It is well known that Thom–Boardman symbols are realized by nonincreasing sequences of nonnegative integers. A natural question is whether the converse is also true. In this paper we answer this question affirmatively, that is, for any nonincreasing sequence of nonnegative integers, there is at least one map-germ with the prescribed sequence as its Thom–Boardman symbol.

Consider the simultaneous real differential equations , where the dot indicates differentiation with respect to time, and a, b, c, d are functions about which the only thing that we know is that they are uniformly bounded, .

Characterizations of collectively precompact and collectively semi-precompact sets of operators on topological vector spaces are obtained. These lead to the characterization of totally bounded sets of semi-precompact operators on locally convex spaces.

1980 Mathematics subject classification (Amer. Math. Soc): primary 47 B 05, 47 D 15; secondary 46 A 05, 46 A 15.

Let σan be an infinite series, with sequence of partial sums {un}. Let {pn} be a sequence of constants, real or complex, and write Pn = po+p1+ … +pn The sequence-to-sequence transformation defines the sequence {tn} of Nörlund means of the sequence {u}, generated by the sequence {pn}. The series σan is said to be surnmable (N, pn), to sum s, if limn→∞ tn = s. It is said to be absolutely sum.mable (N, pn), or summable |N, pn|, if {tn} ∈BV.

Let be regular and unvivalent in the unit disk D and map D onto a region R. A point w0∈ R is called a star-center point of f(z), or of R, if tf(z)+(1−t)w0∈ R for z∈D and 0≦ t≦ 1. In this paper we consider only functions of the form (1) where w = 0 is a star-center point, i.e., those functions that are starlike with respect to the origin.

In this paper the concept of length as defined for groups by Lausch–Nöbauer in their book Algebra of Polynomials (North Holland, Amsterdam, 1973) is generalized in several ways. It turns out that the main results of Lausch-Nöbauer concerning it remain valid for this generalization.

Let $A=\bigoplus _{i\in \mathbb{Z}}A_{i}$ be a finite-dimensional graded symmetric cellular algebra with a homogeneous symmetrizing trace of degree $d$. We prove that if $d\neq 0$ then $A_{-d}$ contains the Higman ideal $H(A)$ and $\dim H(A)\leq \dim A_{0}$, and provide a semisimplicity criterion for $A$ in terms of the centralizer of $A_{0}$.

A map f: A→B in category is called monic if fg = fh implies that g = h for all maps g, h: C → A; it is called epic if gf = hf implies that g = h for all maps g, h: B → C. An object A ∈ is called an S-object if every monic map f: A → A is also epic; it is called a Q-object if every epic map f: A → A is also monic. If A is both an S-object and a Q-object then A is called an SQ-object. In the category of sets the SQ-sets are the finite sets. In the category of vector spaces over a field F the SQ-spaces are precisely the finite dimensional spaces. In the light of these simple examples, it seems reasonable to view the SQ-objects of a category as being of ‘finite type’. We shall be chiefly concerned with investigating the SQ-objects in certain subcategories of the category of locally compact abelian groups.

This paper introduces a tensor that contains the Riemannian curvature tensor and the conformal curvature tensor as special examples in the Riemannian space (Mn, g), and by using this tensor we define C-semi-symmetric space. In this paper, we have the following main result: if there is a non-trivial concircular transformation between two C-semi-symmetric spaces, then both spaces are of quasi-constant curvature.

A number of completions have been applied to p.o.-groups — the Dede kind-Macneille completion of archimedean l.o. groups; the lateral completion of l.o. groups (Conrad [2]); and the orthocompletion of l.o. groups (Bernau [1]). Fuchs in [3] has considered a completion of p.o. groups having a non-trivial open interval topology — the only l.o. groups of this form being fully ordered. He applies an ordering, which arises from the original partial order, to the group of round Cauchy filters over this topology; Kowaisky in [6] has shown that group, imbued with a suitable topology, is in fact the topological completion of the original group under its open interval topology. In this paper a slightly different ordering, also arising from the original order, is proposed for the group of round Cauchy filters; Fuchs' ordering can be obtained from this one as the associated order.

Let be a higher rank Exel–Laca algebra generated by an alphabet . If contains d commuting isometries corresponding to rank d and the transition matrices do not have finite rows, then is trivial and is isomorphic to K0 of the abelian subalgebra of generated by the source projections of .

We present a sufficient and necessary condition for a function module space $X$ to have the approximate hyperplane series property (AHSP). As a consequence, we have that the space ${\mathcal{C}}_{0}(L,E)$ of bounded and continuous $E$-valued mappings defined on the locally compact Hausdorff space $L$ has AHSP if and only if $E$ has AHSP.

A finite group is called repetition-free if its conjugacy classes have distinct sizes. It is known that a supersolvable repetition-free group is necessarily isomorphie to Sym(3). the symmetric group on three symbols. Thus the question arises as to whether Sym (3) is the only repetition-free group. In this paper it is proved that if mk denotes the minimum of the orders of the centralizers of elements of a repetition-free group G and mk ≦ 4 then G is isomorphie to Sym (3).

First, we introduce the following condition (C1) for semigroups S. Let n, k be fixed positive integers such that n > k. (C1) For any given sequence of n elements of S, there existK elements, where 1 ≦ i1 < i2…, < ik ≦ n, such that where

In this article, we give new proofs of two of Ramanujan’s $1/\unicode[STIX]{x1D70B}$ formulae

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2\sqrt{2}}{99^{2}}\mathop{\sum }_{m=0}^{\infty }(26390m+1103)\frac{(4m)!}{396^{4m}(m!)^{4}}\end{eqnarray}$$

$$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70B}}=\frac{2}{84^{2}}\mathop{\sum }_{m=0}^{\infty }(21460m+1123)\frac{(-1)^{m}(4m)!}{(84\sqrt{2})^{4m}(m!)^{4}}\end{eqnarray}$$

$1/\unicode[STIX]{x1D70B}$

Throughout this note Rn denotes the n-dimensional Euclidean space. Addition and multiplication in Rn are defined component-wise. If k ≦ n is a positive integer and x ∈ Rn, we write xk for the k-th component of x. The set {x ∈ Rn: xk ≠ 0 for each k ≦ n} is designated by R#.

Let K be an algebraically closed field of characteristic zero, and S a nonempty subset of K such that S Q = Ø and card S < card K, where Q is the field of rational numbers. By Zorn's Lemma, there exist subfields F of K which are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.

We introduce a new homology theory for infinite graphs in order to generalize some results of Willis and Woodward on translation invariant functionals. We also extend some theorems of Gerl and Gromov.

We classify the irreducible unitary representations of closed simple groups of automorphisms of trees acting $2$-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, we confirm Claudio Nebbia’s CCR conjecture on trees for $(d_0,d_1)$-semi-regular trees such that $d_0,d_1\in \Theta $, where $\Theta $ is an asymptotically dense set of positive integers.

This paper is concerned with the bifurcation of limit cycles from a quadratic reversible system under polynomial perturbations. It is proved that the cyclicity of the period annulus is two, and also a linear estimate of the number of zeros of the Abelian integral for the system under polynomial perturbations of arbitrary degree nis given.