Differential structure associated to axiomatic Sobolev spaces

The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure.     

On a family of lie algebras related to homogeneous surfaces

Real affine homogeneous hypersurfaces of general position in three-dimensional complex space ?³ are studied. The general position is defined in terms of the Taylor coefficients of the surface equation and implies, first of all, that the isotropy groups of the homogeneous manifolds under consideration are discrete. It is this case that has remained unstudied after the author’s works on the holomorphic (in particular, affine) homogeneity of real hypersurfaces in three-dimensional complex manifolds. The actions of affine subgroups G ? Aff(3, ?) in the complex tangent space T ^? _p M of a homogeneous surface are considered. The situation with homogeneity can be described in terms of the dimensions of the corresponding Lie algebras. The main result of the paper eliminates “almost trivial” actions of the groups G on the spaces T _p ^? M for affine homogeneous strictly pseudoconvex surfaces of general position in ?³ that are different from quadrics.     

Homogeneity Rank and Atoms of Actions

We introduce two related concepts for smooth actions of compact Lie groups:The homogeneity rank is a simple numerical invariant of the action.As one of our results we determine the precise range of this invariantfor isometric actions on compact Riemannian manifolds with positivesectional curvature and exhibit special properties of the actionswith maximal homogeneity rank.Atoms are special components of fixed point sets. They inherit actionswith the same cohomogenity and homogeneity rank as the original action,but with trivial principal isotropy group. Other properties of the originalaction like polarity are reflected in the atoms.We determine the atoms in some interesting concrete cases.Not only for this purpose we give a detailed treatise on the structureof fixed point sets, in particular in cohomogeneity one manifolds.     

Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

A Molecular Logic of Chords and Their Internal Harmony

Chords are not pure sets of tones or notes. They are mainly characterized by their matrices. A chord matrix is the pattern of all the lengths of intervals given without further context. Chords are well-structured invariants. They show their inner logical form. This opens up the possibility to develop a molecular logic of chords. Chords are our primitive, but, nevertheless, already interrelated expressions. The logical space of internal harmony is our well-known chromatic scale represented by an infinite line of integers. Internal harmony is nothing more than the pure interrelatedness of two or more chords. We consider three cases: (a) chords inferentially related to subchords, (b) pairs of chords in the space of major–minor tonality and (c) arbitrary chords as arguments of unary chord operators in relation to their outputs. One interesting result is that chord negation transforms any pure major chord into its pure minor chord and vice versa. Another one is the fact that the negation of chords with symmetric matrices does not change anything. A molecular logic of chords is mainly characterized by combining general rules for chord operators with the inner logical form of their arguments.     

On quantification of weak sequential completeness

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.     

On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse

This paper proposes a general study of (I,T)-interval-valued fuzzy rough sets on two universes of discourse integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. Some primary properties of interval-valued fuzzy logical operators and the construction approaches of interval-valued fuzzy T-similarity relations are first introduced. Determined by an interval-valued fuzzy triangular norm and an interval-valued fuzzy implicator, a pair of lower and upper generalized interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy relation on two universes of discourse is then defined. Properties of I-lower and T-upper interval-valued fuzzy rough approximation operators are examined based on the properties of interval-valued fuzzy logical operators discussed above. Connections between interval-valued fuzzy relations and interval-valued fuzzy rough approximation operators are also established. Finally, an operator-oriented characterization of interval-valued fuzzy rough sets is proposed, that is, interval-valued fuzzy rough approximation operators are characterized by axioms. Different axiom sets of I-lower and T-upper interval-valued fuzzy set-theoretic operators guarantee the existence of different types of interval-valued fuzzy relations which produce the same operators.     

Phantom without pathologies in the ultraviolet domain

We present a simple model in which the weak energy condition is violated for spatially homogeneous, slowly evolving fields. The excitations in Minkowski space in an external field with violation of the Lorentz invariance do not contain ghosts, tachyons, or superluminal modes at spatial momenta ranging from some low scale ε to the ultraviolet cutoff scale, while tachyons and possibly ghosts do exist for three-dimensional momenta less than ε. We show that in the absence of other matter, a slow-roll cosmological regime is possible. In this regime, the weak energy condition is violated, and yet homogeneity and isotropy are not completely spoiled (at the cost of fine tuning), because for a given conformal momentum, the tachyon mode increases during a sufficiently short time. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 409–426, December, 2006.     

