Indian Journal of Pure and Applied Mathematics最新文献

In the present paper, we introduce a new sequence of (alpha -)Bernstein-Kantorovich type operators, which fix constant and preserve Korovkin’s other test functions in a limiting sense. We extend the natural Korovkin and Voronovskaja type results into a sequence of probability measure spaces. Then, we establish the convergence properties of these operators using the Ditzian-Totik modulus of smoothness for Lipschitz-type space and functions with derivatives of bounded variations.

This research article introduces the concept of the clear graph associated with a ring ({mathcal {R}}) with identity, denoted as (Cr({mathcal {R}})). This graph comprises vertices of the form ({(x,u):) x is a unit regular element of R and u is a unit of ({mathcal {R}})} and two distinct vertices (x, u) and (y, v) are adjacent if and only if either (xy=yx=0) or (uv=vu=1). This research article also focuses on a specific subgraph of (Cr({mathcal {R}})) denoted as (Cr_2({mathcal {R}})), which is formed by vertices ({(x,u) :x) is a nonzero unit regular element of (R }). The significance of (Cr_2({mathcal {R}})) within the context of (Cr{({mathcal {R}})}) is explored in the article. Taken (Cr_2({mathcal {R}})) into consideration, we found connectedness, regularity, planarity, and outer planarity. Moreover, we characterized the ring ({mathcal {R}}) for which (Cr_2({mathcal {R}})) is unicyclic, a tree and a split graph. Finally, we have found genus one of (Cr_2({mathcal {R}})).

Let k be an algebraically closed field of characteristic (p > 3) and S be a smooth projective surface over k with k-rational point x. For (n ge 2), let (S^{[n]}) denote the Hilbert scheme of n points on S. In this note, we compute the fundamental group scheme (pi ^{text {alg}}(S^{[n]}, {tilde{nx}})) defined by the Tannakian category of stratified bundles on (S^{[n]}).

In this paper, we prove some congruences involving the coefficients ({A_{n}}_{n=0,1,2,ldots }) of the analytic solution (y_0(z)=sum _{n=0}^infty A_nz^n) of certian differential eqution ({mathcal {D}}y=0) normalized by the condition (y_0(0)=A_0=1), where ({mathcal {D}}) is a 4th-order linear differential operator.

We study the following critical Schrödinger-Possion system with steep potential well

where (lambda >0) is a positive parameter, (V:{mathbb {R}}^{3}rightarrow {mathbb {R}}) is a continuous function and f is a continuous subcritical nonlinearity. Under some certain assumptions on V and f, for any (lambda ge lambda _0>0), we prove the existence of a ground state solution via variational methods. Moreover, the concentration behavior of the ground state solution is also described as (lambda rightarrow infty ). Our results extends that in Jiang[11](J. Differ. Equ. 2011) and Zhao[21](J. Differ. Equ. 2013) to the critical growth case.

In this paper, we prove that the study of the subgraph (T(Z^*(L))) of the total graph T(L) of a lattice L is essentially the study of the zero-divisor graph of a poset. Also, we prove that the graph (T^c(Z^*(L))) is weakly perfect whereas (T(Z^*(L))) is not weakly perfect. The graph (T(Z^*(L))) and its complement (T^c(Z^*(L))) are shown to be a perfect graph if and only if L has at most four atoms. In the concluding section, we establish that, in the context of a commutative reduced ring R, the total graph, the annihilating ideal graph, the complement of the co-annihilating ideal graph, and the complement of the comaximal ideal graph coincide.

Abstract

Let G be a group and let S be a subset of (G setminus {e}) with (S^{-1} subseteq S) , where e is the identity element of G. The Cayley graph (mathrm {{{,textrm{Cay},}}}(G,S)) is a graph whose vertices are the elements of G and two distinct vertices (g,hin G) are adjacent if and only if (g^{-1} hin S) . Let (S subseteq Z(G)) . Then the relation ( sim ) on G, given by (asim b) if and only if (Sa=Sb) , is an equivalence relation. Let (G_E) be the set of equivalence classes of (sim ) on G and let [a] be the equivalence class of the element a in G. Then (G_E) is a group with operation ([a].[b]=[ab]) . Also, let (S_E) be the set of equivalence classes of the elements of S. The compressed Cayley graph of G is introduced as the Cayley graph ({{,textrm{Cay},}}(G_E,S_E)) , which is denoted by ({{,textrm{Cay},}}_E(G,S)) . In this paper, we investigate some relations between (mathrm {{{,textrm{Cay},}}}(G,S)) and ({{,textrm{Cay},}}_E(G,S)) . Also, we prove that (mathrm {{{,textrm{Cay},}}}(G,S)) is a ({{,textrm{Cay},}}_E(G,S)) -generalized join of certain empty graphs. Moreover, we describe the structure of the compressed Cayley graph of (mathbb {Z}_n) by introducing a subset S such that ({{,textrm{Cay},}}_E(mathbb {Z}_n,S)) and ({{,textrm{Cay},}}(mathbb {Z}_n,S)) are not isomorphic, and we describe the Laplacian spectrum of ({{,textrm{Cay},}}(mathbb {Z}_n,S)) .

For a simple graph (mathcal {G}= (mathcal {V}, mathcal {E})), an L(2, 1)-labeling is an assignment of non-negative integer labels to vertices of (mathcal {G}). An L(2, 1)-labeling of (mathcal {G}) must satisfy two conditions: adjacent vertices in (mathcal {G}) should get labels which differ by at least two, and vertices at a distance of two from each other should get distinct labels. The (lambda )-number of (mathcal {G}), denoted by (lambda (mathcal {G})), represents the smallest positive integer (ell ) for which an L(2, 1)-labeling exists, the vertices of (mathcal {G}) are provided labels from the set ({0, 1, dots , ell }). Let (Gamma (R)) be a zero-divisor graph of a finite commutative ring R with unity. In (Gamma (R)), vertices represent zero-divisors of R, and two vertices x and y are adjacent if and only if (xy = 0) in R. The methodology of the research involves a detailed investigation into the structural aspects of zero-divisor graphs associated with specific classes of local and mixed rings, such as (mathbb {Z}_{p^n}), (mathbb {Z}_{p^n} times mathbb {Z}_{q^m}), and (mathbb {F}_{q}times mathbb {Z}_{p^n}). This exploration leads us to compute the exact value of L(2, 1)-labeling number of these graphs.

Let G be a graph and (mathcal {H}) be a set of connected graphs. A spanning subgraph H of G is called an (mathcal {H})–factor if each component of H is isomorphic to a member of (mathcal {H}). In this paper, we first present a lower bound on the size (resp. the spectral radius) of G to guarantee that G has a ({P_2,, C_n: nge 3})–factor (or a perfect k–matching for even k) and construct extremal graphs to show all this bounds are best possible. We then provide a lower bound on the signless laplacian spectral radius of G to ensure that G has a ({K_{1,j}:1le jle k})–factor, where (kge 2 ) is an integer. Moreover, we also provide some Laplacian eigenvalue (resp. toughness) conditions for the existence of ({P_2,, C_{n}:nge 3})–factor, (P_{ge 3})–factor and ({K_{1,j}: 1le jle k})–factor in G, respectively. Some of our results extend or improve the related existing results.

We prove several Ramanujan-type congruences modulo powers of 5 for partition k-tuples with 5-cores, for (k=2, 3, 4). We also prove some new infinite families of congruences modulo powers of primes for k-tuples with p-cores, where p is a prime.