A New Strong Laplacian on Differential Forms

We construct a strong Laplacian D ^* D by using the third operator in the basis {d,d ^*,D} of the space of natural first-order operators acting on the differential forms of a Riemannian manifold (M,g). We study the properties of the Laplacian D ^* D and obtain Weitzenbock's formula relating the three strong Laplacians dd ^*, d ^* d, and D ^* D to the curvature of the manifold (M,g).     

On the diophantine equation y2 − D = 2k

The equation of the title is studied for 1 ≤ D ≤ 100. It is shown that for such values of D the above equation is really interesting only if D = 17, 41, 73, 89, 97. Then, for these values of D, (i) necessary conditions are given for the solvability of the diophantine equations y² = 2x⁴ + D and y² = 8x⁴ + D, and (ii) y² ? D = 2^k is solved.     

On commutativity of two unary digraph operations: Subdividing and line-digraphing

For a digraph D, let L(D) and S(D) denote its line digraph and subdivision digraph, respectively. The motivation of this paper is to solve the digraph equation L(S(D))=S(L(D)). We show that L(S(D)) and S(L(D)) are cospectral if and only if D and L(D) have the same number of arcs. Further, we characterize the situation that L(S(D)) and S(L(D)) are isomorphic. Our approach introduces the new notion, the proper image D^* of a digraph D, and a new type of connectedness for digraphs. The concept D^* plays an important role in the main result of this paper. It is also useful in other aspects of the study of line digraphs. For example, L(D) is connected if and only if D^* is connected; L(D) is functional (contrafunctional) if and only if D^* is functional (contrafunctional). Some related results are also presented.     

A class of (4,2)-disk pairs which are trivial

Suppose Δ?S³ is a ribbon disk and let D (Δ) denote the cononical properly embedded 2-disk obtained by pushing the interior of Δ into B⁴. A well-known conjecture states that the disk pair (B⁴, D(Δ)) is trivial provided the sphere pair ?(B⁴, D(Δ)) is trivial. We show here that the conjecture is true for those D(Δ) with the property that there is an embedded 2-disk, D²?S³, whose boundary is ?D(Δ) and which intersects Δ in ‘transverse double points’.     

Testing holomorphy on curves

For a domain D ? ? ⁿ we construct a continuous foliation of D into one-real-dimensional curves such that any function f ∈ C ¹(D) which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on D.     

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.     

A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

We establish that a pair A, B, of nonsingular matrices over a commutative domain R of principal ideals can be reduced to their canonical diagonal forms D ^A and D ^B by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices U, V _A, and V _B over R such that UAV _a=D^A and UAV _B=D^B if and only if the matrices B _*A and D _* ^B D^A where B _* 0 is the matrix adjoint to B, are equivalent.     

The error bound of the perturbation of the Drazin inverse

Let A and E be n×n matrices and B = A + E. Denote the Drazin inverse of A by A^D. In this paper we give an upper bound for the relative error ∥B^D ? A^D∥/∥A^D∥₂ and a lower bound for ∥B^D∥₂ under certain circumstances. The continuity properties and the derivative of the Drazin inverse are also considered.     

Augmentation quotients for burnside rings of generalized dihedral groups

Let H be a finite abelian group of odd order, D be its generalized dihedral group, i.e., the semidirect product of C₂ acting on H by inverting elements, where C₂ is the cyclic group of order two. Let Ω (D) be the Burnside ring of D, Δ(D) be the augmentation ideal of Ω (D). Denote by Δⁿ(D) and Q_n(D) the nth power of Δ(D) and the nth consecutive quotient group Δⁿ(D)/Δⁿ⁺¹(D), respectively. This paper provides an explicit Z-basis for Δⁿ(D) and determines the isomorphism class of Q_n(D) for each positive integer n.     

