Search

In this paper, we explain changes made in an introductory Real Analysis classroom that address common challenges and that may allow for all levels of students to meet a high standard of learning and written work. We aim for this goal by fusing elements of a hybrid course with scaffolded collaborative work to improve student learning. The hybrid structure allows students to engage new material at their pace, while the scaffolding allows for more challenging coursework and models for students some typical arguments in analysis. As a result, students that initially submitted proofs needing significant improvement in coherence and organization show positive gains in self-confidence and are enabled to progress quickly to submitting mathematical writing of a high calibre. Our case study occurred at a large institution serving many underrepresented and non-traditional student populations. The approach may be effective at schools with similar goals and challenges. [ABSTRACT FROM AUTHOR]

In high school, unit discs were all circular. Squares and diamonds were added to the list of shapes in a Real Analysis course and although more bizarre shapes appeared in graduate school, the list of possible shapes didn't appear endless. In this note, we'll show that unit discs can have almost any shape imaginable. [ABSTRACT FROM AUTHOR]

Inequalities play a fundamental role in many branches of mathematics and particularly in real analysis. By using inequalities, we can find extrema, point of inflection, and monotonic behavior of real functions. Subordination and quasisubordination are important tools used in complex analysis as an alternate of inequalities. In this article, we introduce and systematically study certain new classes of meromorphic functions using quasisubordination and Bessel function. We explore various inequalities related with the famous Fekete-Szego inequality. We also point out a number of important corollaries. [ABSTRACT FROM AUTHOR]

In this note, we report on an implementation of discovery-oriented problems in courses on Real Analysis and Differential Equations. We explain a type of task design that gives students the opportunity to conjecture, refute and prove. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework assignments. In addition to several concrete examples, we also discuss feedback and assessment outcomes of our students. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate two parameters hk,n and h ' k,n of some P-Q type Theta functions ϕ(q) for any positive real numbers k and n. During this process, we also evaluate Ramanujan's cubic continued fraction. [ABSTRACT FROM AUTHOR]

In this paper we give a representation formula for the general solution to the following two-dimensional system of difference equations ... where parameters a,b and initial values x-2,x-1,x0,y-2,y-1,y0 are real numbers. We also give some theoretical explanations related to the representation. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the form of the solutions and the periodicity character of the following systems of nonlinear difference equations of order two Zn+1 = Zntn-1/±tn ± tn-1, tn+1 = tnZn-1/±Zn ± Zn-1, where the initial conditions Z-1, Z0, t-1 and t0 are nonzero real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear di¤erence equation... where the parameters α,β,γ, a; b; c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l,k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to investigate the dynamics of a system of fourth-order rational difference equations... where the parameter A is arbitrary positive real number and the initial conditions x-3,x-2,x-1,x0,y-3,y-2,y-1,y0 are arbitrary nonnegative real numbers. By using new iteration method for the more general nonlinear difference equations and inequality skills, we establish some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for this nonlinear system. Numerical examples to the difference system are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this article we study the existence of solutions and some of their qualitative behavior of the following rational nonlinear difference equation xn+1 = axn-(2k+1)/ b + cxn-kxn-(2k+1) , n = 0, 1, ..., where a, b and c are real numbers, k is a non-negative integer number and the initial conditions x-2k-1, x-2k , ..., x-1, x0 are arbitrary non-negative real numbers. Also, the solutions of some special cases of the equation under consideration will be obtained. [ABSTRACT FROM AUTHOR]

The stationary Stokes problem in a n-dimensional domain with a rapidly oscillating (n − 1) dimensional boundary prescribed with Neumann boundary condition and periodicity along the lateral sides is considered. We identify the limit equation in a fixed domain with a one dimensional Laplacian-type problem, satisfied on the region covered by the oscillating part of the boundary, coupled with steady Stokes in the complement. The existence and uniqueness of solution for the coupled limit problem is established and, finally, the weak convergences of velocities are improved to strong convergence by introducing corrector terms. [ABSTRACT FROM AUTHOR]

In 1971 Onnewer and Waterman establish a sufficient condition which guarantees uniform convergence of Vilenkin-Fourier series of continuous function. In this paper we consider different classes of functions of generalized bounded oscillation and in the terms of these classes there are established sufficient conditions for uniform convergence of Cesàro means of negative order. [ABSTRACT FROM AUTHOR]

