Cardinal Number Research Articles

Defining Collective Nouns: How Cognitive Linguistics Can Help Hebrew Grammarians

In previous studies of the collective noun in Hebrew, collectives have been treated as a homogeneous group of nouns. Collectives have been defined as a noun that refers to a group of real-world entities and is grammatically singular. Sometimes included in this discussion are mass nouns and proper nouns. This article argues that mass nouns and proper nouns should not be included as part of the discussion about collective nouns. Then it seeks to apply the research of Joosten et al. on Dutch collective nouns to Biblical Hebrew. Joosten identifies three types of collective nouns. First, Type 1 nouns focus on the collection as a whole. On the other end of the spectrum are Type 3 nouns which focus on the members of a collection. Type 2 nouns have the flexibility to refer to the collection in one context, while in others it can have a member focus. Using a sampling of collectives that appear in Deuteronomy, this article uses attributives, demonstratives, and cardinal numbers to help identify collective nouns in Biblical Hebrew using Joosten’s categories. Observing these syntactical elements alongside collectives this article shows that Type 1 nouns will have agreement between adjectives and collectives. Type 3 nouns will have disagreement. Type 2 nouns will, depending on the focus, have either agreement or disagreement.

Computation of Metric Dimension of Certain Subdivided Convex Polytopes

The distance d(z1, z2) from vertex z1 ∈ V(G) to z2 ∈ V(G) is minimum length of (z1, z2)‐path in a given connected graph G having E(G) and V(G) edges and vertices’/nodes’ sets, respectively. Suppose Z = {z1, z2, z3, …, zm}⊆V(G) is an order set and c ∈ V(G), and the code of c with reference to Z is the m‐tuple {d(c, z1), d(c, z2), d(c, z13), …, d(c, zk)}. Then, Z is named as the locating set or resolving set if each node of G has unique code. A locating set of least cardinality is described as a basis set for the graph G, and its cardinal number is referred to as metric dimension symbolized by dim(G). Metric dimension of certain subdivided convex polytopes STn has been computed, and it is concluded that just four vertices are sufficient for unique coding of all nodes belonging to this family of convex polytopes.

Supporting the understanding of cardinal number knowledge in preschoolers: Evidence from instructional practices based on finger patterns

The acquisition of cardinal numbers represents a crucial milestone in the development of early numerical skills and more advanced math abilities. However, relatively few studies have investigated how children's grasping of the cardinality principle can be supported. It has been suggested that the richness of number inputs children receive influences the acquisition of cardinal numbers. The present study was designed to investigate whether canonical finger patterns representing numbers may contribute to this acquisition. Fifty-one 3-year-olds were randomly assigned to 1 of 2 training conditions: (a) a condition that involved counting and labeling, which has shown efficacy to support the acquisition of cardinality, and (b) a condition in which counting and labeling were enriched with finger patterns. Crucially, we aimed at providing evidence of both training programs in a real-life learning environment where teachers incorporated the training as a group-based activity into their regular schedule of daily activities. Children assigned to the finger-based condition outperformed those who received the counting-and-label training. Findings suggest that finger patterns may have a role in children's cardinality understanding. Furthermore, our study shows that instructional approaches for improving cardinality understanding can be easily and successfully implemented into real-life learning settings.

NON-CLASSICAL FOUNDATIONS OF SET THEORY

AbstractIn this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories.

«Раз, два – дерева…»: лічилки у практиці викладання української мови іноземним студентам

