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The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra 𝕀 n : = K 〈 x 1 , ... , x n , ∂ ∂ x 1 , ... , ∂ ∂ x n , ∫ 1 , ... , ∫ n 〉 of polynomial integro-differential operators and the Jacobian algebra 𝔸 n is equal to n (over a field of characteristic zero). The algebras 𝕀 n and 𝔸 n are neither left nor right Noetherian and 𝕀 n ⊂ 𝔸 n . Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert's Syzygy Theorem is proven for the algebras 𝕀 n , 𝔸 n and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras 𝕀 n and 𝔸 n is n and the weak global dimension of all the factor algebras of 𝕀 n and 𝔸 n is n. [ABSTRACT FROM AUTHOR]

In this paper, we explicitly work out the subfactor planar algebra P (N ⊂ Q) for an intermediate subfactor N ⊂ Q ⊂ M of an irreducible subfactor N ⊂ M of finite index. We do this in terms of the subfactor planar algebra P (N ⊂ M) by showing that if T is any planar tangle, the associated operator Z T (N ⊂ Q) can be read off from Z T (N ⊂ M) by a formula involving the so-called biprojection corresponding to the intermediate subfactor N ⊂ Q ⊂ M and a scalar α (T) carefully chosen so as to ensure that the formula defining Z T (N ⊂ Q) is multiplicative with respect to composition of tangles. Also, the planar algebra of Q ⊂ M can be obtained by applying these results to M ⊂ M 1 . We also apply our result to the example of a semi-direct product subgroup-subfactor. [ABSTRACT FROM AUTHOR]

Two edges and in a graph are said to be adjacent if they have a vertex in common, and distance two apart if they are nonadjacent but both are adjacent to a common edge. An -edge-labeling of a graph is an assignment of nonnegative integers, called labels, to the edges of such that the difference between labels of any two adjacent edges is at least , and the labels of any two edges that are distance two apart are different. The span of an -edge-labeling of a graph is the difference between the maximum and minimum labels. The minimum span over all -edge-labelings of a graph is called the -edge-labeling number of , denoted by . For , the edge-path-replacement of a graph , denoted by , is a graph obtained by replacing each edge of with a path on vertices. This paper investigates the -edge-labeling number of the edge-path-replacement of a graph for . We get the following main results: Let be a graph with maximum degree and be an integer not less than , then if is odd, and otherwise . Let be a graph with maximum degree . Then when is even, and when is odd. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: Here is a domain (bounded or unbounded) in which is convex in -direction. is the nonlocal fractional Laplacian operator which is defined as Under various conditions on and on a solution it is shown that is strictly increasing in in the left half of , or in . Symmetry (in ) of some solutions is proved. [ABSTRACT FROM AUTHOR]

Identifying the top influential spreaders in a network has practical significance. In this paper, we propose a novel centrality to identify influential spreaders based on expansion factor. Nodes with high expansion factor centrality (EFC) have strong spreading capability. During the course of the work, an improved strategy is proposed to reduce the time complexity of EFC. We discuss the correlations between EFC and the other five classical indicators. Simulation results on the Susceptible-Infected-Removed (SIR) model manifest that EFC can identify influential nodes and find some critical influential nodes neglected by other indicators. [ABSTRACT FROM AUTHOR]

In this paper, we study the cover-avoid property of 픉-injectors on chief factors of a group G for a Fitting class 픉 in some universe 픘 with partial solubility. [ABSTRACT FROM AUTHOR]