On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?ⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.     

On the perturbation of the Q‐factor of the QR factorization

This paper gives normwise and componentwise perturbation analyses for the Q‐factor of the QR factorization of the matrix A with full column rank when A suffers from an additive perturbation. Rigorous perturbation bounds are derived on the projections of the perturbation of the Q‐factor in the range of A and its orthogonal complement. These bounds overcome a serious shortcoming of the first‐order perturbation bounds in the literature and can be used safely. From these bounds, identical or equivalent first‐order perturbation bounds in the literature can easily be derived. When A is square and nonsingular, tighter and simpler rigorous perturbation bounds on the perturbation of the Q‐factor are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

CR Structures on the 3‐Sphere and the Blow‐Up of the Projective Plane

Motivated by a problem of characterizing CR‐structures on the 3‐sphere, we give a geometric construction of formal deformations of a complex surface, which is the complement of a ball in the projective plane. They are described by cohomology groups of the blow‐up X of the projective plane. Moreover it will be shown that the space of these formal deformations is an infinite dimensional space with a natural stratification by finite dimensional subspaces. This stratification re ects algebro‐geometric properties of X. It is expected that our construction will clarify the complex geometric nature of the space of non‐embeddable CR‐structures on the 3‐sphere.     

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

We establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures. To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C² test functions, are confined in a compact set in H^?1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation. A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.     

The play operator,the truncated variation and the generalisation of the Jordan decomposition

The play operator minimalizes the total variation on intervals, [0,T],T > 0, of functions approximating uniformly given regulated function with given accuracy and starting from a given point. In this article, we link the play operator with the so‐called truncated variation functionals, introduced recently by the second‐named author, and provide a semi‐explicit expression for the play operator in terms of these functionals. Generalisation for time‐dependent boundaries is also considered. This gives the best possible lower bounds for the total variation of the outputs of the play operator and its Jordan‐like decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

On the Cauchy problem for the two‐component b‐family system

In this paper, we establish the local well‐posedness for the two‐component b‐family system in a range of the Besov space. We also derive the blow‐up scenario for strong solutions of the system. In addition, we determine the wave‐breaking mechanism to the two‐component Dullin–Gottwald–Holm system. Copyright © 2013 John Wiley & Sons, Ltd.     

Bounding the Clique‐Width of H‐Free Chordal Graphs

A graph is H‐free if it has no induced subgraph isomorphic to H. Brandstädt, Engelfriet, Le, and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique‐width. Brandstädt, Le, and Mosca erroneously claimed that the gem and co‐gem are the only two 1‐vertex P₄‐extensions H for which the class of H‐free chordal graphs has bounded clique‐width. In fact we prove that bull‐free chordal and co‐chair‐free chordal graphs have clique‐width at most 3 and 4, respectively. In particular, we find four new classes of H‐free chordal graphs of bounded clique‐width. Our main result, obtained by combining new and known results, provides a classification of all but two stubborn cases, that is, with two potential exceptions we determine all graphs H for which the class of H‐free chordal graphs has bounded clique‐width. We illustrate the usefulness of this classification for classifying other types of graph classes by proving that the class of ‐free graphs has bounded clique‐width via a reduction to K₄‐free chordal graphs. Finally, we give a complete classification of the (un)boundedness of clique‐width of H‐free weakly chordal graphs.     

A note on the convergence of the MSMAOR method for linear complementarity problems

Modulus‐based splitting, as well as multisplitting iteration methods, for linear complementarity problems are developed by Zhong‐Zhi Bai. In related papers (see Bai, Z.‐Z., Zhang, L.‐L.: Modulus‐Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems. Numerical Linear Algebra with Applications 20 (2013) 425–439, and the references cited therein), the problem of convergence for two‐parameter relaxation methods (accelerated overrelaxation‐type methods) is analyzed under the assumption that one parameter is greater than the other. Here, we will show how we can avoid this assumption and, consequently, improve the convergence area. Copyright © 2013 John Wiley & Sons, Ltd.     

Quantum logics with the Riesz Interpolation Property

We study the quantum logics which satisfy the Riesz Interpolation Property. We call them the RIP logics. We observe that the class of RIP logics is considerable large—it contains all lattice quantum logics and, also, many (infinite) non‐lattice ones. We then find out that each RIP logic can be enlarged to an RIP logic with a preassigned centre. We continue, showing that the “nearly” Boolean RIP logics must be Boolean algebras. In a somewhat surprising contrast to this, we finally show that the attempt for the σ‐complete formulation of this result fails: We show by constructing an example that there is a non‐Boolean nearly Boolean σ‐RIP logic. As a result, there are interesting σ‐RIP logics which are intrinsically close to Boolean σ‐algebras. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Size‐Ramsey Number of Hypergraphs

The size‐Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2‐edge‐coloring of H yields a monochromatic copy of G. Size‐Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size‐Ramsey numbers for k‐uniform hypergraphs. Analogous to the graph case, we consider the size‐Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size‐Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.     