The geometric associative algebras of Euclidean space

We deduce two associative algebra structures arising from the homogeneity and isotropy of three dimensional space with an Euclidean geometry. These are the Clifford algebrasCl(3,0) andCl(0,3). We define a bivector as the geometric product of two vectors, a definition that differs from the usual. There is a choice of whether the bivectors are constructed tail to tail or head to head leading respectively to a positive definite or negative definite Euclidean metric. The origin of the two metric choices is not identified in the usual approach. Thus we arrive at a definite geometric answer to the question of Crumeyrolle, “What is a Bivector?”.     

Regularity of the sample paths of a class of second-order spde's

We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (Electron. J. Probab. 8 (2003) 1), we prove that the solutions belong to a fractional L²-Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the stochastic integral used in the rigourous formulation of the spde, as introduced by Dalang and Mueller. For spatial covariances given by Riesz kernels, we show that our results are optimal.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

On isotropy conditions in second gradient materials

In gradient elasticity, isotropy and frame-indifference requirements are sensitive to the homogeneity of the applied rotation field Q ∈ SO(3). This is in contrast to standard elasticity, where only first gradients of the deformation are under consideration. We use a diffeomorphism to show the effect of inhomogeneous coordinate transformation to the form-invariance requirement of elastic energy. From a classical geometric rigidity result follows that the appearance of a right-local SO(3)-invariance condition is not the general condition for isotropy. The correct statement for isotropy in second gradient elasticity should be a right-global SO(3)-invariance condition. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

General logical metatheorems for functional analysis

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.

     

On the geometry of Urysohn's universal metric space

In recent years, much interest was devoted to the Urysohn space U and its isometry group; this paper is a contribution to this field of research. We mostly concern ourselves with the properties of isometries of U, showing for instance that any Polish metric space is isometric to the set of fixed points of some isometry φ. We conclude the paper by studying a question of Urysohn, proving that compact homogeneity is the strongest homogeneity property possible in U.     

Overlapping Schwarz Waveform Relaxation for the Heat Equation in N Dimensions

We analyze overlapping Schwarz waveform relaxation for the heat equation in n spatial dimensions. We prove linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In both cases the convergence rates are shown to depend on the size of the overlap. The linear convergence result depends also on the number of subdomains because it is limited by the classical steady state result of overlapping Schwarz for elliptic problems. However the superlinear convergence result is independent of the number of subdomains. Thus overlapping Schwarz waveform relaxation does not need a coarse space for robust convergence independent of the number of subdomains, if the algorithm is in the superlinear convergence regime. Numerical experiments confirm our analysis. We also briefly describe how our results can be extended to more general parabolic problems.     

Axiomatic Potential Theory,Asymptotic Values and Elliptic Differential Operators

In [14] Fernández, Heinonen and Llorente extend the Hornblower's results, about boundary behaviour of subharmonic functions in the unit disc of the complex plane, to subharmonic functions in the unit ball or the upper half space in higher dimensions. In this paper we establish that those results are also valid in the much more general setting of linear axiomatic potential theory. The interest of our general formulation relies on the applications to differential operators. We apply our result to Laplace–Beltrami operator and some uniformly elliptic second order operators in divergence form.     

An axiomatic characterization of the Owen–Shapley spatial power index

We present an axiomatic characterization of the Owen–Shapley spatial power index for the case where issues are elements of two-dimensional space. This characterization employs a version of the transfer condition, which enables us to unravel a spatial game into spatial games connected to unanimity games. The other axioms include two conditions concerned particularly with the spatial positions of the players, besides spatial versions of anonymity and dummy. The last condition says that dummy players can be left out in a specific way without changing the power of the other players. We show that this condition can be weakened to requiring dummies to have zero power if we add a condition of positional continuity. We also show that the axioms in our characterization(s) are logically independent.     

Formal systems of fuzzy logic and their fragments

Formal systems of fuzzy logic (including the well-known Łukasiewicz and Gödel–Dummett infinite-valued logics) are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction (an essential connective in the usual axiomatic system of fuzzy predicate logics).