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This paper shows that the group SLn(R)${\rm SL}_n(R)$ is boundedly elementary generated for n⩾3$n\geqslant 3$ and R$R$ the ring of algebraic integers in a global function field. Contrary to previous statements for number fields and earlier statements for global function fields, the bounds proven in this paper for elementary bounded generation are independent of the underlying global function field and only depend on the integer n$n$. Combining our main result with earlier results, we further establish that elementary bounded generation always has bounds independent from the global field in question, only depending on n$n$. [ABSTRACT FROM AUTHOR]

Famous Zaremba's conjecture (1971) states that for each positive integer q⩾2$q\geqslant 2$, there exists a positive integer 1⩽a

Let V $V$ be a vector space over the finite field Fq ${{\mathbb{F}}}_{q}$. A q $q$‐Steiner system, or an S(t,k,V)q $S{(t,k,V)}_{q}$, is a collection ℬ ${\rm{{\mathcal B}}}$ of k $k$‐dimensional subspaces of V $V$ such that every t $t$‐dimensional subspace of V $V$ is contained in a unique element of ℬ ${\rm{{\mathcal B}}}$. A large set of q $q$‐Steiner systems, or an LS(t,k,V)q $LS{(t,k,V)}_{q}$, is a partition of the k $k$‐dimensional subspaces of V $V$ into S(t,k,V)q $S{(t,k,V)}_{q}$ systems. In the case that V $V$ has infinite dimension, the existence of an LS(t,k,V)q $LS{(t,k,V)}_{q}$ for all finite t,k $t,k$ with 1

In this paper, we investigate the minimum number of triangles, denoted by t(n,k) $t(n,k)$, in n $n$‐vertex k $k$‐regular graphs, where n $n$ is an odd integer and k $k$ is an even integer. The well‐known Andrásfai–Erdős–Sós Theorem has established that t(n,k)>0 $t(n,k)\gt 0$ if k>2n5 $k\gt \frac{2n}{5}$. In a striking work, Lo has provided the exact value of t(n,k) $t(n,k)$ for sufficiently large n $n$, given that 2n5+12n5

Given two graphs G , H $G,H$ and a positive integer q $q$, an ( H , q ) $(H,q)$‐coloring of G $G$ is an edge‐coloring of G $G$ such that every copy of H $H$ in G $G$ receives at least q $q$ distinct colors. The bipartite Erdős–Gyárfás function r ( K n , n , K s , t , q ) $r({K}_{n,n},{K}_{s,t},q)$ is defined to be the minimum number of colors needed for K n , n ${K}_{n,n}$ to have a ( K s , t , q ) $({K}_{s,t},q)$‐coloring. For balanced complete bipartite graphs K p , p ${K}_{p,p}$, the function r ( K n , n , K p , p , q ) $r({K}_{n,n},{K}_{p,p},q)$ was studied systematically in Axenovich et al. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs K s , t ${K}_{s,t}$ that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi. [ABSTRACT FROM AUTHOR]

This paper details a literature review of mathematics vocabulary intervention studies for students with mathematics difficulty. The primary aim was to identify instructional practices that support mathematics vocabulary development. We conducted a database search to identify mathematics intervention studies either focused exclusively on vocabulary or with an embedded vocabulary component. Ultimately, 13 studies with participants from kindergarten to Grade 8 were included in the review. The majority of included studies had dual foci on vocabulary and other mathematics content, including whole number computation, word problem solving, fractions, algebra, or geometry. All studies that measured mathematics vocabulary performance indicated positive student outcomes. In addition, multiple studies indicated positive effects on measures of other mathematics content beyond mathematics vocabulary knowledge. We noted six instructional practices across studies that bolstered the mathematics vocabulary performance of students with mathematics difficulty. From most to least common, these practices included formal vocabulary use, explicit instruction, use of representations, repeated exposures, pre‐teaching, and graphic organizers. Implications are addressed for both researchers and practitioners. [ABSTRACT FROM AUTHOR]

