Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

Fundamental sets of continuous functions on spheres

LetS ^m andS ^∞ denote the unit spheres inR m+1 ande ², respectively. We look for functionsf inC[?1, 1] such that the family of functionsx →f(<s,v>) is fundamental in the spaceC(S ^m). Herev runs overS ^m. There is a similar question forC(S ^∞), when this space is given the topology of uniform convergence on compact sets.     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

The Zygmund Morse-Sard Theorem

The classical Morse-Sard Theorem says that the set of critical values off:R ^n+k →R ⁿ has Lebesgue measure zero iff ∈C ^k+1. We show theC ^k+1 smoothness requirement can be weakened toC ^k+Zygmund. This is corollary to the following theorem: For integersn >m >r > 0, lets = (n ?r)/(m ?r); iff:R ⁿ →R ^m belongs to the Lipschitz class Λ _s andE is a set of rankr forf, thenf(E) has measure zero.     

The structure of {\phi}-stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as ${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}Suppose (N ⁿ , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as P(g)=R^m_￥(g)=R(g)-2D_glog f-|?_glog f|_g²{P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}, where dm=f dvol(g){dm=\phi\,dvol(g)} and R(g) is the scalar curvature of (N ⁿ , g). In this paper, under a technical assumption on f{\phi}, we prove that f{\phi}-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane \mathbbC{\mathbb{C}} or the cylinder \mathbbR×\mathbbS¹{\mathbb{R}\times\mathbb{S}^1}.     

Edge irregular total labellings for graphs of linear size

As an edge variant of the well-known irregularity strength of a graph G=(V,E) we investigate edge irregular total labellings, i.e. functions f:V∪E→{1,2,…,k} such that f(u)+f(uv)+f(v)≠f(u^′)+f(u^′v^′)+f(v^′) for every pair of different edges uv,u^′v^′∈E. The smallest possible k is the total edge irregularity strength of G. Confirming a conjecture by Ivan?o and Jendrol’ for a large class of graphs we prove that the natural lower bound is tight for every graph of order n, size m and maximum degree Δ with m>111000Δ. This also implies that the probability that a random graph from G(n,p(n)) satisfies the Ivan?o-Jendrol’ Conjecture tends to 1 as n→∞ for all functions p∈[0,1]^N. Furthermore, we prove that is an upper bound for every graph G of order n and size m≥3 whose edges are not all incident to a single vertex.     

Injectivity and sections

We investigate injectivity in a comma-category C/B using the notion of the “object of sections” S(f) of a given morphism f:X→B in C. We first obtain that f:X→B is injective in C/B if and only if the morphism 〈1_X,f〉:X→X×B is a section in C/B and the object S(f) of sections of f is injective in C. Using this approach, we study injective objects f with respect to the class of embeddings in the categories ContL/B (AlgL/B) of continuous (algebraic) lattices over B. As a result, we obtain both topological (every fiber of f has maximum and minimum elements and f is open and closed) and algebraic (f is a complete lattice homomorphism) characterizations.     

On covariant functions and distributions under the action of a compact group

Let G be a compact subgroup of GLn(R) acting linearly on a finite dimensional complex vector space E. B. Malgrange has shown that the space C^∞G(Rn,E) of C^∞ and G-covariant functions is a finite module over the ring C^∞G(Rn) of C^∞ and G-invariant functions. First, we generalize this result for the Schwartz space SG(Rn,E) of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in [4].     

Representations for eigenfunctions of expected value operators in the Wishart distribution

Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the ‘EP’ functions f(S) for which Ef(S) for which E(f(S)) = λ_nf(Σ) for some λ_n ? $\text{R}$, whenever the m × m matrix S has the Wishart distribution W(m, n, Σ). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain L^p spaces.     

Linear transformations of monotone functions on the discrete cube

For a function f:{0,1}ⁿ→R and an invertible linear transformation L∈GL_n(2), we consider the function Lf:{0,1}ⁿ→R defined by Lf(x)=f(Lx). We raise two conjectures: First, we conjecture that if f is Boolean and monotone then I(Lf)≥I(f), where I(f) is the total influence of f. Second, we conjecture that if both f and L(f) are monotone, then f=L(f) (up to a permutation of the coordinates). We prove the second conjecture in the case where L is upper triangular.     

Orthogonally additive polynomials on spaces of continuous functions

We show that, for every orthogonally additive homogeneous polynomial P on a space of continuous functions C(K) with values in a Banach space Y, there exists a linear operator S:C(K)→Y such that P(f)=S(fn). This is the C(K) version of a related result of Sundaresam for polynomials on Lp spaces.     

