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We study the arithmetic three-spheres theorems for subsolutions of subelliptic PDEs of p p -harmonic type in Carnot groups of Heisenberg type for 1 > p > ∞ 1>p>\infty . In the presentation we exhibit the special cases of sub-Laplace equations ( p = 2 p=2 ) and the case p p is equal to the homogeneous dimension of a Carnot group. Corollaries include asymptotic behavior of subsolutions for small and large radii and the Liouville-type theorems.

We characterize the set of diagonals of the unitary orbit of a self-adjoint operator with a finite spectrum. Our result extends the Schur-Horn theorem from a finite dimensional setting to an infinite dimensional Hilbert space, analogous to Kadison’s theorem for orthogonal projections, and the second author’s result for operators with three point spectrum.

In this paper, we are concerned with the existence and multiplicity of positive periodic solutions for first-order vector differential equations. By using the Leray-Schauder alternative theorem and the Kransnosel’skii fixed point theorem, we show that the differential equations under the periodic boundary value conditions have at least two positive periodic solutions.

For the smooth normalization f : X ¯ → X f : {\overline X} \to X of a singular variety X X over a field k k of characteristic zero, we show that for any conducting subscheme Y Y for the normalization, and for any i ∈ Z i \in \mathbb {Z} , the natural map K i ( X , X ¯ , n Y ) → K i ( X , X ¯ , Y ) K_i(X, {\overline X}, nY) \to K_i(X, {\overline X}, Y) is zero for all sufficiently large n n . As an application, we prove a formula for the Chow group of zero cycles on a quasi-projective variety X X over k k with Cohen-Macaulay isolated singularities, in terms of an inverse limit of the relative Chow groups of a desingularization X ~ \widetilde X relative to the multiples of the exceptional divisor. We use this formula to verify a conjecture of Srinivas about the Chow group of zero cycles on the affine cone over a smooth projective variety which is arithmetically Cohen-Macaulay.

The theorem in question gives a sufficient condition for a quadratic space over a local ring R to contain a hyperbolic plane over R.

Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdos and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey's Theorem, and Konig's Lemma.

We prove a fixed point theorem that is a very simple generalization of the Banach contraction principle and characterizes the metric completeness of the underlying space. We also discuss the Meir-Keeler fixed point theorem.

In this paper, an hp-local discontinuous Galerkin method is applied to a class of quasilinear elliptic boundary value problems which are of nonmonotone type. On hp-quasiuniform meshes, using the Brouwer fixed point theorem, it is shown that the discrete problem has a solution, and then using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in broken H 1 norm and L 2 norm which are optimal in h, suboptimal in p are derived. These results are exactly the same as in the case of linear elliptic boundary value problems. Numerical experiments are provided to illustrate the theoretical results.

In the context of Lebesgue integration, we derive the divergence theorem for unbounded vector fields that can have singularities at every point of a compact set whose Minkowski content of codimension greater than two is finite. The resulting integration by parts theorem is applied to removable sets of holomorphic and harmonic functions.

We extend the “matricial corona theorem” of M. Andersson to general algebras of functions which satisfy a corona theorem.

We extend some fixed point theorems in L-spaces, obtaining extensions of the Banach fixed point theorem to partially ordered sets. © 2007 American Mathematical Society Reverts to public domain 28 years from publication.

We work in an o-minimal expansion of a real closed field. Using piecewise smoothness of definable functions we define the topological degree for definable continuous functions. Using this notion of the degree we obtain a new proof for the existence of torsion points in a definably compact group, and also a new proof of an o-minimal analogue of the Brouwer fixed point theorem.

In this paper we present some generalization (at the same time a new and a short proof in the Banach algebra context) of the Weak Spectral Mapping Theorem (WSMT) for non-quasianalytic representations of one-parameter groups.

The Kato spectrum of an operator is deployed to give necessary and sufficient conditions for Browder’s theorem to hold.

We use Levy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.

W. T. Gowers' c 0 theorem asserts that for every Lipschitz function F : S c0 → R and e > 0, there exists an infinite-dimensional subspace Y of c 0 such that the oscillation of F on Sy is at most e. The proof of this theorem has been reduced by W. T. Cowers to the proof of a new Ramsey type theorem. Our aim is to present a proof of the last result.

