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The
$l_{0}$
-minimisation problem has attracted much attention in recent years with the development of compressive sensing. The spark of a matrix is an important measure that can determine whether a given sparse vector is the solution of an
$l_{0}$
-minimisation problem. However, its calculation involves a combinatorial search over all possible subsets of columns of the matrix, which is an NP-hard problem. We use Weyl’s theorem to give two new lower bounds for the spark of a matrix. One is based on the mutual coherence and the other on the coherence function. Numerical examples are given to show that the new bounds can be significantly better than existing ones.
Given a poset
$P$
and a standard closure operator
$\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$
, we give a necessary and sufficient condition for the lattice of
$\unicode[STIX]{x1D6E4}$
-closed sets of
${\wp}(P)$
to be a frame in terms of the recursive construction of the
$\unicode[STIX]{x1D6E4}$
-closure of sets. We use this condition to show that, given a set
${\mathcal{U}}$
of distinguished joins from
$P$
, the lattice of
${\mathcal{U}}$
-ideals of
$P$
fails to be a frame if and only if it fails to be
$\unicode[STIX]{x1D70E}$
-distributive, with
$\unicode[STIX]{x1D70E}$
depending on the cardinalities of sets in
${\mathcal{U}}$
. From this we deduce that if a poset has the property that whenever
$a\wedge (b\vee c)$
is defined for
$a,b,c\in P$
it is necessarily equal to
$(a\wedge b)\vee (a\wedge c)$
, then it has an
$(\unicode[STIX]{x1D714},3)$
-representation.
Robin’s criterion states that the Riemann hypothesis is true if and only if
$\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$
for every positive integer
$n\geq 5041$
. In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For this purpose, we use a precise approximation for Chebyshev’s
$\unicode[STIX]{x1D717}$
-function.
For any finite abelian group
$G$
with
$|G|=m$
,
$A\subseteq G$
and
$g\in G$
, let
$R_{A}(g)$
be the number of solutions of the equation
$g=a+b$
,
$a,b\in A$
. Recently, Sándor and Yang [‘A lower bound of Ruzsa’s number related to the Erdős–Turán conjecture’, Preprint, 2016, arXiv:1612.08722v1] proved that, if
$m\geq 36$
and
$R_{A}(n)\geq 1$
for all
$n\in \mathbb{Z}_{m}$
, then there exists
$n\in \mathbb{Z}_{m}$
such that
$R_{A}(n)\geq 6$
. In this paper, for any finite abelian group
$G$
with
$|G|=m$
and
$A\subseteq G$
, we prove that (a) if the number of
$g\in G$
with
$R_{A}(g)=0$
does not exceed
$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$
, then there exists
$g\in G$
such that
$R_{A}(g)\geq 6$
; (b) if
$1\leq R_{A}(g)\leq 6$
for all
$g\in G$
, then the number of
$g\in G$
with
$R_{A}(g)=6$
is more than
$\frac{7}{32}m-\frac{1}{2}\sqrt{10m}-1$
.
Let
$R$
be a finite commutative ring of odd characteristic and let
$V$
be a free
$R$
-module of finite rank. We classify symmetric inner products defined on
$V$
up to congruence and find the number of such symmetric inner products. Additionally, if
$R$
is a finite local ring, the number of congruent symmetric inner products defined on
$V$
in each congruence class is determined.
This is an addendum to a recent paper by Zaïmi, Bertin and Aljouiee [‘On number fields without a unit primitive element’, Bull. Aust. Math. Soc.93 (2016), 420–432], giving the answer to a question asked in that paper, together with some historical connections.
This work generalises the short resolution given by Pisón Casares [‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081–1091] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen–Macaulay property for simplicial affine semigroups.
In characteristic two, some criteria are obtained for a symmetric square-central element of a totally decomposable algebra with orthogonal involution, to be contained in an invariant quaternion subalgebra.
Let
$G$
be a finite solvable group and let
$p$
be a prime. In this note, we prove that
$p$
does not divide
$\unicode[STIX]{x1D711}(1)$
for every irreducible monomial
$p$
-Brauer character
$\unicode[STIX]{x1D711}$
of
$G$
if and only if
$G$
has a normal Sylow
$p$
-subgroup.
