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The notion of a descent polynomial, a function in enumerative combinatorics that counts permutations with specific properties, enjoys a revived recent research interest due to its connection with other important notions in combinatorics,…
Define $||n||$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. The set $\mathscr{D}$ of defects, differences $\delta(n):=||n||-3\log_3 n$, is known…
As it follows from the theory of almost periodic functions the set of integer solutions $q$ to the Kronecker system $|\omega_{j} q - \theta_{j}| < \varepsilon \pmod 1$, $j=1,\ldots,m$, where $1,\omega_{1},\ldots,\omega_{m}$ are linearly…
It is shown that for finding rational approximates to m'th root of any integer to any accuracy one only needs the ability to count and to distinguish between m different classes of objects. To every integer N can be associated a…
We show there is an uncountable number of parallel total perfect codes in the integer lattice graph ${\Lambda}$ of $\R^2$. In contrast, there is just one 1-perfect code in ${\Lambda}$ and one total perfect code in ${\Lambda}$ restricting to…
Let $n$ be a positive integer and let $0 < \alpha < n.$ In this paper, we continue our study of the integral equation $$ u(x) = \int_{R^{n}} \frac{u(y)^{(n+\alpha)(n-\alpha)}{|x - y|^{n-\alpha}}dy.$$ We mainly consider singular solutions in…
For given positive integers $m$ and $n$ with $m<n$, the Prouhet-Tarry-Escott problem asks if there exist two disjoint multisets of integers of size $n$ having identical $k$th moments for $1\leq k\leq m$; in the ideal case one requires…
We describe an apparently new measure of multivariate goodness-of-fit between sets of quantitative results from a model (simulation, analytical, or multiple regression), paired with those observed under corresponding conditions from the…
We study some divisibility properties of quasiperfect numbers. We show that if $N=(p_1 p_2 \cdots p_t)^{2a}=m^2$ is quasiperfect, then $2a+1$ is divisible by $3$ and $N$ has at least one prime factor smaller than $\exp 716.7944$. Moreover,…
There are numbers k and s and a URM program A(n,m) satisfying the following conditions. 1. If A(n,m) halts, then Cn(m) diverges. 2. For all n, C_k(n) = A(n,n) and C_s(n) = C_k(s). 3. A(k,s) halts and for all n, A(s,n) diverges. Here C_n(_)…
In this paper based on a sort of linear function, a deterministic and simple algorithm with an algebraic structure is presented for calculating all (and only) $k$-almost primes ($where$ $\exists n\in {\rm N} $, $1{\le} k {\le}n$) in certain…
We introduce and study the defect function associated to a pair of filtrations of ideals, which generalizes the symbolic defect of ideals. Under the assumption that the Rees algebra of one filtration is Noetherian and that a natural graded…
We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…
An almost complete intersection ideal can be seen as a $d$-sequence ideal with the minimal number of generators being one more than its height. In this paper, we give exact formulas for the regularity of powers of graded almost complete…
A sufficient statistic is a deterministic function that captures an essential property of a probabilistic function (channel, kernel). Being a sufficient statistic can be expressed nicely in terms of string diagrams, as Tobias Fritz showed…
We show that the problem of counting perfect matchings remains #P-complete even if we restrict the input to very dense graphs, proving the conjecture in [5]. Here "dense graphs" refer to bipartite graphs of bipartite independence number…
Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with discriminant and regulator $\Delta_K, R_K$ respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary…
Let $\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and…
We study the Lagarias inequality, an elementary criterion equivalent to the Riemann Hypothesis. Using a continuous extension of the harmonic numbers, we show that the sequence $B_n=\frac{H_n+e^{H_n}\log(H_n)}{n}$ is strictly increasing for…
Data classification, the process of analyzing data and organizing it into categories, is a fundamental computing problem of natural and artificial information processing systems. Ideally, the performance of classifier models would be…