By a well-known theorem of Lohwater and Piranian, for any set E E on | z | = 1 |z|=1 of type F σ F_\sigma and of measure zero there exists a bounded analytic function in | z | > 1 |z|>1 which fails to have radial limits exactly at the points of E E . We show that this theorem is an immediate corollary of Fatou’s interpolation theorem of 1906.

The Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem plays an important role not only in harmonic analysis and approximation theory, but also in communication engineering since it enables engineers to reconstruct analog signals from their samples at a discrete set of data points. The main aim of this paper is to derive a q q -analogue of the Whittaker-Shannon-Kotel’nikov sampling theorem. The proof uses recent results in the theory of q q -orthogonal polynomials and basic hypergeometric functions, in particular, new results on the addition theorems for q q -exponential functions.

The theorem of Jordan which I want to discuss here dates from 1872. It is an elementary result on finite groups of permutations. I shall first present its translations in Number Theory and Topology. 1. Statements 1.1. Number theory. Let f = ∑n m=0 amx m be a polynomial of degree n, with coefficients in Z. If p is prime, let Np(f) be the number of zeros of f in Fp = Z/pZ. Theorem 1. Assume (i) n ≥ 2, (ii) f is irreducible in Q[x]. Then (a) There are infinitely many p’s with Np(f) = 0. (b) The set P0(f) of p’s with Np(f) = 0 has a density c0 = c0(f) which is > 0. [Recall that a subset P of the set of primes has density c if lim X→∞ number of p ∈ P with p ≤ X π(X) = c, where π(X) is as usual the number of primes ≤ X .] Moreover, Theorem 2. With the notation of Theorem 1, one has c0(f) ≥ 1 n , with strict inequality if n is not a power of a prime. Example. Let f = x + 1. One has p ∈ P0(f) if and only if p ≡ −1 (mod 4); this set is well-known to have density 1/2. We shall see more interesting examples in §5. 1.2. Topology. Let S1 be a circle. Let f : T → S be a finite covering of a topological space S. Assume: (i) f has degree n (i.e. every fiber of f has n elements), with n ≥ 2, (ii) T is arcwise connected and not empty. Theorem 3. There exists a continuous map φ : S1 → S which cannot be lifted to the covering T (i.e. there does not exist any continuous map ψ : S1 → T such that φ = f ◦ ψ). Received by the editors March 1, 2003. 2000 Mathematics Subject Classification. Primary 06-XX, 11-XX, 11F11. This text first appeared in Math Medley 29 (2002), 3–18. The writing was done with the help of Heng Huat Chan. c ©2002 Singapore Mathematical Society. Reprinted with permission.

Our main result is a characterization of open Seifert fibered 3 3 -manifolds in terms of the fundamental group and large-scale geometric properties of a triangulation. As an application, we extend the Seifert Fiber Space Theorem and the Torus Theorem to a class of 3 3 -orbifolds.

The pseudospectrum has become an important quantity for analyzing stability of nonnormal systems. In this paper, we prove a mapping theorem for pseudospectra, extending an earlier result of Trefethen. Our result consists of two relations that are sharp and contains the spectral mapping theorem as a special case. Necessary and sufficient conditions for these relations to collapse to an equality are demonstrated. The theory is valid for bounded linear operators on Banach spaces. For normal matrices, a special version of the pseudospectral mapping theorem is also shown to be sharp. Some numerical examples illustrate the theory.

The Bartle-Graves theorem extends the Banach open mapping principle to a family of linear and bounded mappings, thus showing that surjectivity of each member of the family is equivalent to the openness of the whole family. In this paper we place this theorem in the perspective of recent concepts and results, and present a general Bartle-Graves theorem for set-valued mappings. As application, we obtain versions of this theorem for mappings defined by systems of inequalities, and for monotone variational inequalities.

In this paper, a theorem on the asymptotic property of a summation of digits in a k k -adic representation is presented.

