On the Behaviour of the spectral characteristic of Feigenbaum’s map

Let T _g : [?1, 1] ?? [?1, 1] be the Feigenbaum map. It is well known that T _g has a Cantor-type attractor F and a unique invariant measure ??₀ supported on F. The corresponding unitary operator (U _g ??)(x) = ??(g(x)) has pure point spectrum consisting of eigenvalues ?? _n,r , n ?? 1, 0 ?? r ?? 2 ^n?1 ? 1 with eigenfunctions e _r ⁽ⁿ⁾ (x). Suppose that f ?? C ¹([?1, 1]), f?? is absolutely continuous on [?1, 1] and f?? ?? L ^p ([?1, 1], d??₀), p > 1. Consider the sum of the amplitudes of the spectral measure of f: $$ Sn(f): = \sum\limits_{r = 0}^{2^n - 1} {|\rho _r^{(n)} |^2 ,\rho _r^{(n)} = \int\limits_{ - 1}^1 {f(x)\overline {e_r^{(n)} (x)} d\mu _o } } (x). $$ Using the thermodynamic formalism for T _g we prove that S _n (f) ?? 2^?n q ⁿ , as n ?? ??, where the constant q ?? (0, 1) does not depend on f.     

On the Moments of the Absorption Time of Kingman’s Coalescent

Simple integral representations are derived for the moments of the absorption time of Kingman’s coalescent Kingman (J Appl Probab 19:27-43, (1982a)). Their computational efficiency versus known representations is established.     

Convergence of the Lavrent’ev method

The convergence of the Lavrent’ev method, which is a well-known regularization method for integral equations of the first kind, is analyzed as applied to equations with arbitrary linear bounded operators. A theorem concerning necessary and sufficient conditions for this convergence is proved. It is shown that these conditions are satisfied for two classes of integral equations that do not possess the properties required by the classical Lavrent’ev method.     

Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.     

On the Poisson geometry of the Adler-Gel’fand-Dikii brackets

In this paper, we consider a symplectic leaf that goes through a singular point of the Adler-Gel’fand-Dikii Poisson bracket associated to SL(n,R). We find a finite-dimensional transverse section2 at the singular point and we prove that one can induce a Poisson structure on2 (the transverse structure) that is linearizable and equivalent to the Lie-Poisson structure on sl(n,R)*. This problem is closely related to finding normal forms for nth order scalar differential operators with periodic coefficient. We partially generalize a well-known result for Hill’s operators to the higher order case.     

On the complexity of the closed fragment of Japaridze’s provability logic

We consider the well-known provability logic GLP. We prove that the GLP-provability problem for polymodal formulas without variables is PSPACE-complete. For a number n, let \({L^{n}_0}\) denote the class of all polymodal variable-free formulas without modalities \({\langle n \rangle,\langle n+1\rangle,...}\) . We show that, for every number n, the GLP-provability problem for formulas from \({L^{n}_0}\) is in PTIME.     

On the Kernel of the Borel’s Characteristic Map of Lie Groups

For compact and connected Lie group $G$ with a maximal torus $T$ the quotient space $G/T$ is canonically a smooth projective manifold, known as the complete flag manifold of the group $G.$ The cohomology ring map $c^∗: H^∗ (B_T) → H^∗ (G/T)$ induced by the inclusion $c:G/T→B_T$ is called the Borel’s characteristic map of the group $G [7, 8],$ where $B_T$ denotes the classifying space of $T.$ Let $G$ be simply-connected and simple. Based on the Schubert presentation of the cohomology $H^∗ (G/T)$ of the flag manifold $G/T$ obtained in $[10, 11],$ we develop a method to find a basic set of explicit generators for the kernel ker$c^∗ ⊂ H^∗ (B_T)$ of the characteristic map $c.$     

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.     

Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative

The purpose of this study was to gain insight into 30, first year calculus students’ understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up group interviews. Students’ personal meanings of the vertex, including misconceptions, were explored, along with students’ understanding to solve problems pertaining to the derivative of a quadratic function. Results give evidence of students’ weak schema of the vertex, lack of connection between different problem types and the importance of linguistics in relation to levels of APOS theory. A preliminary genetic decomposition was developed based on the results. Future research is suggested as a continuation to improve student understanding of the relationship between the vertex of quadratic functions and the derivative.     

