We continue the investigation of the algebraic and topological structure of the algebra of Colombeau generalized functions with the aim of building up the algebraic basis for the theory of these functions. This was started in a previous work of Aragona and Juriaans, where the algebraic and topological structure of the Colombeau generalized numbers were studied. Here, among other important things, we determine completely the minimal primes of and introduce several invariants of the ideals of (Ω). The main tools we use are the algebraic results obtained by Aragona and Juriaans and the theory of differential calculus on generalized manifolds developed by Aragona and co-workers. The main achievement of the differential calculus is that all classical objects, such as distributions, become C∞-functions. Our purpose is to build an independent and intrinsic theory for Colombeau generalized functions and place them in a wider context.

The concepts of d- and nd-Frattini chief factors of a finite group are introduced. Their ingrainment into that of the extended Frattini dual subgroup becomes the natural dual to Frattini and supplemented chief factors. Not only does a dual of the strengthened form of the Jordan–Hölder theorem arise, but also the $p$-nilpotent radical becomes the intersection of the centralizers of the nd-Frattini chief factors. As a result, a class $\mathfrak{F}$ of groups is a full integrated local formation $\mathrm{LF}(f)$ if and only if each nd-Frattini chief factor in $G\in\mathfrak{F}$ is $f$-central.

Let $K$ be a field of characteristic $p$ and let $G$ be a finite group of order divisible by $p$. The regularity conjecture states that the Castelnuovo–Mumford regularity of the cohomology ring $H^*(G,K)$ is always equal to 0. We prove that if the regularity conjecture holds for a finite group $H$, then it holds for the wreath product $H\wr\mathbb{Z}/p$. As a corollary, we prove the regularity conjecture for the symmetric groups $\varSigma_n$. The significance of this is that it is the first set of examples for which the regularity conjecture has been checked, where the difference between the Krull dimension and the depth of the cohomology ring is large. If this difference is at most 2, the regularity conjecture is already known to hold by previous work.

For more general wreath products, we have not managed to prove the regularity conjecture. Instead we prove a weaker statement: namely, that the dimensions of the cohomology groups are polynomial on residue classes (PORC) in the sense of Higman.

In this paper we state and prove a new general Hilbert-type inequality in $\mathbb{R}^{n}$ with $k\geq2$ non-conjugate exponents. Using Selberg's integral formula, this result is then applied to obtain explicit upper bounds for the doubly weighted Hardy–Littlewood–Sobolev inequality and some further Hilbert-type inequalities for $k$ non-negative functions and non-conjugate exponents.

The nonlinear eigenvalue problem

for 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified, is studied under conditions on q related to those of Brinck and Molanov. Topics include Sturmian results, connections between problems on finite intervals and the half-line, and variational principles.

This paper is concerned with the study of a class of closed linear operators densely defined on a Hilbert space $H$ and called B-Fredholm operators. We characterize a B-Fredholm operator as the direct sum of a Fredholm closed operator and a bounded nilpotent operator. The notion of an index of a B-Fredholm operator is introduced and a characterization of B-Fredholm operators with index $0$ is given in terms of the sum of a Drazin closed operator and a finite-rank operator. We analyse the properties of the powers $T^m$ of a closed B-Fredholm operator and we establish a spectral mapping theorem.

The loop homology ring of an oriented closed manifold, defined by Chas and Sullivan, is interpreted as a fibrewise homology Pontrjagin ring. The basic structure, particularly the commutativity of the loop multiplication and the homotopy invariance, is explained from the viewpoint of the fibrewise theory, and the definition is extended to arbitrary compact manifolds.

We introduce a geometric condition of Bloch type which guarantees that a subset of a bounded convex domain in several complex variables is degenerate with respect to every iterated function system. Furthermore, we discuss the relations of such a Bloch-type condition with the analogous hyperbolic Lipschitz condition.

Let be the self-adjoint operator associated with the Dirichlet form

where ϕ is a positive C2 function, dλϕ = ϕdλ and λ denotes Lebesgue measure on ℝd. We study the boundedness on Lp(λϕ) of spectral multipliers of . We prove that if ϕ grows or decays at most exponentially at infinity and satisfies a suitable ‘curvature condition’, then functions which are bounded and holomorphic in the intersection of a parabolic region and a sector and satisfy Mihlin-type conditions at infinity are spectral multipliers of Lp(λϕ). The parabolic region depends on ϕ, on p and on the infimum of the essential spectrum of the operator on L2(λϕ). The sector depends on the angle of holomorphy of the semigroup generated by on Lp(λϕ).

