PHANTOM MAPS AND SPACES OF THE SAME {\tf n}-TYPE FOR ALL {\tf n}

After reviewing the theory of phantom maps and SNT, the author gives several general results which relate the theory of phantom maps and SNT and which extend that of Harper and Roitberg.     

Localization,completion and detecting equivariant maps on skeletons

In the category of equivariant spaces with base point, we prove the injectivity of the induced map between homotopy sets under some conditions. We study some relations between the localization and the completion. By using these results, we characterize continuous maps which are homotopic on skeletons, and obtain a generalization of the theory of phantom maps.     

Persistence of Simple Substances

In this paper, we argue for a novel three-dimensionalist (3D'ist) solution to the problem of persistence, i.e. cross-temporal identity. We restrict the discussion of persistence to simple substances, which do not have other substances as their parts. The account of simple substances employed in the paper is a trope-nominalist strong nuclear theory (SNT), which develops Peter Simons' trope nominalism. Regarding the distinction between three dimensionalism (3D) and four dimensionalism (4D), we follow Michael Della Rocca's formulation, in which 3D explains persistence in virtue of same entities and 4D in virtue of distinct entities (temporal parts). SNT is a 3D'ist position because it accounts for the persistence of simple substances in virtue of diachronically identical ‘nuclear’ tropes. The nuclear tropes of a simple substance are necessary for it and mutually rigidly dependent but distinct. SNT explains qualitative change by tropes that are contingent to a simple substance. We show that it avoids the standard problems of 3D: temporal relativization of ontic predication, Bradley's regress and coincidence, fission and fusion cases. The temporal relativization is avoided because of the analysis of temporary parts that SNT gives in terms of temporal sub-location, which is atemporal part–whole relation.     

Rationalization and the Gray index of phantom maps

In this paper we study a homotopy invariant of phantom maps called the Gray index. We give a new interpretation of the Gray index of a phantom map f:X→Y, in terms of the rationalization of X. We use this interpretation, in order to detect phantom maps of a specific Gray index. Finally, we examine the set of phantom maps with infinite Gray index in a tower theoretic way.     

Grasping the phantom – a new approach to nonlinear spectral theory

A new definition of a spectrum for nonlinear operators is suggested, which is called phantom. The phantom naturally divides into two subsets which in the linear case correspond to the point spectrum, and to the union of the continuous and residual spectrum. There is also a natural nonlinear analogue to the approximate point spectrum. The phantom may be considered as a variant of the spectrum of M. Furi, M. Martelli, and A. Vignoli (Ann. Mat. Pura Appl. 118 (1978), 229–294), which reflects the behavior of the operator on bounded sets and which is based on a new definition of an eigenvalue that was suggested earlier by the authors (Nonlinear Anal. 40 (2000), 565–576).?To define the phantom, the class of stably 0-epi maps is introduced and studied. This is a subclass of 0-epi maps satisfying a Rouché type theorem. Received: November 15, 1999?Published online: October 2, 2001     

On the Gray index conjecture for phantom maps

We study the Gray index, a numerical invariant for phantom maps. It has been conjectured that the only phantom map between finite-type spaces with infinite Gray index is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of the phantom maps are restricted to being simply connected finite complexes.As a result of the counter example, we can show that SNT^∞(X) can be non-trivial for some space X of finite type.     

On the Gray index of phantom maps

We study a homotopy invariant of phantom maps called the Gray index. In particular, it is conjectured that the Gray index of an essential phantom map between finite-type spaces is always finite. We obtain some partial results on this conjecture, using a tower-theoretic interpretation of the Gray index.     

Phantom Maps from a Strong Homology Viewpoint

Let X and Y be connected nilpotent CW-complexes of finite type. In this paper we deal with another complete analysis of a set of phantom maps in terms of strong homology by using the derived functors. We also prove that the set of homotopy classes of maps is independent of the choice of both rational homotopy equivalent classes and completion genera.     

A comparative study on the mechanical and barrier characteristics of polyimide nanocomposite films filled with nanoparticles of planar and tubular morphology

Polyimide (PI) films based on poly(pyromellitic dianhydride-co-4,4′-oxydianiline) (PI-PM) were filled with different nanoparticles, such as organically modified montmorillonite (MMT), vapor-grown carbon nanofibers (VGCF), and silicate nanotubes (SNT) of different concentration.. Rheological measurements and structural investigations showed a relatively good dispersion of the nanoparticles in the PI matrix to an extent that depended on the type and morphology of the nanoparticles used. The mechanical (tensile modulus, strength, and deformation at break) and the barrier (oxygen permeability) properties of PI-PM nanocomposite films were investigated. The polyimide nanocomposites filled with SNT and tubular VGCF nanoparticles showed an increased tensile modulus with increasing volume concentration of the nanoparticles without a catastrophic decrease in the elongation at break. In addition, the MMT particles, chemically modified with 4,4'-bis-(4′′-aminophenoxy)diphenylsulfone, significantly improved the barrier properties of the PI-PM films, which exceeded those of the nanocomposites filled with VGCF or SNT. The relative poor oxygen barrier and mechanical properties of the PI-PM/VGCF nanocomposite films are ascribed to the relative weak adhesion between the VGCF and the polyimide matrix, which was confirmed by scanning electron microscopy of the fracture surface of these films.     

