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This study investigates the relationship between executive functions, the number of siblings, and Theory of Mind (ToM) performance in German preschoolers. Prior research has consistently demonstrated that children with more advanced executive function (EF) skills tend to perform better on ToM tasks (e.g. Fujita et al., 2022). In addition to these cognitive factors, the social environment, particularly the number of siblings has also been shown to influence ToM development positively (Shahaeian et al., 2011). Based on these findings, the present study aims to examine whether executive functions and the number of siblings independently predict children's ToM performance, and whether there is an interaction between these two predictors. Specifically, we hypothesize that both EF and the number of siblings will be positively associated with ToM abilities, and that the effect of EF on ToM may be moderated by sibling count. Children will complete standardized tasks designed to assess their executive functions and Theory of Mind. Demographic information, including the number of siblings, age, and gender, will also be collected.

Intended to aid theory building in the study of influences on news content, this paper examines how five basic theoretical approaches to studying news content can be integrated into J. Herbert Altschull's assertion that mass media content reflects the ideology of those who finance the media. The paper notes that Altschull's theory accounts for the between-nation effects predicted by broad-scale sociological theories such as hegemony or the mass manipulative model, and shows that the theory also can explain the null effects which may result from studies looking for hegemonic effects. The first major section of the paper shows how the existing literature on media content can be analyzed according to five hierarchical levels of analysis related to Altschull's economic theory: (1) content accurately reflects social reality, (2) content is a function of media routines, (3) content is influenced by journalists' socialization and attitudes, (4) content results from social and institutional forces working on it, and (5) content is a function of ideological positions and a tool of the status quo. The second section outlines assumptions and theoretical statements resulting from this synthesis on the influence of funding source on content and other variables. The concluding section suggests that researchers integrate the five theoretical approaches into an economic model that explains media content as the product of a more complex set of ideological forces held by those who fund the media. (Six pages of notes are included.) (HTH)

Born in 1707, Leonhard Euler was the son of a Protestant minister from the vicinity of Basel, Switzerland. With the aim of pursuing a career in theology, Euler entered the University of Basel at the age of thirteen, where he was tutored in mathematics by Johann Bernoulli (of the famous Bernoulli family of mathematicians). He developed an interest in mathematics and eventually authored more than 700 books and papers. This paper discusses Euler's work on the solution to Diophantine equations (marking the early history of algebraic number theory), analytic number theory, arithmetic functions, and his theory of partitions.

This monthly publication from Canada highlights a particular subject area of the curriculum in each issue. The March, 1971 newsletter is devoted to the social studies and contains the following articles: The Language of Social Studies, by Edna C. Wilson-Press; Some Questions About Values, by Joseph M. Kirman; Valuing in Elementary Social Studies: An Annotated Bibliography, by John L. Newton; A Simulation Game for Attitude Change, by Rex Vogel and Charles Chamberlin; and a final article on oral reading and speech communication by Jean P. McIntyre, So To Speak. A former issue, April, 1970, is also concerned with social studies in the curriculum. Articles included are: The Social Study: An Investigation of a Problem, by Edna C. Wilson; The Nature of Social Studies, by Hal Chalmers; Beyond Facts: Developing Broad Understanding in the Social Studies, by Charles R. Chamberlin; Social Studies Enters the Space Age, by J. M. Kirman; and Selected Annotated Bibliography for Social Studies Teachers, by Hal Chalmers, R. W. Vogel, and F. M. Wollum. Future issues of this newsletter are available only by subscription. (Author/JSB)

The topics covered are very broad, although there are two main themes under which most of the material can be classified. A Smarandache Partition Function is an operation where a set or number is split into pieces and together they make up the original object. For example, a Smarandache Repeatable Reciprocal partition of unity is a set of natural numbers where the sum of the reciprocals is one. The first chapter of the book deals with various types of partitions and their properties and partitions also appear in some of the later sections.

150 p. 19 cm

A set of new Smarandache functions, sequences and conjectures in number theory by Russo, Felice

A set of new Smarandache functions, sequences and conjectures in number theory by Russo, Felice

A set of new Smarandache functions, sequences and conjectures in number theory by Russo, Felice

We cover some useful techniques in computational aspects of analytic number theory, with specific emphasis on ideas relevant to the evaluation of L-functions. These techniques overlap considerably with basic methods from analytic number theory. On the elementary side, summation by parts, Euler Maclaurin summation, and Mobius inversion play a prominent role. In the slightly less elementary sphere, we find tools from analysis, such as Poisson summation, generating function methods, Cauchy's residue theorem, asymptotic methods, and the fast Fourier transform. We then describe conjectures and experiments that connect number theory and random matrix theory.