The hull and geodetic numbers of orientations of graphs

For an oriented graph D, let I_D[u,v] denote the set of all vertices lying on a u-v geodesic or a v-u geodesic. For S⊆V(D), let I_D[S] denote the union of all I_D[u,v] for all u,v∈S. Let [S]_D denote the smallest convex set containing S. The geodetic number g(D) of an oriented graph D is the minimum cardinality of a set S with I_D[S]=V(D) and the hull number h(D) of an oriented graph D is the minimum cardinality of a set S with [S]_D=V(D). For a connected graph G, let O(G) be the set of all orientations of G, define g⁻(G)=min{g(D):D∈O(G)}, g⁺(G)=max{g(D):D∈O(G)}, h⁻(G)=min{h(D):D∈O(G)}, and h⁺(G)=max{h(D):D∈O(G)}. By the above definitions, h⁻(G)≤g⁻(G) and h⁺(G)≤g⁺(G). In the paper, we prove that g⁻(G)<h⁺(G) for a connected graph G of order at least 3, and for any nonnegative integers a and b, there exists a connected graph G such that g⁻(G)−h⁻(G)=a and g⁺(G)−h⁺(G)=b. These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256-262].     

Twisted unknots

Let K be a knot in the 3-sphere S³, and D a disk in S³ meeting K transversely in the interior. For non-triviality we assume that |D∩K|?2 over all isotopies of K in S³??D. Let K_D,n(?S³) be the knot obtained from K by n twisting along the disk D. If the original knot is unknotted in S³, we call K_D,n a twisted unknot. We describe for which pairs (K,D) and integers n, the twisted unknot K_D,n is a torus knot, a satellite knot or a hyperbolic knot. To cite this article: M. A??t Nouh et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Obtaining graph knots by twisting unknots

Let K be a knot in the 3-sphere S³ and D a disk in S³ meeting K transversely more than once in the interior. For nontriviality we assume that |D∩K|⩾2 over all isotopies of K in S³−∂D. Let K_D,n (⊂S³) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and K_D,n is a graph knot, then |n|⩽1 or K and D form a special pair which we call an “exceptional pair”. As a corollary, if (K,D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner.     

On Bounded Positive (m,p)-Circle Domains

Let D be a bounded positive (m, p)-circle domain in ?². The authors prove that if dim(Iso(D)⁰) = 2, then D is holomorphically equivalent to a Reinhardt domain; if dim(Iso(D)⁰) = 4, then D is holomorphically equivalent to the unit ball in ?². Moreover, the authors prove the Thullen’s classification on bounded Reinhardt domains in ?² by the Lie group technique.     

Inversion formulas for complex radon transform on projective varieties and boundary value problems for systems of linear PDEs

Let G ? ?P ⁿ be a linearly convex compact set with smooth boundary, D = ?P ⁿ \ G, and let D* ? (?P ⁿ )* be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety V of dimension d we construct an explicit inversion formula for the complex Radon transform R _V : H ^d,d?1(V ∩ D) → H ^1,0(D*) and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on D*.     

Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains

Let D be a bounded symmetric domain. We calculate operator norm of the multiplication operator on the Hardy space H^p(D), as well as of the weighted composition operator from H^p(D) to a weighted-type space.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ?_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ? ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ?_q,1(B _n )- module D^λ remains irreducible on restriction to ?_q(D _n ).     

Quasirecognition by prime graph of 2 D n (3α) where n = 4m + 1 ≥ 21 and α is odd

Let G be a finite group. The prime graph of G is denoted by Γ(G). In this paper, as the main result, we show that if G is a finite group such that Γ(G) = Γ(² D _n (3^α)), where n = 4m+ 1 and α is odd, then G has a unique non-Abelian composition factor isomorphic to ² D _n (3^α). We also show that if G is a finite group satisfying |G| = |² D _n (3^α)|, and Γ(G) = Γ(² D _n (3^α)), then G ? ² D _n (3^α). As a consequence of our result, we give a new proof for a conjecture of Shi and Bi for ² D _n (3^α). Application of this result to the problem of recognition of finite simple groups by the set of element orders are also considered. Specifically, it is proved that ² D _n (3^α) is quasirecognizable by the spectrum.     

A theorem concerning certain sign symmetric matrices whose inverses are Morishima

Suppose A is an invertible sign symmetric matrix whose associated digraph D(A) is a tree. Then A^-1 will be Morishima iff a^?? ? 0 for all interior points ? in D(A). A^-1 will be anti-Morishima iff a^?? ? 0 for all interior points ? in D(A).     

Drazin inverse of partitioned matrices in terms of Banachiewicz-Schur forms

Let be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A^D. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S=D-CA^DB, group invertible, can be expressed in terms of a matrix in the Banachiewicz-Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank(M)=rank(A^D)+rank(S^D), and give conditions under which the group inverse of M exists and a formula for its computation.