A class of vectorfields on S2 that are topologically equivalent to polynomial vectorfields

Let X be a C¹ vectorfield on S² = {(x, y, z)?R: x² + y² + z² = 1} such that no open subset of S² is the union of closed orbits of X. If X has only a finite number of singular orbits and satisfies one additional condition, then it is shown that X is topologically equivalent to a polynomial vectorfield. A corollary is that a foliation $\text{F}$ of the plane is topologically equivalent to a foliation by orbits of a polynomial vectorfield if and only if $\text{F}$ has only a finite number of inseparable leaves.     

A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II

Let A be an n × n matrix; write A = H+iK, where i²=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if $\text{f(x,y,z) = [g(x,y,z)]}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}$, where g has degree 2, then for some unitary matrix U, the matrix U¹AU is the direct sum of $\text{n}\text{2}$ copies of a 2×2 matrix A₁, where A₁ is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)^r[g(x,y,z)]^s, where g(x,y,z) is irreducible of degree 2.     

Construction of translation planes from t-spread sets

A t-spread set [1] is a set $\text{C}$ of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = q^t+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of $\text{C}$. The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let $\text{G}$ be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|a^t+1, and det (G ? H) ≠ 0 if G and H are distinct elements of $\text{G}$. Let A₁, A₂, …, A_n?GL(t + 1, q) such that det(A_i ? G) ≠ 0 for i = 1, …, n and all G?G, and det(A_i ? A_jG) ≠ 0 for i > j and all G?G. Let $\text{C = \&{0\&} ∪ G ∪ A}{}_1\text{G ∪ … ∪ A}{}_{\mathrm{n}}\text{G}$, and ∥C∥ = q^t+1. Then $\text{C}$ is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of $\text{C}$ with x = eM(x). Theorem: Let$\text{C}$be a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: For$\text{C}$constructed as aboveG ? {M(x) | x?N_λ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥N_λ∥ = 6, and ∥N_μ∥ = 4. This system coordinatizes a new projective plane.     

Some generic properties of α-nonexpansive mappings

Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, $\text{E}$ a complete metric space formed by all α-nonexpansive mappings fC → A and $\text{M}$ a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in $\text{E}$ (2)the subset of $\text{E}$ formed by all α-contractive mappings is of Baire first category in $\text{E}$; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in $\text{M}$. Some applications to the fixed point theory and calculation of the topological degree are given.     

Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems

In this paper we study linear differential systems (1) $\text{x′ =}\text{A}\text{?}\text{(θ + ωt)x,}\text{where}\text{A}\text{?}\text{(θ)}\text{is an}\text{(n × n)}$ matrix-valued function defined on the k-torus T^k and (θ, t) → θ + ωt is a given irrational twist flow on T^k. First, we show that if A ? C^N(T^k), where N ? {0, 1, 2,…; ∞; ω}, then the spectral subbundles are of class C^N on T^k. Next we assume that à is sufficiently smooth on T^k and ω satisfies a suitable “small divisors” inequality. We show that if (1) satisfies the “full spectrum” assumption, then there is a quasi-periodic linear change of variables x = P(t)y that transforms (1) to a constant coefficient system y′ =By. Finally, we study the case where the matrix $\text{A}\text{?}\text{(θ + ωt)}$ in (1) is the Jacobian matrix of a nonlinear vector field $\text{?(x)}$ evaluated along a quasi-periodic solution x = φ(t) of (2) $\text{x′ = ?(x)}$. We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system (z, ?) defined in the vicinity of $\text{Ω = H(φ)}$, the hull of φ, so that the linearized system (1) can be represented in the form z′ = Dz, ?′ = ω, where D is a constant matrix. Our results represent substantial improvements over known methods because we do not require that à be “close to” a constant coefficient system.     

