Recently Howe [4] has considered the oscillation theory for the two-parameter eigenvalue problem

1a

1b

subjected to the boundary conditions

2a

2b

where for i = 1, 2, — ∞<ai<bi<∞, and qi are real-valued, continuous functions in [ai, bi], pi is positive in [ai, biz], and pi(ai)=pi(bi). Furthermore, it is also assumed that (A1B2—A2B1)≠0 for all values of x1 and x2 in their respective intervals.

We show that every compactum has cohomological dimension 1 with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension 1 with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\text{CW}$-complex $M$ with finitely generated homotopy groups over all of $X$.

A Banach space is fully 2-rotund if (xn) converges whenever ‖xn + xm‖ converges as m, n → ∞ and locally uniformly rotund if xn → x whenever ‖xn‖ and ‖(xn + x)/2‖ → ‖x‖.

We show that I2 with the equivalent norm

is fully 2-rotund but not locally uniformly rotund, thus answering in the negative a question first raised by Fan and Glicksberg in 1958.

It is a commonplace that F is continuous on the cartesian square of the range of f if f is continuous and satisfies

1

say, for all real x, y (cf. e.g. [2]). A.D. Wallace has kindly called my attention to the fact, that this is trivial only if f is (constant or) strictly monotonic and asked for a simple proof of the strict monotonicity of f. The following could serve as such: if on an interval f is continuous, nonconstant and satisfies (1), then f is strictly monotonic there.

The Tarry-Escott problem in Diophantine analysis is the following: consider the system of equations

1

for the unknowns a1, …, ak, b1, …, bk, what is the smallest integer K = K(n) in the set of all k! s for which the system (1) possesses a non - trivial solution in integers ? By an integer we mean a rational integer and a solution is called trivial if the sets { a1, …, ak} and {b1, …, bk} are permutations of each other. By the Tarry-Escott problem one may also mean the problem of finding an actual solution of (1) or finding all such solutions but we shall be concerned with the bound K(n) only. However, the method given here is constructive, that is to say, with each estimate K for K it leads to finding actual non - trivial solutions of (1) with k = K1.

This article is a short introduction to super-Brownian motion. Some of its properties are discussed but our main objective is to describe a number of limit theorems which show super-Brownian motion is a universal limit for rescaled spatial stochastic systems at criticality above a critical dimenson. These systems include the voter model, the contact process and critical oriented percolation.

We study the behavior of the eigenvalues of a sublaplacian ${{\Delta }_{b}}$ on a compact strictly pseudoconvex $\text{CR}$ manifold $M$, as functions on the set ${{\mathcal{P}}_{+}}$ of positively oriented contact forms on $M$ by endowing ${{\mathcal{P}}_{+}}$ with a natural metric topology.

One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to $\mathbb{C}{{\mathbb{P}}^{2}}$ and a projection of the image curve froman appropriate point $p\in \mathbb{C}{{\mathbb{P}}^{2}}$ to the pencil of lines through $p$ . We introduce a natural stratification of Hurwitz spaces according to the minimal degree of a plane curve such that a given meromorphic function can be represented in the above way and calculate the dimensions of these strata. We observe that they are closely related to a family of Severi varieties studied earlier by J. Harris, Z. Ran, and I. Tyomkin.

In an earlier paper [2], we raised the question of determining the minimum span of the kth derivative of a polynomial with real zeros having a given span. More precisely let πn, s denote the class of polynomials , with x1 ≤ x2 ≤ … ≤ xn, and the span σ(P) ≡xn - x1 = 2s (fixed).

We show that a semiperfect right FPF-ring is right self-injective if and only if J(R) — Z(RR), extending a well-known result due to Carl Faith on semiperfect right FPF-rings with nil Jacobson radical.

In this paper, we investigate the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of $\Gamma \left( {R}/{I}\; \right)$ and ${{\Gamma }_{1}}\left( R \right)$, and the domination numbers of $\Gamma \left( R \right)$ and $\Gamma \left( R\left[ x,\,\alpha ,\,\delta \right] \right)$, where $R\left[ x,\,\alpha ,\,\delta \right]$ is the Ore extension of $R$, are studied.

