Chladni figures are formed when particles scattered across a plate move due to an external harmonic force resonating with one of the natural frequencies of the plate. Chladni figures are precisely the nodal set of the vibrational mode corresponding to the frequency resonating with the external force. We propose a plausible model for the movement of the particles that explains the formation of Chladni figures in terms of the stochastic stability of the equilibrium solutions of stochastic differential equations.

This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C *-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

Whittaker and Ruse have developed forms of Gauss's theorem in general relativity, their theorems connecting integrals of normal force taken over a closed 2-space V2 with integrals involving the distribution of matter taken over an open 3-space bounded by V2. The definition of force employed by them involves the introduction of a normal congruence (with unit tangent vector λi), the “force” relative to the congruence being the negative of the first curvature vector of the congruence (– δλi/δs). This appears at first sight a natural enough definition, because – δλi/δs at an event P represents the acceleration relative to the congruence of a free particle travelling along a geodesic tangent to the congruence at P. In order to give physical meaning to this definition of force it is necessary to specify the congruence λi physically, and it would seem most natural to choose the congruence of world-lines of flow of the medium. Supposing certain conditions satisfied by this congruence (cf. Ruse, loc. cit.), the theory of Ruse is applicable, and from this follows a form of Gauss's theorem.

A well-known fact is that every automorphism of the symmetric group on a set must be inner (whether the set is finite or infinite) unless the set has exactly six elements (4, § 13). A long-standing conjecture concerns the analogue of this fact for the group A(S) of all order-preserving permutations of a totally ordered set S. The group A(S) is lattice ordered (l-group) by defining, for f, g ∈ A (S), f ≦ g whenever xy ≦ xg for all x ∈ S. From the standpoint of l-groups, A(S) is of considerable interest because of the analogue of Cayley's theorem proved in (2), namely every l-group may be embedded in some A(S). Unlike the non-ordered symmetric groups, which must be highly transitive, A(S) is severely restricted if assumed transitive. Of special interest are those cases when A(S) is doubly transitive (relative to the order), since the building blocks (primitive) of all transitive A(S) are either doubly transitive or uniquely transitive. It is easily seen that each inner automorphism of A(S) must preserve the lattice ordering (that is, must be an l-automorphism). It is tempting to conjecture that every l-automorphism of A(S) must be inner. However, an easy counterexample is at hand. If S consists of two copies of the ordered set of integers, one entirely above the other, then A(S) is the direct product of two copies of the l-group of integers, which has an outer l-automorphism exchanging the two factors. The conjecture referred to above is that if A(S) is transitive on S, then every l-automorphism of A(S) is inner.

We define fractal interpolation on unbounded domains for a certain class of topological spaces and construct local fractal functions. In addition, we derive some properties of these local fractal functions, consider their tensor products, and give conditions for local fractal functions on unbounded domains to be elements of Bochner–Lebesgue spaces.

In the theory of Electrostatics, or of the Newtonian potential, there exists between two systems of potentiating matter, a wellknown reciprocal relation, analytically expressed in the proposition known as Green's Theorrn. By applying his theorem to the case when one of the systems is of the simplest possible character, namely, a mass concentrated at a single point, Green deduced a general method of solving the equation for the potential. The idea of a similar general method of dealing with the equations of Elasticity is due to Professor Betti, of Pisa, who has proved a reciprocal relation between two states of strain of an elastic solid, analogous to the relation in Electrostatics referred to.

We show that Brin's generalizations 2V and 3V of the Thompson–Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.

Let E be a nuclear space provided with a topology different from the weak topology. Let {Ai: i ∈ I} be a fundamental system of equicontinuous subsets of the topological dual E' of E. If {Fi: i ∈ I} is a family of infinite dimensional Banach spaces with separable predual, there is a fundamental system {Bi: i ∈ I} of weakly closed absolutely convex equicontinuous subsets of E'such that is norm-isomorphic to Fi, for each i ∈ I. Other results related with the one above are also given.

