We discuss a general method for classifying certain subcategories of the category of finite-dimensional modules over a finite-dimensional co-commutative Hopf algebra B. Our method is based on that of Benson–Carlson–Rickard [BCR1], who classify such subcategories when B = kG, the group ring of a finite group G over an algebraically closed field k. We get a similar classification when B is a finite sub-Hopf algebra of the mod 2 Steenrod algebra, with scalars extended to the algebraic closure of F2. Along the way, we prove a Quillen stratification theorem for cohomological varieties of modules over any B, in terms of quasi-elementary sub-Hopf algebras of B.

For bond percolation on the two-dimensional triangular lattice with arbitrary retention parameter p ∈ [0, 1], we show that the number of infinite rigid components is a.s. at most 1. This proves a conjecture by Holroyd. Further results, concerning simultaneous uniqueness, and continuity (in p) of the probability that a given edge is in an infinite rigid component, are also obtained.

The classical account of the invariants and covariants of a pair of quadric surfaces is due to Salmon. The actual determination, in explicit form, of the complete system of concomitants of two quaternary quadratic forms is much later in date, and is due to Turnbull. This system, which includes mixed concomitants of various kinds, is complicated. As originally determined, it comprised 125 forms, three of which were later shown to be reducible. In the accounts of both these authors, however, the forms are considered as belonging to a particular pair of quadric surfaces, and the problem of determining these covariant forms which are invariant, not merely under change of coordinate system but also under change of base in the pencil denned by the two quadrics, is not alluded to. Such forms, which are of obvious geometrical interest, are called combinants. It is rather surprising to find that very little seems to be known about the combinants attaching to a pencil of quadric surfaces; the Encyklopädie scarcely mentions them, and I have been unable to trace any references in the literature except to the two invariants whose vanishing expresses the condition that the parameters of the four cones in the pencil form an equianharmonic or a harmonic set.

Let p be an odd prime number. Let k be a [pfr ]-number field and [ofr ] the ring of all integers of k with a prime element π. Let K/k be a cyclic extension with Galois group G, and [Afr ] the associated order of the ring [Ofr ] of all integers in K:

[Afr ]={f∈kG[mid ]f[Ofr ]⊆[Ofr ]}.

F. Bertrandias and M-J. Ferton [1] obtained necessary and sufficient conditions that [Ofr ] is [Afr ]-free in the case K/k is of degree p. The purpose of this paper is to study such conditions in case K/k is a cyclic totally ramified Kummer extension of degree n=pm.

It is well known that the eigen-functions corresponding to the free electrons in a one-dimensional lattice are of the form

where C (x) is a periodic function with period c (the lattice constant), and Kis a continuous variable.

For a principal fibre bundle (p, q, B, G) with structure group G, the group of G-equivariant self equivalences of the total space P is investigated by using bundle map theory. Computations are given for well-known principal fibre bundles.

The consideration of the statistical aspect of the second law of thermodynamics of a previous paper is extended. The influence of fluctuations in the wind-wood model of the Ehrenfests is investigated, and the relative occurrence, mean life and time of recurrence of such fluctuations are calculated. These are found to agree with the results obtained for the simpler models considered previously.

Let G be a connected Lie group and let {λi} be a sequence of probability measures on G converging (in the usual weak topology) to a probability measure λ. Suppose that {αi} is a sequence of affine automorphisms of G such that the sequence {αi,(λi)} also converges, say to a probability measure μ. What does this imply about the sequence {αi}? It is a classical observation that if G = ℝn for some n, and neither of λ and μ is supported on a proper affine subspace of ℝn, then under the above condition, {αi} is relatively compact in the group of all affine automorphisms of ℝn.