We present a systematic method for proving non-terminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson's formula that implies the Rogers–Ramanujan identities.

We prove the following two new optimal immersion results for complex projective space. First, if $n\equiv3\,\Mod 8$ but $n\not\equiv3\,\Mod 64$, and $\alpha(n)=7$, then $CP^{n}$ can be immersed in $\mathbb{R}^{4n-14}$. Second, if $n$ is even and $\alpha(n)=3$, then $CP^n$ can be immersed in $\mathbb{R}^{4n-4}$. Here $\alpha(n)$ denotes the number of 1s in the binary expansion of $n$. The first contradicts a result of Crabb, which said that such an immersion does not exist, apparently due to an arithmetical mistake. We combine Crabb's method with that developed by the author and Mahowald.

We give a characterization of composition operators between algebras of analytic functions on a Banach space. We show (under fairly general conditions) that they are precisely the multiplicative operators that are transposes of operators of between the preduals of the algebras. The special cases of $H^\infty(U)$ and $H_{\mathrm{b}}(U)$ are considered. In these cases, the composition operators are those which are pointwise-to-pointwise continuous and/or $\tau_0$-to-$\tau_0$ continuous (where $\tau_0$ is the compact-open topology). We obtain Banach–Stone-type theorems for these algebras.

Asymptotic approximations to the Green's functions of Sturm–Liouville boundary-value problems on graphs are obtained. These approximations are used to study the regularized traces of the differential operators associated with these boundary-value problems. Various inverse spectral problems for Sturm–Liouville boundary-value problems on graphs resembling those considered in Halberg and Kramer's ‘A generalization of the trace concept' (Duke Mathematics Journal27 (1960), 607–617), for Sturm–Liouville problems, and Pielichowski's ‘An inverse spectral problem for linear elliptic differential operators' (Universitatis Iagellonicae Acta Mathematica27 (1988), 239–246), for elliptic boundary-value problems, are solved.

In 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.

A generalization of the odd Bernoulli polynomials related to the quantum Euler top is introduced and investigated. This is applied in order to compute the coefficients of the spectral polynomials for the classical Lamé operator.

We give a Dirichlet form approach for the construction and analysis of elliptic diffusions in $\bar{\varOmega}\subset\mathbb{R}^n$ with reflecting boundary condition. The problem is formulated in an $L^2$-setting with respect to a reference measure $\mu$ on $\bar{\varOmega}$ having an integrable, $\mathrm{d} x$-almost everywhere (a.e.) positive density $\varrho$ with respect to the Lebesgue measure. The symmetric Dirichlet forms $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ we consider are the closure of the symmetric bilinear forms

\begin{gather*} \mathcal{E}^{\varrho,a}(f,g)=\sum_{i,j=1}^n\int_{\varOmega}\partial_ifa_{ij} \partial_jg\,\mathrm{d}\mu,\quad f,g\in\mathcal{D}, \\ \mathcal{D}=\{f\in C(\bar{\varOmega})\mid f\in W^{1,1}_{\mathrm{loc}}(\varOmega),\ \mathcal{E}^{\varrho,a}(f,f)\lt\infty\}, \end{gather*}

