Let the random variables X, Y, Z have respectively the extreme value densities as

1

2

3

where θ>0, λ>0 and μ are real numbers.

In another paper [l] measures determined from base functions, that have finite right and left limits everywhere and are of generalized bounded variation in the restricted sense, are studied and used to define non absolutely convergent integrals of Denjoy type. In this paper base functions of bounded variation and the corresponding measures are studied as a background for that paper. The results supplement parts of [2].

This note proves (in the theorem below) a conjecture made by the author last year through the pages of the Departmental Problem Book. This arose in connection with some other investigations of arithmetic functions.

The main purpose of this paper is to prove, that the topology of any (non-complete) algebra norm on a JB* -algebra is stronger than the topology of the usual norm. The proof of this theorem consists of an adaptation of the recent Rodriguez proof [8] that every homomorphism from a complex normed (associative) Q-algebra onto a B*-algebra is continuous.

Repeated application of the unitary reflections

to the point (1,0) and line x + iy = 1 yields 24 points and 24 lines. These are the vertices and edges of the regular complex polygon 4 {3} 4 whose group has the abstract definition R4 = I, RSR = SRS [l]. The purpose of this note is to introduce the notion of regular quaternion polygon and give an example analogous to the preceding one.

We embed the moduli space $Q$ of 5 points on the projective line ${{S}_{5}}$-equivariantly into $\mathbb{P}\left( V \right)$, where $V$ is the 6-dimensional irreducible module of the symmetric group ${{S}_{5}}$. This module splits with respect to the icosahedral group ${{A}_{5}}$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of ${{A}_{5}}$ on $Q$ to those on the regular icosahedron.

Il me fait un très grand plaisir de vous présenter aujourd'hui la conférence Jeffrey-Williams nommée ainsi en l'honeur des distingués fondateurs de la Société Mathématique du Canada.

Il y a beaucoup d'évidences indirectes d'un vif échange d'idées entre Herman Minkowski et David Hilbert, pendant la période de temps entre la soutenance de thèse de Hilbert (décembre 1884) et la mort prématurée de Minkowski (le 12 janvier 1909).

In this note, we collect various properties of Seifert homology spheres from the viewpoint of Dehn surgery along a Seifert fiber. We expect that many of these are known to various experts, but include them in one place, which we hope will be useful in the study of concordance and homology cobordism.

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$. Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$. We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

We characterize surjective linear maps on $B\text{(X)}$ that are spectrally bounded and spectrally bounded below.

This note is concerned with the number C(n) of ordered non-trivial factorizations of an integer n in the special case where n is square free. If F (m) denotes C(p1…Pm) where pi. are distinct primes, it is shown that

and that

A Newton polygon is introduced for a ramified point of a Galois covering of curves over a finite field. It is shown to be determined by the sequence of higher ramification groups of the point. It gives a blowing up of the wildly ramified part which separates the branches of the curve. There is also a connection with local reciprocity.

All rings are associative. A ring T is said to be radical over a subring R if for every t ∈ T, there exists a natural number n(t) such that tn(t) ∈ R.

In [1] Faith showed that if T is radical over R and T is primitive, then R is primitive. We might then ask if the same is true if prime is substituted for primitive.

By an ortholattice we mean a lattice with 0 and 1 and a complementation operation which is an involutorial antiautomorphism. The free modular ortholattice on two generators has 96 elements—cf. J. Kotas [8].

Let ${{A}_{n}}\,=\,\{{{a}_{0}}+{{a}_{1}}z+\,.\,.\,.\,+\,{{a}_{n-1}}{{z}^{n-1}}\,:\,{{a}_{j}}\,\in \,\{0,1\}\}$ , whose elements are called zero-one polynomials and correspond naturally to the ${{2}^{n}}$ subsets of $\left[ n \right]\,:=\,\{0,\,1,\,.\,.\,.\,,\,n-1\}$. We also let ${{A}_{n,m\,}}\,=\,\{\alpha \left( z \right)\,\in \,{{A}_{n}}\,:\,\alpha \left( 1 \right)\,=\,m\}$ , whose elements correspond to the $\left( _{m}^{n} \right)$ subsets of $\left[ n \right]$ of size $m$, and let ${{B}_{n}}\,=\,{{A}_{n+1}}\,\backslash \,{{A}_{n}}$ , whose elements are the zero-one polynomials of degree exactly $n$.

