Mean convex hulls and least area disks spanning extreme curves

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in such that for any asymptotic curve , there is a canonical mean convex hull containing all minimal planes spanning Γ. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C³-smooth curve in the boundary of C³-smooth mean convex domain in ℝ³, there exist a unique least area disk spanning the curve.     

Locally boundedly A‐p‐convex algebras

We consider locally p‐convex algebras the bornologies of which coincide with the ones of p‐normed or multiplicatively p‐convex topologies. Two new classes of these algebras are introduced here. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the convolution and subordination of convex functions

In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order α is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order α is preserved under convolution with a prestarlike function of the same order. Moreover, we find a subordinating factor sequence for the class of convex functions. The work deals with several ideas and techniques used in geometric function theory, contained in the book Convolutions in Geometric Function Theory by Ruscheweyh (1982).     

Applying geometric K‐cycles to fractional indices

A geometric model for twisted K‐homology is introduced. It is modeled after the Mathai–Melrose–Singer fractional analytic index theorem in the same way as the Baum–Douglas model of K‐homology was modeled after the Atiyah–Singer index theorem. A natural transformation from twisted geometric K‐homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted K‐homology in this model is an isomorphism for torsion twists on a finite ‐complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforementioned fractional invariants, we study T‐duality for geometric cycles.     

Dynamical transition for S‐K‐T biological competing model with cross‐diffusion

The main objective of this paper is to study the dynamical transition for the S‐K‐T biological competition system with cross‐diffusion. Based on the spectral analysis, the principle of exchange of stabilities conditions for eigenvalues are obtained. By using the dynamical transition theory, 2 different types of dynamical transition for the S‐K‐T model are also derived. In addition, an example is given to illustrate our main results.     

On the K‐sat model with large number of clauses

We show that in the K‐sat model with N variables and αN clauses, the expected ratio of the smallest number of unsatisfied clauses to the number of variables is up to smaller order terms as uniformly in N, where is the expected normalized maximum energy of some specific mixed p‐spin spin glass model. The formula for the limit of is well known in the theory of spin glasses.     

Space‐time spectral collocation method for the one‐dimensional sine‐Gordon equation

In this article, we introduce a new space‐time spectral collocation method for solving the one‐dimensional sine‐Gordon equation. We apply a spectral collocation method for discretizing spatial derivatives, and then use the spectral collocation method for the time integration of the resulting nonlinear second‐order system of ordinary differential equations (ODE). Our formulation has high‐order accurate in both space and time. Optimal a priori error bounds are derived in the L²‐norm for the semidiscrete formulation. Numerical experiments show that our formulation have exponential rates of convergence in both space and time. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 670–690, 2015     

Hermite‐Hadamard‐type inequalities for interval‐valued coordinated convex functions involving generalized fractional integrals

In this paper, we define interval‐valued left‐sided and right‐sided generalized fractional double integrals. We establish inequalities of Hermite‐Hadamard like for coordinated interval‐valued convex functions by applying our newly defined integrals.     

Estimates of the first Neumann eigenvalue and the log‐Sobolev constant on non‐convex manifolds

In this paper a number of explicit lower bounds are presented for the first Neumann eigenvalue on non‐convex manifolds. The main idea to derive these estimates is to make a conformal change of the metric such that the manifold is convex under the new metric, which enables one to apply known results obtained in the convex case. This method also works for more general functional inequalities. In particular, some explicit lower bounds are presented for the log‐Sobolev constant on non‐convex manifolds. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Selection by rank in K‐dimensional binary search trees

In this work we show how to augment general purpose multidimensional data structures, such as K‐d trees, to efficiently support search by rank (that is, to locate the i‐th smallest element along the j‐th coordinate, for given i and j) and to find the rank of a given item along a given coordinate. To do so, we introduce two simple, practical and very flexible algorithms – Select‐by‐Rank and Find‐Rank – with very little overhead. Both algorithms can be easily implemented and adapted to several spatial indexes, although their analysis is far from trivial. We are able to show that for random K‐d trees of size n the expected number of nodes visited by Find‐Rank is for or , and for (with ), where depends on the dimension K and the variant of K‐d tree under consideration. We also show that Select‐by‐Rank visits nodes on average, where is the given rank and the exponent α is as above. We give the explicit form of the functions and , both are bounded in [0, 1] and they depend on K, on the variant of K‐d tree under consideration, and, eventually, on the specific coordinate j for which we execute our algorithms. As a byproduct of the analysis of our algorithms, but no less important, we give the average‐case analysis of a partial match search in random K‐d trees when the query is not random. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 14–37, 2014     