Abstract. The article is devoted to the research of linguodidactic potential of counting-out rhymes in classes of Ukrainian as a foreign language. Now there are many works in Ukrainian methodology of teaching language to foreign students concerning the formation of a set of texts for studying purposes. The authors of selected scientific articles suggest including folklore texts (fairy tales, songs, legends, proverbs and sayings, tongue twisters) in the teaching of the Ukrainian language. The aim of our article is to explore the place and role of the counting-out rhyme as a supplementary text for foreign students of beginner level. For this purpose, we describe the definition of the term «counting-out rhyme», define the linguistic and methodological criteria for selecting texts and how they should be presented to foreign audiences. Results of the research. The counting-out rhyme is a genre of children's playful folklore, a small-form poetic text with a clear rhyme-rhythmic structure. For to study Ukrainian as a foreign language at the beginning stage not all the texts of counting-out rhymes are suitable, but only those that folklorist G. Vinogradov called «chislovka» or «proper chislovka". This type of counting-out rhymes have cardinal numbers and ordinal numbers as a part of their structure. We consider that the main criteria for selecting and presenting texts of this genre as educational units should be: 1) a simple structure, clear and understandable content of the text; 2) the common vocabulary of counting-out rhymes, which corresponds as much as possible to communicative topics and grammatical material of the beginner level of Ukrainian language learning; 3) the texts of the counting rhymes should be chosen with the aim of practising certain speech skills and abilities; 4) limited amount of the material. In classes with foreign students, we primarily use counting-out rhymes to introduce the category of numerals and to teach counting. Children's folklore play texts act as very important supplementary linguistic material. Rational, accurate counting (mostly from one to ten) and the rhythm and rhythmics of the short poetic text help a foreign student to learn to count in Ukrainian quickly and easily. In addition, the expressive reading of counting-out rhymes helps to improve the phonetic aspect of foreign language speech - clear pronunciation of separate sounds, intonation, accents, training of rhythm, speed of speech. Working with the texts of counting-out rhymes in UFL classes also actualises word meanings within the boundaries of communicative topics of the first year of language learning, enriches vocabulary, and optimises the rapid memorisation of typical lexical-grammatical constructions with their subsequent inclusion in the active vocabulary. Moreover, tasks with counting-out rhymes engage an element of a game in the learning process, making the lesson more exciting and interesting. Conclusions. The results of the analysis show that the counting-out rhymes have linguodidactic potential and can be used as supplementary teaching material in UFL classes (beginner level).

The Zariski covering number for vector spaces and modules

Given a module M over a commutative unital ring R, let denote the covering number, i.e. the smallest (cardinal) number of proper submodules whose union covers M; this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, Khare–Tikaradze [Comm. Algebra, in press] showed in several cases that where SM is the set of maximal ideals with Our first main result extends this equality to all R-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce and study a topological counterpart for finitely generated R-modules M over rings R, whose ‘some’ residue fields are infinite, which we call the Zariski covering number To do so, we first define the “induced Zariski topology” τ on M, and now define to be the smallest (cardinal) number of proper τ-closed subsets of M whose union covers M. We then show our next main result: for all finitely generated R-modules M for which (a) the dual Goldie dimension is finite, and (b) whenever is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number of the aforementioned finitely generated modules. Finally, we discuss the notion of κ-Baire spaces, and show that the inequalities again become equalities when the image of M under the continuous map (with appropriate Zariski-type topologies) is a κM -Baire subspace of the product space.

CHANGES IN MARI NUMERAL PHRASES UNDER THE INFLUENCE OF MARI-RUSSIAN LANGUAGE CONTACTS

This article presents the results of a study of the Mari numeral phrases in terms of the influence of the Russian language. The aim of this work is to trace the role of Russian borrowings in the formation of Mari numeral phrases, primarily in the expression of their components, and to reveal other changes that have arisen under the influence of similar phrases and structures of the Russian language. The study was conducted on the basis of the lexical card index of the MarNIIYALI (Mari Scientific Research Institute of Language, Literature and History), which is based on written sources of the meadow-eastern literary norm, namely, its electronic part in the amount of about one thousand author's sheets. During the collection and analysis of material, elements and techniques of the following research methods were applied: descriptive and analytical (observation with the identification of the studied facts in sources, their generalization, interpretation and classification, description), comparative (regular comparison of Mari models with Russian ones for identity and non-identity), comparative-historical (in other cases, indications of the origin of the words), quantitative (counting models of various groups containing Russianisms). According to the results of the research, Russian borrowings may play a role of a head word (3 units in 4 models) and a dependent component (mainly substantive case forms and postpositional constructions, numerals, as well as some pronouns and adverbs of degree in 14 models). 3 models with cardinal numbers as a head very rarely can be represented by phrases with Russianisms in both components. As a result the syntactic units in some models and the models of numeral phrases themselves were replenished, the last ones by 3 units. Also the shifts in the forms of grammatical number of dependent nouns in some models appeared.