Symmetric inverse monoids of objects in arbitrary categories are studied. Necessary and sufficient conditions are given for such monoids to be E-unitary or else form (complete) inverse algebras. Particular attention is given to symmetric inverse monoids of objects in free categories.

We prove formulas of different types that allow us to calculate the Gerstenhaber bracket on the Hochschild cohomology of an algebra using some arbitrary projective bimodule resolution for it. Using one of these formulas, we give a new short proof of the derived invariance of the Gerstenhaber algebra structure on Hochschild cohomology. We also give some new formulas for the Connes differential on the Hochschild homology that lead to formulas for the Batalin–Vilkovisky (BV) differential on the Hochschild cohomology in the case of symmetric algebras. Finally, we use one of the obtained formulas to provide a full description of the BV structure and, correspondingly, the Gerstenhaber algebra structure on the Hochschild cohomology of a class of symmetric algebras.

During an investigation into the existence of Gauss-type quadrature formulae for the numerical solution of Fredholm integral equations with weakly singular kernels an intermediate result was found which is of independent interest.

We define a new height function $\mathcal{R}(\alpha)$, the Remak height of an algebraic number $\alpha$. We give sharp upper and lower bounds for $\mathcal{R}(\alpha)$ in terms of the classical Mahler measure $M(\alpha)$. Study of when one of these bounds is exact leads us to consideration of conjugate sets of algebraic numbers of norm $\pm 1$ lying on two circles centred at 0. We give a complete characterization of such conjugate sets. They turn out to be of two types: one related to certain cubic algebraic numbers, and the other related to a non-integer generalization of Salem numbers which we call extended Salem numbers.

AMS 2000 Mathematics subject classification: Primary 11R06

The Boros–Moll polynomials Pm (a) arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that Pm (a) is 2-log-concave for any m ≥ 2. Let di (m) be the coefficient of ai in Pm (a). We also show that the sequence is log-concave.

In an earlier paper, which this note is intended to supplement and in some respects improve, the writer gave a general theorem of duality relating to isobaric determinants with elements Cr and Hr, the elementary and the complete homogeneous symmetric functions of a set of variables. The result was shewn to include as special cases the dual forms of “bi-alternant” symmetric functions given by Jacobi and Naegelsbach, as well as two equivalent forms of isobaric determinant used by MacMahon as a generating function in an important problem of permutations.

On a separable C*-algebra A every (completely) bounded map which preserves closed two-sided ideals can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C*-algebras of continuous sections vanishing at ∞ of locally trivial C*-bundles of finite type.

Necessary and sufficient conditions are found on an ideal a⊲ℤ[x] for the additive group [a]+ of ℤ[x]/a to be finite and cyclic. As a consequence, the abelianizations of certain cyclically-presented groups are computed explicitly.

In [7] Plemelj established some fundamental results in two- and three-dimensional potential theory about the eigenvalues of both the double layer potential operator and its adjoint, the normal derivative of the single layer potential operator. In [3] Blumenfeld and Mayer established some additional results concerning the eigenvalues of these integral operators in the case of ℝ2. The spectral properties established by Plemelj [7] and by Blumenfeld and Mayer [3] have had a profound effect in the area of integral equation methods in scattering and potential theory in both ℝ2 and ℝ3.

Dirac (2) and Plummer (5) independently investigated the structure of minimally 2-connected graphs G, which are characterized by the property that for any line x of G, G–x is not 2-connected. In this paper we investigate an analogous class of strongly connected digraphs D such that for any arc x, D–x is not strong. Not surprisingly, these digraphs have much in common with the minimally 2-connected graphs, and a number of theorems similar to those in (2) and (5) are proved, notably our Theorems 9 and 12.

We investigate both left and right cancellation in the Stone–Čech compactification βS of a discrete semigroup S, obtaining several results for arbitrary semigroups S and others for more restricted semigroups. In particular, if S is the semigroup of injective functions from a set to itself we determine precisely which pairs x, y and S have some p∈βS with px = py. We also obtain several new results about right cancellation in (βℕ, +).