Generalizations of Arnold’s version of Euler’s theorem for matrices

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has ${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k}). We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A,  B are congruent modulo p ^k then the characteristic polynomials of A ^p and B ^p are congruent modulo p ^k+1, and then we show that Arnold’s conjecture follows from it easily. Using this result, we prove the following generalization of Euler’s theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A ^Φ(n) and A ^Φ(n)-ϕ(n) are congruent modulo n. Here ϕ is the Euler function, ?_i=1^l p_i^a_i\prod_{i=1}^{l} p_i^{\alpha_i} is a prime factorization of n and $\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2.     

Lewis’ Reduction of Modality

I start by reconsidering two familiar arguments against modal realism. The argument from epistemology relates to the issue whether we can infer the existence of concrete objects by a priori means. The argument from pragmatics purports to refute the analogy between the indispensability of possible worlds and the indispensability of unobserved entities in physical science and of numbers in mathematics. Then I present two novel objections. One focusses on the obscurity of the notion of isolation required by modal realism. The other stresses the arbitrary nature of the rules governing the behaviour of Lewisean universes. All four objections attack the reductive analysis of modality that is supposed to be the chief merit of modal realism.     

Continuous approximations of Gol’dshtik’s model

We consider continuous approximations to the Gol’dshtik problem for separated flows in an incompressible fluid. An approximated problem is obtained from the initial problem by small perturbations of the spectral parameter (vorticity) and by approximating the discontinuous nonlinearity continuously in the phase variable. Under certain conditions, using a variational method, we prove the convergence of solutions of the approximating problems to the solution of the original problem.     

Dimensions of students’ views of themselves as learners of mathematics

Students’ views of themselves as learners of mathematics are a decisive parameter for their engagement and success in school. We are interested in students’ experiences with mathematics encompassing cognitive, emotional and motivational aspects. In particular, we focus on capturing the structural properties of affect related to mathematics. Participants in our study were 1,436 randomized chosen students of secondary schools from overall Finland. In the Finnish upper secondary school, there are two different syllabi for mathematics: the general and the advanced one. Schools were invited to organize the survey by one of their year 2 general syllabus courses and one of their year 2 advanced syllabus courses in grade 11. By means of factor analysis, we obtained seven dimensions in which students’ hold beliefs and emotions about mathematics partly intertwined with their motivational orientations. These dimensions are described by reliable scales, which allow outlining an average image of Finnish students’ views of themselves as learners of mathematics. Moreover, we analyzed relations between the seven dimensions and what kind of structure they generate. Thereby, a core of three high correlating dimensions could be identified, yielding different accentuations with regard to course choice.     

Stationary points of O’Hara’s knot energies

In this article we study the regularity of stationary points of the knot energies E ^(α) introduced by O’Hara (Topology 30(2):241–247, 1991; Topol Appl 48(2):147–161, 1992; Topol Appl 56(1):45–61, 1994) in the range ${\alpha\in(2,3)}$ . In a first step we prove that E ^(α) is C ¹ on the set of all regular embedded curves belonging to ${{H^{(\alpha+1)/2,2}(\mathbb {R}{/}\mathbb {Z}, \mathbb {R}^n)}}$ and calculate its derivative. After that we use the structure of the Euler-Lagrange equation to study the regularity of stationary points of E ^(α) plus a positive multiple of the length. We show that stationary points of finite energy are of class C ^∞—so especially all local minimizers of E ^(α) among curves with fixed length are smooth.     

Geometry of D’Atri spaces of type k

A Riemannian n-dimensional manifold M is a D’Atri space of type k (or k-D’Atri space), 1 ≤ k ≤ n ? 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D’Atri spaces for all possible k ≥ 1 and the property 1-D’Atri is the D’Atri condition in the usual sense. In this article we study some aspects of the geometry of k-D’Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D’Atri spaces for all k = 1, . . ., l satisfy that ${{\rm{tr}}(R_{v}^{k})}$ , v a unit vector in TM, is invariant under the geodesic flow for all k = 1, . . ., l. Further, if M is k-D’Atri for all k = 1, . . ., n ? 1, then the eigenvalues of Jacobi operators are constant functions along geodesics. In the case of spaces of Iwasawa type, we show that k-D’Atri spaces for all k = 1, . . ., n ? 1 are exactly the symmetric spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D’Atri.     

Generalizations of Sherman’s inequality

The concept of majorization is a powerful and useful tool which arises frequently in many different areas of research. Together with the concept of Schur-convexity it gives an important characterization of convex functions. The well known Majorization theorem plays a very important role in majorization theory—it gives a relation between one-dimensional convex functions and n-dimensional Schur-convex functions. A more general result was obtained by S. Sherman. In this paper, we get generalizations of these results for n-convex functions using Taylor’s interpolating polynomial and the ?eby?ev functional. We apply the exponentially convex method in order to interpret our results in the form of exponentially, and in the special case logarithmically convex functions. The outcome is some new classes of two-parameter Cauchy-type means.     

