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After Strassen presented the first sub-cubic matrix multiplication algorithm, many Strassen-like algorithms are presented. Most of them with low asymptotic cost have large hidden leading coefficient which are thus impractical. To reduce the leading coefficient, Cenk and Hasan give a general approach reducing the leading coefficient of $$-algorithm to $5$ but increasing IO complexity. In 2017, Karstadt and Schwartz also reduce the leading coefficient of $$-algorithm to $5$ by the Alternative Basis Matrix Multiplication method. Meanwhile, their method reduces the IO complexity and low-order monomials in arithmetic complexity. In 2019, Beniamini and Schwartz generalize Alternative Basis Matrix Multiplication method reducing leading coefficient in arithmetic complexity but increasing IO complexity. In this paper, we propose a new matrix multiplication algorithm which reduces leading coefficient both in arithmetic complexity and IO complexity. We apply our method to Strassen-like algorithms improving arithmetic complexity and IO complexity (the comparison with previous results are shown in Tables 1 and 2). Surprisingly, our IO complexity of $$-algorithm is $14n^{\log_323}M^{-\frac{1}{2}} + o(n^{\log_323})$ which breaks Ballard's IO complexity low bound ($Ω(n^{\log_323}M^{1-\frac{\log_323}{2}})$) for recursive Strassen-like algorithms., 24 pages and one figure

The concept of almost interior Γ-hyperideals (A−I−Γ-hyperideals) in ordered Γ-semihypergroups is a generalization of the concept of interior Γ-hyperideals (I−Γ-hyperideals). In this study, the connections between I−Γ-hyperideals and A−I−Γ-hyperideals in ordered Γ-semihypergroups were presented. Also, we define the notion of m,n-Γ-hyperfilters of ordered Γ-semihypergroups and provide useful results on it. Moreover, we use the weak pseudoorders to construct quotient ordered Γ-semihypergroups by using ordered regular equivalence relations. These concepts lead us to a new research direction in ordered Γ-semihypergroups.

In this work, we consider a generalization of the nonlinear Langevin equation of fractional orders with boundary value conditions. The existence and uniqueness of solutions are studied by using results of the fixed point theory. Moreover, the previous results of fractional Langevin equations are a special case of our problem.

For a nondecreasing sequence of integers $S=(s_1, s_2, \ldots)$ an $S$-packing $k$-coloring of a graph $G$ is a mapping from $V(G)$ to $\{1, 2,\ldots,k\}$ such that vertices with color $i$ have pairwise distance greater than $s_i$. By setting $s_i = d + \lfloor \frac{i-1}{n} \rfloor$ we obtain a $(d,n)$-packing coloring of a graph $G$. The smallest integer $k$ for which there exists a $(d,n)$-packing coloring of $G$ is called the $(d,n)$-packing chromatic number of $G$. In the special case when $d$ and $n$ are both equal to one we speak of the packing chromatic number of $G$. We determine the packing chromatic number of the base-3 Sierpi\'{n}ski graphs $S_k$ and provide new results on $(d,n)$-packing chromatic colorings, $d \le 2$, for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for $H$-graphs $H(r)$.

For a nondecreasing sequence of integers $S=(s_1, s_2, \ldots)$ an $S$-packing $k$-coloring of a graph $G$ is a mapping from $V(G)$ to $\{1, 2,\ldots,k\}$ such that vertices with color $i$ have pairwise distance greater than $s_i$. By setting $s_i = d + \lfloor \frac{i-1}{n} \rfloor$ we obtain a $(d,n)$-packing coloring of a graph $G$. The smallest integer $k$ for which there exists a $(d,n)$-packing coloring of $G$ is called the $(d,n)$-packing chromatic number of $G$. In the special case when $d$ and $n$ are both equal to one we speak of the packing chromatic number of $G$. We determine the packing chromatic number of the base-3 Sierpi��ski graphs $S_k$ and provide new results on $(d,n)$-packing chromatic colorings, $d \le 2$, for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for $H$-graphs $H(r)$.

A double Roman dominating function of a graph $G$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that for each vertex $v$ with $f(v)=0$, there exists $u\in N(v)$ with $f(u)=3$, or there are $u,w\in N(v)$ with $f(u)=f(w)=2$, and if $f(v)=1$, then $v$ is adjacent to a vertex assigned at least $2$ under $f$. The double Roman domination number $\gamma_{dR}(G)$ is the minimum weight $f(V(G))=\sum_{v\in V(G)}f(v)$ among all double Roman dominating functions of $G$. An outer independent double Roman dominating function is a double Roman dominating function $f$ for which the set of vertices assigned $0$ under $f$ is independent. The outer independent double Roman domination number $\gamma_{oidR}(G)$ is the minimum weight taken over all outer independent double Roman dominating functions of $G$. In this work, we present some contributions to the study of outer independent double Roman domination in graphs. Characterizations of the families of all connected graphs with small outer independent double Roman domination numbers, and tight lower and upper bounds on this parameter are given. We moreover bound this parameter for a tree $T$ from below by two times the vertex cover number of $T$ plus one. We also prove that the decision problem associated with $\gamma_{oidR}(G)$ is NP-complete even when restricted to planar graphs with maximum degree at most four. Finally, we give an exact formula for this parameter concerning the corona graphs.