Shooting at the poachers while the rhinos drown: Managing short‐ and long‐term threats to endangered wildlife in conservation reserves

This paper addresses management challenges associated with conserving endangered wildlife facing multiple threats from illegal poaching, habitat encroachment, and climate and land‐use change‐induced flooding. While poaching and encroachment challenges in conservation parks are of immediate nature, climate‐related risks exist in the long term. The park manager faces a utility function that includes as its arguments local community’s incomes, benefits to the larger society from preserving threatened species and the financial costs of monitoring and land‐use change efforts. Using the case of single‐horned rhinos in the Kaziranga National Park, India, an optimal mix of monitoring and land‐use changes is designed in presence of tradeoffs between short‐ and long‐term management efforts. As monitoring only addresses immediate challenges associated with poaching and encroachment, long‐term climatic risks remain ignored. Land‐use management offers risk‐protection as well as risk‐insurance benefits with respect to climate change‐induced flooding of the park. Recommendations for Resource Managers

• It is important to incorporate both short‐ and long‐term risks posed to endangered wildlife while investing in conservation efforts. There may exist a tradeoff between mitigating short‐ and long‐run risks due to financial and physical resource constraints. However, ignoring long‐term risks to wildlife habitats can jeopardize past conservation efforts.
• Land‐use management, both within and outside of conservation reserves, enhances resilience to climatic shocks through reducing flooding risks and must be an essential part of wildlife conservation efforts.
• Conservation efforts ignoring local community welfare considerations can become suboptimal as they lead to reduced cooperation and potential conflicts. When wildlife conservation efforts account for local community welfare implications, optimal management plans could result in lower species abundance in the short term. However, increasing the park size through additional land enrollment can mitigate some of this tradeoff.

     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Quasi‐complements of the cappable degrees

Say that a nonzero c. e. degree b is a quasi‐complement of a c. e. degree a if a ∩ b = 0 and a ∪ b is high. It is well‐known (due to Shore) that each cappable degree has a high quasi‐complement. However, by the existence of the almost deep degrees, there are nonzero cappable degrees having no low quasi‐complements. In this paper, we prove that any nonzero cappable degree has a low₂ quasi‐complement. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the weak solutions to a shallow water equation

We obtain the existence of global‐in‐time weak solutions to the Cauchy problem for a one‐dimensional shallow‐water equation that is formally integrable and can be obtained by approximating directly the Hamiltonian for Euler's equation in the shallow‐water regime. The solution is obtained as a limit of viscous approximation. The key elements in our analysis are some new a priori one‐sided supernorm and space‐time higher‐norm estimates on the first‐order derivatives. © 2000 John Wiley & Sons, Inc.     

A generalization of the vector product

The vector product, often called the cross‐product, is well known in classical mechanics; it is generally assumed to be characteristic of three‐dimensional space. In this paper a vector product is defined on spaces of arbitrary dimension (excluding one‐dimensional spaces) and the definition is shown to coincide with that of the well‐known cross‐product in the case of three‐dimensional spaces; the generalization throws additional light on some of the properties of the classical vector product.     

Unitals in the code of the Hughes plane

A coding‐theoretic characterization of a unital in the Hughes plane is provided, based on and extending the work of Blokhuis, Brouwer, and Wilbrink in PG(2,q²). It is shown that a Frobenius‐invariant unital is contained in the p‐code of the Hughes plane if and only if that unital is projectively equivalent to the Rosati unital. © 2003 Wiley Periodicals, Inc.     

Local Regularity for the Modified SQG Patch Equation

We study the patch dynamics on the whole plane and on the half‐plane for a family of active scalars called modified surface quasi‐geostrophic (SQG) equations. These involve a parameter α that appears in the power of the kernel in their Biot‐Savart laws and describes the degree of regularity of the equation. The values α =0 and α =½ correspond to the two‐dimensional Euler and SQG equations, respectively. We establish here local‐in‐time regularity for these models, for all α ? (0,½) on the whole plane and for all small α > 0 on the half‐plane. We use the latter result in [16], where we show existence of regular initial data on the half‐plane that lead to a finite‐time singularity.© 2016 Wiley Periodicals, Inc.     

Error estimates for mixed finite element approximations of the viscoelasticity wave equation

This paper studies mixed finite element approximations to the solution of the viscoelasticity wave equation. Two new transformations are introduced and a corresponding system of first‐order differential‐integral equations is derived. The semi‐discrete and full‐discrete mixed finite element methods are then proposed for the problem based on the Raviart–Thomas–Nedelec spaces. The optimal error estimates in L²‐norm are obtained for the semi‐discrete and full‐discrete mixed approximations of the general viscoelasticity wave equation. Copyright © 2004 John Wiley & Sons, Ltd.