Students having had a semester course in abstract algebra are exposed to the elegant way in which finite group theory leads to proofs of familiar facts in elementary number theory. In this note we offer two examples of such group theoretical proofs using the action of a group on a set. The first is Fermat's little theorem and the second concerns a well known identity involving the famous Euler phi function. The tools that we use to establish both results are sometimes seen in a second semester algebra course in which group actions are studied. Specifically, we will use the class equation of a group action and Burnside's theorem.

Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.

We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula connecting the respective number of solutions.

We discuss string theory vacua which have the wrong number of spacetime dimensions, and give a crude argument that vacua with more than four large dimensions are improbable. We then turn to two dimensional vacua, which naively appear to violate Bekenstein's entropy principle. A classical analysis shows that the naive perturbative counting of states is unjustified. All excited states of the system have strong coupling singularities which prevent us from concluding that they really exist. A speculative interpretation of the classical solutions suggests only a finite number of states will be found in regions bounded by a finite area. We also argue that the vacuum degeneracy of two dimensional classical string theory is removed in quantum mechanics. The system appears to be in a Kosterlitz-Thouless phase. This leads to the conclusion that it is also improbable to have only two large spacetime dimensions in string theory. However, we note that, unlike our argument for high dimensions, our conclusions about the ground state have neglected two dimensional quantum gravitational effects, and are at best incomplete.

We study spin-1/2 Heisenberg XXX antiferromagnet. The spectrum of the Hamiltonian was found by Hans Bethe in 1931. We study the probability of formation of ferromagnetic string in the antiferromagnetic ground state, which we call emptiness formation probability P(n). This is the most fundamental correlation function. We prove that for the short strings it can be expressed in terms of the Riemann zeta function with odd arguments, logarithm ln 2 and rational coefficients. This adds yet another link between statistical mechanics and number theory. We have obtained an analytical formula for P(5) for the first time. We have also calculated P(n) numerically by the Density Matrix Renormalization Group. The results agree quite well with the analytical ones. Furthermore we study asymptotic behavior of P(n) at finite temperature by Quantum Monte-Carlo simulation. It also agrees with our previous analytical results.

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by \O{}. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields.

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by \O{}. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields.

We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by \O{}. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields.

In this paper we show a lethargy result in the non-Arquimedian context, for general ultrametric approximation schemes and, as a consequence, we prove the existence of p-adic transcendental numbers whose best approximation errors by algebraic p-adic numbers of degree less than or equal to n decays slowly.

In this paper we show a lethargy result in the non-Arquimedian context, for general ultrametric approximation schemes and, as a consequence, we prove the existence of p-adic transcendental numbers whose best approximation errors by algebraic p-adic numbers of degree less than or equal to n decays slowly.

scan of covers + pdf of book Number theory (2.2) Chapter D2 Structure in mathematics Block D MS221 Exploring Mathematics 2nd edition by The MS221 Course Team [2006] {510--oclc}

Abstract for OSF Preprints: We present two independent analytical proofs of Goldbach’s Conjecture, demonstrating that every even integer greater than 2 can be expressed as the sum of two prime numbers. The Rigorous Analytical Proof is based on a differential equation derived from the distribution of prime numbers: 𝑑 2 𝐺 𝑑 𝑁 2 + 𝐺 ( 𝑁 ) = 0.183 𝑁 dN 2 d 2 G ​ +G(N)=0.183N The solution ensures that 𝐺 ( 𝑁 ) > 0 G(N)>0 for all even 𝑁 > 2 N>2, thereby proving the conjecture. The Alternative Proof using the Extended Number 𝑥 x introduces a novel mathematical structure and the sum-of-squares operation ⊙ ⊙ to formulate an equivalent differential equation. This framework provides an independent verification of Goldbach’s statement through a new algebraic approach. Both proofs have been extensively verified through numerical simulations up to 𝑁 = 10 9 N=10 9 . This work contributes to the field of number theory by offering new mathematical techniques applicable to prime distributions and additive problems. If you need any modifications or additional details, let me know!