Associate an optimal normal distribution with a finite numerical discrete data set via extended spline functions

Aim: Given a finite set of cardinal numbers, we approximate the data by an optimal normal distribution based on some criteria. Methods: There are mainly two parts: interpolating the given data via natural spline functions and searching the optimal normal distribution that would minimize the absolute area of the difference function derived from the interpolated density function and the Gaussian distributions. The whole procedures are divided into 10 steps. Results: Our experimental results show the recovered population mean and standard derivation highly depend on the chosen samples. Nonetheless, the interpolated density functions are in the similar shapes of the normal distribution. In addition, the increase of the standard derivation doesn’t affect the approximation, though the optimal standard derivation is largely proportional to it. Conclusions: A method with a set of procedures is devised to search the optimal normal distribution for a given set of data. The proposed method takes the absolute area of the difference function into consideration. The interpolated density function is used to match the optimal normal distribution. This matching is much intuitive and straightforward. Other comparative results also show this approach is stable and intuitively justifiable.

Cantor's Abstractionism and Hume's Principle

Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the concept of cardinal number. Neologicism would therefore be threatened by the ‘good company' HP is kept by such alternative assignments. In his review of Mancosu's book, Bob Hale argues, however, that ‘getting different numerosities for different countable infinite collections depends on taking the groups in a certain order – but it is of the essence of cardinal numbers that the cardinal size of a collection does not depend upon how its members are ordered'. This paper's goal is to implement Hale's response to the Good Company problem by producing a Cantorian argument for HP. In Section 2, we present the Heck-Mancosu argument against neologicism. In Section 3, we discuss Hale's defence of Hume's Principle. In Section 4, we discuss Cantor's abstractionist definitions of number. In Section 5, we argue that good abstraction must comply with what we call ‘Gödel’s Minimal Account of ion’ (GMAA). We finally show (Sections 5 and 6) that non-Cantorian theories of cardinality fail to satisfy GMAA.

Investigation of Counting Skills of Pre-School Children

The research aims at examining the counting skills of children in the preschool period. For this purpose, this study has been planned according to the “survey model”. The study group of the research is comprised of 108 children with ages between 60th-72nd months, who attend kindergartens. The “Counting Skills Test" has been used, which has been developed by the researcher as a data gathering tool. The research concluded that the preschool children possessed good levels in the skills of “rhythmic counting, stable order, counting the next number, abstraction, cardinal number and recognizing numbers” and that they had no problems therewith.

Un enfoque cognitivo en la comprensión de la comparación entre los cardinales del conjunto de los números naturales y el conjunto de los números enteros en estudiantes de secundaria

This research studies the mental structures and mechanisms of 164 Chilean students of third and fourth half (17 and 18 years old), when comparing the cardinal of the set of natural numbers (ℕ) and the cardinal of integers (ℤ). It is based on a provisional genetic decomposition of infinity within the framework of APOS theory, and shows that students establish only an injective function from ℕ to ℤ to coordinate the infinite iterative processes and, furthermore, they do not manage to present the structure of a single infinite iterative process that would allow them to conclude that they have the same cardinal number.

Covering modules by proper submodules

A classical problem in the literature seeks the minimal number of proper subgroups whose union is a given finite group. A different question, with applications to error-correcting codes and graph colorings, involves covering vector spaces over finite fields by (minimally many) proper subspaces. In this note we cover R-modules by proper submodules for commutative rings R, thereby subsuming and recovering both cases above. Specifically, we study the smallest cardinal number possibly infinite, such that a given R-module is a union of -many proper submodules. (1) We completely characterize when is a finite cardinal; this parallels for modules a 1954 result of Neumann. (2) We also compute the covering (cardinal) numbers of finitely generated modules over quasi-local rings and PIDs, recovering past results for vector spaces and abelian groups respectively. (3) As a variant, we compute the covering number of an arbitrary direct sum of cyclic monoids. Our proofs are self-contained.

Factorization of maps from products in uniform spaces

Results in this paper concern finding a cardinal number κ, as small as possible, such that every uniformly or proximally continuous mapping from a given subspace X of a product ∏IXi into some A factorizes via a subspace of ∏JXi with |J|≤κ. In some cases we substantially improve Vidossich theorem from [12]. Influenced by investigating locally co-presentable categories in [8], also cardinals κ are found such that the above factorization hold for all products ∏IXi from an epireflective subclass C of Unif2, their closed subspaces X and spaces A generating C. That situation is close to factorizing maps on limits of inverse systems via limits of smaller inverse subsystems. We define and investigate a new cardinal function on uniform spaces helping to find a convenient factorization.