On the generalization of Faltings’ Annihilator Theorem

Let R be a commutative Noetherian ring, and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the invariants ${\inf\{i \in \mathbb{N}_0|\, \rm{dim\, Supp}(\mathfrak{b}^t H_{\mathfrak{a}}^i(M)) \geq n\, \rm{for\, all}\, t \in \mathbb{N}_0\}}$ and ${\inf\{\lambda_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\, \mathfrak{p} \in {\rm Spec} \, R\, \rm{and\, dim}\, R/ \mathfrak{p} \geq n\}}$ are equal, for every finitely generated R-module M and for all ideals ${\mathfrak{a}, \mathfrak{b}}$ of R with ${\mathfrak{b}\subseteq \mathfrak{a}}$ . This generalizes Faltings’ Annihilator Theorem (see [6]).     

On the Stability of Eaton’s Characterization by the Properties of Linear Forms

We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics $(X_{1}+\cdots +X_{k_{1}})/k_{1}^{1/\alpha}We consider a population and a sample X ₁,X ₂,…,X _n of n independent observations drawn from this population. We assume that two suitably chosen linear statistics of X ₁,X ₂,…,X _n are given. The assumption that these statistics are identically distributed or have the same distribution as the monomial X ₁ can be used to characterize various populations. This is an object of the so-called characterization theorems. But if the assumptions of a characterization theorem are fulfilled only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems concerning problems of this type are called stability theorems. By Eaton’s theorem, if, under additional conditions, two linear statistics and have the same distribution as the monomial X ₁, then this monomial has a symmetric stable distribution of order α. The stability estimation in this theorem is investigated in the λ ₀-metric.      

A proof of Terras’ conjecture on the radius of convergence of the Ihara zeta function

We prove the conjecture by Audrey Terras (2011) on the inequalities among the spectral radius ${\rho }_X$ of a finite graph $X$, the radius of convergence $R$ of its Ihara zeta function $Z_X\left(u\right)$, and the average degree $\overline{d_X}$ of $X$. Relating to Terras’ conjecture, we propose a new conjecture between $R$ and certain Rayleigh quotients.     

Modifications of the Hurwicz’s decision rule

The Hurwicz’s criterion is one of the classical decision rules applied in decision making under uncertainty as a tool enabling to find an optimal pure strategy both for interval and scenarios uncertainty. The interval uncertainty occurs when the decision maker knows the range of payoffs for each alternative and all values belonging to this interval are theoretically probable (the distribution of payoffs is continuous). The scenarios uncertainty takes place when the result of a decision depends on the state of nature that will finally occur and the number of possible states of nature is known and limited (the distribution of payoffs is discrete). In some specific cases the use of the Hurwicz’s criterion in the scenarios uncertainty may lead to quite illogical and unexpected results. Therefore, the author presents two new procedures combining the Hurwicz’s pessimism-optimism index with the Laplace’s approach and using an additional parameter allowing to set an appropriate width for the ranges of relatively good and bad payoffs related to a given decision. The author demonstrates both methods on the basis of an example concerning the choice of an investment project. The methods described may be used in each decision making process within which each alternative (decision, strategy) is characterized by only one criterion (or one synthetic measure).     

On the equivalence of Lagrange’s axiom to the Lotschnittaxiom

We prove that, in Hilbert’s plane absolute geometry, an axiom used by Lagrange in a proof of the Euclidean parallel postulate in a paper read on 3 February 1806 at the Institut de France, which states that “If a and b are two parallels from P to g, then the reflection of a in b is parallel to g as well”, is equivalent to F. Bachmann’s Lotschnittaxiom, which states that “The perpendiculars on the sides a right angle always intersect.”     

On the solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation

We study the solvability of the Gellerstedt problem for the Lavrent??ev-Bitsadze equation under an inhomogeneous boundary condition on the half-circle of the ellipticity domain of the equation, homogeneous boundary conditions on external, internal, and parallel side characteristics of the hyperbolicity domain of the equation, and the transmission conditions on the type change line of the equation.     

Exploring US textbooks’ treatment of the estimation of linear measurements

Learning to estimate a linear measurement is critical in becoming a successful measurer. Research indicates that the teaching of the estimation of linear measurement is quite open and that instruction does not make explicit to students how to carry out estimation work. Because written curriculum has been identified as one of the main sources affecting teachers’ instruction and students’ learning, this study examined how estimation of linear measurement tasks were presented to students in three US elementary mathematics curricula to see how much and in what ways these tasks were presented in an open manner. The principal result was that the length estimation tasks were frequently not explicit about which attribute of the object to measure and the requested level of precision of the estimate. Length estimation tasks were also left more open than other measurement tasks like measuring length with rulers.