Abstract It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time when all elements are represented by so-called power words, i.e., words of form $$p_1^{z_1} p_2^{z_2} \cdots p_k^{z_k}$$ p 1 z </mml:...

Abstract Given finite configurations $P_1, \dots , P_n \subset \mathbb {R}^d$ let us denote by $\mathbf {m}_{\mathbb {R}^d}(P_1, P_n)$ the maximum density a set $A \subseteq can have without containing congruent copies of any $P_i$ . We will initiate study this geometrical parameter, called independence considered configurations, and give several results we believe are interesting. For instance...

In their famous paper on partitions, Hardy and Ramanujan also raised the question of behaviour number $p_s(n)$ partitions a positive integer~$n$ into $s$-th powers gave some preliminary results. We give first an asymptotic formula to all orders, then exact formula, describing corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root unity, generali...

A {\em theta} is a graph made of three internally vertex-disjoint chordless paths $P_1 = \dots b$, $P_2 $P_3 b$ length at least~2 and such that no edges exist between the except incident to $a$ $b$. pyramid} b_1$, b_2$, b_3$ least~1, two which have least 2, $a$, $b_1b_2b_3$ triangle those to~$a$. An \emph{even hole} cycle even length. For non-negative integers $i\leq j\leq k$, let $S_{i,j,k}$ b...

As part of an All-Sky Automated Survey for SuperNovae (ASAS-SN) search sources with large flux decrements, we discovered a transient where the quiescent, stellar source, ASASSN-V J192114.84+624950.8, rapidly decreased in by $\sim55\%$ ($\sim0.9$ mag) g-band. The \textit{TESS} light curve revealed that source is highly eccentric, eclipsing binary. Fits to using \textsc{phoebe} find binary orbit ...

In CRYPTO’03, Patarin conjectured a lower bound on the number of distinct solutions $$(P_1, \ldots , P_{q}) \in (\{0,1\}^{n})^{q}$$ satisfying system equations form $$X_i \oplus X_j = \lambda _{i,j}$$ such that $$P_1, P_2, $$ $$P_{q}$$ are pairwise distinct. This result is known as “ $$P_i P_j$$ Theorem for any $$\xi _{\max }$$ ” or alternatively Mirror Theory general which was later proved by ...

We address the discretization of two-phase Darcy flows in a fractured and deformable porous medium, including frictional contact between matrix-fracture interfaces. Fractures are described as network planar surfaces leading to so-called mixed- or hybrid-dimensional models. Small displacements linear elastic behavior considered for matrix. Phase pressures supposed be discontinuous at interfaces,...

We consider the vertex cover problem with multiple coverage constraints in hypergraphs. In this problem, we are given a hypergraph \(G=(V,E)\) maximum edge size f, cost function \(w: V\rightarrow {\mathbb {Z}}^+\), and subsets \(P_1,P_2,\ldots ,P_r\) of E along covering requirements \(k_1,k_2,\ldots ,k_r\) for each subset. The objective is to find minimum subset S V such that, \(P_i\), at least...

This paper separately evaluates the effects of inclination and asymmetry solar coronal loops on resonant absorption kink magnetohydrodynamic (MHD) oscillations. We modelled a typical loop by straight axisymmetric cylindrical magnetic flux tube filled with cold plasma. solved dispersion relation numerically for different values longitudinal mass density stratification. show that, in inclined asy...