When almost Cohen–Macaulay algebras map into big Cohen–Macaulay modules

In this article, we show that almost Cohen–Macaulay algebras are solid. Moreover, we seek for the conditions when (a) an almost Cohen–Macaulay algebra is a phantom extension and (b) when it maps into a balanced big Cohen–Macaulay module.     

The Space of Loops on a Homogeneous Space and Rational Equivalence

We shall construct a map which is a rational equivalence with applications to phantom maps,spaces of the same n-type and genus sets. This answers in theaffirmative a question of C. McGibbon. 1991 Mathematics SubjectClassification 55Q05.     

Effect of the dynamic pressure on the similarity solution of cylindrical shock waves in a rarefied polyatomic gas

The similarity solution for a strong cylindrical shock wave in a rarefied polyatomic gas is analyzed on the basis of Rational Extended Thermodynamics with six independent fields; the mass density, the velocity, the pressure and the dynamic pressure. A new ODE system for the similarity solution is derived in a systematic way by using the method based on the Lie group theory proposed in the context of the spherical shock wave in a rarefied monoatomic gas in Donato and Ruggeri (J Math Anal Appl 251:395, 2000). The boundary conditions are also specified from the Rankine–Hugoniot conditions for the sub-shock. The derived similarity solution is characterized by only one dimensionless parameter \(\alpha \) related to the relaxation time for the dynamic pressure. The numerical analysis of the similarity solution is also performed. The solution agrees with the well-known Sedov–von Neumann–Taylor (SNT) solution when \(\alpha \) is small. When \(\alpha \) is larger, due to the presence of the dynamic pressure, the deviation from the SNT solution is evident; the strength of a peak near the shock front becomes smaller and the profile becomes broader.

     

Coincidence theory for multivalued maps satisfying compactness conditions on countable sets

ABSTRACT

We present a coincidence theory for set-valued maps which satisfy certain compactness-type conditions on countable sets. Our theory is based on fixed point results for compositions of set-valued self maps.     

A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

A variational inequality theory for demicontinuous S-contractive maps in Hilbert spaces is established by employing the ideas of Granas' topological transversality. Such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward S-contractive maps and to those of fixed point index for condensing maps. The variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear second-order elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical Sobolev exponent. There has been little discussion for such elliptic inequalities involving the critical Sobolev exponent in the literature.     

Phantom depth and stable phantom exactness

Phantom depth, phantom nonzerodivisors, and phantom exact sequences are analogues of the non-``phantom' notions which have been useful in tackling the (very difficult) localization problem in tight closure theory. In the present paper, these notions are developed further and partially reworked. For instance, although no analogue of a long exact sequence arises from a short stably phantom exact sequence of complexes, we provide a method for recovering the kind of information obtainable from such a long sequence. Also, we give alternate characterizations of the notion of phantom depth, including one based on Koszul homology, which we use to show that with very mild conditions on a finitely generated module , any two maximal phantom -regular sequences in an ideal have the same length. In order to do so, we prove a ``Nakayama lemma for tight closure', which is of independent interest. We strengthen the connection of phantom depth with minheight, we explore several analogues of ``associated prime' in tight closure theory, and we discuss a connection with the problem of when tight closure commutes with localization.

     

有序拓扑向量空间中的共鸣定理

本文研究有序拓扑向量空间中非线性映照的共鸣定理．对于取值于有序拓扑向量空间中的映照，利用序关系，引入了一类广泛的非线性映照．对于这类非线性映照，应用纲理论，并给出了关于点态序有界蕴涵一致序有界的共鸣定理．     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Random fixed points of discontinuous random maps

We prove some random fixed-point theorems for random maps which are not necessarily continuous. This may lead to the discovery of some new results in random fixed-point theory for discontinuous maps.     

Embeddings with multiple regularity

We introduce (k,l)-regular maps, which generalize two previously studied classes of maps: affinely k-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a (k,l)-regular map. The problem can be regarded as an extension of embedding theory to embeddings with certain non-degeneracy conditions imposed, and is related to approximation theory.     

A regularity theory for intrinsic minimising fractional harmonic maps

We define and develop an interior partial regularity theory for intrinsic energy minimising fractional harmonic maps from Euclidean space into smooth compact Riemannian manifolds for fractional powers strictly between zero and one. Intrinsic fractional harmonic maps are critical points of an energy whose first variation is a Dirichlet to Neumann map for the harmonic map problem on a half-space with a Riemannian metric which can degenerate/become singular along the boundary, depending on the fractional power. Similarly to the approach used to prove regularity for stationary intrinsic semi-harmonic maps, we take advantage of the connection between fractional harmonic maps and free boundary problems for harmonic maps in order to develop a partial regularity theory for the fractional harmonic maps we consider. In particular, we prove partial regularity for locally minimising harmonic maps with (partially) free boundary data on half-spaces with the aforementioned metrics up to the boundary; fractional harmonic maps then inherit this regularity. As a by-product of our methods we shed some new light on the monotonicity of the average energy of solutions of the degenerate linear elliptic equation related to fractional harmonic functions.