Error analysis for solving the Korteweg‐de Vries equation by a Legendre pseudo‐spectral method

A Legendre pseudo‐spectral method is proposed for the Korteweg‐de Vries equation with nonperiodic boundary conditions. Appropriate base functions are chosen to get an efficient algorithm. Error analysis is given for both semi‐discrete and fully discrete schemes. The numerical results confirm to the theoretical analysis. © (2000) John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 513–534, (2000)     

A projected‐steepest‐descent potential‐reduction algorithm for convex programming problems

A recent work of Shi (Numer. Linear Algebra Appl. 2002; 9 : 195–203) proposed a hybrid algorithm which combines a primal‐dual potential reduction algorithm with the use of the steepest descent direction of the potential function. The complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. In this paper, we make efforts to further reduce the computational costs. We notice that in order to obtain the steepest descent direction of the potential function, the Hessian matrix of second order partial derivatives of the objective function needs to be computed. To avoid this, we in this paper propose another hybrid algorithm which uses a projected steepest descent direction of the objective function instead of the steepest descent direction of the potential function. The complexity of the original potential reduction algorithm still remains valid but the overall computational cost is further reduced. Our numerical experiments are also reported. Copyright © 2004 John Wiley & Sons, Ltd.     

Semi‐implicit spectral collocation methods for reaction‐diffusion equations on annuli

In this article, we develop numerical schemes for solving stiff reaction‐diffusion equations on annuli based on Chebyshev and Fourier spectral spatial discretizations and integrating factor methods for temporal discretizations. Stiffness is resolved by treating the linear diffusion through the use of integrating factors and the nonlinear reaction term implicitly. Root locus curves provide a succinct analysis of the A‐stability of these schemes. By utilizing spectral collocation methods, we avoid the use of potentially expensive transforms between the physical and spectral spaces. Numerical experiments are presented to illustrate the accuracy and efficiency of these schemes. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1113–1129, 2011     

Convergence of a spectral projection of the Camassa‐Holm equation

A spectral semi‐discretization of the Camassa‐Holm equation is defined. The Fourier‐Galerkin and a de‐aliased Fourier‐collocation method are proved to be spectrally convergent. The proof is supplemented with numerical explorations that illustrate the convergence rates and the use of the dealiasing method. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

F‐products and nonstandard hulls for semigroups

Derndinger [2] and Krupa [5] defined the F‐product of a (strongly continuous one‐parameter) semigroup (of linear operators) and presented some applications (e. g. to spectral theory of positive operators, cf. [3]). Wolff (in [7] and [8]) investigated some kind of nonstandard analogon and applied it to spectral theory of group representations. The question arises in which way these constructions are related. In this paper we show that the classical and the nonstandard F‐product are isomorphic (Theorem 2.6). We also prove a little “classical” corollary (2.7.). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Least‐squares spectral collocation method for the Stokes equations

First‐order system least‐squares spectral collocation methods are presented for the Stokes equations by adopting the first‐order system and modifying the least‐squares functionals in 2 . Then homogeneous Legendre and Chebyshev (continuous and discrete) functionals are shown to be elliptic and continuous with respect to appropriate product weighted norms. The spectral convergence is analyzed for the proposed methods with some numerical experiments. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 128–139, 2004     

Non‐autonomous forms and invariance

We generalize the Beurling–Deny–Ouhabaz criterion for parabolic evolution equations governed by forms to the non‐autonomous, non‐homogeneous and semilinear case. Let be Hilbert spaces such that V is continuously and densely embedded in H and let be the operator associated with a bounded H‐elliptic form for all . Suppose is closed and convex and the orthogonal projection onto . Given and , we investigate when the solution of the non‐autonomous evolutionary problem remains in and show that this is the case if for a.e. . Moreover, we examine necessity of this condition and apply this result to a semilinear problem.     

Threshold values of random K‐SAT from the cavity method

Using the cavity equations of Mézard, Parisi, and Zecchina 23 ; Mézard and Zecchina, 24 we derive the various threshold values for the number of clauses per variable of the random K‐satisfiability problem, generalizing the previous results to K ≥ 4. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large K. The stability of the solution is also computed. For any K, the satisfiability threshold is found to be in the stable region of the solution, which adds further credit to the conjecture that this computation gives the exact satisfiability threshold.© 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006