Congruences on ockham algebras with pseudocomplementation

The variety pOconsists of those algebras (L;?,?,f,^*,0,1) where (L;?,?,f,0,1) is an Ockham algebra, (L;?,?,f,^*,0,1) is a p-algebra, and the unary operations fand ^*. commute. For an algebra in pK _ωwe show that the compact congruences form a dual Stone lattice and use this to determine necessary and sufficient conditions for a principal congruence to be complemented. We also describe the lattice of subvarieties of pK _1,1identifying therein the biggest subvariety in which every principal congruence is complemented, and the biggest subvariety in which the intersection of two principal congruences is principal.     

A note on a global invertibility of mappings on Rn

We provide sufficient conditions for a mapping f : Rⁿ → Rⁿ to be a global di?eomorphism in case f need not be continuously di?erentiable. Instead it is assumed to be strictly (Hadamard-like) and Fréchet di?erentiable. We use classical local invertibility conditions together with the non-smooth critical point theory.     

Exhausters af a positively homogeneous function *

Notions of upper exhauster and lower exhauster of a positively homogeneous (of the first degree) function h: ? ⁿ →? are introduced. They are linked to exhaustive families of upper convex and lower concave approximations of the function h. The pair of an upper exhauster and a lower exhauster is called a biexhauster of h. A calculus for biexhausters is described (in particular, a composition theorem is formulated). The problem of minimality of exhausters is stated. Necessary and sufficient conditions for a constrained minimum and a constrained maximum of a directionally differentiable function f: ? ⁿ →? are formulated in terms of exhausters of the directional derivative of f. In general, they are described by means of exhausters of the Hadamard upper and lower directional derivatives of the function f. To formulate conditions for a minimum, an upper exhauster is employed while conditions for a maximum are formulated via a lower exhauster of the respective directional derivative (the Hadamard lower derivative for a minimum and the Hadamard upper derivative for a maximum).

If a point x _o is not stationary then directions of steepest ascent and descent can also be calculated by means of exhausters.     

A note on the complexity and tractability of the heat equation

We wish to solve the heat equation u_t=Δu-qu in I^d×(0,T), where I is the unit interval and T is a maximum time value, subject to homogeneous Dirichlet boundary conditions and to initial conditions u(·,0)=f over I^d. We show that this problem is intractable if f belongs to standard Sobolev spaces, even if we have complete information about q. However, if f and q belong to a reproducing kernel Hilbert space with finite-order weights, we can show that the problem is tractable, and can actually be strongly tractable.     

Harnack estimate for curvature flows depending on mean curvature

The maximum principle is applied to prove the Harnack estimate of curvature flows of hypersurfaces in Rn＋1,where the normal velocity is given by a smooth function f depending only on the mean curvature.By use of the estimate,some corollaries are obtained including the integral Harnack inequality.In particular,the conditions are given with which the solution to the flows is a translation soliton or an expanding soliton.     

Some new bounds for the maximum number of vertex colorings of a (v,e)-graph

Let f(v, e, λ) denote the maximum number of proper vertex colorings of a graph with v vertices and e edges in λ colors. In this paper we present some new upper bounds for f(v, e, λ). In particular, a new notion of pseudo-proper colorings of a graph is given, which allows us to significantly improve the upper bounds for f(v, e, 3) given by Lazebnik and Liu in the case where e > v²/4. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 115–128, 1998     

Second-order necessary conditions in semismooth optimization

First- and second-order conditions are given which are necessary for a functionf to have a local minimal value atx ^* inR ⁿ. It is assumed thatf is locally Lipschitzian nearx ^* and semismooth atx ^*. The necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second-order directional derivatives. The method of proof bears no resemblance to standard methods. Three special cases are discussed here, but applications to constrained problems are made elsewhere.     

On the number of nodal solutions to a singularly perturbed Neumann problem

We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ² and f grows superlinearly. (A typical f(u) is f(u)= a₁ u₊^p – a₁ u_-^p, a₁, a₂ >0, p, q>1.) More precisely, for any positive integer K, there exists ε_K>0 such that for 0<ε<ε_K, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.     

Extrema Constrained by a Family of Curves and Local Extrema

This paper considers the connections between the local extrema of a function f:DR and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C ¹ or C ²-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C ¹ or C ²-curves. Surprisingly, what is true for C ¹-curves fails to be true in part for C ²-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.     

A counterexample to a ‘simplex algorithm’ for convex bodies

A counterexample, in E ³, is given to the following conjecture. Suppose f ^* is a linear functional, and e an exposed point of a convex body K such that f ^* does not attain its maximum on K at e; then there is an f ^*-strictly increasing path in the one-skeleton of K emanating from e. The counterexample shows that a certain generalized simplex algorithm fails. Furthermore for a different linear functional f, there are no three disjoint f-strictly increasing paths in the one-skeleton of K leading to e.     