MAPSOC.ORG // 00:14:00 - Alchemy and John Dee // 01:13:00 - (A Long Transition into) Number Theory of Reality Linkswww.MAPSOC.orgFollow Sumo on TwitterAlternate Current RadioSupport the Show!Subscribe to the Podcast on GumroadSubscribe to the Podcast on PatreonBuy Us a Tibetan Herbal TeaSumo's SubstacksHoly is He Who WrestlesModern Pulp

We experiment with some topics in elementary number theory. For matrices defined by Gaussian primes we observe a circular spectral law for the eigenvalues. We look at matrices defined by Gaussian primes and look at the growth of the determinant, trace. We experiment with various Goldbach conjectures for Gaussian primes, Eisenstein primes, Hurwitz primes or Octavian primes. These conjectures relate with Landau or Bunyakovsky or Andrica type conjectures for rational primes. We also look at some graphs finite simple defined by primes. Finally we look at some spectra of almost periodic pseudo random matrices defined by Diophantine irrational rotations, where fractal spectral phenomena occur. The matrix is the real part of a van der Monde matrix.

We study numerically phonon modes of the classical one-dimensional Frenkel-Kontorova chain, in the regime of pinned phase characterized by the phonon gap and devil's staircase, as well as by a large number of states (configurational excitations), which energy splitting from the ground state is exponentially small. We demonstrate, these states behave like disorder media: their phonon modes are {\it exponentially} localized, in contrast to the phonon modes in the ground state, where phonons are {\it prelocalized} only. We demonstrate also, the phonon frequency spectrum of the ground state has an hierarchical structure, a direct manifestation of hierarchical spatial structure, found for the ground state of the FK chain in our recent work.

We review and develop the formalism of ghost number cohomologies, outlined in our previous work, to classify the quantum states of M-theory. We apply this formalism to the matrix formulation of M-theory to obtain NSR superstring action from dimensionally reduced matrix model. The BPS condition of the matrix theory is related to the worldsheet reparametrizational invariance in superstring theory, underlining the connection between unbroken supersymmetries in M-theory and superstring gauge symmetries.

We review and develop the formalism of ghost number cohomologies, outlined in our previous work, to classify the quantum states of M-theory. We apply this formalism to the matrix formulation of M-theory to obtain NSR superstring action from dimensionally reduced matrix model. The BPS condition of the matrix theory is related to the worldsheet reparametrizational invariance in superstring theory, underlining the connection between unbroken supersymmetries in M-theory and superstring gauge symmetries.

We review and develop the formalism of ghost number cohomologies, outlined in our previous work, to classify the quantum states of M-theory. We apply this formalism to the matrix formulation of M-theory to obtain NSR superstring action from dimensionally reduced matrix model. The BPS condition of the matrix theory is related to the worldsheet reparametrizational invariance in superstring theory, underlining the connection between unbroken supersymmetries in M-theory and superstring gauge symmetries.

We review and develop the formalism of ghost number cohomologies, outlined in our previous work, to classify the quantum states of M-theory. We apply this formalism to the matrix formulation of M-theory to obtain NSR superstring action from dimensionally reduced matrix model. The BPS condition of the matrix theory is related to the worldsheet reparametrizational invariance in superstring theory, underlining the connection between unbroken supersymmetries in M-theory and superstring gauge symmetries.

In this paper, we analyze some equations linked to the meaning of DN Constant. We describe new possible mathematical connections with the Ramanujan Recurring Numbers and some parameters of Number Theory, Theoretical Cosmology and String Theory

The Pseudo-Smarandache function has a simple definition: Given any integer n > 0, the value of the Pseudo-Smarandache function is the smallest integer m such that n evenly divides the sum 1 + 2 + 3 + ... + m. In this paper, several problems concerning this function will be presented and solved. Most will involve the standard number theory functions such as Euler's phi function and the sum of divisors function. [1]

The Pseudo-Smarandache function has a simple definition: Given any integer n > 0, the value of the Pseudo-Smarandache function is the smallest integer m such that n evenly divides the sum 1 + 2 + 3 + ... + m. In this paper, several problems concerning this function will be presented and solved. Most will involve the standard number theory functions such as Euler's phi function and the sum of divisors function. [1]

This research aims to describe and analyze result of applying Problem-Based Learning and Cognitive Conflict Strategy (PBLCCS) in increasing students' Mathematical Critical Thinking (MCT) ability and Mathematical Curiosity Attitude (MCA). Adopting a quasi-experimental method with pretest-posttest control group design and using mixed method with sequential explanatory strategy, the study involves undergraduate students of Mathematics Education Program at a private university in Riau, Indonesia, academic year 2015/2016 enrolled in Number Theory course. Quantitative data were obtained from essay test and questionnaire whereas qualitative data were from interview and observation. The findings show that; (1) there is a difference between MCT ability taught by PBLCCS and Explicit Direct Instruction (EDI), (2) there is no significant difference between MCA taught by PBLCCS and EDI, (3) students still face obstacles in developing their critical thinking skills when solving mathematical problems.