Many researchers have studied norms of polynomials with restricted coefficients. Using $\|\alpha {{\|}_{p}}$ to denote the usual ${{L}_{p}}$ norm of $\alpha$ on the unit circle, one easily sees that $\alpha \left( z \right)\,=\,{{a}_{0}}+{{a}_{1}}z+.\,.\,.+{{a}_{N}}{{z}^{N}}\,\in \,\mathbb{R}\left[ z \right]$ satisfies $\|\alpha \|_{2}^{2}\,=\,{{c}_{0}}$ and $\|\alpha \|_{4}^{4}\,=\,c_{0}^{2}\,+\,2\left( c_{1}^{2}\,+\,.\,.\,.\,+\,c_{N}^{2} \right)$ , where ${{c}_{k}}\,:=\,\sum _{j=0}^{N-k}\,{{a}_{j}}{{a}_{j+k}}\,\text{for}\,\text{0}\le \,\text{k}\,\le N$.

If $\alpha \left( z \right)\,\in \,{{A}_{n,m}}$ , say $\alpha \left( z \right)\,=\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{1}}}}\,+\,.\,.\,.\,+\,{{z}^{{{\text{ }\!\!\beta\!\!\text{ }}_{m}}}}$ where ${{\beta }_{1}}\,<\,.\,.\,.\,<\,{{\beta }_{m}}$ , then ${{c}_{k}}$ is the number of times $k$ appears as a difference ${{\beta }_{i}}\,-\,{{\beta }_{j}}$ . The condition that $\alpha \,\in \,{{A}_{n,m}}$ satisfies ${{c}_{k}}\,\in \,\{0,1\}$ for $1\,\le \,k\,\le \,n\,-\,1$ is thus equivalent to the condition that $\{{{\beta }_{1}},\,.\,.\,.\,,\,{{\beta }_{m}}\}$ is a Sidon set (meaning all differences of pairs of elements are distinct).

In this paper, we find the average of $\left\| \alpha \right\|_{4}^{4}$ over $\alpha \,\in \,{{A}_{n}}$, $\alpha \,\in \,{{B}_{n}}$ and $\alpha \,\in \,{{A}_{n,m}}$. We further show that our expression for the average of $\left\| \alpha \right\|_{4}^{4}$ over ${{A}_{n,m}}$ yields a new proof of the known result: if $m\,=\,o\left( {{n}^{1/4}} \right)$ and $B\left( n,\,m \right)$ denotes the number of Sidon sets of size $m$ in $\left[ n \right]$, then almost all subsets of $\left[ n \right]$ of size $m$ are Sidon, in the sense that ${{\lim }_{n\to \infty }}\,B\left( n,\,m \right)/\left( _{m}^{n} \right)\,=\,1$.

We give a simple proof of the value of the resultant of two Chebyshev polynomials (of the first or the second kind), values lately obtained by D. P. Jacobs, M. O. Rayes, and V. Trevisan.

It is a well-known trivial fact that for a given integer n there exists at most n — 2 pairwise orthogonal idempotent latin squares. In the following note we prove that for n a prime power there always exists n—2 such squares.

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus, and 3-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, 4-genus, and 3-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of 4-genus one.

In this paper, we apply the saddle-point method in conjunction with the theory of the Nörlund–Rice integrals to derive precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have

$${{\lambda }_{F}}(n)\,=\,\frac{{{d}_{F}}}{2}n\,\log \,n\,+\,{{c}_{F}}n\,+\,O(\sqrt{n}\,\log \,n),$$

with

$${{c}_{F}}\,=\,\frac{{{d}_{F}}}{2}(\gamma \,-\,1)\,+\,\frac{1}{2}\log (\lambda \text{Q}_{F}^{2}),\,\,\lambda \,=\,\prod\limits_{j=1}^{r}{\lambda _{j}^{2{{\lambda }_{j}}}},$$

where $\gamma $ is the Euler's constant and the notation is as below.

In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction:

Given ∊ > 0, there exists δ > 0 such that

(A)