Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

Asymptotic cones of finitely presented groups

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.

(a)

If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.

(b)

If CH fails, then Γ has 2^2ω asymptotic cones up to homeomorphism.

     

The strong maximum principle revisited

In this paper we first present the classical maximum principle due to E. Hopf, together with an extended commentary and discussion of Hopf's paper. We emphasize the comparison technique invented by Hopf to prove this principle, which has since become a main mathematical tool for the study of second order elliptic partial differential equations and has generated an enormous number of important applications. While Hopf's principle is generally understood to apply to linear equations, it is in fact also crucial in nonlinear theories, such as those under consideration here.In particular, we shall treat and discuss recent generalizations of the strong maximum principle, and also the compact support principle, for the case of singular quasilinear elliptic differential inequalities, under generally weak assumptions on the quasilinear operators and the nonlinearities involved. Our principal interest is in necessary and sufficient conditions for the validity of both principles; in exposing and simplifying earlier proofs of corresponding results; and in extending the conclusions to wider classes of singular operators than previously considered.The results have unexpected ramifications for other problems, as will develop from the exposition, e.g.

(i)

two point boundary value problems for singular quasilinear ordinary differential equations (Sections 3 and 4);

(ii)

the exterior Dirichlet boundary value problem (Section 5);

(iii)

the existence of dead cores and compact support solutions, i.e. dead cores at infinity (Section 7);

(iv)

Euler-Lagrange inequalities on a Riemannian manifold (Section 9);

(v)

comparison and uniqueness theorems for solutions of singular quasilinear differential inequalities (Section 10).

The case of p-regular elliptic inequalities is briefly considered in Section 11.     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered:

•

we improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets;

•

we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances;

•

we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte.

We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

•

T contains all weakly Lindelöf Banach spaces;

•

l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

•

T is stable under weak homeomorphisms;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Optimum quantization and its applications

Minimum sums of moments or, equivalently, distortion of optimum quantizers play an important role in several branches of mathematics. Fejes Tóth's inequality for sums of moments in the plane and Zador's asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d?2. In this article these results are generalized in the form of asymptotic formulae for minimum sums of moments, resp. distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In addition, we provide geometric and analytic information on the structure of optimum configurations. Our results are then used to obtain information on

(i)

the minimum distortion of high-resolution vector quantization and optimum quantizers,

(ii)

the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures,

(iii)

the minimum error of numerical integration formulae for classes of Hölder continuous functions and optimum sets of nodes,

(iv)

best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes, and

(v)

the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the form of the minimizing polytopes.

     

The Mukai pairing—II: the Hochschild-Kostant-Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are:

•

we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds;

•

we give a proof of the fact that the natural Chern character map K₀(X)→HH₀(X) becomes, after the HKR isomorphism, the usual one ; and

•

we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.

     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.

     

The Fréchet-Urysohn property of Pixley-Roy hyperspaces

Let F[X] be the Pixley-Roy hyperspace of a regular space X. In this paper, we prove the following theorem.

Theorem. For a space X, the following are equivalent:

(1)

F[X]is a k-space;

(2)

F[X]is sequential;

(3)

F[X]is Fréchet-Urysohn;

(4)

Every finite power of X is Fréchet-Urysohn for finite sets;

(5)

Every finite power ofF[X]is Fréchet-Urysohn for finite sets.

As an application, we improve a metrization theorem onF[X].     

Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field

In this paper, we develop a set of differential equations describing the steady flow of an Oldroyd 6-constant magnetohydrodynamic fluid. The fluid is electrically conducting in the presence of a uniform transverse magnetic field. The developed non-linear differential equation takes into account the effect of the material constants and the applied magnetic field. We presented the solution for three types of steady flows, namely,

(i)

Couette flow

(ii)

Poiseuille flow and

(iii)

generalized Couette flow.

Homotopy analysis method (HAM) is used to solve the non-linear differential equation analytically. It is found from the present analysis that for steady flow the obtained solutions are strongly dependent on the material constants (non-Newtonian parameters) which is different from the model of Oldroyd 3-constant fluid. Numerical solutions are also given and compared with the solutions by HAM.     

Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:

(i)

Characterize those base spaces X and Y for which all isometries are weighted composition maps.

(ii)

Give a condition independent of base spaces under which all isometries are weighted composition maps.

(iii)

Provide the general form of an isometry, both when it is a weighted composition map and when it is not.

In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.     

D-spaces and thick covers

The following results are obtained.

-

An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.

-

Every hereditarily thickly covered space is aD and linearly D.

-

Every t-metrizable space is a D-space.

-

X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.

     

Green bundles,Lyapunov exponents and regularity along the supports of the minimizing measures

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as:

•

the Lyapunov exponents of minimizing measures;     

Spaces with large weak extent

In this paper, we show the following statements:

(1)

For any cardinal κ, there exists a pseudocompact centered-Lindelöf Tychonoff space X such that we(X)?κ.

(2)

Assuming ℵ₀²=ℵ₁², there exists a centered-Lindelöf normal space X such that we(X)?ω₁.

     

On the minimal members of convex expectations

In this paper, we show that for a convex expectation E[⋅] defined on L¹(Ω,F,P), the following statements are equivalent:

(i)

E is a minimal member of the set of all convex expectations defined on L¹(Ω,F,P);

(ii)

E is linear;

(iii)

two-dimensional Jensen inequality for E holds.

In addition, we prove a sandwich theorem for convex expectation and concave expectation.     

On some contributions of John Horváth to the theory of distributions

Our main task is a presentation of J. Horváth's results concerning

•

singular and hypersingular integral operators,

•

the analytic continuation of distribution-valued meromorphic functions, and

•

a general definition of the convolution of distributions.

At some instances minor supplements to his results are given.     

Canonical matrices of bilinear and sesquilinear forms

Canonical matrices are given for

(i)

bilinear forms over an algebraically closed or real closed field;

(ii)

sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and

(iii)

sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F; the canonical matrices are based on any given set of canonical matrices for similarity over F.

A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (1988) 481-501].