Uncertainty quantification of a technical system can be done using density estimation. We usually start with a stochastic model, which is fitted to the technical system, and the density estimation is done using data from this stochastic model. However, in any application, such a stochastic model will not be perfect, and the estimation of the density should take into account the inadequacy of the stochastic model. In this paper, we show how observed data of the real system together with an imperfect simulation model can be used to derive confidence bands for the density of the technical system. Our main result is that the newly introduced confidence bands allow to derive lower and upper bounds on the probability of intervals in the technical system. Furthermore, we present an upper bound on the area of the confidence band in case of a smooth density. The results are illustrated by applying the estimates to simulated and real data. [ABSTRACT FROM AUTHOR]

In this paper, we show that the abstract framework developed in [G. Pagès and C. Rey, Recursive computation of the invariant distribution of Markov and Feller processes, preprint 2017, https://arxiv.org/abs/1703.04557] and inspired by [D. Lamberton and G. Pagès, Recursive computation of the invariant distribution of a diffusion, Bernoulli 8 2002, 3, 367–405] can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean-reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth. [ABSTRACT FROM AUTHOR]

Singular spectrum analysis (SSA) or extended empirical orthogonal function methods are powerful, commonly used data‐driven techniques to identify modes of variability in time series and space‐time data sets. Due to the time‐lagged embedding, these methods can provide inaccurate reconstructions of leading modes near the endpoints, which can hinder the use of these methods in real time. A modified version of the traditional SSA algorithm, referred to as SSA with conditional predictions (SSA‐CP), is presented to address these issues. It is tested on low‐dimensional, approximately Gaussian data, high‐dimensional non‐Gaussian data, and partially observed data from a multiscale model. In each case, SSA‐CP provides a more accurate real‐time estimate of the leading modes of variability than the traditional reconstruction. SSA‐CP also provides predictions of the leading modes and is easy to implement. SSA‐CP is optimal in the case of Gaussian data, and the uncertainty in real‐time estimates of leading modes is easily quantified. Plain Language Summary: Singular spectrum analysis (SSA) is a powerful, commonly used technique to identify prominent patterns in observed data. However, SSA has some difficulty in providing accurate estimates near the endpoints of the time series, which can hinder its use in real time. A modified version of the SSA algorithm, referred to as SSA with conditional predictions, is presented to address these issues. SSA with conditional predictions provides a more accurate real‐time estimate of the leading modes of variability than the traditional method in a variety of tests. It can also be used to predict these patterns, and it is easy to implement. The uncertainty in the real‐time estimates of leading patterns is easily quantified as well. Key Points: Singular spectrum analysis (SSA) and extended empirical orthogonal function (EEOF) methods suffer from endpoint issuesSSA with conditional predictions (SSA‐CP) is presented as a simple modification to improve real‐time estimates near endpointsForecasts are also possible, including error estimates, and are optimal for Gaussian data and shown to also be skillful for non‐Gaussian data [ABSTRACT FROM AUTHOR]

Abstract Let M ⊂ C N be a generic real submanifold of class C 4. In case M is Levi non-degenerate in the sense Tumanov, we construct stationary discs for M. If furthermore M satisfies an additional non-degeneracy condition, we apply the method of stationary discs to obtain 2-jet determination of CR automorphisms of M. [ABSTRACT FROM AUTHOR]

Let d(N) (resp., p(N)) be the number of summands in the determinant (resp., permanent) of an N × N circulant matrix A = (aij) given by aij = Xi+j where i + j should be considered mod N. This short note is devoted to proving that d(N) = p(N) if and only if N is a prime power. We then give an application to homogeneous monomial ideals failing the Weak Lefschetz property. [ABSTRACT FROM AUTHOR]