If M C = ( A 0 C B ) is a 2 × 2 upper triangular matrix on the Hilbert space H ○+ K, then a-Weyl's theorem for A and B need not imply a-Weyl's theorem for M C , even when C = 0. In this note we explore how a-Weyl's theorem and a-Browder's theorem survive for 2 x 2 operator matrices on the Hilbert space.

Dydak and Kozlowski (1991) obtained a generalization of the Vietoris-Begle theorem for the cohomology theories induced by CW spectra. In this note we prove a dual of their theorem involving the pro-homology theories induced by CW spectra.

This article provides a short and elementary proof of the key theorem of reduced K-theory, namely Platonov's Congruence theorem. Our proof is based on Wedderburn's factorization theorem.

This paper presents a generalization of the invariant subspace theorem of Helson and Lowdenslager along the lines of de Branges’ generalization of Beurling’s theorem.

Let A A be a TAF-algebra, Z ( A ) Z(A) the centre of A , I d ( A ) A, Id(A) the ideal lattice of A A , and M i r ( A ) Mir(A) the space of meet-irreducible elements of I d ( A ) Id(A) , equipped with the hull-kernel topology. It is shown that M i r ( A ) Mir(A) is a compact, locally compact, second countable, T 0 T_0 -space, that I d ( A ) Id(A) is an algebraic lattice isomorphic to the lattice of open subsets of M i r ( A ) Mir(A) , and that Z ( A ) Z(A) is isomorphic to the algebra of continuous, complex functions on M i r ( A ) Mir(A) . If A A is semisimple, then Z ( A ) Z(A) is isomorphic to the algebra of continuous, complex functions on P r i m ( A ) Prim(A) , the primitive ideal space of A A . If A A is strongly maximal, then the sum of two closed ideals of A A is closed.

A classical theorem of Pick gives a criterion for interpolation by analytic functions in the open unit disc D subject to an H∞-norm bound. This result has substantial generalizations in two different directions. On the one hand, one can replace H∞ by the multiplier algebras of certain Hilbert function spaces, some of them having no connection with analyticity [Ag1, Ag2]. On the other hand, one can obtain a criterion for interpolation by meromorphic functions with a prescribed number of poles in D and with an L∞-norm bound on the unit circle T; this is a classical result of Akhiezer [Ak], now better known in the form of the far-reaching generalizations due to Adamyan, Arov and Krein [AAK]. In each case the criterion is in terms of the signature of a “Pick matrix” constructed from the interpolation data and the reproducing kernel of the appropriate Hilbert function space (i.e. H in the case of the AAK theorem). It is therefore conceivable that there might be a common generalization which would hold for a significant class of function spaces. After all, the analogue of Pick’s theorem is true for the Dirichlet space D of analytic functions in D with finite Dirichlet integral [Ag1] and for the space W [a, b] of L functions f on [a, b] for which f ′ ∈ L(a, b) [Ag2]. Might not an analogue of the Akhiezer-Adamyan-Arov-Krein theorem hold for these spaces? This natural question was posed in [Q2]. Pick’s theorem has long held the attention of analysts as one of the most elegant of all interpolation results. However, there are grounds beyond the aesthetic ones for its continued prominence. Knowledge that the analogue of Pick’s theorem holds for a particular Hilbert function space gives a powerful tool for the conversion of L∞-type problems to Hilbert space problems; this principle is brought out in [MS], where among other things it is used to give a relatively simple proof of Carleson’s theorem on interpolation sequences for H∞ and also to characterize geometrically the interpolation sequences for multipliers of the Dirichlet space. Since about 1980 the Pick property has played an important role in linear control theory. The feedback controllers which internally stabilize a given linear system can be described by analytic functions satisfying a finite set of interpolation conditions, so

A selector theorem for non-convex orientor fields on closed manifolds is given and the Lefschetz fixed point theorem is used to establish an existence result for these ones.