Measuring Emerging Number Knowledge in Toddlers.

Recent evidence suggests that infants and toddlers may recognize counting as numerically relevant long before they are able to count or understand the cardinal meaning of number words. The Give-N task, which asks children to produce sets of objects in different quantities, is commonly used to test children’s cardinal number knowledge and understanding of exact number words but does not capture children’s preliminary understanding of number words and is difficult to administer remotely. Here, we asked whether toddlers correctly map number words to the referred quantities in a two-alternative forced choice Point-to-X task (e.g., “Which has three?”). Two- to three-year-old toddlers (N = 100) completed a Give-N task and a Point-to-X task through in-person testing or online via videoconferencing software. Across number-word trials in Point-to-X, toddlers pointed to the correct image more often than predicted by chance, indicating that they had some understanding of the prompted number word that allowed them to rule out incorrect responses, despite limited understanding of exact cardinal values. No differences in Point-to-X performance were seen for children tested in-person versus remotely. Children with better understanding of exact number words as indicated on the Give-N task also answered more trials correctly in Point-to-X. Critically, in-depth analyses of Point-to-X performance for children who were identified as 1- or 2-knowers on Give-N showed that 1-knowers do not show a preliminary understanding of numbers above their knower-level, whereas 2-knowers do. As researchers move to administering assessments remotely, the Point-to-X task promises to be an easy-to-administer alternative to Give-N for measuring children’s emerging number knowledge and capturing nuances in children’s number-word knowledge that Give-N may miss.

Parent Math Anxiety Predicts Early Number Talk

ABSTRACT Differences in children’s math knowledge emerge as early as the start of kindergarten, and persist throughout schooling. Previous research implicates the importance of early parent number talk in the development of math competency. Yet we understand little about the factors that relate to variation in early parent number talk. The current study examined the relation of parent math anxiety and family socioeconomic status (SES) to parent number talk with children under the age of three (n = 36 dyads). For the first time, we show preliminary evidence that parent math anxiety (MA) predicts the amount of number talk children hear at home, beyond differences accounted for by SES. We also found a significant SES by parent MA interaction such that parent MA was predictive of higher-SES parents’ number talk but not that of lower-SES parents. Furthermore, we found that these relations were specific to parents’ cardinal number talk (but not counting), which has been shown to be particularly important in children’s math development.

On a Rigidity Problem of Beardon and Minda

In this paper, we give a positive answer to a rigidity problem of maps on the Riemann sphere related to cross-ratios, posed by Beardon and Minda (Proc Am Math Soc 130(4):987–998, 2001). Our main results are: (I) Let Enot subset {hat{mathbb {R}}} be an arc or a circle. If a map f:{hat{mathbb {C}}}mapsto {hat{mathbb {C}}} preserves cross-ratios in E, then f is a Möbius transformation when {bar{E}}not =E and f is a Möbius or conjugate Möbius transformation when {bar{E}}=E, where {bar{E}}={{bar{z}}|zin E}. (II) Let Esubset {hat{mathbb {R}}} be an arc satisfying the condition that the cardinal number #(Ecap {0,1,infty })<2. If f preserves cross-ratios in E, then f is a Möbius or conjugate Möbius transformation. Examples are provided to show that the assumption #(Ecap {0,1,infty })<2 is necessary.