in $L^2(\bar{\varOmega},\mu)$, where $a$ is a symmetric, elliptic, $n\times n$-matrix-valued measurable function on $\bar{\varOmega}$. Assuming that $\varOmega$ is an open, relatively compact set with boundary $\partial\varOmega$ of Lebesgue measure zero and that $\varrho$ satisfies the Hamza condition, we can show that $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and diffusion process $\bm{M}^{\varrho,a}$ (i.e. an associated strong Markov process with continuous sample paths). Furthermore, since $1\in D(\mathcal{E}^{\varrho,a})$ (due to the Neumann boundary condition) and $\mathcal{E}^{\varrho,a}(1,1)=0$, we obtain a conservative process $\bm{M}^{\varrho,a}$ (i.e. $\bm{M}^{\varrho,a}$ has infinite lifetime). Additionally, assuming that $\sqrt{\varrho}\in W^{1,2}(\varOmega)\cap C(\bar{\varOmega})$ or that $\varrho$ is bounded, $\varOmega$ is convex and $\{\varrho=0\}$ has codimension at least 2, we can show that the set $\{\varrho=0\}$ has $\mathcal{E}^{\varrho,a}$-capacity zero. Therefore, in this case we can even construct an associated conservative diffusion process in $\{\varrho>0\}$. This is essential for our application to continuous $N$-particle systems with singular interactions. Note that for the construction of the self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and the Markov process $\bm{M}^{\varrho,a}$ we do not need to assume any differentiability condition on $\varrho$ and $a$. We obtain the following explicit representation of the generator for $\sqrt{\varrho}\in W^{1,2}(\varOmega)$ and $a\in W^{1,\infty}(\varOmega)$:

$$ L^{\varrho,a}=\sum_{i,j=1}^n\partial_i(a_{ij}\partial_j)+\partial_i(\log\varrho)a_{ij}\partial_j. $$

Note that the drift term can be singular, because we allow $\varrho$ to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift that is not integrable with respect to the Lebesgue measure.

There has been much recent progress in the study of arithmetic progressions in various sets, such as dense subsets of the integers or of the primes. One key tool in these developments has been the sequence of Gowers uniformity norms $U^d(G)$, $d=1,2,3,\dots$, on a finite additive group $G$; in particular, to detect arithmetic progressions of length $k$ in $G$ it is important to know under what circumstances the $U^{k-1}(G)$ norm can be large.

The $U^1(G)$ norm is trivial, and the $U^2(G)$ norm can be easily described in terms of the Fourier transform. In this paper we systematically study the $U^3(G)$ norm, defined for any function $f:G\to\mathbb{C}$ on a finite additive group $G$ by the formula

\begin{multline*} \qquad\|f\|_{U^3(G)}:=|G|^{-4}\sum_{x,a,b,c\in G}(f(x)\overline{f(x+a)f(x+b)f(x+c)}f(x+a+b) \\ \times f(x+b+c)f(x+c+a)\overline{f(x+a+b+c)})^{1/8}.\qquad \end{multline*}

We give an inverse theorem for the $U^3(G)$ norm on an arbitrary group $G$. In the finite-field case $G=\mathbb{F}_5^n$ we show that a bounded function $f:G\to\mathbb{C}$ has large $U^3(G)$ norm if and only if it has a large inner product with a function $e(\phi)$, where $e(x):=\mathrm{e}^{2\pi\ri x}$ and $\phi:\mathbb{F}_5^n\to\mathbb{R}/\mathbb{Z}$ is a quadratic phase function. In a general $G$ the statement is more complicated: the phase $\phi$ is quadratic only locally on a Bohr neighbourhood in $G$.

As an application we extend Gowers's proof of Szemerédi's theorem for progressions of length four to arbitrary abelian $G$. More precisely, writing $r_4(G)$ for the size of the largest $A\subseteq G$ which does not contain a progression of length four, we prove that

$$ r_4(G)\ll|G|(\log\log|G|)^{-c}, $$

where $c$ is an absolute constant.

We also discuss links between our ideas and recent results of Host, Kra and Ziegler in ergodic theory.

In future papers we will apply variants of our inverse theorems to obtain an asymptotic for the number of quadruples $p_1\ltp_2\ltp_3\ltp_4\leq N$ of primes in arithmetic progression, and to obtain significantly stronger bounds for $r_4(G)$.

We show that if G is an infinitely generated locally (polycyclic-by-finite) group with cohomology almost-everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere.

Let $\mu$ be a positive Radon measure on $\mathbb{R}^d$ which satisfies $\mu(B(x,r))\le Cr^{n}$ for any $x\in\mathbb{R}^d$ and $r>0$ and some fixed constants $C>0$ and $n\in(0,d]$. In this paper, a new characterization of the space $\rbmo(\mu)$, which was introduced by Tolsa, is given. As an application, it is proved that the $L^p(\mu)$-boundedness with $p\in(1,\infty)$ of Calderón–Zygmund operators is equivalent to various endpoint estimates.

It is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).