DSpace collection: 第47卷第3期

TWIN SIGNED ROMAN DOMINATION NUMBERS IN DIRECTED GRAPHS

title: TWIN SIGNED ROMAN DOMINATION NUMBERS IN DIRECTED GRAPHS abstract: Let DD be a finite simple digraph with vertex set V(D)V(D) and arc set A(D)A(D). A twin signed Roman dominating function (TSRDF) on the digraph DD is a function f:V(D)→{−1,1,2}f:V(D)→{−1,1,2} satisfying the conditions that (i) ∑x∈N−[v]f(x)≥1∑x∈N−[v]f(x)≥1 and ∑x∈N+[v]f(x)≥1∑x∈N+[v]f(x)≥1 for each v∈V(D)v∈V(D), where N−[v]N−[v] (resp. N+[v]N+[v]) consists of vv and all in-neighbors (resp. out-neighbors) of vv, and (ii) every vertex uu for which f(u)=−1f(u)=−1 has an in-neighbor vv and an out-neighbor ww for which f(v)=f(w)=2f(v)=f(w)=2. The weight of an TSRDF ff is ω(f)=∑v∈V(D)f(v)ω(f)=∑v∈V(D)f(v). The twin signed Roman domination number γ∗sR(D)γsR∗(D) of DD is the minimum weight of an TSRDF on DD. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on γ∗sR(D)γsR∗(D). In addition, we determine the twin signed Roman domination number of some classes of digraphs.

INEQUALITIES FOR SOME CLASSICAL INTEGRAL TRANSFORMS

title: INEQUALITIES FOR SOME CLASSICAL INTEGRAL TRANSFORMS abstract: By using a form of the Cauchy-Bunyakovsky-Schwarz inequality, we establish new inequalities for some classical integral transforms such as Laplace transform,Fourier transform, Fourier cosine transform, Fourier sine transform, Mellin transform and Hankel transform.

ON CERTAIN INTEGRAL FORMULAS INVOLVING THE PRODUCT OF BESSEL FUNCTION AND JACOBI POLYNOMIAL

title: ON CERTAIN INTEGRAL FORMULAS INVOLVING THE PRODUCT OF BESSEL FUNCTION AND JACOBI POLYNOMIAL abstract: In the present paper, we establish some interesting integrals involving the product of Bessel function of the first kind with Jacobi polynomial, which are expressed in terms of Kampe de Feriet and Srivastava and Daoust functions. Some other integrals involving the product of Bessel (sine and cosine) function with ultraspherical polynomial, Gegenbauer polynomial, Tchebicheff polynomial, and Legendre polynomial are also established as special cases of our main results. Further, we derive an interesting connection between Kampe de Feriet and Srivastava and Daoust functions.

QUENCHING PROBLEMS FOR A SEMILINEAR REACTION-DIFFUSION SYSTEM WITH SINGULAR BOUNDARY OUTFLUX

title: QUENCHING PROBLEMS FOR A SEMILINEAR REACTION-DIFFUSION SYSTEM WITH SINGULAR BOUNDARY OUTFLUX abstract: In this paper, we study two quenching problems for the following semilinear reaction-diffusion system: ut=uxx+(1−v)−p1,0

HERMITE-HADAMARD TYPE INEQUALITIES FOR (P1,H1)-(P2,H2)-CONVEX FUNCTIONS ON THE CO-ORDINATES

title: HERMITE-HADAMARD TYPE INEQUALITIES FOR (P1,H1)-(P2,H2)-CONVEX FUNCTIONS ON THE CO-ORDINATES abstract: In this paper, we establish some Hermite-Hadamard type inequalities for (p1,h1)(p1,h1)-(p2,h2)(p2,h2)-convex function on the co-ordinates. Furthermore, some inequalities of Hermite-Hadamard type involving product of two convex functions on the co-ordinates are also considered. The results presented here would provide extensions of those given in earlier works.

ON SOLVABILITY OF COUPLED HYBRID SYSTEM OF QUADRATIC FRACTIONAL INTEGRAL EQUATIONS

title: ON SOLVABILITY OF COUPLED HYBRID SYSTEM OF QUADRATIC FRACTIONAL INTEGRAL EQUATIONS abstract: Of concern is studying solvability of the hybrid systems of quadratic fractional integral equations. To this aim applying hybrid fixed point theory due to DhageDhage, existence of at least one positive solution for mentioned systems via so called D-Lipschitzian mappings will be concluded . We illustrate the obtained results by presenting an example.

A NOTE ON THE LEAST (NORMALIZED) LAPLACIAN EIGHVA;UE OF SIGNED GRAPHS

title: A NOTE ON THE LEAST (NORMALIZED) LAPLACIAN EIGHVA;UE OF SIGNED GRAPHS abstract: Let Γ=(G,σ)Γ=(G,σ) be a connected signed graph, and L(Γ)L(Γ) be its Laplacian and L(Γ)L(Γ) its normalized Laplacian with eigenvalues λ1≥λ2≥⋯≥λnλ1≥λ2≥⋯≥λn and μ1≥μ2≥⋯≥μnμ1≥μ2≥⋯≥μn, respectively. It is known that a signed graph ΓΓ is balanced if and only if λn=0λn=0 (or μn=0μn=0). We show that λnλn and μnμn measure how much ΓΓ is far from being balanced by proving that μn(Γ)λn(Γ)≤min{2ϵ(Γ)m,ν(Γ)ν(Γ)+ν1(Γ)},≤min{λ1(Γ′):Γ−Γ′isbalanced}, μn(Γ)≤min{2ϵ(Γ)m,ν(Γ)ν(Γ)+ν1(Γ)},λn(Γ)≤min{λ1(Γ′):Γ−Γ′isbalanced}, where ν(Γ)ν(Γ) (resp. ϵ(Γ)ϵ(Γ)) denotes the frustration number (resp. the frustration index) of ΓΓ, that is the minimum number of vertices (resp. edges) to be deleted such that the signed graph is balanced.

A GCD AND LCM-LIKE INEQUALITY FOR MULTIPLICATIVE LATTICES

title: A GCD AND LCM-LIKE INEQUALITY FOR MULTIPLICATIVE LATTICES abstract: Let A1,…,AnA1,…,An (n≥2)(n≥2) be elements of an commutative multiplicative lattice. Let G(k)G(k) (resp., L(k)L(k)) denote the product of all the joins (resp., meets) of kk of the elements. Then we show that L(n)G(2)G(4)⋯G(2⌊n/2⌋)≤G(1)G(3)⋯G(2⌈n/2⌉−1). L(n)G(2)G(4)⋯G(2⌊n/2⌋)≤G(1)G(3)⋯G(2⌈n/2⌉−1). In particular this holds for the lattice of ideals of a commutative ring. We also consider the relationship between G(n)L(2)L(4)⋯L(2⌊n/2⌋) and L(1)L(3)⋯L(2⌈n/2⌉−1) G(n)L(2)L(4)⋯L(2⌊n/2⌋) and L(1)L(3)⋯L(2⌈n/2⌉−1) and show that any inequality relationships are possible.