Let G $G$ be a simple graph. Let Δ(G) ${\rm{\Delta }}(G)$ and χ′(G) $\chi ^{\prime} (G)$ be the maximum degree and the chromatic index of G $G$, respectively. We call G $G$overfull if ∣E(G)∣∕⌊∣V(G)∣∕2⌋>Δ(G) $| E(G)| \unicode{x02215}\lfloor | V(G)| \unicode{x02215}2\rfloor \gt {\rm{\Delta }}(G)$, and critical if χ′(H)<χ′(G) $\chi ^{\prime} (H)\lt \chi ^{\prime} (G)$ for every proper subgraph H $H$ of G $G$. Clearly, if G $G$ is overfull then χ′(G)=Δ(G)+1 $\chi ^{\prime} (G)={\rm{\Delta }}(G)+1$. The core of G $G$, denoted by GΔ ${G}_{{\rm{\Delta }}}$, is the subgraph of G $G$ induced by all its maximum degree vertices. We believe that utilizing the core degree condition could be considered as an approach to attack the overfull conjecture. Along this direction, we in this paper show that for any integer k≥2 $k\ge 2$, if G $G$ is critical with Δ(G)≥23n+3k2 ${\rm{\Delta }}(G)\ge \frac{2}{3}n+\frac{3k}{2}$ and δ(GΔ)≤k $\delta ({G}_{{\rm{\Delta }}})\le k$, then G $G$ is overfull. [ABSTRACT FROM AUTHOR]

This paper introduces a modified sequential version method for optimizing a linear function over an integer efficient set, as well as a new exact parallel algorithm. The performance of parallel programming in this context is clear and shown through different instances with different sizes. Each procedure builds a finite monotonous sequence of values for the main criterion to be optimized, in a reasonable amount of CPU execution time. This latter remains much better. For the first time, the Algerian IBNBADIS cluster—CERIST—was used with this type of problem. Significant results are obtained by both proposed techniques, particularly with the parallel one. [ABSTRACT FROM AUTHOR]

Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field Q(d)$\mathbb {Q}(\sqrt {d})$ where d<0$d<0$. We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non‐zero frieze patterns for any given height. We then show that apart from the cases d∈{−1,−2,−3,−7,−11}$d\in \lbrace -1,-2,-3,-7,-11\rbrace$ all non‐zero frieze patterns over the rings of integers Od$\mathcal {O}_d$ for d<0$d<0$ have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns. [ABSTRACT FROM AUTHOR]

In this paper, we use theoretical and computational tools to continue our investigation of K2 ${K}_{2}$‐hamiltonian graphs, that is, graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, and their interplay with K1 ${K}_{1}$‐hamiltonian graphs, that is, graphs in which every vertex‐deleted subgraph is hamiltonian. Perhaps surprisingly, there exist graphs that are both K1 ${K}_{1}$‐ and K2 ${K}_{2}$‐hamiltonian, yet non‐hamiltonian, for example, the Petersen graph. Grünbaum conjectured that every planar K1 ${K}_{1}$‐hamiltonian graph must itself be hamiltonian; Thomassen disproved this conjecture. Here we show that even planar graphs that are both K1 ${K}_{1}$‐ and K2 ${K}_{2}$‐hamiltonian need not be hamiltonian, and that the number of such graphs grows at least exponentially. Motivated by results of Aldred, McKay, and Wormald, we determine for every integer n $n$ that is not 14 or 17 whether there exists a K2 ${K}_{2}$‐hypohamiltonian, that is non‐hamiltonian and K2 ${K}_{2}$‐hamiltonian, graph of order n $n$, and characterise all orders for which such cubic graphs and such snarks exist. We also describe the smallest cubic planar graph which is K2 ${K}_{2}$‐hypohamiltonian, as well as the smallest planar K2 ${K}_{2}$‐hypohamiltonian graph of girth 5. We conclude with open problems and by correcting two inaccuracies from the first article. [ABSTRACT FROM AUTHOR]

Let k $k$ be a positive integer. A graph is said to be uniformly k $k$‐connected if between any pair of vertices the maximum number of independent paths is exactly k $k$. Dawes showed that all minimally 3‐connected graphs can be constructed from K4 ${K}_{4}$ such that every graph in each intermediate step is also minimally 3‐connected. In this paper, we generalize Dawes' result to uniformly 3‐connected graphs. We give a constructive characterization of the class of uniformly 3‐connected graphs which differs from the characterization provided by Göring et al., where their characterization requires the set of all 3‐connected and 3‐regular graphs as a starting set, the new characterization requires only the graph K4 ${K}_{4}$. Eventually, we obtain a tight bound on the number of edges in uniformly 3‐connected graphs. [ABSTRACT FROM AUTHOR]

