We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.

Nous voulons montrer que le théorème de Bernstein est une conséquence simple de la formule de Taylor et d' un théoréme de Helly. Rappelons une définition: une fonction réelle f définie sur (0, ∞) est dite complètement monotone si f est indéfiniment derivable, si f ainsi que toutes ses dérivées paires sont positive set si toutes les dérivées impaires sont négatives.

We construct vector-valued modular forms of weight 2 associated to Jacobi-like forms with respect to a symmetric tensor representation of $\Gamma$ by using the method of Kuga and Shimura as well as the correspondence between Jacobi-like forms and sequences of modular forms. As an application, we obtain vector-valued modular forms determined by theta functions and by pseudodifferential operators.

B. Grünbaum, J. N. Lillington and lately R. J. Gardner, S. Kwapien and D. P. Laurie have considered inequalities defined by three concurrent straight lines in the interior of a planar compact convex set. In this note we prove two elegant conjectures by R. J. Gardner, S. Kwapien and D. P. Laurie.

A homeomorphism f of a space M is pointwise periodic if for each m ∈ M there exists an integer k such that fk(m) =m, where fk is the kth iterate of f. Montgomery proves [5] that if M is a connected topological manifold, then f is periodic; i.e., there exists an integer n such that fn = id. Noting this, Weaver [7] proves that if M is an orientable 2-manifold of class C1, U⊆M open and C ⊆ U a compact connected set and if g : U → M of class C1 is such that (i) g(C) = C and (ii) whenever the derivative of g at points x∈C has rank 2 it is orientation preserving, then f = g|c : C → C periodic implies that all but a finite number of points of C have as least period the period of f.

The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof using the notion of essential extension was given by Eckmann-Schopf [ES]. Both proofs require the p reliminary construction of some injective overmodule. In [F] I showed how the latter proof could be freed from this requirement by exhibiting a set F in which every essential extension could be embedded. Subsequently J. M. Maranda pointed out that F has minimal cardinality. It follows that F is equipotent with the injective hull. Below Icon struct the injective hull by equipping Fit self with a module strucure.

We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

We show that if the general real method ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for ${{\left( \cdot \,,\,\cdot \right)}_{\Gamma }}$.

Fix an irreducible (finite) root system $R$ and a choice of positive roots. For any algebraically closed field $k$ consider the almost simple, simply connected algebraic group ${{G}_{k}}$ over $k$ with root system $k$. One associates with any dominant weight $\lambda $ for $R$ two ${{G}_{k}}$-modules with highest weight $\lambda $, the Weyl module $V{{(\lambda )}_{k}}$ and its simple quotient $V{{(\lambda )}_{k}}$. Let $\lambda $ and $\mu $ be dominant weights with $\mu <\lambda $ such that $\mu $ is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists $k$ such that $L{{(\mu )}_{k}}$ is a composition factor of $V{{(\lambda )}_{k}}$, and they exhibit an example in type ${{E}_{8}}$ where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs $(\lambda ,\mu )$, and another that relies only on the classiûcation of root systems.

We write ω for the set of natural numbers (including zero) and for the set of all finite sequences of natural numbers. If n ∊ ω we write . If x is a function which takes its values in cu and whose domain of definition contains then we write for the element (x(0), …, x(n)) of .

Let A be an ideal of the commutative ring R with identity. There is a canonical homomorphism ϕA from the polynomial ring R[X] onto (R/A)[X], obtained by reducing all coefficients modulo A. If fR[X], then we say that f is reducible (irreducible) modulo A if ϕA(f) is reducible (irreducible) in (R/A)[X].

The paper proves that the group of infinite bounded Nielsen transformations is generated by elementary simultaneous Nielsen transformations modulo the subgroup of those transformations which are equivalent to the identical transformation while acting in a free abelian group. This can be formulated somewhat differently: the group of bounded automorphisms of a free abelian group of countably infinite rank is generated by the elementary simultaneous automorphisms. This proves D. Solitar's conjecture for the abelian case.

A set of sentences T is called independent if for every . It is countably independent if every countable subset is independent. In flnitary first order logic, Lωω, the two notions coincide because of compactness. This is not the case for infinitary logic.

Upper bounds of second order Dehn functions of asynchronously automatic groups are obtained.

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1 (C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

The cover product of disjoint graphs $G$ and $H$ with fixed vertex covers $C\left( G \right)$ and $C\left( H \right)$, is the graph $G\circledast H$ with vertex set $V\left( G \right)\cup V\left( H \right)$ and edge set

$$E\left( G \right)\,\cup \,E\left( H \right)\,\cup \,\left\{ \left\{ i,\,j \right\}\,:\,i\,\in \,C\left( G \right),\,j\,\in \,C\left( H \right) \right\}.$$

We describe the graded Betti numbers of $G\circledast H$ in terms of those of $G$ and $H$. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph $G$ such that $\text{reg}\,R/I\left( G \right)\,=\,{{\mu }_{s}}\left( G \right)\,+\,k$, where, $I\left( G \right)$ denotes the edge ideal of $G$, $\text{reg}\,\text{R/I}\left( G \right)$ is the Castelnuovo–Mumford regularity of $\text{R/I}\left( G \right)$ and ${{\mu }_{s}}\left( G \right)$ is the induced or strong matching number of $G$; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The $h$-vector of $R/I\left( G\circledast H \right)$ is described in terms of the $h$-vectors of $\text{R/I}\left( G \right)$ and $R/I\left( H \right)$. Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

It is shown that the operator of integration Fy(x) = ∫x0y(t) dt defined on the space C(—∞, ∞) of all continuous real valued functions on (—∞, ∞) is a contraction relative to a certain family of seminorms generating the topology of uniform convergence on compacta. However, as a contrast to this it is proved that F is not contractive with respect to any metric on C(—∞, ∞) inducing the above topology on C(—∞, ∞).

Leeming [3] has defined a sequence of polynomials {Q4n(x)} and a sequence of integers {Q4n} by means of

1.1

and

1.2

Thus

1.3

and equivalently

1.4

In this paper, we consider the behavior of the commutators of convolution operators on the Triebel–Lizorkin spaces ${{\dot{F}}_{p}}^{s,q}$.