In this paper we complete the work begun in [2] on the subgroup structure of the Lyons simple group Ly of order

We expand the notion of core to cl-core for Nakayama closures cl. In the characteristic p > 0 setting, when cl is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that *-core(I) = core(I), if I is an ideal in a one-dimensional domain with infinite residue field or if I is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a Cohen–Macaulay normal local domain with infinite perfect residue field, if the analytic spread, ℓ, is equal to the *-spread and I is Gℓ and weakly-(ℓ − 1)-residually S2. This last is dependent on our result that generalizes the notion of general minimal reductions to general minimal *-reductions. We also determine that the *-core of a tightly closed ideal in certain one-dimensional semigroup rings is tightly closed and therefore integrally closed.

In general, the class of a nilpotent linear algebra of dimension n is at most n + 1, and the index, or derived length, of a solvable linear algebra of dimension n is at most n. In this note it is shown that, for a nilpotent linear algebra of dimension n satisfying x2 = 0 for all x, the class is at most n; and bounds are obtained for the indices of solvable Lie algebras.

Let E[τ] be a locally convex Hausdorif topological vector space, with a Schauder basis {xi, x′j where

for each x ∈ E. The partial summation operator Sn, defined by

is a linear operator on E, whose definition extends at once to a linear operator mapping (E′)* into E, where (E′)* is the algebraic dual of E′. The dual of Sn is the operator S′n, mapping E* into E′, defined by

and 〈Snx, x′〉 = 〈x, S′nx′〉 for each x ∈ (E′)*. It is easy to see that S′nx′ → x′ with respect to the weak topology σ(E′, E) for each x′ ∈ E′.

It is shown that apart from

there are only finitely many arithmetic progressions with given differences of equal lengths ≥ 2 and with equal products and that they can be effectively determined.

For ξ ε (0, ½) let Eξ be the perfect symmetric set of constant ratio ξ and set

It was shown by the first author that if T is a contraction on the Hilbert space H with spectrum contained in Eξ, and if log ∥T−n∥ = O(nα) as n → ∞ for some α < b(ξ), then T is unitary. In the other direction, we show here that there exists a (non-unitary) contraction T on H such that SpT = Eξ, log ∥T-n∥ = O(nb(ξ)) as n → ∞, and lim supn → ∞ ∥T−n∥ = ∞.

Let ψ be a distribution function on [−1,1] from the Szegö-class, which contains in particular all Jacobi weights, and let (pn) be the monic polynomials orthogonal with respect to dψ. Let m(n)∈ℕ, n∈ℕ, be non-decreasing with limn → ∞ (n − m(n)) = ∞, l(n)∈ℕ with 0 ≤ l(n) ≤ m(n), and μj, n ∈ℝ for j = 0, …, m(n), n∈ℕ. It is shown that for each sufficiently large n, has n−l(n) simple zeros in (−1, 1)and l(n) zeros in ℂ\[−1,1] if for n ≥ n0, has m(n) − l(n) zeros in the disc |z| ≤ r < 1, l(n) zeros outside of the disc |z| ≥ R > 1 and where q > 2 max {r, 1/R}. If m(n) is constant for n ≥ n0 then the statement holds even for such polynomials (pn) orthogonal with respect to a distribution dψ satisfying the weak assumption ψ′ > 0 a.e. on [−1, 1]. For linear combinations of polynomials orthogonal on the unit circle corresponding results are derived.

We study the scaling scenery of Gibbs measures for subshifts of finite type on self-conformal fractals and applications to Falconer's distance set problem and dimensions of projections. Our analysis includes hyperbolic Julia sets, limit sets of Schottky groups and graph-directed self-similar sets.

Throughout this paper we shall fix an algebraically closed field k. Consider the following:

Problem. Let (Y, L) be a normal polarized variety over k, i.e. a normal projective variety Y over k together with an ample line bundle L on Y. Then one may ask under which conditions the following statement holds:

(*) Every normal projective variety X containing Y as an ample Cartier divisor such that the normal bundle of Y in X is L, is isomorphic to the projective cone over Y.

