Timo Tossavainen

Different aspects of the monotonicity of a function

In this paper, we investigate which aspects are overriding in the concept images of monotonicity of Finnish tertiary mathematics students, i.e., on which aspects of monotonicity they base their argument in different types of exercises related to that concept. Further, we examine the relationship between the quality of principal aspects and the success in solving monotonicity exercises and a few other standard problems in calculus. Our findings indicate that a mathematics students conception about monotone functions is often restricted to continuous or differentiable functions and the algebraic aspect – the nearest one to the formal definition of monotonicity – is rare.

Boundary behavior of conformal deformations

We study conformal deformations of the Euclidean metric in the unit ball B. We assume that the density associated with the deformation satisfies a Harnack inequality and an arbitrary volume growth condition on the isodiametric profile. We establish a Hausdorff (gauge) dimension estimate for the set E ⊂ ∂B where such a deformation mapping can “blow up”. We also prove a generalization of Gerasch’s theorem in our setting.

Means and the mean value theorem

Let I be a real interval. We call a continuous function μ : I × I → ℝ a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I → ℝ be a differentiable and strictly convex or strictly concave function. If a, b ∈ I with a ≠ b, then there exists a unique number ξ between a and b such that f(b) − f(a) = f ′(ξ)(b − a). We study under what conditions ξ is a proper mean of a and b, and what kind of means are obtained by applying certain f s. We also study the converse problem: Given a proper mean μ(a, b), does there exist f such that f(b) − f(a) = f ′(μ(a, b))(b − a) for all a, b ∈ I with a ≠ b?

PATHWISE CONNECTIVITY OF A CONFORMAL BOUNDARY

We show that, in dimensions n ≥ 3, the metric boundary of a conformal deformation of the unit ball is pathwise connected, and even of bounded turning, provided the conformal scaling factor satisfies a Harnack inequality and the volume growth of the deformed space is at most euclidean. Following [1] we consider conformal deformations of the unit ball of R, n ≥ 2, of the following type. Let ρ : B → (0,∞) be a continuous function satisfying for some constants A ≥ 1 and B > 0, HI(A) 1/A ≤ ρ(x) ρ(y) ≤ A whenever x, y ∈ Bz = B(z, 1 2 (1 − |z|)) for any z ∈ B, and V G(B) μρ(Bρ(x, r)) ≤ Br for all x ∈ B, and r > 0. Here μρ(E) = ∫ E ρdmn and Bρ(x, r) is a ball center centered at x with radius r with respect to the metric dρ, defined for x, y ∈ B by the formula dρ(x, y) = inf γ ∫

Fooling Newton's Method as Much as One Can

6. G. L. Mullen and C. Mummert, Finite Fields and Applications, American Mathematical Society, Providence, RI, 2007. 7. I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, 3rd ed., John Wiley, New York 1972. 8. J. Silverman, The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. 9. D. Vella and A. Vella, Cycles in the generalized Fibonacci sequence modulo a prime, this MAGAZINE 75 (2002) 294–299. 10. D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly 67 (1960) 525–532. 11. M. Ward, The arithmetical theory of linear recurring series, Trans. Amer. Math. Soc. 35 (1933) 600–628.

ON NONNEGATIVE MATRICES WITH GIVEN ROW AND COLUMN SUMS

where suB denotes the sum of the entries of a matrix B and m ≥ 0 (define 0 = 1). Hoffman’s proof was based on certain properties of stochastic matrices. Much later, in 1985, Sidorenko [9], without knowing Hoffman’s work, gave an independent proof as an elementary application of Holder’s inequality. In 1990, Virtanen [10] generalized (1.1) to the nonsymmetric case. He proved, using majorization, that

On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix

Abstract Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.

Perpendicularity in an Abelian Group

We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in and and in certain other groups. Our approach provides a justification for the use of the symbol denoting relative primeness in number theory and extends the domain of this convention to some degree. Related to that, we also consider parallelism from an axiomatic perspective.

Explorations of Finnish mathematics students' beliefs about the nature of mathematics

According to certain surveys made in Germany, mathematics can be seen either as a stagnant structure (formalism-related orientation), a collection of rules, formulas and methods (scheme-related ori ...

Student teachers’ concept definitions of area and their understanding about two-dimensionality of area

ABSTRACT We examine what kind of concept definitions of area a group of Finnish primary and lower secondary student teachers (N = 82) use, and how the quality of the definitions is associated with the participants’ success in seven exercises involving area. We are especially interested in how the understanding of the two-dimensionality of area appears in the participants’ responses. Only six student teachers were able to give a mathematically precise and correct definition of area. Altogether 26 participants defined it as ‘the size of a figure’ and 20 respondents required that a figure must be bounded. Further, 22 of them associated area with a formula or an example and eight respondents gave an incorrect or nonsensical definition. On average, student teachers master rather well the area formulae of a circle and a rectangle but already the relationship between the surface area of a cube and its volume is less commonly perceived. Most student teachers associate the area of an irregular domain with the method of exhaustion but clearly fewer of them acknowledge the difference between the area and an approximation of it. Surprisingly, there is only a weak Spearman correlation between the participants’ scores in the test exercises and the qualitatively ordered categories of concept definitions.