We establish existence and uniqueness conditions for the solutions of difference equations with contracting compact operators in the metric space of two-sided sequences. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the form of the solutions and the periodicity character of the following systems of nonlinear difference equations of order two zn+1=zn+tn-l/±tn±tn-1,tn+1=tnzn-l/±zn±zn-l where the initial conditions z-1, z0, t-1 and t0 are nonzero real numbers. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate the global stability of the solutions, the boundedness and the periodic character of the nonlinear difference equationxn+1 = βxn+βxn-l+γxn-k + axnxn-k/bxn + cxn-l + dxn-k, n = 0, 1, ..., where the parameters α, β, γ, a, b, c and d are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1, x0 are positive real numbers where s = max{l, k}. Some numerical examples will be given to explicate our results. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to investigate the dynamics of a system of fourth-order rational difference equations xn+1=xn-3-yn-1/A+xn-3yn-1,yn+1=yn-3-xn-1/A+yn-3xn-1, n=0,1,..., where the parameter A is arbitrary positive real number and the initial conditions x-3,x-2,x-1,x0,y-3,y-2,y-1,y0 are arbitrary nonnegative real numbers. By using new iteration method for the more general nonlinear difference equations and inequality skills, we establish some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for this nonlinear system. Numerical examples to the difference system are given to verify our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the local stability of the equilibrium points, global attractor, boundedness and the form of the solutions for the following equations xn+1 = Axnxn-3/Bxn + Cxn-2 and xn+1 =Axnxn-3/ Bxn - Cxn-2 , where the coefficients A, B and C are real positive numbers, and the initial conditions x-3,x-2, x-1, x0 are arbitrary non zero real numbers. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the dynamics of the rational difference equation xn+1 =xn-1/A + Bxnxn-1 with arbitrary initial conditions A and B as nonzero real numbers. The solution is obtained and analytical study and asymptotic behavior are investigated. The forbidden set is determined. The existence of periodic and oscillatory solutions are discussed. Our results are illustrated with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, third and fourth order of accuracy stable difference schemes for approximately solving multipoint nonlocal boundary value problems for hyperbolic equations with the Neumann boundary conditions are considered. Stability estimates for the solutions of these difference schemes are presented. Finite difference method is used to obtain numerical solutions. Numerical results of errors and CPU times are presented and are analyzed. [ABSTRACT FROM AUTHOR]

The aim of this note is to study the determinant inequalities for operators in a finite von Neumann algebra. In particular, some determinant inequalities due to Hua, Lin and Zhang are extended to the case of operators in a finite von Neumann algebra. [ABSTRACT FROM AUTHOR]

This paper describes a research project implemented in a course of Real Analysis II. The aim of this project is to analyse students' comprehension of the limiting behaviour of a function from R2 to R at a point focusing on the formal definitions. Alternative definitions and other statements are discussed, some of which meet both sufficient and necessary conditions, while others meet only sufficient conditions; some involve neighbourhoods and others 'paths'. Emphasis is placed on the criteria according to which students choose one of the given definitions, equivalence or not of these and their overall influence on the concept images formed by the students for the limit concept. In particular, the paper discusses how students come to a conclusion about the existence of the limit of a function in R2 through the consideration of contour curves. We also present and analyse data from a written test as well as from the conducted interviews. [ABSTRACT FROM AUTHOR]

In this work, we define a new class of functions of the Bernoulli type using the Riemann-Liouville fractional integral operator and derive a generating function for these class generalized functions. Then, these functions are employed to derive formulas for certain Dirichlet series. [ABSTRACT FROM AUTHOR]

The sequence of Jacobi polynomials {Pn(α,β)}n=0∞ is orthogonal on (− 1,1) with respect to the weight function (1 − x)α(1 + x)β provided α > − 1,β > − 1. When the parameters α and β lie in the narrow range − 2 < α, β < − 1, the sequence of Jacobi polynomials {Pn(α,β)}n=0∞ is quasi-orthogonal of order 2 with respect to the weight function (1 − x)α+ 1(1 + x)β+ 1 and each polynomial of degree n,n ≥ 2, in such a Jacobi sequence has n real zeros. We show that any sequence of Jacobi polynomials {Pn(α,β)}n=0∞ with − 2 < α, β < − 1, cannot be orthogonal with respect to any positive measure by proving that the zeros of Pn− 1(α,β) do not interlace with the zeros of Pn(α,β) for any n∈ℕ,n ≥ 2, and any α,β lying in the range − 2 < α, β < − 1. We also investigate interlacing properties satisfied by the zeros of equal degree Jacobi polynomials Pn(α,β),Pn(α,β+ 1) and Pn(α+ 1,β+ 1) where − 2 < α, β < − 1. Upper and lower bounds for the extreme zeros of quasi-orthogonal order 2 Jacobi polynomials Pn(α,β) with − 2 < α, β < − 1 are derived. [ABSTRACT FROM AUTHOR]

