Asymptotics of the number of threshold functions on a two-dimensional rectangular grid

Let $m,n\ge 2$, $m\le n$. It is well-known that the number of (two-dimensional) threshold functions on an $m\times n$ rectangular grid is {eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})= \frac{6}{\pi^2}(mn)^2+O(mn^2\log{m}). {eqnarray*} We improve the error term by showing that $$ t(m,n)=\frac{6}{\pi^2}(mn)^2+O(mn^2). $

The arithmetic derivative and Leibniz-additive functions

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$f(mn)=f(m)h_f(n)+f(n)h_f(m)$$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function $f$ is totally determined by the values of $f$ and $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions

Improving eigenvalue bounds using extra bounds

AbstractBounds for various functions of the eigenvalues of a Hermitian matrix A, based on the traces of A and A2, are improved. A technique is presented whereby these bounds can be improved by combining them with other bounds. In particular, the diagonal of A, in conjunction with majorization, is used to improve the bounds. These bounds all require O(n2) multiplications

Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

Let $pinmathbb P$ and $sinmathbb R$, and suppose that$emptysetne Psubsetmathbb P$ is finite.Given $ninmathbb Z_+$, let $n\u27$, $n\u27_p$, and $n\u27_P$ denote respectively its arithmetic derivative, arithmetic partial derivative with respect to~$p$,and arithmetic subderivative with respect to~$P$. We study the asymptotics of $$sum_{1le nle x}frac{n\u27}{n^s},,sum_{1le nle x}frac{n\u27_p}{n^s},quad{rm and},,sum_{1le nle x}frac{n\u27_P}{n^s}.$$ We also show that the abscissa of convergence of the corresponding Dirichlet series equals~two

Asymptotics of partial sums of the Dirichlet series of the arithmetic derivative

Let $pinmathbb P$ and $sinmathbb R$, and suppose that$emptysetne Psubsetmathbb P$ is finite.Given $ninmathbb Z_+$, let $n\u27$, $n\u27_p$, and $n\u27_P$ denote respectively its arithmetic derivative, arithmetic partial derivative with respect to~$p$,and arithmetic subderivative with respect to~$P$. We study the asymptotics of $$sum_{1le nle x}frac{n\u27}{n^s},,sum_{1le nle x}frac{n\u27_p}{n^s},quad{rm and},,sum_{1le nle x}frac{n\u27_P}{n^s}.$$ We also show that the abscissa of convergence of the corresponding Dirichlet series equals~two

Peruskoulun yläluokkien matematiikan opetus on eriytettävä

Kirjoittajien mielestä matematiikan opetus pitäisi eriyttää peruskoulun 7.–9. luokilla. Jäykkään tasoryhmittelyyn ei ole paluuta, mutta näiden luokkien matematiikan opetus on jaettava kahteen linjaan, joiden ”työnimet” voisivat olla matematiikkalinja ja laskentolinja. Niillä olisi eri opetussuunnitelmat ja oppimateriaalit