On tensor products of certain group C1-algebras

If A and B are C¹-algebras there is, in general, a multiplicity of C¹-norms on their algebraic tensor product A ⊙ B, including maximal and minimal norms ν and α, respectively. A is said to be nuclear if α and ν coincide, for arbitrary B. The earliest example, due to Takesaki [11], of a nonnuclear C¹-algebra was $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, the C¹-algebra generated by the left regular representation of the free group on two generators F₂. It is shown here that W¹-algebras, with the exception of certain finite type I's, are nonnuclear.If $\text{C}{}^1\text{(F}{}_2\text{)}$ is the group C¹-algebra of F₂, there is a canonical homomorphism λ_l of $\text{C}{}^1\text{(F}{}_2\text{)}$ onto $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$. The principal result of this paper is that there is a norm ζ on $\text{C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$, distinct from α, relative to which the homomorphism $\text{λ ⊙ λ}{}_{\mathrm{l}}\text{: C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{) → C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{) ⊙ C}{}_{\mathrm{l}}{}^1\text{(F}{}_2\text{)}$ is bounded ($\text{C}{}^1\text{(F}{}_2\text{) ⊙ C}{}^1\text{(F}{}_2\text{)}$ being endowed with the norm α). Thus quotients do not, in general, respect the norm α; a consequence of this is that the set of ideals of the α-tensor product of C¹-algebras A and B may properly contain the set of product ideals {$\text{I ? B + A ? J: I ? A, J ? B}$}.Let A and B be C¹-algebras. If A or B is a W¹-algebra there are on A ⊙ B certain C¹-norms, defined recently by Effros and Lance [3], the definitions of which take account of normality. In the final section of the paper it is shown by example that these norms, with α and ν, can be mutually distinct.     

Symmetric positive systems in corner domains

Consider the symmetric positive system of n equations in m + 2 variables,

$\text{A}\text{?u}\text{?x}\text{+ B}\text{?u}\text{?y}\text{+}\text{∑}\text{i=1}\text{m}\text{C}{}_{\mathrm{i}}\text{?u}\text{?z}{}_{\mathrm{i}}\text{+ Du = ?}$

in the corner domain x > 0, y > 0, ? ∞ < z_i < ∞, with homogeneous data on x = 0 and y = 0. The n × n matrices A, B, C_i are symmetric and D is sufficiently positive. On the boundary surfaces the matrix coefficients A, B, C_i satisfy certain “torsion” conditions. For ? with square integrable first-order derivatives, the strong solution with first-order strong derivatives is derived for the boundary value problem. For less restricted ?, the partially differentiable strong solution is established, provided more severe torsion conditions are satisfied on the boundaries. Also, the partially differentiable strong solution is obtained for the case that the torsion conditions are satisfied on one side of the boundary only.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.     

C0-semigroups on a locally convex space

A necessary and sufficient condition that a densely defined linear operator A in a sequentially complete locally convex space X be the infinitesimal generator of a quasi-equicontinuous C₀-semigroup on X is that there exist a real number β ? 0 such that, for each λ > β, the resolvent (λI ? A)^?1 exists and the family {(λ ? β)^k(λI ? A)^?k; λ > β, k = 0, 1, 2,…} is equicontinuous. In this case all resolvents (λI ? A)^?1, λ > β, of the given operator A and all exponentials exp(tA), t ? 0, of the operator A belong to a Banach algebra $\text{B}{}_{\mathrm{г}}\text{(X)}$ which is a subspace of the space L(X) of all continuous linear operators on X, and, for each t ? 0 and for each x?X, one has lim_{k → z} (I ? k^?1tA)^?kx = exp(tA) x. A perturbation theorem for the infinitesimal generator of a quasi-equicontinuous C₀-semigroup by an operator which is an element of $\text{B}{}_{\mathrm{г}}\text{(X)}$ is obtained.     

The number of positive integers ≤x and free of prime factors >y

The number defined by the title is denoted by Ψ(x, y). Let $\text{u =}\text{log}\text{x}\text{log}\text{y}$ and let ?(u) be the function determined by ?(u) = 1, 0 ≤ u ≤ 1, u?′(u) = ? ?(u ? 1), u > 1. We prove the following:Theorem. For x sufficiently large and log y ≥ (log log x)², Ψ(x,y) ? x?(u) while for 1 + log log x ≤ log y ≤ (log log x)², and ε > 0, $\text{Ψ(x, y) ?}{}_{\mathrm{\epsilon }}\text{x?(u)}\text{exp}\text{(?u e}\text{xp}\text{(?(}\text{log}\text{y)}{}^{\mathrm{<mtext>(</mtext><mtext>3</mtext><mtext>5</mtext><mtext>\; ?\; \epsilon )</mtext>}}\text{))}$.The proof uses a weighted lower approximation to Ψ(x, y), a reinterpretation of this sum in probability terminology, and ultimately large-deviation methods plus the Berry-Esseen theorem.     