We consider an approximation method based on Steklov functions of the first and second order. We obtain estimates for the norms in the space C of continuous periodic functions and clarify how they connect with numerical differentiation formulas. [ABSTRACT FROM AUTHOR]

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is {-div[(1+uq)∇u]+u=γ∇u2/1+u1-q+f  in  Ω,  u=0  on  ∂Ω,}, where 00 and f belongs to some appropriate Lebesgue space. We give assumptions on f with respect to q and γ to show the existence and regularity results for the solutions of previous equation. [ABSTRACT FROM AUTHOR]

Abstract We give a full classification, up to equivalence, of finite-dimensional graded division algebras over the field of real numbers. The grading group is any abelian group. [ABSTRACT FROM AUTHOR]

We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers. Further, we introduce a concept of double lacunary strong (A, φ) -convergence with respect to a modulus function. In addition, we also study some relationships between double lacunary strong (A, φ) -convergence with respect to a modulus and double lacunary statistical convergence. [ABSTRACT FROM AUTHOR]

Applying techniques of real analysis and weight functions, we study some equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular nonhomogeneous kernel. The constant factors are related to the Riemann zeta function and are proved to be best possible. In the form of applications, we deduce a few equivalent conditions of two kinds of the reverse Hardy-type integral inequalities with a particular homogeneous kernel. We also consider some corollaries as particular cases. [ABSTRACT FROM AUTHOR]

This work is concerned with the construction of solutions for a class of first-order dynamic equations with nonlocal initial conditions on time scales. We also give an example to illustrate our results. [ABSTRACT FROM AUTHOR]

Let XN be a set of N elements and F1, F2,... be a sequence of random independent equiprobable mappings XN → N. For a subset S0 ⊂ XN, |S0|=m, we consider a sequence of its images St=Ft(...F2(F1(S0))...), t=1,2... An approach to the exact recurrent computation of distribution of |St| is described. Two-sided inequalities forM{|St|||S0|=m} such that the difference between the upper and lower bounds is o(m)for m, t, N → ∞, mt=o(N) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms. [ABSTRACT FROM AUTHOR]

By means of the technique of real analysis and the weight functions, a few equivalent statements of a Hilbert-type integral inequality with the nonhomogeneous kernel in the whole plane are obtained. The constant factor related to the beta function is proved to be the best possible. As applications, the case of the homogeneous kernel, the operator expressions, and a few corollaries are considered. [ABSTRACT FROM AUTHOR]

The notions of the L-convolution operator and the ℒ-Wiener-Hopf operator are introduced by replacing the Fourier transform in the definition of the convolution operator by a spectral transformation of the self-adjoint Sturm-Liouville operator on the axis ℒ. In the case of the zero potential, the introduced operators coincide with the convolution operator and theWiener-Hopf integral operator, respectively. A connection between the ℒ-Wiener-Hopf operator and singular integral operators is revealed. In the case of a piecewise continuous symbol, a criterion for the Fredholm property and a formula for the index of the ℒ-Wiener-Hopf operator in terms of the symbol and the elements of the scattering matrix of the operator ℒ are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present some uniform relative perturbation bounds for eigenvalues and eigenspaces of diagonalizable matrices under additive and multiplicative perturbations. Some existing perturbation bounds can be improved based on the new bounds. Numerical experiments are given to demonstrate the advantage of the new bounds. [ABSTRACT FROM AUTHOR]

