In this paper, we consider signal recovery via $l_{1}$-analysis optimisation. The signals we consider are not sparse in an orthonormal basis or incoherent dictionary, but sparse or nearly sparse in terms of some tight frame $D$. The analysis in this paper is based on the restricted isometry property adapted to a tight frame $D$ (abbreviated as $D$-RIP), which is a natural extension of the standard restricted isometry property. Assuming that the measurement matrix $A\in \mathbb{R}^{m\times n}$ satisfies $D$-RIP with constant ${\it\delta}_{tk}$ for integer $k$ and $t>1$, we show that the condition ${\it\delta}_{tk}<\sqrt{(t-1)/t}$ guarantees stable recovery of signals through $l_{1}$-analysis. This condition is sharp in the sense explained in the paper. The results improve those of Li and Lin [‘Compressed sensing with coherent tight frames via $l_{q}$-minimization for $0<q\leq 1$’, Preprint, 2011, arXiv:1105.3299] and Baker [‘A note on sparsification by frames’, Preprint, 2013, arXiv:1308.5249].

We prove that if a space $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ with a rank 2-diagonal either has the countable chain condition or is star countable then the cardinality of $X$ is at most $\mathfrak{c}$.

It is shown that meromorphic solutions of certain first-order nonlinear differential equations do not have wandering domains.

We give a new q-analogue of the (A.2) supercongruence of Van Hamme. Our proof employs Andrews’ multiseries generalisation of Watson’s $_{8}\phi _{7}$ transformation, Andrews’ terminating q-analogue of Watson’s $_{3}F_{2}$ summation, a q-Watson-type summation due to Wei–Gong–Li and the creative microscoping method, developed by the author and Zudilin [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358]. As a conclusion, we confirm a weaker form of Conjecture 4.5 by the author [‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct. 28 (2017), 888–899].

In this note we prove that for a monotone function that fixes the origin, the complementarity problem for Cn always admits a solution. If, moreover, the function is strictly monotone, then zero is the unique solution. These results are stronger than known results in this direction for two reasons: firstly, there is no condition on the nature of the cone and secondly, no feasibility assumptions are made.

In this article we prove that there is a positive solution in of the equation −Δu + λu = |u|p−2u in Ω where Ω is the half space with a hole, λ > 0 and 2 < p < 2N/(N−2).

We give the representation formula of a function in a disc in terms of its arithmetic means on the boundary. We also give applications of this formula to the solutions of some elementary differential equations with prescribed means of boundary or initial data.

We show that the null limit hypothesis, in the definition of a barrier, can be relaxed for normal boundary points that satisfy a mild additional condition. We also give a simple necessary and sufficient condition for the regularity of semi-singular boundary points.

In this paper, we obtain all rational points (x,y) on the superelliptic curves

Two theorems for Lebesgue integrals, namely the Gauss-Green Theorem relating surface and volume integrals, and the integration-by-parts formula, are shown to possess generalizations where the integrands take values in a Banach space, the integrals are Bochner integrals, and derivatives are Fréchet derivatives. For integration-by-parts, the integrand consists of a continuous linear map applied to a vector-valued function. These results were required for a generalization of the calculus of variations, given in another paper.

In 1919, Adolf Hurwitz formed the quaternion ring R composed of elements whose coordinates were either all integers or halves of odd integers. The objective of this paper is to examine the (two-sided) ideal structure in the hurwitzean polynomial ring R[x], formed by taking all polynomials with coefficients in R. The maximal and prime ideals of R[x] will be characterized with results surprisingly analogous to those in Z[x]. In addition, a canonical basis, of the type developed by G. Szekeres, 1952, for polynomial domains, will be developed for the ideals of R[x].

For a finite group G and a subset S of G with 1 ∉ S, the Cayley graph Cay(G, S) is the digraph with vertex set G such that (x, y) is an arc if and only if yx−1 ∈ S. The Cayley graph Cay(G, S) is called a CI-graph if, for any T ⊂ G, whenever Cay (G, S) ≅ Cay(G, T) there is an element a σ ∈ Aut(G) such that Sσ = T. For a positive integer m, G is called an m-DCI-group if all Cayley graphs of G of valency at most m are CI-graphs; G is called a connected m-DCI-group if all connected Cayley graphs of G of valency at most m are CI-graphs. The problem of determining Abelian m-DCI-groups is a long-standing open problem. It is known from previous work that all Abelian m-DCI-groups lie in an explicitly determined class of Abelian groups. First we reduce the problem of determining Abelian m-DCI-groups to the problem of determining whether every subgroup of a member of is a connected m-DCI-group. Then (for a finite group G, letting p be the least prime divisor of |G|,) we completely classify Abelian connected (p + 1)-DCI-groups G, and as a corollary, we completely classify Abelian m-DCI-groups G for m ≤ p + 1. This gives many earlier results when p = 2.

Let be a real quadratic field. It is well known that if 3 divides the class number of k, then 3 divides the class number of , and thus it divides B1,χω−1, where χ and ω are characters belonging to the fields k and respectively. In general, the main conjecture of Iwasawa theory implies that if an odd prime p divides the class number of k, then p divides B1,χω−1, where ω is the Teichmüller character for p.

The aim of this paper is to examine its converse when p splits in k. Let k∞ be the ℤp-extension of k = k0 and hn be the class number of kn, the n th layer of the ℤp-extension. We shall prove that if p |B1,χω−1, then p | hn for all n ≥ 1. In terms of Iwasawa theory, this amounts to saying that if M∞/k∞, is nontrivial, then L∞/k∞ is nontrivial, where M∞ and L∞ are the maximal abelian p-extensions unramified outside p and unramified everywhere respectively.

In the present note the equation yn = x1-m ym is reduced, under appropriate conditions, to a quadratic autonomous system of differential equations in the plane. In pursuance of this new approach, the main geometric features of this autonomous system are determined and a method of solving it is outlined.

It is shown that any sheaf of R˜-modules, all of whose stalks are injective, is necessarily a flabby sheaf. This generalizes the result or Grothendieck that the sheaf M˜ determined by an injective module M over a commutative noetherian ring with 1 is flabby.

We develop a generalised polynomial formalism which captures the concept of an algebra of piece-wise denned polynomials. The formalism is based on the Boolean power construction of universal algebra. A generalisation of the theory of substitution homomorphisms is developed. The abstract operation of composition of generalised polynomials in one variable is denned and shown to correspond to function composition.

For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,…,xk} such that for some positive integer d, xi + 1 − xi ∈ {d, 2d,…, md} for i = 1,…, k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2, …, n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r ≤ m, hm(m + r) ≤ 2c(m + r − 1) + 1, where c = ⌈m/(m − r)⌉. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

Starters with adders in abelian groups of odd order have been used extensively in the construction and study of Room squares. It is possible to define these concepts in non-abelian groups, with similar applicability to Room squares. Examples are given of adders for the “patterned starter” in the non-abelian groups of order pq, p and q primes larger than 3, p ≡ q ≡ 3 (mod 4), q ≡ 1 (mod p).

The group of an arbitrary companion knot is determined using the theory of braids. This seems to be a new result as far as the resulting group is concerned. The latter part of the paper considers infinite necklaces of knots, their groups and in special cases their Alexander power series.