Recurrent points and non-wandering points of graph maps

Let G be a graph and f:G→G be a continuous map. Denote by P(f), R(f) and Ω(f) the sets of periodic points, recurrent points and non-wandering points of f, respectively. In this paper we show that: (1) If L=(x,y) is an open arc contained in an edge of G such that {fm(x),fk(y)}⊂(x,y) for some m,k∈N, then R(f)∩(x,y)≠∅; (2) Any isolated point of P(f) is also an isolated point of Ω(f); (3) If x∈Ω(f)−Ω(fn) for some n∈N, then x is an eventually periodic point. These generalize the corresponding results in W. Huang and X. Ye (2001) [9] and J. Xiong (1983, 1986) [17] and [19] on interval maps or tree maps.     

On a property of functions on the sphere and its application

Given a continuous function f:S^m+n−2→R^m, and n points u₁,u₂,…,u_n∈S^m+n−2; does there exist a rotation r∈SO(m+n−1) such that f(ru₁)=f(ru₂)=?=f(ru_n)? In this paper, we study the property of a continuous map from a sphere to a Euclidean space by using the theory of Smith periodic transformation and Brouwer degree of map theorem. The conjecture is proved under the case of n=2 and m being even. Furthermore, this conjecture is proved for the case when u_j⋅u_j+1=λ and the dimension of the sphere is not less than m+n−2. This paper generalizes the Borsuk-Ulam theorem and then presents its application.     

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.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

André-Quillen homology of algebra retracts

Given a homomorphism of commutative noetherian rings ?:R→S, Daniel Quillen conjectured in 1970 that if the André-Quillen homology functors D_n(S∣R;−) vanish for all n?0, then they vanish for all n?3. We prove the conjecture under the additional hypothesis that there exists a homomorphism of rings ψ:S→R such that ?°ψ=id_S. More precisely, in this case we show that ψ is a complete intersection at for every prime ideal  of S. Using these results, we describe all algebra retracts S→R→S for which the algebra Tor_•^R(S,S) is finitely generated over Tor₀^R(S,S)=S.     

On the decomposition of the tensor algebra of the classical Lie algebras

Let $\text{g}$_n be the Lie algebra gl_n(C), let S($\text{g}$_n) be the symmetric algebra of $\text{g}$_n, and let T($\text{g}$_n) be the tensor algebra of $\text{g}$_n. In a recent paper, R. K. Gupta studied certain sequences of representations R_∞ = (R_n)^∞_{n = 1}, where R_n is a representation of $\text{g}$_n. These sequences have the property that every irreducible representation occurring in S($\text{g}$_n) is in exactly one of these sequences. Fixing f, she considers s(R_∞, f) which is the limit on n of the multiplicity of R_n in S^f($\text{g}$_n), the fth-graded piece of S($\text{g}$_n). She and R. P. Stanley independently showed that the limit s(R_∞, f) exists and is given by an amazingly elegant formula. They call s(R_∞, f) the stable multiplicity of R_n in S^f($\text{g}$_n). In this paper, an entirely different approach is used to extend the above result in several directions. Appropriately defined sequences R_∞ for all of the classical Lie algebras $\text{g}$_n are studied, and a simple formula for the stable multiplicity m(R)_∞, ψ, f, $\text{g}$_∞) of R_n in the ψ-isotypic component of T^f($\text{g}$_n), where ψ is any irreducible character of the symmetric group tS_f, is obtained. As in the work of Gupta and Stanley, the expressions for m(R)_∞, ψ, f, $\text{g}$_∞) are amazingly simple. Special cases include the stable decomposition of the tensor algebra, the symmetric algebra and the exterior algebra of $\text{g}$_n. As a byproduct of our proof, a “stable” decomposition of every isotypic component of T($\text{g}$_n) is obtained. This combinatorial decomposition is in some sense a generalization of Kostant's decomposition of S($\text{g}$_n) into direct sum of the harmonics and the ideal generated by the invariants of positive degree. To be precise, for f <n the combinatorial decomposition of T^f($\text{g}$_n) projects onto Kostant's decomposition of S^f($\text{g}$_n).     

Strong commutativity preserving maps on Lie ideals

Let A be a prime ring and let R be a noncentral Lie ideal of A. An additive map f:R→A is called strong commutativity preserving (SCP) on R if [f(x),f(y)]=[x,y] for all x,y∈R. In this paper we show that if f is SCP on R, then there exist λ∈C, λ²=1 and an additive map μ:R→Z(A) such that f(x)=λx+μ(x) for all x∈R where C is the extended centroid of A, unless charA=2 and A satisfies the standard identity of degree 4.