A Tighter Erdős‐Pósa Function for Long Cycles

We prove that there exists a bivariate function f with such that for every natural k and ?, every graph G has at least k vertex‐disjoint cycles of length at least ? or a set of at most vertices that meets all cycles of length at least ?. This improves a result by Birmelé et al. (Combinatorica, 27 (2007), 135–145), who proved the same result with .     

Increasing returns economics and generalized Pólya processes

This article extends the traditional Pólya scheme consisting of one urn with two colors to the schemes where multiple independent and/or interdependent urns with multiple additions and/or withdraws and several independent and/or interdependent colors are considered. It also argues that with these schemes many complex economic systems subject to increasing returns can be formalized mathematically, for they allow for positive and negative feedbacks among many variables, “jumps,” “bad” behaved dynamics, dis continuities, and interrelation among systems. © 2013 Wiley Periodicals, Inc. Complexity 19: 21–37, 2013     

Pósa's conjecture for graphs of order at least 2 × 108

In 1962 Pósa conjectured that every graph G on n vertices with minimum degree \begin{align*}\delta(G)\ge \frac{2}{3}n\end{align*} contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Pósa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than \begin{align*}\frac{2}{3}n\end{align*}. Still in 1996, Komlós, Sárközy, and Szemerédi proved Pósa's Conjecture, using the Regularity and Blow‐up Lemmas, for graphs of order n ≥ n₀, where n₀ is a very large constant. Here we show without using these lemmas that n₀:= 2 × 10⁸ is sufficient. We are motivated by the recent work of Levitt, Sárközy and Szemerédi, but our methods are based on techniques that were available in the 90's. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Long cycles through prescribed vertices have the Erdős‐Pósa property

We prove that for every graph, any vertex subset S, and given integers : there are k disjoint cycles of length at least ℓ that each contain at least one vertex from S, or a vertex set of size that meets all such cycles. This generalizes previous results of Fiorini and Herinckx and of Pontecorvi and Wollan. In addition, we describe an algorithm for our main result that runs in time, where s denotes the cardinality of S.     

Squares of Hamiltonian cycles in 3‐uniform hypergraphs

We show that every 3‐uniform hypergraph H = (V,E) with |V(H)| = n and minimum pair degree at least (4/5 + o(1))n contains a squared Hamiltonian cycle. This may be regarded as a first step towards a hypergraph version of the Pósa‐Seymour conjecture.     

Approximation of Schrödinger operators with δ‐interactions supported on hypersurfaces

We show that a Schrödinger operator with a δ‐interaction of strength α supported on a bounded or unbounded C²‐hypersurface , can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator with a singular interaction is regarded as a self‐adjoint realization of the formal differential expression , where is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.     

Fully discrete A‐ϕ finite element method for Maxwell's equations with nonlinear conductivity

This article is devoted to the study of a fully discrete A ‐ finite element method to solve nonlinear Maxwell's equations based on backward Euler discretization in time and nodal finite elements in space. The nonlinearity is owing to a field‐dependent conductivity with the power‐law form . We design a nonlinear time‐discrete scheme for approximation in suitable function spaces. We show the well‐posedness of the problem, prove the convergence of the semidiscrete scheme based on the boundedness of the second derivative in the dual space and derive its error estimate. The Minty–Browder technique is introduced to obtain the convergence of the nonlinear term. Finally, we discuss the error estimate for the fully discretized problem and support the theoretical result by two numerical experiments. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 2083–2108, 2014     

A compact local one‐dimensional scheme for solving a 3D N‐carrier system with Neumann boundary conditions

We consider a mathematical model for thermal analysis in a 3D N‐carrier system with Neumann boundary conditions, which extends the concept of the well‐known parabolic two‐step model for micro heat transfer. To solve numerically the complex system, we first reduce 3D equations in the model to a succession of 1D equations by using the local one‐dimensional (LOD) method. The obtained 1D equations are then solved using a fourth‐order compact finite difference scheme for the interior points and a second‐order combined compact finite difference scheme for the points next to the boundary, so that the Neumann boundary condition can be applied directly without discretizing. By using matrix analysis, the compact LOD scheme is shown to be unconditionally stable. The accuracy of the solution is tested using two numerical examples. Results show that the solutions obtained by the compact LOD finite difference scheme are more accurate than those obtained by a Crank‐Nicholson LOD scheme, and the convergence rate with respect to spatial variables is about 2.6. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A gray model for increasing sequences with nonhomogeneous index trends based on fractional‐order accumulation

By comparing the class ratio deviation and restoring error of first‐order accumulation with that of fractional‐order accumulation, a gray model for monotonically increasing sequences can obtain optimal simulation accuracy via selecting a proper cumulative order. In this study, a gray model for increasing sequences with nonhomogeneous index trends based on fractional‐order accumulation is proposed. To reduce the modeling error caused by the background value and to improve the prediction accuracy of the model, an optimized model using the 3/8 Simpson formula is constructed. Finally, the 2 proposed models are used to predict the total energy consumption in China and the monthly sales of new products in an enterprise. Compared with the GM(1,1) model based on fractional‐order accumulation, the proposed model exhibits better simulation and prediction accuracy.     

Nonelement boundary representation with Bézier surface patches for 3D linear elasticity problems in parametric integral equation system (PIES) and its solving using Lagrange polynomials

In this article, we present a strategy of using rectangular and triangular Bézier surface patches for nonelement representation of 3D boundary geometries for problems of linear elasticity. The boundary generated in this way is directly incorporated in the parametric integral equation system (PIES), which has been developed by the authors. The boundary values on each surface patch are approximated by Lagrange polynomials. Three illustrative examples are presented to confirm the effectiveness of the proposed boundary representation in connection with PIES and to show good accuracy of numerical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 51–79, 2018