out-of-sample performance, due to the relatively large biases introduced in the estimates of tracking portfolio weights by shrinking the parameter estimates toward to zero. On the other hand, non-convex penalties could be used to improve the bias issue of LASSO-type penalty. However, the resulting problem is non-convex optimization and thus is computationally intensive, especially in high-dimensional settings. Aimed at ameliorating bias introduced by LASSO-type penalty while preserving computational efficiency, this paper proposes a multi-step convex optimization approach based on the multi-step weighted LASSO (MSW-LASSO) for sparse index tracking. Empirical results show that the proposed method can achieve smaller out-of-sample tracking errors than those based on LASSO-type penalties and have performance competitive to those based on non-convex penalties. [ABSTRACT FROM AUTHOR]

The authors assessed a battery of number skills in a sample of over 500 preschoolers, including both monolingual and bilingual/multilingual learners from households at a range of socio-economic levels. Receptive vocabulary was measured in English for all children, and also in Spanish for those who spoke it. The first goal of the study was to describe entailment relations among numeracy skills by analyzing patterns of co-occurrence. Findings indicated that transitive and intransitive counting skills are jointly present when children show understanding of cardinality and that cardinality and knowledge of written number symbols are jointly present when children successfully use number lines. The study’s second goal was to describe relations between symbolic numeracy and language context (i.e., monolingual vs. bilingual contexts), separating these from well-documented socio-economic influences such as household income and parental education: Language context had only a modest effect on numeracy, with no differences detectable on most tasks. However, a difference did appear on the scaffolded number-line task, where bilingual learners performed slightly better than monolinguals. The third goal of the study was to find out whether symbolic number knowledge for one subset of children (Spanish/English bilingual learners from low-income households) differed when tested in their home language (Spanish) vs. their language of preschool instruction (English): Findings indicated that children performed as well or better in English than in Spanish for all measures, even when their receptive vocabulary scores in Spanish were higher than in English.

Summary: In the article, we devise streaming algorithms for maximization of a monotone submodular function subject to a cardinality constraint on the integer lattice. Based on the observation that lattice submodularity is not equivalent to diminishing return submodularity on the integer lattice but rather a weaker condition, we propose a one‐pass streaming algorithm with a modified binary search as subroutine of each step. Finally, we show that the algorithm is with approximation ratio 1/2−ε, memory complexity O(ε−1klogk), and per‐element query complexity O(ε−2log2k). [ABSTRACT FROM AUTHOR]

Previous research suggests that subset-knowers have an approximate understanding of small numbers. However, it is still unclear exactly what subset-knowers understand about small numbers. To investigate this further, we tested 133 participants, ages 2.6–4 years, on a newly developed eye-tracking task targeting cardinal recognition. Participants were presented with two sets differing in cardinality (1–4 items) and asked to find a specific cardinality. Our main finding showed that on a group level, subset-knowers could identify all presented targets at rates above chance, further supporting that subset-knowers understand several of the basic principles of small numbers. Exploratory analyses tentatively suggest that 1-knowers could identify the targets 1 and 2, but struggled when the target was 3 and 4, whereas 2-knowers and above could identify all targets at rates above chance. This might tentatively suggest that subset-knowers have an approximate understanding of numbers that is just (i.e. +1) above their current knower level. We discuss the implications of these results at length.

There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math. 106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman's work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to advance understanding of intellectually virtuous honesty, by examining the relationship between a recent account of intellectual honesty and a recent account of intellectual transparency. The account of intellectual honesty comes from Nathan King, who adapts the work of Christian Miller on moral honesty, while the account of intellectual transparency comes from T. Ryan Byerly. After introducing the respective accounts, I identify four potential differences between intellectual honesty and intellectual transparency as understood by these accounts. I then turn to the question of how to think about the relationship between these traits in light of these potential differences. I make the case that intellectual transparency can either be regarded as an exceptionally strong or ideal variety of intellectual honesty, or it can be regarded as a distinct virtue from intellectual honesty which is a more cardinal virtue than the latter. Along the way, I also note some places where a case can be made that Miller's and King's accounts of honesty and intellectual honesty are in need of refinement or clarification. [ABSTRACT FROM AUTHOR]

This paper examines the syntax of additive compound numerals in Modern Standard Arabic (MSA), uncovering their unique properties related to number morphology, definiteness, and Case assignment within numeral–noun constructions. These properties necessitate a constituency analysis which reveals that compound numerals have the structure of copulative compounds in MSA, and they are phrases, not functional heads. Drawing on the distinction between inherent, lexical, and structural Cases, this paper posits that the accusative Case on the numerals is an inherent Case, inaccessible to syntactic transformations. Furthermore, the analysis of numeral–noun constructions as numerically quantified phrases (NQPs) explains the assignment of a structural accusative Case or the inherent genitive Case on the quantified noun, based on the overtness of NQ0. Finally, the paper addresses the intriguing question of how NQPs in MSA, despite lacking a nominative Case, can assume the subject position and govern agreement in both verbal and verbless sentences.

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existence of the number sense is still pending the investigation of a further issue, namely, whether the mechanisms the brain uses to assess numerosity qualify as numerical or nonnumerical. [ABSTRACT FROM AUTHOR]

Recent studies point to an embodied representation of numbers. We reconcile these ideas with traditional influential accounts of numerical cognition. We identify symbolic number representations of visual, verbal, and sensorimotor form. Number symbols evoke three distinct semantic concepts: magnitude, ordinality, and cardinality. We propose a sensorimotor perspective of numerical cognition in which number meaning emerges from differently grounding these concepts in sensorimotor experiences. Numbers are present in every part of modern society and the human capacity to use numbers is unparalleled in other species. Understanding the mental and neural representations supporting this capacity is of central interest to cognitive psychology, neuroscience, and education. Embodied numerical cognition theory suggests that beyond the seemingly abstract symbols used to refer to numbers, their underlying meaning is deeply grounded in sensorimotor experiences, and that our specific understanding of numerical information is shaped by actions related to our fingers, egocentric space, and experiences with magnitudes in everyday life. We propose a sensorimotor perspective on numerical cognition in which number comprehension and numerical proficiency emerge from grounding three distinct numerical core concepts: magnitude, ordinality, and cardinality. [ABSTRACT FROM AUTHOR]

Formal reasoning about finite sets and cardinality is important for many applications, including software verification, where very often one needs to reason about the size of a given data structure. The Constraint Logic Programming tool $$\{ log\} $$ provides a decision procedure for deciding the satisfiability of formulas involving very general forms of finite sets, although it does not provide cardinality constraints. In this paper we adapt and integrate a decision procedure for a theory of finite sets with cardinality into $$\{ log\} $$. The proposed solver is proved to be a decision procedure for its formulas. Besides, the new CLP instance is implemented as part of the $$\{ log\} $$ tool. In turn, the implementation uses Howe and King's Prolog SAT solver and Prolog's CLP(Q) library, as an integer linear programming solver. The empirical evaluation of this implementation based on +250 real verification conditions shows that it can be useful in practice. Under consideration in Theory and Practice of Logic Programming (TPLP) [ABSTRACT FROM AUTHOR]