Suppose that D is a plane domain and that f(z), g(z) are meromorphic in D and f(z) ╪ g(z) for all z in D. Then following Rubel and Yang [11], we say that f(z) avoids g(z) in D. A system of functions g1(z), …, gn(z) is said to be unavoidable if, whenever f is meromorphic in D, at least one of the equations f(z) = gv(z) has a root in D. Rubel and Yang [11] proved that if D is the open plane, then any two functions form an avoidable system, but three distinct polynomials a1, a2, a3 such that a1−a2 and a2−a3 are not both constant form an unavoidable system.

The metric projection mapping πX plays an important role in nonlinear approximation theory. Usually X is a closed subset of a Banach space [ ] and, for each e∈[ ], πX(e) is the set, perhaps empty, of all points in X which are nearest to e. From a classical theorem due to Stečkin [7] it is known that, when [ ] is uniformly convex, the metric projection πX(e) is single valued at each typical point e of [ ] (in the sense of the Baire categories), i.e. at each point e of a residual subset of [ ]. More recently Zamfirescu [8] has proven that, if X is a typical compact set in ℝn (in the sense of Baire categories) and n[ges ]2, then the metric projection πX(e) has cardinality at least 2 at each point e of a dense subset of ℝn. This result has been extended in several directions by Zhivkov [9, 10], who has also considered the case of the metric antiprojection mapping νX (which associates with each e∈[ ] the set νX(e), perhaps empty, of all ∈X which are farthest from e). For this mapping De Blasi [2] has shown that, if [ ] is a real separable Hilbert space with dim[ ]=+∞ and n is an arbitrary natural number not less than 2, then, for a typical compact convex set X⊂[ ], the metric antiprojection νX(e) has cardinality at least n at each point e of a dense subset of [ ]. A systematic discussion of the properties of the maps πX and νX, and additional bibliography, can be found in Singer [5, 6] and Dontchev and Zolezzi [3].

In the present paper we consider some further properties of the metric projection mapping πX, with X a compact set in a real separable Hilbert space [ ]. If dim[ ]=n and 2[ges ]n<+∞, it is proven that for a typical compact set X⊂[ ], the metric projection πX(e) has cardinality exactly n+1 at each point e of a dense subset of [ ], while the set of those points e∈[ ] where πX(e) has cardinality at least n+2 is empty. Furthermore it is shown that, if dim[ ]=+∞, then for a typical compact set X⊂[ ] the metric projection πX(e) has cardinality at least n (for arbitrary n[ges ]2) at each point e of a dense subset of [ ]. Incidentally we obtain a characterization of the dimension of the space [ ] by means of a typical property holding in the space of the compact subsets of [ ].

Let ${\cal A}$ be the mod 2 Steenrod algebra and $D_k$ the Dickson algebra of $k$ variables. We study the Lannes–Zarati homomorphisms \[ \varphi_k:{\rm Ext}_{{\cal A}}^{k, k+i}({\bb F}_2,{\bb F}_2)\rightarrow ({\bb F}_2\otimes_{{\cal A}}D_k)^{*}_{i}, \] which correspond to an associated graded of the Hurewicz map $H:\pi^s_*(S^0)\cong\pi_*(Q_0S^0)\rightarrow H_*(Q_0S^0)$. An algebraic version of the long-standing conjecture on spherical classes predicts that $\varphi_k = 0$ in positive stems, for $k > 2$. That the conjecture is no longer valid for $k = 1$ and $2$ is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes.

This conjecture has been proved for $k = 3$ by Hu'ng ‘9’. It has been shown that $\varphi_k$ vanishes on decomposable elements for $k > 2$ ‘14’ and on the image of Singer's algebraic transfer for $k > 2$ ‘9, 12’. In this paper, we establish the conjecture for $k = 4$. To this end, our main tools include (1) an explicit chain-level representation of $\varphi_k$ and (2) a squaring operation ${\rm Sq}^0$ on $({\bb F}_2\otimes_{{\cal A}}D_k)^*$, which commutes with the classical ${\rm Sq}^0$ on ${\rm Ext}_{{\cal A}}^k({\bb F}_2,{\bb F}_2)$ through the Lannes–Zarati homomorphism.