On topological and metric critical point theory

Starting from the concept of Morse critical point, introduced in [19], we propose a possible approach to critical point theory for continuous functionals defined on topological spaces, which includes some classical results, also in an infinite-dimensional setting.     

The formal theory of hopf algebras Part I: Hopf monoids in a monoidal category

Abstract

The category Hopf ? of Hopf monoids in a symmetric monoidal category ?, assumed to be locally finitely presentable as a category, is analyzed with respect to its categorical properties. Assuming that the functors “tensor squaring” and “tensor cubing” on ? preserve directed colimits one has the following results: (1) If, in ?, extremal epimorphisms are stable under tensor squaring, then Hopf C is locally presentable, coreflective in the category of bimonoids in ? and comonadic over the category of monoids in C. (2) If, in ?, extremal monomorphisms are stable under tensor squaring, then Hopf ? is locally presentable as well, reflective in the category of bimonoids in C and monadic over the category of comonoids in ?.     

The application of the nonsmooth critical point theory to the stationary electrorheological fluids

In this paper, we prove the existence of variational solutions to systems modeling electrorheological fluids in the stationary case. Our method of proof is based on the nonsmooth critical point theory for locally Lipschitz functional and the properties of the generalized Lebesgue–Sobolev space.     

Nonsmooth critical point theory and applications to the spectral graph theory

Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings,because the notions of critical sets could be either very vague or too large.To overcome these difficulties,we develop the critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons.This yields natural extensions of classical results in the critical point theory,such as the Liusternik-Schnirelmann multiplicity theorem.More importantly,eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes,which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.     

Topological geometrodynamics,Part I: General theory

The recent status of topological geometrodynamics (TGD) is reviewed. One can end up with TGD either by starting from the energy problem of general relativity or from the need to generalize hadronic or superstring models. The basic principle of the theory is `Do not quantize!' meaning that quantum physics is reduced to Kähler geometry and spinor structure of the infinite-dimensional space of 3-surfaces in 8-dimensional space H=M⁴₊×CP₂ with physical states represented by classical spinor fields. General coordinate invariance implies that classical theory becomes an exact part of the quantum theory and configuration space geometry and that space-time surfaces are generalized Bohr orbits. The uniqueness of the infinite-dimensional Kähler geometric existence fixes imbedding space and the dimension of the space-time highly uniquely and implies that superconformal and supercanonical symmetries acting on the lightcone boundary δM⁴₊×CP₂ are cosmologies symmetries.The work with the p-adic aspects of TGD, the realization of the possible role of quaternions and octonions in the formulation of quantum TGD, the discovery of infinite primes, and TGD inspired theory of consciousness encouraged the vision about TGD as a generalized number theory. The vision leads to a considerable generalization of TGD and to an extension of the symmetries of the theory to include superconformal and Super-Kac-Moody symmetries associated with the group P×SU(3)×U(2)_ew (P denotes the Poincaré group) acting as the local symmetries of the theory. Quantum criticality, which can be seen as a prediction of the theory, fixes the value spectrum for the coupling constants of the theory.The proper mathematical and physical interpretation of the p-adic numbers has remained a long-lasting challenge. Both TGD inspired theory of consciousness and the vision about physics as a generalized number theory suggest that p-adic space-time regions obeying p-adic counterparts of the field equations are geometric correlates of mind in the sense that they provide cognitive representations for the physics in the real space-time regions representing matter. Evolution identified as a gradual increase of the infinite p-adic prime characterizing the entire Universe is basic prediction of the theory.S-matrix elements can be identified as Glebsch–Gordan coefficients between interacting and free Super-Kac-Moody algebra representations and it is now possible to give Feynmann rules for the S-matrix in the approximation that elementary particles correspond to the so-called CP₂ type extremals.     