Let
$w$
be a group-word. For a group
$G$
, let
$G_{w}$
denote the set of all
$w$
-values in
$G$
and let
$w(G)$
denote the verbal subgroup of
$G$
corresponding to
$w$
. The group
$G$
is an
$FC(w)$
-group if the set of conjugates
$x^{G_{w}}$
is finite for all
$x\in G$
. It is known that if
$w$
is a concise word, then
$G$
is an
$FC(w)$
-group if and only if
$w(G)$
is
$FC$
-embedded in
$G$
, that is, the conjugacy class
$x^{w(G)}$
is finite for all
$x\in G$
. There are examples showing that this is no longer true if
$w$
is not concise. In the present paper, for an arbitrary word
$w$
, we show that if
$G$
is an
$FC(w)$
-group, then the commutator subgroup
$w(G)^{\prime }$
is
$FC$
-embedded in
$G$
. We also establish the analogous result for
$BFC(w)$
-groups, that is, groups in which the sets
$x^{G_{w}}$
are boundedly finite.
We prove that a finite coprime linear group
$G$
in characteristic
$p\geq \frac{1}{2}(|G|-1)$
has a regular orbit. This bound on
$p$
is best possible. We also give an application to blocks with abelian defect groups.
In this paper, we prove some new reverse dynamic inequalities of Renaud- and Bennett-type on time scales. The results are established using the time scales Fubini theorem, the reverse Hölder inequality and a time scales chain rule.
In this paper, we first give a description of the holomorphic automorphism group of a convex domain which is a simple case of the so-called generalised minimal ball. As an application, we show that any proper holomorphic self-mapping on this type of domain is biholomorphic.
In a general setting of an ergodic dynamical system, we give a more accurate calculation of the speed of the recurrence of a point to itself (or to a fixed point). Precisely, we show that for a certain
$\unicode[STIX]{x1D709}$
depending on the dimension of the space,
$\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,x)=0$
almost everywhere and
$\liminf _{n\rightarrow +\infty }(n\log \log n)^{\unicode[STIX]{x1D709}}d(T^{n}x,y)=0$
for almost all
$x$
and
$y$
. This is done by assuming the exponential decay of correlations and making a weak assumption on the invariant measure.
where
$f$
satisfies the condition
$f(\sum _{\unicode[STIX]{x1D706}\in K}\unicode[STIX]{x1D706}x)\neq 0$
for all
$x\in S$
,
$(S,+)$
is an abelian semigroup and
$K$
is a subgroup of the automorphism group of
$S$
.
Let
$G$
be a locally compact amenable group and
$A(G)$
and
$B(G)$
be the Fourier and the Fourier–Stieltjes algebras of
$G,$
respectively. For a power bounded element
$u$
of
$B(G)$
, let
${\mathcal{E}}_{u}:=\{g\in G:|u(g)|=1\}$
. We prove some convergence theorems for iterates of multipliers in Fourier algebras.
(a) If
$\Vert u\Vert _{B(G)}\leq 1$
, then
$\lim _{n\rightarrow \infty }\Vert u^{n}v\Vert _{A(G)}=\text{dist}(v,I_{{\mathcal{E}}_{u}})\text{ for }v\in A(G)$
, where
$I_{{\mathcal{E}}_{u}}=\{v\in A(G):v({\mathcal{E}}_{u})=\{0\}\}$
.
(b) The sequence
$\{u^{n}v\}_{n\in \mathbb{N}}$
converges for every
$v\in A(G)$
if and only if
${\mathcal{E}}_{u}$
is clopen and
$u({\mathcal{E}}_{u})=\{1\}.$
(c) If the sequence
$\{u^{n}v\}_{n\in \mathbb{N}}$
converges weakly in
$A(G)$
for some
$v\in A(G)$
, then it converges strongly.
We present refined and reversed inequalities for the weighted arithmetic mean–harmonic mean functional inequality. Our approach immediately yields the related operator versions in a simple and fast way. We also give some operator and functional inequalities for three or more arguments. As an application, we obtain a refined upper bound for the relative entropy involving functional arguments.
We establish a Schwarz lemma for
$V$
-harmonic maps of generalised dilatation between Riemannian manifolds. We apply the result to obtain corresponding results for Weyl harmonic maps of generalised dilatation from conformal Weyl manifolds to Riemannian manifolds and holomorphic maps from almost Hermitian manifolds to quasi-Kähler and almost Kähler manifolds.
We study a special class of quasi-cyclic codes, obtained from a cyclic code over an extension field of the alphabet field by taking its image on a basis. When the basis is equal to its dual, the dual code admits the same construction. We give some examples of self-dual codes and LCD codes obtained in this way.