Applications of weighted estimates of Calderon’s commutators

The paper introduces singular integral operators of a new type defined in the space L ^p with the weight function on the complex plane. For these operators, norm estimates are derived. Namely, if V is a complex-valued function on the complex plane satisfying the condition |V(z) ? V(??)| ?? w|z ? ??| and F is an entire function, then we put $$P_F^* f(z) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int\limits_{\left| {\zeta - z} \right| > \varepsilon } {F\left( {\frac{{V(\zeta ) - V(z)}} {{\zeta - z}}} \right)\frac{{f(\zeta )}} {{\left( {\zeta - z} \right)^2 }}d\sigma (\zeta )} } \right|.$$ It is shown that if the weight function ?? is a Muckenhoupt A _p weight for 1 < p < ??, then $$\left\| {P_F^* f} \right\|_{p,\omega } \leqslant C(F,w,p)\left\| f \right\|_{p,\omega } .$$ .     

Students’ ratings of teacher practices

In this paper, we explore a novel approach for assessing the impact of a professional development programme on classroom practice of in-service middle school mathematics teachers. The particular focus of this study is the assessment of the impact on teachers’ employment of strategies used in the classroom to foster the mathematical habits of mind and mathematical self-efficacy of their students. We describe the creation and testing of a student survey designed to assess teacher classroom practice based primarily on students’ ratings of teacher practices.     

Unirationality of Ueno-Campana’s threefold

We shall prove that the threefold studied in the paper “ Remarks on an Example of K. Ueno” by F. Campana is unirational. This gives an affirmative answer to a question posed in the paper above and also in the book by K. Ueno, “Classification theory of algebraic varieties and compact complex spaces”.     

Generalizations of girstmair’s formulas

In 1994 and 1995 GIRSTMAIR gave (relative) class number formulas for the imaginary quadratic field $\mathbb{Q}(\sqrt { - p} )$ , P an odd prime with p ≡ 3 (mod 4) and p ≥ 7, using the coefficients of the digit expression of 1/p and z/p, respectively, where z is an integer with 1 ≤ z ≤p - 1. We extend the formulas to an imaginary abelian number field.     

Theo’s reinvention of the logic of conditional statements’ proofs rooted in set-based reasoning

This report documents how one undergraduate student used set-based reasoning to reinvent logical principles related to conditional statements and their proofs. This learning occurred in a teaching experiment intended to foster abstraction of these logical relationships by comparing the relationships between predicates within the conditional statements and inference structures among various proofs (in number theory and geometry). We document the progression of Theo’s set-based emergent model (Gravemeijer, 1999) from a model-of the truth of statements to a model-for logical relationships. This constitutes some of the first evidence for how students can abstract such logical concepts in this way and provides evidence for the viability of the learning progression that guided the instructional design.     

Rebuttal of Kowalenko’s Paper As Concerns the Irrationality of Euler’s Constant γ

We rebut Kowalenko??s claims in this journal that he proved the irrationality of Euler??s constant ??, and that his rational series for ?? is new.     

Prospective teachers’ unified understandings of the structure of identities

United States curriculum standards advise mathematics teachers to teach students to attend to structure and understand how mathematical concepts are related. This requires teachers to have a structural perspective and a coherent, unified understanding of mathematical structures that span curricula. This study explores Prospective Secondary Mathematics Teachers’ (PSMTs) unified understandings of identities and characterizes the structural features of identities that PSMTs attend to. I contribute a theoretical framework of three ways in which PSMTs reason about identities: a do-nothing element, a result of undoing something, and a coordination with inverse, binary operation, and set. I classify the level of coherence of their identity schemas demonstrated as they reasoned about the structural connections among additive, multiplicative, and compositional identities. I illustrate how having unified, coherent understandings of identities can lead PSMTs to reason productively about inverse and identity functions, while having incoherent understandings of identities can lead to inaccurate reasoning about inverse and identity functions. I conclude with teaching implications for fostering PSMTs’ unified understandings of algebraic concepts.     

A Critique of Armstrong’s Truthmaking Account of Possibility

In this paper I argue against Armstrong’s recent truthmaking account of possibility. I show that the truthmaking account presupposes modality in a number of different ways, and consequently that it is incapable of underwriting a genuine reduction of modality. I also argue that Armstrong’s account faces serious difficulties irrespective of the question of reduction; in particular, I argue that his Entailment and Possibility Principles are both false.

Uniform distribution of generalized Kakutani’s sequences of partitions

We consider a generalized version of Kakutani’s splitting procedure where an arbitrary starting partition π is given and in each step all intervals of maximal length are split into m parts, according to a splitting rule ρ. We give conditions on π and ρ under which the resulting sequence of partitions is uniformly distributed.