We shall provide conditions on the functionf(t,u1,⋯,un−1)f(t,u_{1},\cdots , u_{n-1}). The higher order boundary value problem({BVP}){(E) u(n)(t)+f(t,u(t),u(1)(t),⋯,u(n−2)(t))=0 for t∈(0,1) and n≥2, (BC) {u(i)(0)=0, 0≤i≤n−3, αu(n−2)(0)−βu(n−1)(0)=0, γu(n−2)(1)+δu(n−1)(1)=0\begin{equation*}\begin {cases}(E)~~ u^{(n)}(t)+ f(t, u(t),u^{(1)}(t),\cdots ,u^{(n-2)}(t))=0~~~~~\mathrm {~for~}~~~~~t\in (0,1)~~~~\mathrm {and}~~~~~~n\ge 2,\ (BC)~~ \begin {cases}u^{(i)}(0)=0,~~~~~0\le i \le n-3,\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\ \gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=0\end{cases} \end{cases} \tag {{BVP}}\end{equation*}has at least one solution.

We show that a theorem of Kisyński on the generation of Banach-algebra homomorphisms of certain convolution algebras is equivalent to the Hille-Yosida theorem on the generation of operator-valued one-parameter semigroups.

Time-dependent flows of viscous incompressible immiscible fluids are studied in the limit of vanishing Reynolds numbers. The velocity fields associated to each fluid solve Stokes equations in a time-dependent domain. Classical immiscibility conditions on the varying fluids interfaces are taken into account by a new formulation of the problem: the viscosity solves a transport equation and the velocity field solves a Stokes problem with this nonconstant viscosity. This formulation, based on the use of a pseudoconcentration function, has already been used for numerical computations (see [9] and [4]). For this nonlinear system of equations, existence of solutions is proved, using the Schauder fixed point theorem and the concept of renormalized solutions introduced recently by R. J. DiPerna and P. L. Lions.

Let A A be a separable simple A F AF -algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element x ∈ M ( A ) x\in M(A) , i.e., π ( x ) \pi (x) is normal ( π : M ( A ) → M ( A ) / A \pi : M(A)\to M(A)/A is the quotient map), having the form y + a y+a for some normal element y ∈ M ( A ) y\in M(A) and a ∈ A . a\in A. We also show that a normal element x ∈ M ( A ) x\in M(A) can be quasi-diagonalized if and only if the Fredholm index i n d ( λ − x ) = 0 ind(\lambda -x)=0 for all λ ∉ s p ( π ( x ) ) . \lambda \not \in sp(\pi (x)). In the case that A A is a simple C ∗ C^* -algebra of real rank zero, with stable rank one and with continuous scale, K 1 ( A ) = 0 , K_1(A)=0, and K 0 ( A ) K_0(A) has countable rank, we show that a normal element x ∈ M ( A ) x\in M(A) with zero Fredholm index can be written as x = ∑ n = 1 ∞ λ n ( e n − e n − 1 ) + a , \begin{equation*} x=\sum _{n=1}^{\infty }\lambda _n(e_n-e_{n-1})+a, \end{equation*} where { e n } \{e_n\} is an (increasing) approximate identity for A A consisting of projections, { λ n } \{\lambda _n\} is a bounded sequence of numbers and a ∈ A a\in A with ‖ a ‖ > ϵ \|a\|>\epsilon for any given ϵ > 0. \epsilon >0.

This paper establishes a generalization of the Lusin-Privalov radial uniqueness theorem which applies to subharmonic functions in all dimensions. In particular, it answers a question of Rippon by showing that no subharmonic function on the upper half-space can have normal limit − ∞ -\infty at every boundary point.

We study the a.e. convergence of orthogonal series defined over a general measure space. We give sufficient conditions which contain the Menshov-Rademacher theorem as an endpoint case. These conditions turn out to be necessary in the particular case where the measure space is the unit interval [0, 1] and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality.

