Related papers: On Symmetric Invertible Binary Pairing Functions
In the literature, few $n$-variable rotation symmetric bent functions have been constructed. In this paper, we present two infinite classes of rotation symmetric bent functions on $\mathbb{F}_2^{n}$ of the two forms: {\rm (i)}…
Given $d_1,\ldots,d_k$ in the field $F$, there is a weighted trace function $F^k\rightarrow F$ given by $tr(x_1,\ldots,x_k)=\sum d_ix_i$. We prove that $F^k$ satisfies trace identities of the forms $\alpha(d_1,\ldots,d_k) x^N=$ a linear…
Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_\mathcal{A}(n)$ be the number of ways to write $n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. We consider asymptotics of the summatory…
This paper is devoted to the investigation of harmonic and biharmonic functions on vector bundles equipped with spherically symmetric metrics. We will study the biharmonicity of vertical lifts of functions as well as $r$-radial functions on…
In this paper, we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres,…
Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew…
We show that double sums of the form $$ \sum_{i,j=-n} ^{n} |i^sj^t(i^k-j^k)^\beta| \binom {2n} {n+i} \binom {2n} {n+j} $$ can always be expressed in terms of a linear combination of just four functions, namely $\binom {4n}{2n}$, ${\binom…
For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric function $G\left( k,m\right)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $G\left( k,m\right)$ a…
It is well known that if a function from set A to set B has a right inverse then the function is a surjection and the right inverse is an injection. For finite sets, the number of functions, injections, and surjections can also be counted.…
In this paper, we consider the characterization of the bentness of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1} Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$ is even and $c_i\in…
A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B),…
Let $m$ be an even positive integer. A Boolean bent function $f$ on $\GF{m-1} \times \GF {}$ is called a \emph{cyclic bent function} if for any $a\neq b\in \GF {m-1}$ and $\epsilon \in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+\epsilon)$ is always…
We define a "shifted analogue" $\mathrm{SH}_n$ of the parking function symmetric function $\mathrm{PF}_n$. The expansion of $\mathrm{SH}_n$ in terms of three bases for shifted symmetric functions is explicitly described. We don't know a…
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
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…
In a basic framework of a complex Hilbert space equipped with a complex conjugation and an involution, linear operators can be real, quaternionic, symmetric or anti-symmetric, and orthogonal projections can furthermore be symplectic. This…
We characterize pairs of rational functions $A$, $B$ such that $A$ is semiconjugate to $B$, and $B$ is semiconjugate to $A$.
In this paper, some left-symmetric algebras are constructed from linear functions. They include a kind of simple left-symmetric algebras and some examples appearing in mathematical physics. Their complete classification is also given, which…
We determine all pairs $(f,g)$ of meromorphic functions that share four pairs of values $(a_\nu,b_\nu)$, $1\le\nu\le 4$, and a fifth pair $(a_5,b_5)$ under some mild additional condition.
An \emph{$(r,k)$-parking function} of length $n$ may be defined as a sequence $(a_1,\dots,a_n)$ of positive integers whose increasing rearrangement $b_1\leq\cdots\leq b_n$ satisfies $b_i\leq k+(i-1)r$. The case $r=k=1$ corresponds to…