A \emph{perfect Roman dominating function} (PRDF) on a graph $G = (V, E)$ is a function $f : V \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The weight of a PRDF is the value $w(f) = \sum_{u \in V}f(u)$. The minimum weight of a PRDF on a graph $G$ is called the \emph{perfect Roman domination number $\gamma_R^p(G)$} of $G$. A graph $G$ is perfect Roman domination domination stable if the perfect Roman domination number of $G$ remains unchanged under the removal of any vertex. In this paper, we characterize all trees that are perfect Roman domination stable.

To improve the flame retardancy of polycarbonate (PC), a novel and environmentally friendly flame retardant was synthesized by combining boron-doped silicone resin (BSR) with a layered double hydroxide (LDH) modified with sodium dodecyl benzenesulfonate (SDBS) which was denoted as DBS-LDH/BSR. The structure of the hybrid was characterized by Fourier transform infrared (FTIR) spectroscopy and X-ray photoelectron spectroscopy (XPS), which indicated that BSR was successfully combined with DBS-LDH. X-ray diffraction (XRD) studies showed that the reaction of BSR occurred only on the surface of DBS-LDH. In addition, scanning electron microscopy (SEM) was used to further verify the combination of DBS-LDH with BSR. PC exhibited the optimum flame retardancy following the incorporation of 10 wt% DBS-LDH/BSR (5 wt% DBS-LDH and 5 wt% BSR). Based on thermogravimetric analysis, the char residue of this PC composite in air at 750 °C increased to 3.60 wt%. Mechanical test showed that the DBS-LDH/BSR could affect the mechanical properties after incorporation into PC. According to the UL-94 vertical burning test, the flame retardant rating of the PC composite improved to V-0. Furthermore, the limiting oxygen index (LOI) value of the PC composite increased to 34%. According to the cone calorimeter test, the peak heat release rate (PHRR) dramatically decreased by 44%. The morphology of the PC composite after combustion was characterized by SEM, which revealed that the pores of the composite were smaller than those of pure PC. This result was attributed to the limited spread of oxygen and heat permeation. Thus, both DBS-LDH and BSR contributed to the synergistic effects of reducing the fire hazard of PC.

For a function $f : V(G ) \rightarrow \{0, 1, 2\}$ we denote by $V_i$ the set of vertices to which the value $i$ is assigned by $f$, i.e. $V_i = \{ x \in V (G ) : f(x ) = i \}$. If a function $f: V(G) \rightarrow \{0,1,2\}$ satisfying the condition that $V_i$ is independent for $i \in \{1,2\}$ and every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = i$ for each $i \in \{1,2\}$, then $f$ is called a 2-rainbow independent dominating function (2RiDF). The weight $w(f)$ of a 2RiDF $f$ is the value $w(f) = |V_1|+|V_2|$. The minimum weight of a 2RiDF on a graph $G$ is called the \emph{2-rainbow independent domination number} of $G$. A graph $G$ is 2-rainbow independent domination stable if the 2-rainbow independent domination number of $G$ remains unchanged under removal of any vertex. In this paper, we characterize 2-rainbow independent domination stable trees and we study the effect of edge removal on 2-rainbow independent domination number in trees.

An L(2,1)-labeling of a graph G = (V, E) is a function f from the vertex set V(G) to the set of nonnegative integers such that the labels on adjacent vertices differ by at least two and the labels on vertices at distance two differ by at least one. The span of f is the difference between the largest and the smallest numbers in f(V). The ƛ-number of G, denoted by ƛ(G), is the minimum span over all L(2,1)-labelings of G. We consider the ƛ-number of Pn ☒ Cm and for n ≤ 11 the ƛ-number of Cn ☒ Cm. We determine ƛ-numbers of graphs of interest with the exception of a finite number of graphs and we improve the bounds on the ƛ-number of Cn ☒ Cm, m ≥ 24 and n ≥ 26.

For given graphs G1,G2,G3, the three-color Ramsey number R(G1,G2,G3) is defined to be the least positive integer n such that every 3-coloring of the edges of complete graph Kn contains a monochromatic copy of Gi colored with i, for some 1≤i≤3. In this paper, we prove that R(P4,P5,C3)=11, R(P4,P5,C4)=7, R(P4,P5,C5)=11, R(P4,P5,C7)=11, R(P4,P5,Ck)=k+2 for k≥23; R(P4,P6,C4)=8, R(P4,P6,C3)=R(P4,P6,C5)=R(P4,P6,C7)=13, R(P4,P6,Ck)=k+3 for k≥18.