In this paper, we analyze two classes of fully discrete spectral volume schemes (SV) for solving the one‐dimensional scalar hyperbolic equation. These two schemes are constructed by using the forward Euler (EU) method or the second‐order Runge–Kutta (RK2) method in time‐discretization, and by letting a piecewise kth degree(k≥1$$ k\ge 1 $$ is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with k$$ k $$ Gauss–Legendre points (LSV) or right‐Radau points (RRSV). We prove that for the EU‐SV schemes, the weak(2) stability holds and the L2$$ {L}_2 $$ norm errors converge with optimal orders 풪(hk+1+τ), provided that the CFL condition τ≤Ch2$$ \tau \le C{h}^2 $$ is satisfied. While for the RK2‐SV schemes, the weak(4) stability holds and the L2$$ {L}_2 $$ norm errors converge with optimal orders 풪(hk+1+τ2), provided that the CFL condition τ≤Ch43$$ \tau \le C{h}^{\frac{4}{3}} $$ is satisfied. Here h$$ h $$ and τ$$ \tau $$ are, respectively, the spacial and temporal mesh sizes and the constant C$$ C $$ is independent of h$$ h $$ and τ$$ \tau $$. Our theoretical findings have been justified by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a model that combines the movement of a multivisit drone with a limited endurance and a base vehicle that can move freely in the continuous space is considered. The mothership is used to charge the battery of the drone, whereas the drone performs the task of visiting multiple targets of distinct shapes: points and polygonal chains. For polygonal chains, it is required to traverse a given fraction of its lengths that represent surveillance/inspection activities. The goal of the problem is to minimize the overall weighted distance traveled by both vehicles. A mixed integer second‐order cone program is developed and strengthened using valid inequalities and giving good bounds for the Big‐M constants that appear in the model. A refined matheuristic that provides reasonable solutions in short computing time is also established. The quality of the solutions provided by both approaches is compared and analyzed on an extensive battery of instances with different number and shapes of targets, which shows the usefulness of our approach and its applicability in different situations. [ABSTRACT FROM AUTHOR]

A conjecture proposed by Bermond and Thomassen in 1981 states that every digraph with minimum out‐degree at least 2k−1 $2k-1$ contains k $k$ vertex disjoint directed cycles for any integer k≥1 $k\ge 1$, which has received substantial attention. This conjecture has been confirmed for k≤3 $k\le 3$. In 2014, Lichiardopol raised a very related conjecture that for every integer k≥2 $k\ge 2$, there exists an integer g(k) $g(k)$ such that every digraph with minimum out‐degree at least g(k) $g(k)$ contains k $k$ vertex disjoint directed cycles of different lengths. For a digraph D $D$ and a set of k $k$ vertex disjoint directed cycles C ${\mathscr{C}}$ in D $D$, we denote κk(C) ${\kappa }^{k}({\mathscr{C}})$ to be the maximum number of directed cycles in C ${\mathscr{C}}$ of distinct lengths. Let κk(D)=max{κk(C)∣Cis a set ofkvertexdisjoint directed cycles inD} ${\kappa }^{k}(D)=\max \{{\kappa }^{k}({\mathscr{C}})| {\mathscr{C}}\,\,\text{is a set of}\,\,k\,\text{vertex}\,\text{disjoint directed cycles in}\,\,D\}$. We define κk(D)=0 ${\kappa }^{k}(D)=0$ if D $D$ has no k $k$ vertex disjoint directed cycles. In this paper, we mainly investigate vertex disjoint directed cycles in tournaments and bipartite tournaments. We first show that κk(D)≥2 ${\kappa }^{k}(D)\ge 2$ for every tournament D $D$ with minimum out‐degree at least 2k−1 $2k-1$, where k≥3 $k\ge 3$. We further prove that for k≥1 $k\ge 1$ and γ∈{1,2,...,k} $\gamma \in \{1,2,\ldots ,k\}$, any tournament D $D$ with minimum out‐degree at least γ2−2γ+6k−32 $\frac{{\gamma }^{2}-2\gamma +6k-3}{2}$ satisfies that κk(D)≥γ ${\kappa }^{k}(D)\ge \gamma $. Moreover, we deduce that for any tournament D $D$ with minimum out‐degree at least 7, κ3(D)=3 ${\kappa }^{3}(D)=3$ holds. Additionally, we classify strong bipartite tournaments with minimum out‐degree at least 2k−1 $2k-1$ in which any k $k$ vertex disjoint directed cycles have the same length, where k≥2 $k\ge 2$. That is, for any strong bipartite tournament D $D$ with minimum out‐degree at least 2k−1 $2k-1$, then κk(D)=1 ${\kappa }^{k}(D)=1$ if and only if D $D$ is isomorphic to a member of BT(n1,n2,...,n2k) $BT({n}_{1},{n}_{2},\ldots ,{n}_{2k})$, which is defined in the context. [ABSTRACT FROM AUTHOR]