In 1961, Simon and Ando wrote a classical paper describing the convergence properties of nearly completely decomposable matrices. Basically, their work concerned a partitioned stochastic matrix e.g. \[ A = [ A 1 a m p ; E 1 E 2 a m p ; A 2 ] A = \begin {bmatrix} A_1&E_1\ E_2&A_2\end {bmatrix} \] where A 1 A_1 and A 2 A_2 are square blocks whose entries are all larger than those of E 1 E_1 and E 2 E_2 respectively. Setting \[ A k = [ A 1 ( k ) a m p ; E 1 ( k ) E 2 ( k ) a m p ; A 2 ( k ) ] , A^k=\begin {bmatrix} A^{(k)}_1&E^{(k)}_1\ E^{(k)}_2&A^{(k)}_2\end {bmatrix}, \] partitioned as in A A , they observed that for some, rather short, initial sequence of iterates the main diagonal blocks tended to matrices all of whose rows are identical, e.g. A 1 ( k ) A^{(k)}_1 to F 1 F_1 and A 2 ( k ) A^{(k)}_2 to F 2 F_2 . After this initial sequence, subsequent iterations showed that all blocks lying in the same column as those matrices tended to a scalar multiple of them, e.g. \[ lim k → ∞ A k = [ α F 1 a m p ; β F 2 α F 1 a m p ; β F 2 ] \lim _{k\to \infty }A^k=\begin {bmatrix} \alpha F_1&\beta F_2\ \alpha F_1&\beta F_2\end {bmatrix} \] where α ≥ 0 \alpha \geq 0 and β ≥ 0 \beta \geq 0 . The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.

Recently, we introduced a space [ H μ ( A ) ; B ] [{H_\mu }(A);B] which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (The Hankel transformation of a Banach space-valued generalized function, Proc. Amer. Math. Soc. 119 (1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of [ H μ × A ; B ] [{H_\mu } \times A;B] , it can be uniquely represented by an element of [ H μ ( A ) ; B ] [{H_\mu }(A);B] and hence of [ H μ ; [ A ; B ] ] [{H_\mu };[A;B]] . This is motivated by a generalization of Zemanian (Realizability theory for continuous linear systems, Academic Press, New York, 1972) for the product space D R n × V {D_{{R^n}}} \times V where V is a Fréchet space. His work is based on the facts that the space D R n {D_{{R^n}}} is an inductive limit space and the convolution product is well defined in D K j {D_{{K_j}}} . This is not possible here since the space H μ ( A ) {H_\mu }(A) is not an inductive limit space. Furthermore, D ( A ) D(A) is not dense in H μ ( A ) {H_\mu }(A) . To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of A.

We prove an intermediate value theorem for noncontinuous functions; as consequences, we obtain coincidence and fixed points theorems for nonmonotone and noncontinuous functions defined and with values in a product space R I {\mathbb {R}^I} . Some of them, even when the index set I is a singleton, improve recent statements of S. Schmitd.

Let X be a reflexive Banach space, D an open and bounded subset of X, and T : D ¯ → X T:\bar D \to X a continuous mapping which is locally pseudocontractive on D. Suppose there exists an element z ∈ D z \in D such that ‖ z − T z ‖ > ‖ x − T x ‖ \left \| {z - Tz} \right \| > \left \| {x - Tx} \right \| for all x on the boundary of D. Then under the so-called condition (S), T has a fixed point in D. Although this result was proved earlier by Kirk, we show here a much easier approach.

With a technical assumption (E-k), which is a relaxed version of the condition T n / n → 0 , n → ∞ {T^n}/n \to 0,n \to \infty , where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition ( X = ( I − T ) k X ⊕ ker ⁡ ( I − T ) k ) (X = {(I - T)^k}X \oplus \ker {(I - T)^k}) , of closedness of ( I − T ) k X {(I - T)^k}X , and of quasi-Fredholmness of I − T I - T . One consequence, still assuming (E-k), is that I − T I - T is semi-Fredholm if and only if I − T I - T is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.

Let h be a free homeomorphism with a finite set of fixed points on a compact surface. Then h is, on the complement of the fixed point set, everywhere ’semi-locally’ conjugate to a translation. This generalizes the Brouwer plane translation theorem.

In this paper we show how the quantitative forms of Kronecker’s theorem in Diophantine approximations can be applied to investigate view-obstruction problems. In particular we answer a question in [Yong-Gao Chen, On a conjecture in Diophantine approximation, III, J. Number Theory 39 (1991), 91-103].

In an earlier paper, we began a study of linear random operators which have a certain probability of behaving as continuous operators. In this paper we study the pointwise limit in probability of a sequence of such operators, extending the Banach-Steinhaus theorem in a stochastical sense.