Separating sets by Darboux Baire 1 and approximately continuous functions

We characterize sets A₀, A₁ for which there is a DB1 function f with [f = 0] = A₀ and [f = 1] = A₁. This characterization is a conjunction of necessary conditions for Darboux and for Baire 1 functions. We also characterize sets A^?, A⁺ for which there is a DB₁ function with [f < 0] = A^? and [f > 0] = A⁺. The same characterzations are provided for approximately continuous functions.     

A degenerate Hartman theorem

Consider a real analytic diffeomorphism,f:ℝ²→ℝ², withq as a non-hyperbolic fixed point andDf(q)=Id. Placing sufficient conditions on lowest-order non-linear terms in the expansion off, we show the function is topologically conjugate with a decoupled product map. The impetus for studying such a function arose in the classical three-body problem.     

On the existence and approximation of zeroes

In this paper we consider the problem of finding zeroes of a continuous functionf from a convex, compact subsetU of ℝ ⁿ to ℝ ⁿ . In the first part of the paper it is proved thatf has a computable zero iff:C ⁿ →ℝ ⁿ satisfies the nonparallel condition for any two antipodal points on bdC ⁿ, i.e. if for anyx∈bdC ⁿ ,f(x)≠αf(−x), α≥0, holds. Therefore we describe a simplicial algorithm to approximate such a zero. It is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm, i.e. the number of times the algorithm switches from a face τ on bdC ⁿ to the face −τ. Therefore the index of a terminal simplex σ is defined which equals the local Brouwer degree of the function if σ is full-dimensional. In the second part of the paper the algorithm is used to generate possibly several approximate zeroes off. Two sucessive solutions may have both the same or opposite degrees, again depending on the number of reflection steps. By extendingf:U→ℝ ⁿ to a function g from a cube containingU to ℝ ⁿ , the procedure can be applied to any continuous functionf without having any information about the global and local Brouwer degrees a priori.     

A degenerate parabolic equation with a nonlocal source and an absorption term

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u _t = f(u)(Δu + a∫_Ω u(x, t)dx ? u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s ^p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.     

On splice quotients of the form {zn = f(x,y)}

The splice quotients, defined by W. D. Neumann and J. Wahl, are an interesting class of normal surface singularities with rational homology sphere links. In general, it is difficult to determine whether or not a singularity is analytically isomorphic to a splice quotient, although there are certain necessary topological conditions. Let {zⁿ = f(x, y)} define a surface X_{f, n} with an isolated singularity at the origin in $\mathbb {C}^3$. We show that for irreducible f, if (X_{f, n}, 0) satisfies the necessary topological conditions, then there exists a splice quotient of the form (X_{g, n}, 0), where the plane curve singularity defined by g = 0 has the same topological type as the one defined by f = 0. We also present an example of an (X_{f, n}, 0) that is not a splice quotient, but for which the universal abelian cover is a complete intersection of splice type together with a non‐diagonal action of the discriminant group. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Baire*1, Baire 1 and Zahorski properties of higher order derivatives

Summary It is proved that if f is continuous and the approximate symmetric d.l.V.P. derivatives D_a^n-2f of f of order n-2 exist in (a,b) then under a certain smoothness type condition on f, D_a^n-2f is in Baire*1. Also Zahorski property and Denjoy property for the ordinary symmetric d.l.V.P. derivative Dⁿf are established under certain suitable conditions.     

Ockham algebras with double pseudocomplementation

The variety dpO consists of those algebras (L; ∧, ∨, f, *, ⁺, 0, 1) with ∧, ∨ binary, f, *, ⁺ unary and 0, 1 nullary, and where (L; ∧, ∨, f, 0, 1) is an Ockham algebra and the unary operations f and * commute, f and⁺ commute. We describe completely the structure of the subdirectly irreducible algebras that belong to the subclass dpK_1,1, characterised by the property f³ = f. This paper is dedicated to Walter Taylor. Received September 29, 2004; accepted in final form September 8, 2005.     

On the Minimization of Difference Functions

Simple necessary optimality conditions are formulated for a function f of the form f _ g–h, where g and h are nonsmooth functions. Related sufficient conditions are given for local minimization and global minimization.     

Relaxed Derivatives and Extremality Conditions in Optimal Control

In previous work dating back to the early 1970’s F.H. Clarke and the author had independently derived necessary conditions for minimum including a maximum principle for optimal control problems defined by ordinary differential equations in which the right hand side f(t,·, r) and functions defining side conditions are Lipschitz continuous in their dependence on the state variable. Our results, though not the methods, were similar in the formulation of the maximum principle in which the nonexisting derivative f _v (t, v, σ) was replaced by an unknown element of Clarke’s generalized Jacobian but differed in handling some side conditions. In the present paper we exhibit a maximum principle in which the dual variables and the related functions are limits of appropriate subsequences of computable sequences.