In this paper, we prove a weak form of the abc$abc$ conjecture generalised to algebraic number fields. Given integers satisfying a+b=c$a+b=c$, Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height HK(a,b,c)$H_{K}(a,\,b,\,c)$ in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers a,b,c$a,\,b,\,c$ in a number field K satisfying a+b=c$a+b=c$, we give an upper bound for the logarithm of the projective height HL(a,b,c)$H_{L}(a,\,b,\,c)$ in terms of norms of prime ideals dividing abcOL$abc \mathcal {O}_{L}$, where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical G:=G(a,b,c)$G:=G(a,\,b,\,c)$ as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound logHLa,b,c

In 1977, G. Bennett proved, by means of nondeterministic methods, an inequality that plays a fundamental role in a series of optimization problems. More precisely, Bennett's inequality shows that, for p1,p2∈[1,∞]$p_{1},p_{2} \in [1,\infty ]$ and all positive integers n1,n2$n_{1},n_{2}$, there exists a bilinear form An1,n2:(Rn1,∥·∥p1)×(Rn2,∥·∥p2)⟶R$A_{n_{1},n_{2}}\colon (\mathbb {R}^{n_{1}},\Vert \cdot \Vert _{p_{1}}) \times (\mathbb {R}^{n_{2}},\Vert \cdot \Vert _{p_{2}}) \longrightarrow \mathbb {R}$ with coefficients ±1 satisfying An1,n2⩽Cp1,p2maxn11−1p1n2max12−1p2,0,n21−1p2n1max12−1p1,0$$\begin{eqnarray*} &&\hspace*{13pc}{\left\Vert A_{n_{1},n_{2}}\right\Vert} \leqslant C_{p_{1},p_{2}}\max {\left\lbrace n_{1}^{1-\frac{1}{p_{1}}}n_{2}^{\max {\left\lbrace \frac{1}{2}-\frac{1}{p_{2} },0\right\rbrace} },\right.}\\ &&\hspace*{21pc}{\left.n_{2}^{1-\frac{1}{p_{2}}}n_{1}^{\max {\left\lbrace \frac{1}{2} -\frac{1}{p_{1}},0\right\rbrace} }\right\rbrace} \end{eqnarray*}$$for a certain constant Cp1,p2$C_{p_{1},p_{2}}$ depending just on p1,p2$p_{1},p_{2}$; moreover, the exponents of n1,n2$n_{1},n_{2}$ cannot be improved. In this paper, using a constructive approach, we prove that Cp1,p2⩽8/5$C_{p_{1},p_{2}}\leqslant \sqrt {8/5}$ whenever p1,p2∈[2,∞]$p_{1},p_{2}\in [ 2,\infty ]$ or p1=p2=p∈[1,∞]$p_{1}=p_{2}=p\in [ 1,\infty ]$; our techniques are applied to provide new upper bounds for the constants of the Gale–Berlekamp switching game, improving estimates obtained by Brown and Spencer in 1971 and by Carlson and Stolarski in 2004. [ABSTRACT FROM AUTHOR]