Compact sets of continuity for Borel functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set , where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming -determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented.     

The zero-divisor graph of a reduced ring

In this paper the zero-divisor graph Γ(R) of a commutative reduced ring R is studied. We associate the ring properties of R, the graph properties of Γ(R) and the topological properties of . Cycles in Γ(R) are investigated and an algebraic and a topological characterization is given for the graph Γ(R) to be triangulated or hypertriangulated. We show that the clique number of Γ(R), the cellularity of and the Goldie dimension of R coincide. We prove that when R has the annihilator condition and ; Γ(R) is complemented if and only if is compact. In a semiprimitive Gelfand ring, it turns out that the dominating number of Γ(R) is between the density and the weight of . We show that Γ(R) is not triangulated and the set of centers of Γ(R) is a dominating set if and only if the set of isolated points of is dense in .     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .     

A remark on Ricceri's conjecture for a class of nonlinear eigenvalue problems

Consider the eigenvalue problem : −Δu=λf(x,u) in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN. Denote by the set of all Carathéodory functions f:Ω×R→R such that for a.e. x∈Ω, f(x,⋅) is Lipschitzian with Lipschitz constant L, f(x,0)=0 and , and denote by (resp. ) the set of λ>0 such that has at least one nonzero classical (resp. weak) solution. Let λ₁ be the first eigenvalue for the Laplacian-Dirichlet problem. We prove that and . Our result is a positive answer to Ricceri's conjecture if use f(x,u) instead of f(u) in the conjecture.     

The geometry of the critical set of nonlinear periodic Sturm-Liouville operators

We study the critical set C of the nonlinear differential operator F(u)=−u^″+f(u) defined on a Sobolev space of periodic functions Hp(S¹), p?1. Let be the plane z=0 and, for n>0, let n be the cone x²+y²=tan²z, |z−2πn|<π/2; also set . For a generic smooth nonlinearity f:R→R with surjective derivative, we show that there is a diffeomorphism between the pairs (Hp(S¹),C) and (R³,Σ)×H where H is a real separable infinite-dimensional Hilbert space.     

Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H(Γ) and H₊(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of Γ×Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of RP³. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on for finite graphs Γ.     

The topology of the monodromy map of a second order ODE

We consider the following question: given A∈SL(2,R), which potentials q for the second order Sturm-Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q∈L²([0,2π]) to , the lift to the universal cover of SL(2,R) of the fundamental matrix map ,

     

Topology of dynamical systems in finite groups and number theory

We study and develop a very new object introduced by V.I. Arnold: a monad is a triple consisting of a finite set, a map from that finite set to itself and the monad graph which is the directed graph whose vertices are the elements of the finite set and whose arrows lead each vertex to its image (by the map). We consider the case in which the finite set entering in the monad definition is a finite group G and the map is the Frobenius map, for some k∈Z. We study the Frobenius dynamical system defined by the iteration of the monad fk, and also study the combinatorics and topology (i.e., the discrete invariants) of the monad graph. Our study provides useful information about several structures on the group associated to the monad graph. So, for example, several properties of the quadratic residues of finite commutative groups can be obtained in terms of the graph of the Frobenius monad .     

Note on separate continuity and the Namioka property

A pair 〈B,K〉 is a Namioka pair if K is compact and for any separately continuous , there is a dense A⊆B such that f is ( jointly) continuous on A×K. We give an example of a Choquet space B and separately continuous such that the restriction fΔ_| to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous and for any Baire subspace F of T×K, the set of points of continuity of is dense in F. We say that 〈B,K〉 is a weak-Namioka pair if K is compact and for any separately continuous and a closed subset F projecting irreducibly onto B, the set of points of continuity of fF_| is dense in F. We show that T is a Baire space if the pair 〈T,K〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈B,K〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous which has no dense set of continuity points; in fact, f does not even have the Baire property.     

Universal factorization property of certain polycyclic groups

Let H be a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group. Let G be any group with maximal condition. We show that there exists a torsion-free strongly polycyclic (torsion-free virtually polycyclic, resp.) group and an epimorphism such that for any homomorphism ?:G→H, it factors through , i.e., there exists a homomorphism such that . We show that this factorization property cannot be extended to any finitely generated group G. As an application of factorization, we give necessary and sufficient conditions for N(f,g)=R(f,g) to hold for maps f,g:X→Y between closed orientable n-manifolds where π₁(X) has the maximal condition, Y is an infra-solvmanifold, N(f,g) and R(f,g) denote the Nielsen and Reidemeister coincidence numbers, respectively.     

The phantom cover of a module

A morphism of left R-modules is a phantom morphism if for any morphism , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor₁(X,f)=0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective.If is the category of finitely presented right R-modules modulo projectives, then the association M?Tor₁(−,M) is a functor from the category of left R-modules to that of the flat functors on . The phantom cover is used to characterize when this functor is faithful or full. It is faithful if and only if the flat cover of every module has a pure injective kernel; this is equivalent to the flat cover being the phantom cover. The question of fullness is only reasonable when the functor is restricted to the subcategory of cotorsion modules. This restriction is full if and only if every phantom cover of a cotorsion module is pure injective.     

Density of constant radius normal binary covering codes

A binary code with covering radius R is a subset C of the hypercube Q_n={0,1}ⁿ such that every x∈Q_n is within Hamming distance R of some codeword c∈C, where R is as small as possible. For a fixed coordinate i∈[n], define to be the set of codewords with a b in the ith position. Then C is normal if there exists an i∈[n] such that for any v∈Q_n, the sum of the Hamming distances from v to and is at most 2R+1. We newly define what it means for an asymmetric covering code to be normal, and consider the worst-case asymptotic densities ν^*(R) and of constant radius R symmetric and asymmetric normal covering codes, respectively. Using a probabilistic deletion method, and analysis adapted from previous work by Krivelevich, Sudakov, and Vu, we show that and , giving evidence that minimum size constant radius covering codes could still be normal.