By considering the specialisation ${{s}_{\lambda }}(1,\,q,\,{{q}^{2}},\ldots ,\,{{q}^{n-1}})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $\lambda $ in terms of the contents and hook lengths of the boxes in the Young diagram. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $\lambda $-tableaux.

Let S be a subnormal operator and let be the weak-star closed algebra generated by S and 1. An example of an irreducible cyclic subnormal operator S is found such that there is a T in with S and T quasisimilar but not unitarily equivalent. However, if S is the unilateral shift, T ∈ and S and T are quasisimilar, then S ≅ T.

In this paper we have shown that the existence of some balanced incomplete block (b.i.b. for abbreviation) designs implies the existence of some others in some cases. We have here established the following theorems.

Theorem 1. If there exists a symmetric b.i.b. design with parametersv, b, r, k, λ, (where v-b, r=k), then there exists a b.i.b. design with parameters,

Let ${{G}_{1}}$ and ${{G}_{2}}$ be $p$-adic groups. We describe a decomposition of Ext-groups in the category of smooth representations of ${{G}_{1}}\times {{G}_{2}}$ in terms of Ext-groups for ${{G}_{1}}$ and ${{G}_{2}}$. We comment on $\text{Ext}_{G}^{1}\left( \pi ,\pi \right)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable $E\text{x}{{\text{t}}^{1}}$, of a Jacquet module of certain representations of $p$-adic $\text{G}{{\text{L}}_{2n}}$.

Suppose that $D\subset \mathbb{C}$ is a simply connected subdomain containing the origin and $f(z_{1})$ is a normalized convex (resp., starlike) function on $D$. Let

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$

$p_{j}\geqslant 1$

$N=1+n_{1}+\cdots +n_{k}$

$w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$

$\unicode[STIX]{x1D706}_{D}$

$D$

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$

$\unicode[STIX]{x1D6FA}_{N}(D)$

$D$

$\mathbb{C}^{2}$

$\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$

$\unicode[STIX]{x1D6FC}$

$f$

$\unicode[STIX]{x1D6FC}$

Let R and S be rings, G and H abelian groups, and RG and SH the goup rings of G and H over R and S respectively. In this note we consider what relations must hold between G and H or between R and S if the group rings RG and SH are isomorphic. For example, it is shown that if R and S are integral domains of characteristic zero, G and H torsion abelian groups such that if G has an element of order p then p is not invertible in R, and RG and SH are isomorphic, then the rings R and 5 are isomorphic and the groups G and H are isomorphic.

The number of elements in the sum A + B of two complexes A and B of a group G which have multiple representations a + b = a '+ b' has been investigated by Scherk and Kemperman [1]. Kemperman [2] appealed to transfinite techniques (to order G) to prove:

We answer the following problem posed by Herrlich in the affirmative: “Can the Freudenthal compactification be regarded as a reflection in a sensible way?” This is accomplished by exploiting the one-to-one correspondence between proximities compatible with a given Tihonov space and compactifications of that space. We give topological characterizations of proximally continuous functions for the proximities associated with the Freudenthal and Fan-Gottesman compactifications.

The purpose of the present note is to point out that the results of D. S. Goel, A. S. B. Holland, C. Nasim and B. N. Sahney [1] on best simultaneous approximation are easy consequences of simple facts about convex functions. Given a normed linear space X, a convex subset K of X, and points x1, x2 in X, [1] discusses existence and uniqueness of K* ∈ K such that

Let $\phi$ be a Drinfeld module of generic characteristic, and let $X$ be a sufficiently generic affine subvariety of $\mathbb{G}_{a}^{g}$. We show that the intersection of $X$ with a finite rank $\phi$-submodule of $\mathbb{G}_{a}^{g}$ is finite.

Given an integer $n\,\ge \,3$, a metrizable compact topological $n$-manifold $X$ with boundary, and a finite positive Borel measure $\mu $ on $X$, we prove that for the typical homeomorphism $f\,:\,X\,\to \,X$, it is true that for $\mu $-almost every point $x$ in $X$ the limit set $\omega (f,\,x)$ is a Cantor set of Hausdorff dimension zero, each point of $\omega (f,\,x)$ has a dense orbit in $\omega (f,\,x)$, $f$ is non-sensitive at each point of $\omega (f,\,x)$, and the function $a\,\to \,\omega (f,\,a)$ is continuous at $x$.

Generalizing a result of Soulé we prove that for an elliptic curve E defined over an imaginary quadratic field K with complex multiplication having good ordinary reduction at the prime number p > 3 which is regular for E and the extension F of K contained in K(Ep) the dimensions of the étale K-groups are equal to the numbers predicted by Bloch and Beilinson, i.e.,

In [15], O. Taussky-Todd posed the problem of title, namely to find X, Y, Z when A, B are given. Clearly if X, Y, Z exist then A, B are either both invertible or both noninvertible.

In section 1, the problem is reviewed in case A, B are both invertible. The problem is seen to be fundamentally one of group theory rather than matrix theory. Application of results of Shoda, Thompson, Ree to the general group-theoretical results allows specialization to certain matrix groups.

In Section 2, examples and counterexamples are given in case A, B are noninvertible. A general necessary condition for solvability (involving ranks) is obtained. This condition may or may not be sufficient. For dim A=2, 3 the problem is settled: there is always a solution in the noninvertible case.

All rings are associative but do not necessarily have identities. Definitions and basic results about radical classes can be found in [2]. A radical class is strong [3] if for every ring A, (A) contains all left and right -ideals of A.

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. This paper proves the following results. Except for parallelograms and centrally symmetric hexagons, there are no other convex domains that can form two-, three- or four-fold lattice tilings in the Euclidean plane. However, there are both octagons and decagons that can form five-fold lattice tilings. Whenever $n\geqslant 3$, there are non-parallelohedral polytopes that can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

A subset of the line is called non-averaging if, whenever two points belong to the set, their average does not. This paper provides an example of a closed set which is small in the sense that it is non-averaging and has porosity 1 at each of its points and yet large in the sense that its Hausdorff dimension is 1.

Vector invariants of finite groups (see the introduction for an explanation of the terminology) have often been used to illustrate the difficulties of invariant theory in the modular case: see, e.g., [1], [2], [4], [7], [11] and [12]. It is therefore all the more surprising that the unpleasant properties of these invariants may be derived from two unexpected, and remarkable, nice properties: namely for vector permutation invariants of the cyclic group $Z/p$ of prime order in characteristic $p$ the image of the transfer homomorphism $\text{T}{{\text{r}}^{Z/p}}\,:\,F[V]\,\to \,F{{[V]}^{Z/p}}$ is a prime ideal, and the quotient algebra $F{{[V]}^{Z/p}}/\,\text{IM(T}{{\text{r}}^{Z/p}})$ is a polynomial algebra on the top Chern classes of the action.

In a recent series of papers [3,4,5], H. Zassenhaus considered the structure of those linear transformations T on real 4-space, R4, into itself that preserve the quadratic form . That is,

1.1

Define a function ϕ on R4 to the space M 2 of 2-square matrices over the complex numbers as follows:

1.2

Let G2 be the vector space of matrices generated by all real linear combinations of

1.3

An atomic integral domain with conjugation has unique (in the sense of Theorem 6 below) factorization of atomic factors if it is an LCM domain. If the LCM hypothesis is dropped not even the number of atomic factors in a complete factorization of an element need be unique.

In this paper, we construct, for any open cover of a metric space, a partition of unity consisting of a family of uniformly continuous functions.