THIS BOOKLET, ONE OF A SERIES, HAS BEEN DEVELOPED FOR THE PROJECT, A PROGRAM FOR MATHEMATICALLY UNDERDEVELOPED PUPILS. A PROJECT TEAM, INCLUDING INSERVICE TEACHERS, IS BEING USED TO WRITE AND DEVELOP THE MATERIALS FOR THIS PROGRAM. THE MATERIALS DEVELOPED IN THIS BOOKLET INCLUDE ELEMENTARY IDEAS CONCERNING (1) WHOLE NUMBERS, (2) OPERATIONS WITH SETS, (3) DIVISORS AND MULTIPLES OF A NUMBER, AND (4) ACTIVITIES INVOLVING DIVISORS AND MULTIPLES. ACCOMPANYING THESE BOOKLETS WILL BE A "TEACHING STRATEGY BOOKLET" WHICH WILL INCLUDE A DESCRIPTION OF TEACHER TECHNIQUES, METHODS, SUGGESTED SEQUENCES, ACADEMIC GAMES, AND SUGGESTED VISUAL MATERIALS. (RP)

THIS BOOKLET, ONE OF A SERIES, HAS BEEN DEVELOPED FOR THE PROJECT, A PROGRAM FOR MATHEMATICALLY UNDERDEVELOPED PUPILS. A PROJECT TEAM, INCLUDING INSERVICE TEACHERS, IS BEING USED TO WRITE AND DEVELOP THE MATERIALS FOR THIS PROGRAM. THE MATERIALS DEVELOPED IN THIS BOOKLET INCLUDE ELEMENTARY IDEAS CONCERNING (1) WHOLE NUMBERS, (2) OPERATIONS WITH SETS, (3) DIVISORS AND MULTIPLES OF A NUMBER, AND (4) ACTIVITIES INVOLVING DIVISORS AND MULTIPLES. ACCOMPANYING THESE BOOKLETS WILL BE A "TEACHING STRATEGY BOOKLET" WHICH WILL INCLUDE A DESCRIPTION OF TEACHER TECHNIQUES, METHODS, SUGGESTED SEQUENCES, ACADEMIC GAMES, AND SUGGESTED VISUAL MATERIALS. (RP)

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for $F$ with infinite type equal to a special value of certain $G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l> 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberger-splitting maps in the $l$-primary Quillen localization sequence for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$. We also apply the Stickelberger-splitting maps to construct special elements in the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$, for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general totally real number field, we study the cyclicity of $D(n)_l$ in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if $F$ is CM, special values of our $l$-adic Hecke characters are used to construct Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler system of Gauss sums and of the $K$-theoretic Euler systems constructed in Banaszak-Gajda when $K = \Bbb Q$.

From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field of kernels of the derivative of the map. Homotopically, such a plane field determines two integers. I show that the sum of these integers is the Milnor number of the fibered link. Taking the mirror image of a link exchanges the integers. Various examples are computed. It is noted (proof given elsewhere) that these integers are not determined by the algebraic monodromy (or Seifert form) of the fibered link.

The notion of density of a finite set is introduced. We prove a general theorem of set theory which refines the Gibbs, Bose--Einstein, and Pareto distributions as well as the Zipf law.

Comparative prime number theory is the study of the {\em{discrepancies}} of distributions when we compare the number of primes in different residue classes. This work presents a list of the problems being investigated in comparative prime number theory, their generalizations, and an extensive list of references on both historical and current progresses.

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*f is at least 0.631 \| f \|_1^2 / I. This improves the previous bound of 0.591389 \| f \|_1^2 / I. Consequently, we improve the known bounds on several related number-theoretic problems. For a subset A of {1,2, ..., n}, let g be the maximum multiplicity of any element of the multiset {a+b: a,b in A}. Our main corollary is the inequality gn>0.631|A|^2, which holds uniformly for all g, n, and A.

We have recently extended many-body perturbation theory and coupled-cluster theory performed on top of a Slater determinant breaking rotational symmetry to allow for the restoration of the angular momentum at any truncation order [T. Duguet, J. Phys. G: Nucl. Part. Phys. 42 (2015) 025107]. Following a similar route, we presently extend Bogoliubov many-body perturbation theory and Bogoliubov coupled cluster theory performed on top of a Bogoliubov reference state breaking global gauge symmetry to allow for the restoration of the particle number at any truncation order. Eventually, formalisms can be merged to handle $SU(2)$ and $U(1)$ symmetries at the same time. Several further extensions of the newly proposed many-body formalisms can be foreseen in the mid-term future. The long-term goal relates to the ab initio description of near-degenerate finite quantum systems with an open-shell character.

Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K_{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence to split: these conditions involve coinvariants of twisted p-parts of the p-class groups of certain subfields of the fields F(\mu_{p^n}) for n\in N. We also compare our conditions with the weaker condition WK^{et}_{2i}(F)=0 and give some example.

In this section, we introduce the strategy initiated by Goldston-Y�ld�r�m, and carried out by them and Pintz, for proving new results on the existence of short gaps between primes. Some calculations are postponed to a later unit.

A wind-tunnel study was conducted to determine the capability of a method combining linear theory and shock-expansion theory to design optimum camber surfaces for wings that will fly at high-supersonic/low-hypersonic speeds. Three force models (a flat-plate reference wing and two cambered and twisted wings) were used to obtain aerodynamic lift, drag, and pitching-moment data. A fourth pressure-orifice model was used to obtain surface-pressure data. All four wing models had the same planform, airfoil section, and centerbody area distribution. The design Mach number was 4.5, but data were also obtained at Mach numbers of 3.5 and 4.0. Results of these tests indicated that the use of airfoil thickness as a theoretical optimum, camber-surface design constraint did not improve the aerodynamic efficiency or performance of a wing as compared with a wing that was designed with a zero-thickness airfoil (linear-theory) constraint.

A wind-tunnel study was conducted to determine the capability of a method combining linear theory and shock-expansion theory to design optimum camber surfaces for wings that will fly at high-supersonic/low-hypersonic speeds. Three force models (a flat-plate reference wing and two cambered and twisted wings) were used to obtain aerodynamic lift, drag, and pitching-moment data. A fourth pressure-orifice model was used to obtain surface-pressure data. All four wing models had the same planform, airfoil section, and centerbody area distribution. The design Mach number was 4.5, but data were also obtained at Mach numbers of 3.5 and 4.0. Results of these tests indicated that the use of airfoil thickness as a theoretical optimum, camber-surface design constraint did not improve the aerodynamic efficiency or performance of a wing as compared with a wing that was designed with a zero-thickness airfoil (linear-theory) constraint.

A wind-tunnel study was conducted to determine the capability of a method combining linear theory and shock-expansion theory to design optimum camber surfaces for wings that will fly at high-supersonic/low-hypersonic speeds. Three force models (a flat-plate reference wing and two cambered and twisted wings) were used to obtain aerodynamic lift, drag, and pitching-moment data. A fourth pressure-orifice model was used to obtain surface-pressure data. All four wing models had the same planform, airfoil section, and centerbody area distribution. The design Mach number was 4.5, but data were also obtained at Mach numbers of 3.5 and 4.0. Results of these tests indicated that the use of airfoil thickness as a theoretical optimum, camber-surface design constraint did not improve the aerodynamic efficiency or performance of a wing as compared with a wing that was designed with a zero-thickness airfoil (linear-theory) constraint.

A wind-tunnel study was conducted to determine the capability of a method combining linear theory and shock-expansion theory to design optimum camber surfaces for wings that will fly at high-supersonic/low-hypersonic speeds. Three force models (a flat-plate reference wing and two cambered and twisted wings) were used to obtain aerodynamic lift, drag, and pitching-moment data. A fourth pressure-orifice model was used to obtain surface-pressure data. All four wing models had the same planform, airfoil section, and centerbody area distribution. The design Mach number was 4.5, but data were also obtained at Mach numbers of 3.5 and 4.0. Results of these tests indicated that the use of airfoil thickness as a theoretical optimum, camber-surface design constraint did not improve the aerodynamic efficiency or performance of a wing as compared with a wing that was designed with a zero-thickness airfoil (linear-theory) constraint.

In this unit, we introduce (without proof for now) a formula which relates the distribution of primes to the zeroes of the Riemann zeta function. Given a suitable zero-free region for _(s) in the critical strip, this can be used to prove the prime number theorem with an estimate for the error term.

This course is meant to be an introduction to analytic number theory; this handout provides an overview of what we will be talking about in the course. It also �xes some notation that I�ll be using throughout.

This course is meant to be an introduction to analytic number theory; this handout provides an overview of what we will be talking about in the course. It also �xes some notation that I�ll be using throughout.