Quasi-implication algebras,Part I: Elementary theory

Quasi-implication algebras (QIA's) are intended to generalize orthomodular lattices (OML's) in the same way that implication algebras (J. C. Abbott) generalize Boolean lattices. A QIA is defined to be a setQ together with a binary operation → satisfying the following conditions (a→b is denotedab). (Q1) $$\left( {ab} \right)a = a$$ (Q2) $$\left( {ab} \right)\left( {ac} \right) = \left( {ba} \right)\left( {bc} \right)$$ (Q3) $$\left( {\left( {ab} \right)\left( {ba} \right)} \right)a = \left( {\left( {ba} \right)\left( {ab} \right)} \right)b$$ Every OML induces a QIA, wherea → b=a ^⊥?(a?b). On the other hand, every QIA induces a join semi-lattice with a greatest element 1, where 1=aa,a≤b iffab=1, anda?b=((ab)(ba))a. A bounded QIA is defined to be a QIA with a least element 0 (w.r.t.≤). The QIA associated with any OML is bounded, the zero elements being the same. Conversely, every bounded QIA induces an OML, wherea ^⊥=a0, anda?b=((ab)(a0))0. The relationC of compatibility is defined so thataCb iffa≤ba, and it is shown that every compatible sub-QIA of a QIA is an implication algebra.     

On a notion of unilateral slope for the Mumford-Shah functional

In this paper we introduce a notion of unilateral slope for the Mumford-Shah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional.     

On parallel solution of linear elasticity problems: Part I: theory

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.     

Problems, Models and Complexity. Part I: Theory

The meaning of the term problem in operations research (OR) deviates from the understanding in the theoretical computer sciences: While an OR problem is often conceived to be stated or represented by model formulations, a computer-science problem can be viewed as a mapping from encoded instances to solutions. Such a computer-science problem turns out to be rather similar to an OR model formulation. This ambiguity may cause difficulties if the computer-science theory of computational complexity is applied in the OR context. In OR, a specific model formulation is commonly used in the analysis of the underlying problem and the results obtained for this model are simply lifted to the problem level. But this may lead to erroneous results, if the model used is not appropriate for such an analysis of the problem.To resolve this issue, we first suggest a new precise formal definition of the term problem which is identified with an equivalence class of models describing it. Afterwards, a new definition is suggested for the size of a model which depends on the assumed encoding scheme. Only models which exhibit a minimal size with respect to a reasonable encoding scheme finally turn out to be suitable for the model-based complexity analysis of an OR problem. Therefore, the appropriate choice (or if necessary the construction) of a suitable representative of an OR problem becomes an important theoretical aspect of the modelling process.     

On the application of approximation of the central manifold of a stationary point to one critical case

We establish conditions under which a central manifold can be replaced by its approximation in the reduction principle for ordinary differential equations in a critical case of one zero root. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 3, pp. 430–432, March, 1998.     

Design strategies for libration point missions: theory to application

Spacecraft missions to the libration points in Sun-planet and planet-moon systems are of increasing interest. A number of such missions have already been launched. As the theoretical understanding of the dynamical structure in these regions of space expands, however, trajectory design to support actual mission requirements is more complex. Baseline trajectories for some recent mission scenarios are presented. Example approaches for transition of the designs to higher fidelity models is discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semicontinuity of a functional depending on curvature

In this paper we prove a lower semicontinuity result for a functional , defined on a class of bounded subsets of with a piecewise boundary, with respect to the -convergence of the sets. The functional depends on the curvature of in a linear way and contains a penalizing term which prevents the appearance of thin sets in the symmetric difference , where in an -approximating sequence of . Received: 3 January 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

On a global implicit function theorem for locally Lipschitz maps via non-smooth critical point theory

We prove a non-smooth generalization of the global implicit function theorem. More precisely we use the non-smooth local implicit function theorem and the non-smooth critical point theory in order to prove a non-smooth global implicit function theorem for locally Lipschitz functions. A comparison between several global inversion theorems is discussed. Applications to algebraic equations are given.     

Generalized geometric programming applied to problems of optimal control: Part I,theory

A modification to the formulation in Ref. 1 is given.