We consider Presburger arithmetic (PA) extended by scalar multiplication an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger ($\alpha$-PA). show that the complexity of deciding sentences in $\alpha$-PA is substantially harder than PA. Indeed, when $\alpha$ quadratic $r\geq 4$, with $r$ alternating quantifier blocks at most $c\ r$ variables inequalities requires ...

For a given bounded positive (semidefinite) linear operator $A$ on complex Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$, we consider the semi-Hilbertian \cdot\rangle_A \big)$ where ${\langle x, y\rangle}_A := Ax, y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded $T$ $\mathcal{H}$ is by\[\omega_A(T)=\sup\Big\{\big|{\langle Tx, x\rangle}_A...

The lll algorithm is a polynomial-time for reducing d-dimensional lattice with exponential approximation factor. Currently, the most efficient variant of lll, by Neumaier and Stehlé, has theoretical running time in $$d^4\cdot B^{1+o\left( 1\right) }$$ where B bitlength entries, but never been implemented. This work introduces new asymptotically fast, parallel, yet heuristic, reduction algorithm...

Consider the random subgraph process on a base graph $G$ $n$ vertices: sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of subgraphs obtained by choosing an ordering edges uniformly at random, and sequentially adding to $G_0$, empty vertex set $G$, according chosen ordering. We show that if has one following properties: 1. There is positive constant $\varepsilon > 0$ such $\delta (G) \geq \lef...

If A, B are bounded linear operators on a complex Hilbert space, then we prove that $$\begin{aligned} w(A)\le & {} \frac{1}{2}\left( \Vert A\Vert +\sqrt{r\left( |A||A^*|\right) }\right) ,\\ w(AB \pm BA)\le 2\sqrt{2}\Vert B\Vert \sqrt{ w^2(A)-\frac{c^2(\mathfrak {R}(A))+c^2(\mathfrak {I}(A))}{2} }, \end{aligned}$$ where $$w(\cdot ),\left\| \cdot \right\| $$ , and $$r(\cdot )$$ the numerical radi...

Abstract Here, the existence and multiplicity of weak solutions to a generalized $(p(\cdot ),q(\cdot ))$ ( p ⋅ ) , q -Laplace equation involving Leray–Lions type operators with Hardy potential are studied under Dirichlet boun...

The estimate of the change rate solar gravitational parameter $\mathrm{d}(GM_{\odot})/\mathrm{d}t$ is obtained from processing modern positional observations planets and spacecraft. Observations were processed parameters determined basing on numerical planetary ephemeris EPM2019. annual decrease in mass $M_{\odot}$ accounts for loss through radiation ${\dot M}_{{\odot}\mathrm{rad}}$, outgoing w...

Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has $1$. On the other hand, for any $\varepsilon>0$, show there are $>1-\varepsilon$ which $\mathcal{A}\cdot\mathcal{A}$ $<\varepsilon$. This answers questions Hegyv\'{a}ri, Hennecart Pach.

The orientation morphism $Or(\cdot)(P)\colon \gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\gamma$ ordered sets edges and without multiple one-cycles. It is known that $d$-cocycles $\boldsymbol{\gamma}\in\ker d$ respect to the vertex-expanding differenti...

New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation $$ \dot{x}(t) + \sum_{k=1}^m a_k(t)x(h_k(t)) \int_{g(t)}^t K(t,s) x(s)ds=0, where $h_k(t)\leq t$, $g(t)\leq $a_k(\cdot)$ and kernel $K(\cdot,\cdot)$ oscillatory and, generally, discontinuous functions. The proofs based on establishing boundedness of solutions later u...