Related papers: Multiplicative functions commutable with binary qu…

We determine all pairs of real numbers $(\alpha, \beta)$ such that the dilated floor functions $\lfloor \alpha x\rfloor$ and $\lfloor \beta x\rfloor$ commute under composition, i.e., such that $\lfloor \alpha \lfloor \beta x\rfloor\rfloor =…

Let $q>2$ be a prime power and $f=-{\tt x}+t{\tt x}^q+{\tt x}^{2q-1}$, where $t\in\Bbb F_q^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q$ is even and…

Commuting is an important property in many cases of investigation of properties of operators as well as in various applications, especially in quantum physics. Using the observation that the generalized weighted differential operator of…

Let $f=a\x+\x^{3q-2}\in\Bbb F_{q^2}[\x]$, where $a\in\Bbb F_{q^2}^*$. We prove that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if and only if one of the following occurs: (i) $q=2^e$, $e$ odd, and $a^{\frac{q+1}3}$ is a primitive…

We provide a characterization of integers represented by the positive definite binary quadratic form $ax^2+bxy+cy^2$. In order to prove the main theorem, we define the "relative conductor" of two orders in an imaginary quadratic field. Then…

In this paper, we mainly discuss the characterization of a class of arithmetic functions $f: N \rightarrow C$ such that $f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ $(k, u, v \in N)$. We obtain a characterization with given condition, propose a…

A function in a class $\mathcal{F}(X)$ is said to be subdifferentially determined in $\mathcal{F}(X)$ if it is equal up to an additive constant to any function in $\mathcal{F}(X)$ with the same subdifferential. A function is said to be…

When searching for Calabi.Yau differential equations, often different formulas for the coefficients give the same differential equation. The coefficients are usually sums (simple, double or triple) of products of binomial coefficients. This…

In this paper we derive some identities and inequalities on the M\"obius mu function. Our main tool is phi functions for intervals of positive integers and their unions.

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function, $f: [0,1]\rightarrow [0,\infty]$ is a monotone function…

Two transforms of functions on a half-line are considered. It is proved that their composition gives a concave majorant for every nonnegative function. In particular, this composition is the identity transform on the class of nonnegative…

We prove an asymptotic formula for primes of the shape $f(a,b^2)$ with $a,b$ integers and of the shape $f(a,p^2)$ with $p$ prime. Here $f$ is a binary quadratic form with integer coefficients, irreducible over $\mathbb{Q}$ and has no local…

Let $p>3$ and consider a prime power $q=p^h$. We completely characterize permutation polynomials of $\mathbb{F}_{q^2}$ of the type $f_{a,b}(X) = X(1 + aX^{q(q-1)} + bX^{2(q-1)}) \in \mathbb{F}_{q^2}[X]$. In particular, using connections…

We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.

The well known formula $[X,Y]=\tfrac12\tfrac{\partial^2}{\partial t^2}|_0 (\Fl^Y_{-t}\o\Fl^X_{-t}\o\Fl^Y_t\o\Fl^X_t)$ for vector fields $X$, $Y$ is generalized to arbitrary bracket expressions and arbitrary curves of local diffeomorphisms.

The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate…

This paper extends previous work on linear correlations of representation functions of positive definite binary quadratic forms to allow indefinite forms.

If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz…

We describe a purely-multiplicative method for extending an analytic function. It calculates the value of an analytic function at a point, merely by multiplying together function values and reciprocals of function values at other points…

The four types of homogeneity -- additive, multiplicative, exponential, and logarithmic -- are generalized as transformations describing how a function $f$ changes under scaling or shifting of its arguments. These generalized homogeneity…