Let ${{T}_{n}}$ denote the $n$-th Chebyshev polynomial of the first kind, and let ${{U}_{n}}$ denote the $n$-th Chebyshev polynomial of the second kind. We give an explicit formula for the resultant res$({{T}_{m}},\,{{T}_{n}})$. Similarly, we give a formula for res$({{U}_{m}},\,{{U}_{n}})$.

An integral domain $R$ is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element $a$ in $R$, the ascending chain of non-associate irreducible divisors in $R$ of ${{a}^{n}}$ stabilizes on a finite set as $n$ ranges over the positive integers, while $R$ is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension $S$ of $R$ is a root extension or radical extension if for each $s$ in $S$, there exists a natural number $n\left( s \right)$ with ${{s}^{n\left( s \right)}}$ in $R$. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $\left( R,\,S \right)$ is governed by the relative sizes of the unit groups $\text{U}\left( R \right)$ and $\text{U}\left( S \right)$ and whether $S$ is a root extension of $R$. The following results are deduced from these considerations: An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let $R$ be a Noetherian domain with integral closure $S$. Suppose the conductor of $S$ into $R$ is non-zero. Then $R$ is IDPF if and only if $S$ is a root extension of $R$ and $\text{U}\left( S \right)/\text{U}\left( R \right)$ is finite.

In this paper we obtain quantitative results on the occurrence of consecutive large gaps between $B$-free numbers, and consecutive large gaps between nonzero Fourier coefficients of a class of newforms without complex multiplication.

By a semi-lattice we mean a system <L,.> where L is a set and. is a binary operation in L that is idempotent, commutative and associative. In a recent article [2] D.H. Potts considers the problem of reducing the number of axioms for a semi-lattice. His result was that the following two axioms viz. (1) xx=x, (2) (uv)((wx)(yz)) = ((uv)(xw))(zy) are sufficient to give a semi-lattice. But the second identity contains six elements instead of the original three. In the following we give a set of two simple identities with just three elements for a semi-lattice. This improves the above mentioned result.

In the course of work on factor groups of the modular group, A. O. L. Atkin (private communication) obtained the permutations

which satisfy the relations

(1)

for T = T1, T2. It was evident that neither pair S, T generates either A12 or LF(2, 11), so he suggested that they probably generate the Mathieu group M2.

All spaces in this note are regular, Hausdorff topological spaces.

At the topology conference in Pullman, Washington, in March 1970, E. Michael posed the following question: if X is Lindelôf Y is separable and metrizable and I × Y is paracompact, must X × Y be Lindelöf?

We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.

In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares $\sum{\mathbb{R}{{\left[ \underline{X} \right]}^{2}}}$ in the polynomial ring $\mathbb{R}\left[ \underline{X} \right]\,:=\,\mathbb{R}\left[ {{X}_{1}},\,.\,.\,.\,,\,{{X}_{n}} \right]$ in the topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$, the cone consisting of all polynomials that are non-negative on the hypercube ${{\left[ -1,\,1 \right]}^{n}}$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $~{{\ell }_{1}}$-seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theorem from 2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\,\ge \,1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}{{\left[ \underline{X} \right]}^{2d}}$ in the $\mathbb{R}\left[ \underline{X} \right]$ topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$.

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n\,:=\,\text{grad}{{\text{e}}_{R}}\,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))\,\cong \,\text{Ext}_{R}^{n}(H_{\mathfrak{a}}^{n}(R),\,R)$. We also prove that, for a nonnegative integer $n$ such that $H_{\mathfrak{a}}^{i}(R)\,=\,0$ for every $i\,\ne \,n$, if $\text{Ext}_{R}^{i}({{R}_{z}},\,R)\,=\,0$ for all $i\,>\,0$ and $z\,\in \,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$ is a homomorphic image of $R$, where ${{R}_{z}}$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{{{z}^{j}}|j\,\ge \text{0}\}$ of $R$. Moreover, if $\text{Ho}{{\text{m}}_{R}}({{R}_{z}},R)=0$ for all $z\,\in \,\mathfrak{a}$, then ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is an isomorphism, where ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is the canonical ring homomorphism $R\,\to \,\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$.