The Minkowski question-mark function $?(x)$ is a continuous strictly increasing defined on $[0,1]$ interval. It well known fact that the derivative of this function, if exists, can take only two values: $0$ and $+\infty$. also value $?'(x)$ at point $x=[0;a_1,a_2,\ldots,a_t,\ldots]$ connected with limit behavior arithmetic mean $(a_1+a_2+\ldots+a_t)/t$. Particularly, N. Moshchevitin A. Dushisto...

indeed has $D=b¥cdot c^{*}$ for $b^{*}=(0,¥ldots,0,1)$ and $c^{*}$ with entries $¥frac{1}{2}(¥beta_{k}-a_{k})$ . (As Jan Willems once pointed out, all the entries are constants; however, the zeros and ones are stiff structure constants, while only the last row has “soft” parameters, to be encompassed by a rank-one control matrix.) Section 1 presents a canonic decomposition of the state space of...

We consider the allocation problem in which $m \leq (1-\ve) dn $ items are to be allocated to $n$ bins with capacity $d$. The items $x_1,x_2,\ldots,x_m$ arrive sequentially and when item $x_i$ arrives it is given two possible bin locations $p_i=h_1(x_i),q_i=h_2(x_i)$ via hash functions $h_1,h_2$. We consider a random walk procedure for inserting items and show that the expected time insertion t...

Let $G$ be a simple graph with vertex set ${v_1,v_2,ldots,v_n}$. The common neighborhood graph (congraph) of $G$, denoted by $con(G)$, is the graph with vertex set ${v_1,v_2,ldots,v_n}$, in which two vertices are adjacent if and only they have at least one common neighbor in the graph $G$. The basic properties of $con(G)$ and of its energy are established.

In this paper, we use trace methods to study the algebraic $K$-theory of rings form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. We compute relative $p$-adic $K$ groups for $R$ a perfectoid ring. particular, get integral when is finite field, and $K_*(R[x_1,\ldots, x_d)^2, (x_1,\ldots, x_d))$ perfect $\mathbb{F}_p$-algebra. conclude paper with some other notable computations, including which are n...

the set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $ns_n$‎. ‎a sequence $piin ns_{n}$ is said to be graphic if it is the degree sequence of a simple graph $g$ on $n$ vertices‎, ‎and such a graph $g$ is called a realization of $pi$‎. ‎the set of all graphic sequences in $ns_{n}$ is denoted by $gs_{n}$‎. ‎the complete product split graph on $l‎ ...