Orthogonality and the numerical range. II

For a continuous linear operator A on a Hilbert space X and unit vectors x and y, an investigation of the set $\text{W[x,y]={z}{}^1\text{Az:z}{}^1\text{z=1}$ and z?span{x,y}} reveals several new results about W(A), the numerical range of A. W[x,y] is an elliptical disk (possibly degenerate), and several conditions are given which imply that W[x,y] is a line segment. In particular if x is a reducing eigenvector of A, then W[x,y] is a line segment. A unit vector is called interior (boundary) if x¹Ax is in the interior (boundary) of W(A). It is shown that interior reducing eigenvectorsare orthogonal to all boundary vectors and that boundary eigenvectors are orthogonal to all other boundary vectors y [except possibly when y¹Ay is interior to a line segment in the boundary of W(A) through the given eigenvalue].     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

Calculation of the Shannon information

Let ($\text{Ω, β, μ}{}_{\mathrm{X}}$) and $\text{(?, F, μ}{}_{\mathrm{N}}\text{)}$ be probability spaces, with $\text{f: Ω × ? ? ?}\text{a}\text{β × F|F}$ measurable map. Define μ_XY on β × $\text{F}$ by μ_XY(A) = μ_X ? μ_N{(x, y): (x, f(x, y)) ?A}, and let μ_Y = (μ_X ? μ_N)^of^?1. An expression is determined for computing the Shannon information in the measure μ_XY. This expression is used to compute the information for the non-linear additive Gaussian channel, and can be used to solve the channel capacity problem.     

On the existence and exactness of the associated sheaf functor

Let $\text{C}$ be a category with inverse limits. A category $\text{x}$is called an $\text{A}$-topos if there is a site (?, τ), i.e. a small category ? together with a Grothendieck topology τ such that $\text{x}$is equivalent to the category Sh_τ[$\text{C}$⁰. $\text{A}$] of τ-sheaves on $\text{C}$ with values in $\text{C}$. If $\text{x}$is an $\text{C}$-topos, then so is Sh_τ'[?⁰, $\text{x}$] for any site (?', τ'). It is shown that if for every site (?,τ) the associated sheaf functor from presheaves to τ-sheaves with values in $\text{A}$ exists (and preserves finite inverse limits), then the same holds if $\text{A}$is replaced by any $\text{A}$-topos $\text{x}$. Roughly speaking, the main result is that for a site (?,τ) the associated sheaf functor [?⁰, $\text{A}$] → Sh_τ [?⁰, $\text{A}$] exists and preserves finite inverse limits, provided $\text{A}$ has filtered direct limits which commute with finite inverse limits, e.g. if $\text{A}$ is a Grothendieck category or a category of sheaves with values in a locally finitely presentable category [8. 7.1]. Analogous results hold in the additive case.     

Fourier transforms of smooth functions on certain nilpotent Lie groups

The authors consider irreducible representations $\text{π ?}\text{N}\text{?}$ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_φ(x, y) of the trace class operations π_φ = ∝_Nφ(n)π_ndn, regarding the π as modeled in L²(R^k) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_φ(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ° A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ? f is Schwartz class on Rⁿ. If polynomial operators T?P(N) are transferred to operators $\text{T}\text{?}$ on Rⁿ such that $\text{F(Tφ) =}\text{T}\text{?}\text{(Fφ), P(N)}$ is transformed isomorphically to P(Rⁿ).     

On the existence of finite central groupoids of all possible ranks. I

The problem of determining the number of finite central groupoids (an algebraic system satisfying the identity (x · y) · (y ? z) = y) is equivalent to the problem of determining the number of solutions of the matrix equation A² = J, where A is a 0, 1 matrix and J is a matrix of 1's.The existence of solutions of A² = J of all ranks r, where $\text{n ? r ? [}\text{(n}{}^2\text{+ 1)}\text{2}\text{]}$, and A is n² × n², is proven. Since these are the only possible values, the question of existence solutions of all possible ranks is completely answered. The techniques and proofs are of a constructive nature.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

Lyapunov equations and Gram matrices

If p is a polynomial with all roots inside the unit disc and C its companion matrix, then the Lyapunov equation

$\text{X ? C}{}^1\text{XC = P}$

has a unique solution for every positive semidefinite matrix P. We characterize sets of vectors x₀,…,x_n?1 and y₀,…,y_n?1 such that X = G(x₀,…,x_n?1)= G(y₀,…, y_n?1)^-1. Geometrical connections between such bases and contractions with one- dimensional defect spaces are established.     

Finitely generated submodules of differentiable functions II

Let [$\text{E}$(Ω)]^p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in $\text{R}$ⁿ with itself p-times. The finitely generated submodules of [$\text{E}$(Ω)]^p are of the form im(F) where F: [$\text{E}$(Ω)]^q → [$\text{E}$(Ω)]^p is a p × q matrix of infinitely differentiable functions on Ω. Let $\text{r =}\text{max}\text{{}\text{rank}\text{(F(x)): x ? Ω}}$. The main results of the present paper are that for Ω ? $\text{R}$ⁿ, if the finitely generated submodule im(F) is closed in [$\text{E}$(Ω)]^p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? $\text{R}$¹ the converse also holds.