We construct approximations to the renewal function for a bivariate renewal process. Suppose (X, Y), (X1, Y1), (X2, Y2), . . . denote i.i.d. positive random vectors with common distribution function F(x, y) = P(X ≤ x, Y ≤ y). Let Sn (1) = X1 + X2 + · · · + Xn and Sn (2) = Y1 + Y2 + · · · + Yn denote the partial sums where we set S0¹ = S0² = 0. Associated with {(Xi, Yi)}, we define, respectively, the univariate and bivariate renewal counting processes: Ni(x) = min{n ≥ 1 : Sn (i) > x} (i = 1, 2) and N(x, y) = min{N1(x),N2(y)}. The bivariate renewal function is given by U(x, y) = EN(x, y) = Σ0∞ F*n(x, y). From the practical point of view it is hard to find explicit expressions for the renewal function U(x, y). Recently, Mitov and Omey (2014) introduced a new method to obtain approximations for univariate renewal functions based on expansions of Laplace-Stieltjes transforms. In this paper we generalize these approximations to the bivariate case and apply them to regularly varying increments. We show that the approximation along the diagonal is different from off the diagonal. [ABSTRACT FROM AUTHOR]

Given an integer k ≥ 2 and a real number ʇ ϵ[0,1], which graphs of edge density ʇ contain the largest number of k-edge stars? For k = 2 Ahlswede and Katona proved that asymptotically there cannot be more such stars than in a clique or in the complement of a clique (depending on the value of ʇ). Here we extend their result to all integers k ≥ 2. [ABSTRACT FROM AUTHOR]

We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian. [ABSTRACT FROM AUTHOR]

Abstract: We consider a Dirichlet problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ, and the level of anisotropy of the cell is determined by a diagonal matrix γ with positive diagonal entries. The relative size of each periodic perforation is instead determined by a positive parameter ε. For a given value γ ∼ of γ, we analyze the behavior of the unique solution of the problem as ( ε , δ , γ ) tends to ( 0 , 0 , γ ∼ ) by an approach which is alternative to that of asymptotic expansions and of classical homogenization theory. [ABSTRACT FROM AUTHOR]

The purpose of this research study is to understand how math majors conceptualize and make sense of pointwise convergence of sequence of functions in a series of in-depth qualitative interviews. Framed in APOS ( Dubinsky & McDonald, 2002 ) and VA (visual-analytic) strategy coordination ( Zazkis, Dubinsky & Dautermann, 1996 ) theoretical perspectives, the analysis showed that students demonstrated a diversity of mathematical behaviors associated mostly with the process level of understanding in APOS model. Yet a complete understanding of pointwise convergence at object level did not take place, students still also exhibited significant achievements in progression from process to object. The VA coordination also proved a beneficial mechanism in students’ making sense of pointwise convergence in a dynamic geometry environment. [ABSTRACT FROM AUTHOR]

In this paper, we present a comparative case study of two students with different epistemological frames watching the same real analysis lectures. We show that students with different epistemological frames can interpret the same lecture in different ways. These results illustrate how a student’s interpretation of a lecture is not inherently tied to the lecture, but rather depend on the student and her perspective on mathematics. Thus, improving student learning may depend on more than improving the quality of the lectures, but also changing student’s beliefs and orientations about mathematics and mathematics learning. [ABSTRACT FROM AUTHOR]

In Gordon (1998), Gordon presented alternate proofs of several well known results in elementary real analysis using tagged partitions. In this paper, we provide a set of alternative proofs based on the dyadic partitions. An important difference between tagged and dyadic partitions is that the results based on the dyadic partition can be obtained constructively, i.e. an algorithm is supplied to reach the conclusion of each proof. In addition, the construction involves only concepts and results presented in a first course real analysis, which is the reason why, comparing tagged partition and its theory, the proofs based on dyadic partition are more straightforward and accessible to a wide range of audience. [ABSTRACT FROM AUTHOR]

Let be a real number with . We study the representations of a large integer in the form where is an integer and is a friable number with for . We prove that when and , all sufficiently large integers are thus representable. [ABSTRACT FROM AUTHOR]

For real analytic expanding interval maps, a novel method is given for rigorously approximating the diffusion coefficient of real analytic observables. As a theoretical algorithm, our approximation scheme is shown to give quadratic exponential convergence to the diffusion coefficient. The method for converting this rapid convergence into explicit high precision rigorous bounds is illustrated in the setting of Lanford's map $$x\mapsto 2x +\frac{1}{2}x(1-x) \pmod 1 $$ . [ABSTRACT FROM AUTHOR]