Symmetric edge-decompositions of hypercubes

We call a set of edgesE of the n-cubeQ _n a fundamental set for Q _n if for some subgroupG of the automorphism group ofQ _n , theG-translates ofE partition the edge set ofQ _n .Q _n possesses an abundance of fundamental sets. For example, a corollary of one of our main results is that if |E| =n and the subgraph induced byE is connected, then if no three edges ofE are mutually parallel,E is a fundamental set forQ _n . The subgroupG is constructed explicitly. A connected graph onn edges can be embedded intoQ _n so that the image of its edges forms such a fundamental set if and only if each of its edges belongs to at most one cycle.We also establish a necessary condition forE to be a fundamental set. This involves a number-theoretic condition on the integersa _j (E), where for 1 j n, a _j (E) is the number of edges ofE in thej ^th direction (i.e. parallel to thej ^th coordinate axis).     

On the Kadomtsev-Petviashvili Equation and Associated Constraints

The initial-boundary-value problem for the Kadomtsev-Petviashvili equation in infinite space is considered. When formulated as an evolution equation, found that a symmetric integral is the appropriate choice in the nonlocal term; namely, . If one simply chooses , then an infinite number of constraints on the initial data in physical space are required, the first being . The conserved quantities are calculated, and it is shown that they must be suitably regularized from those that have been used when the constraints are imposed.     

The mean curvature of the influence surface of wave equation with sources on a moving surface

The mean curvature of the influence surface of the space–time point ( x , t) appears in linear supersonic propeller noise theory and in the Kirchhoff formula for a supersonic surface. Both these problems are governed by the linear wave equation with sources on a moving surface. The influence surface is also called the Σ‐surface in the aeroacoustic literature. This surface is the locus, in a frame fixed to the quiescent medium, of all the points of a radiating surface f( x , t)=0 whose acoustic signals arrive simultaneously to an observer at position x and at the time t. Mathematically, the Σ‐surface is produced by the intersection of the characteristic conoid of the space–time point ( x , t) and the moving surface. In this paper, we derive the expression for the local mean curvature of the Σ‐surface of the space–time point ( x , t) for a moving rigid or deformable surface f( x , t)=0. This expression is a complicated function of the geometric and kinematic parameters of the surface f( x , t)=0. Using the results of this paper, the solution of the governing wave equation of high‐speed propeller noise radiation as well as the Kirchhoff formula for a supersonic surface can be written as very compact analytic expressions. Copyright © 1999 John Wiley & Sons, Ltd.     

Robust sparse phase retrieval made easy

In this short note we propose a simple two-stage sparse phase retrieval strategy that uses a near-optimal number of measurements, and is both computationally efficient and robust to measurement noise. In addition, the proposed strategy is fairly general, allowing for a large number of new measurement constructions and recovery algorithms to be designed with minimal effort.     

Analysis of Iterative Sub-structuring Techniques for Boundary Element Approximation of the Hypersingular Operator in Three Dimensions

The article deals with the analysis of Additive Schwarz preconditioners for the h -version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The first preconditioner consists of decomposing into local spaces associated with the subdomain interiors, supplemented with a wirebasket space associated with the subdomain interfaces. The wirebasket correction only involves the inversion of a diagonal matrix, while the interior correction consists of inverting the sub-blocks of the stiffness matrix corresponding to the interior degrees of freedom on each subdomain. It is shown that the condition number of the preconditioned system grows at most as max K H m 1 (1 + log H / h K ) 2 where H is the size of the quasi-uniform subdomains and h K is the size of the elements in subdomain K . A second preconditioner is given that incorporates a coarse space associated with the subdomains. This improves the robustness of the method with respect to the number of subdomains: theoretical analysis shows that growth of the condition number of the preconditioned system is now bounded by max K (1 + log H / h K ) 2 .     

The Most Widely Cited Papers in BIT

The sixteen BIT papers that have been most frequently cited since 1981 are listed, along with citation counts collected from the ISI Web of Science citation database.     

Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.