The solvability cardinal of the class of polynomials

Let G be an Abelian group, and let {{mathbb {C}}}^G denote the set of complex valued functions defined on G. A map D: {{mathbb {C}}}^G rightarrow {{mathbb {C}}}^G is a difference operator, if there are complex numbers a_i and elements b_i in G(i=1,ldots , n) such that (Df)(x)=sum _{i=1}^n a_i f(x+b_i) for every fin {{mathbb {C}}}^G and xin G. By a system of difference equations we mean a set of equations { D_i f=g_i : iin I}, where I is an arbitrary set of indices, D_i is a difference operator and g_i in {{mathbb {C}}}^G is a given function for every iin I, and f is the unknown function. The solvability cardinal mathrm{sc} ,({{mathcal {F}}}) of a class of functions {{mathcal {F}}} subset {{mathbb {C}}}^G is the smallest cardinal number kappa with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality <kappa has a solution in {{mathcal {F}}}, then S itself has a solution in {{mathcal {F}}}. The behaviour of mathrm{sc} ,({{mathcal {F}}}) is rather erratic, even for classes of functions defined on {{mathbb {R}}}. For example, mathrm{sc} ,({{mathbb {C}}}[x])=3, but mathrm{sc} ,({mathcal {TP}}) =omega _1, where {mathcal {TP}} is the set of trigonometric polynomials; mathrm{sc} ,({{mathbb {C}}}^{{mathbb {R}}})=omega , but mathrm{sc} ,({mathcal {DF}}) =(2^omega )^+, where {mathcal {DF}} is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on {{{mathbb {R}}}}^n, on normed linear spaces and, in general, on topological Abelian groups. Let {{mathcal {P}}}_G denote the class of polynomials defined on the group G. After presenting some general estimates we prove that mathrm{sc} ,({{mathbb {C}}}[x_1 ,ldots ,x_n ])=omega if 2le n<infty , and mathrm{sc} ,({{mathcal {P}}}_X)=omega _1 if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that mathrm{sc} ,({{mathcal {P}}}_G)=3 if r_0 (G)le 1, mathrm{sc} ,({{mathcal {P}}}_G)=omega if 2le r_0 (G)<infty , and mathrm{sc} ,({{mathcal {P}}}_G)ge omega _1 if r_0 (G) is infinite, where r_0 (G) denotes the torsion free rank of G. The solvability of systems of difference equations is closely connected to the existence of projections of function classes commuting with translations (see Theorem 7.1). As an application we construct a projection from {{mathbb {C}}}^{{{{mathbb {R}}}}^n} onto {{mathbb {C}}}[x_1 ,ldots ,x_n ] commuting with translations by vectors having rational coordinates (Theorem 7.4).

Emergent Morphology in Child Homesign: Evidence from Number Language

ABSTRACT Human languages, signed and spoken, can be characterized by the structural patterns they use to associate communicative forms with meanings. One such pattern is paradigmatic morphology, where complex words are built from the systematic use and re-use of sub-lexical units. Here, we provide evidence of emergent paradigmatic morphology akin to number inflection in a communication system developed without input from a conventional language, homesign. We study the communication systems of four deaf child homesigners (mean age 8;02). Although these idiosyncratic systems vary from one another, we nevertheless find that all four children use handshape and movement devices productively to express cardinal and non-cardinal number information, and that their number expressions are consistent in both form and meaning. Our study shows, for the first time, that all four homesigners not only incorporate number devices into representational devices used as predicates , but also into gestures functioning as nominals, including deictic gestures. In other words, the homesigners express number by systematically combining and re-combining additive markers for number (qua inflectional morphemes) with representational and deictic gestures (qua bases). The creation of new, complex forms with predictable meanings across gesture types and linguistic functions constitutes evidence for an inflectional morphological paradigm in homesign and expands our understanding of the structural patterns of language that are, and are not, dependent on linguistic input.

A Note on 3×3-valued Łukasiewicz Algebras with Negation

In 2004, C. Sanza, with the purpose of legitimizing the study of \(n\times m\)-valued Łukasiewicz algebras with negation (or \(\mathbf{NS}_{n\times m}\)-algebras) introduced \(3 \times 3\)-valued Łukasiewicz algebras with negation. Despite the various results obtained about \(\mathbf{NS}_{n\times m}\)-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated \(NS_{3 \times 3}\)-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \(\Lambda(\mathbf{NS}_{3\times 3})\) of all subvarieties of \(\mathbf{NS}_{3\times 3}\) and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \(\mathbf{NS}_{3\times 3}\).

Diffi culties in learning the infl ection of numerals in the teaching of Polish as a foreign language on the example of Ukrainian- and Russian-speaking students

The aim of this paper is to describe the most common errors made by Ukrainian- and Russian-speaking people while learning infl ection of numerals for person and their causes. The applied comparative method permits a closer look at formal and syntactic relations as well as collocations of numerals and specifi c lexeme classes (this regards primarily agreement between cardinal numbers and masculine personal verbs, compound structures with the numeral jeden (one) and numerals showing adjectival infl ection). This study concentrates on a comparison of cardinal and ordinal numbers in Ukrainian, Russian and Polish, which allowed an analysis of the causes of errors made while learning the infl ection of numerals by Ukrainian- and Russian-speaking students.