The normalized eigenvalues ${{\Lambda }_{i}}\left( M,\,g \right)$ of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in ${{\mathbb{S}}^{3}}$ or ${{\mathbb{S}}^{4}}$. In this paper a family of extremal metrics induced by minimal immersions in ${{\mathbb{S}}^{5}}$ is investigated.

Van der Waerden [1, 4, 5] proved that if the nonnegative integers are partitioned into a finite number of sets, then at least one set in the partition contains arbitrarily long finite arithmetic progressions. This is equivalent to the result that a strictly increasing sequence of integers with bounded gaps contains arbitrarily long finite arithmetic progressions. Szemerèdi [3] proved the much deeper result that a sequence of integers of positive density contains arbitrarily long finite arithmetic progressions. The purpose of this note is a quantitative comparison of van der Waerden's theorem and sequences with bounded gaps.

We examine the ranks of operators in semi-finite ${{C}^{*}}$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple ${{C}^{*}}$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray–von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first author, this shows that slow dimension growth coincides with $Z$-stability for approximately subhomogeneous algebras.

In his “lost” notebook, Ramanujan stated two results, which are equivalent to the identities

$$\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{n}} \right)}^{5}}}{\left( 1-{{q}^{5n}} \right)}=1-5\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{d} \right)d \right){{q}^{n}}$$

and

$$q\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{5n}} \right)}^{5}}}{\left( 1-{{q}^{n}} \right)}=\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{n/d} \right)d \right){{q}^{n}}.$$

We give several more identities of this type.

In this paper we study the best constant of the Sobolev trace embedding ${{H}^{1}}(\Omega )\,\to \,{{L}^{2}}(\partial \Omega ),$ where $\Omega$ is a bounded smooth domain in ${{\mathbb{R}}^{N}}.$ We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.

Let $b$ be an integer with $b\,>\,1$. In this note, we prove that the number of non-zero digits in the base $b$ representation of $n!$ grows at least as fast as a constant, depending on $b$, times $\log \,n$.

In this paper, under certain conditions on the orthogonal distribution T0, we give a characterization of real hypersurfaces of type A in a complex space form Mn(c), c ≠ 0.

We give an estimate for the error of Lp approximation by reciprocals of polynomials. These estimates are the analogues of the Jackson and Ditzian - Totik estimates for polynomial approximation.

We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz–Krieger uniqueness theorem for graph algebras by means of branching systems.

The local Hopf Bifurcation theorem is extended to implicit differential equations in Rn, of the form ẋ = f(x,ẋ, α), which are not solvable for the variable ẋ. The proof uses the S1 -degree of convex-valued mappings. An example of an implicit differential equation in R3 to which the presented theorem applies is provided.

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type.

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

We give an explicit recipe for the determination of the images associated to the Galois action on $p$-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over $\mathbb{Q}$ without complex multiplication with conductor less than 200 and for each prime number $p$.

If R is an uncountable commutative ring, it is shown that there exists a proper subring of R having the same cardinality as R. It is also shown that if |R| = ω is an uncountable regular cardinal, and if R1 is a subring of R containing an identity of R and such that |R1| < ω, then there exists a proper R1-subalgebra S of R such that |S| = |R|.

Chern and Hamilton considered the integral of the Webster scalar curvature as a functional on the set of CR-structures on a compact 3-dimensional contact manifold. Critical points of this functional can be viewed as Riemannian metrics associated to the contact structure for which the characteristic vector field generates a 1-parameter group of isometries i.e. K-contact metrics. Tanno defined a higher dimensional generalization of the Webster scalar curvature, computed the critical point condition of the corresponding integral functional and found that it is not the K-contact condition. In this paper two other generalizations are given and the critical point conditions of the corresponding integral functionals are found. For the second of these, this is the K-contact condition, suggesting that it may be the proper generalization of the Webster scalar curvature.

In Bernoulli site percolation on Penrose tilings there are two natural definitions of the critical probability. This paper shows that they are equal on almost all Penrose tilings. It also shows that for almost all Penrose tilings the number of infinite clusters is almost surely 0 or 1. The results generalize to percolation on a large class of aperiodic tilings in arbitrary dimension, to percolation on ergodic subgraphs of ${{\mathbb{Z}}^{d}}$, and to other percolation processes, including Bernoulli bond percolation.

We show that, for a finite group G and a prime p, the following facts are equivalent: (i) the p-localization homomorphism l: G —> Gp induces p-localization on integral homology; (ii) the higher homotopy groups of the Bousfield-Kan Zp-completion of a K(G, 1) vanish; (iii) the group G is p-nilpotent.

A seven-parameter extension of Askey and Wilson's four parameter q-beta integral is written in a symmetric form as the sum of multiples of two very-well-poised balanced basic hypergeometric 10Φ9 series. Two special cases are considered in which the evaluation of the integral gives single terms by the q-Dixon formula in one case and by a special case of the Verma-Jain formula in the other. An orthogonal polynomial system is obtained in the first case and a system of biorthogonal rational function is obtained in the second. It is also shown that the biorthogonal system represents a generalization of Rogers’ q-ultraspherical polynomials.

The Abbena-Thurston manifold (M,g) is a critical point of the functional where Q is the Ricci operator and R is the scalar curvature, and then the index of I(g) and also the index of — I(g) are positive at (M,g).

In this paper we provide new proofs of some interesting results of Firey [2] on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon is a plane convex set of constant width whose boundary consists of a finite (odd) number of circular arcs. Equivalently, it is the intersection of a finite number of suitably chosen congruent discs. For more details, see [1, p. 128].

If a Reuleaux polygon has n sides (arcs) of positive length (where n is odd and ≥ 3), we will refer to it as a Reuleaux n-gon, or sometimes just as an n-gon. If all of the sides are equal, it is termed a regular n-gon.