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In this paper, we determine when
$\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $
, the complement of the zero divisor graph
${\Gamma }_{I} (L)$
with respect to a semiprime ideal
$I$
of a lattice
$L$
, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for
${\Gamma }_{I} (L)$
when
$\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $
.
We compute the rings
${H}^{\ast } (N; { \mathbb{F} }_{2} )$
for
$N$
a closed
$ \mathbb{S} {\mathrm{ol} }^{3} $
-manifold, and then determine the Borsuk–Ulam indices
$\text{BU} (N, \phi )$
with
$\phi \not = 0$
in
${H}^{1} (N; { \mathbb{F} }_{2} )$
.
A long-standing conjecture asserts that every finite nonabelian
$p$
-group has a noninner automorphism of order
$p$
. In this paper the verification of the conjecture is reduced to the case of
$p$
-groups
$G$
satisfying
${ Z}_{2}^{\star } (G)\leq {C}_{G} ({ Z}_{2}^{\star } (G))= \Phi (G)$
, where
${ Z}_{2}^{\star } (G)$
is the preimage of
${\Omega }_{1} ({Z}_{2} (G)/ Z(G))$
in
$G$
. This improves Deaconescu and Silberberg’s reduction of the conjecture: if
${C}_{G} (Z(\Phi (G)))\not = \Phi (G)$
, then
$G$
has a noninner automorphism of order
$p$
leaving the Frattini subgroup of
$G$
elementwise fixed [‘Noninner automorphisms of order
$p$
of finite
$p$
-groups’, J. Algebra250 (2002), 283–287].
We consider a recent work of Pascu and Pascu [‘Neighbourhoods of univalent functions’, Bull. Aust. Math. Soc.83(2) (2011), 210–219] and rectify an error that appears in their work. In addition, we study certain analogous results for sense-preserving harmonic mappings in the unit disc
$\vert z\vert \lt 1$
. As a corollary to this result, we derive a coefficient condition for a sense-preserving harmonic mapping to be univalent in
$\vert z\vert \lt 1$
.
We study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number
$n\geq 3$
a set of
$n$
closed subspaces of
${\ell }^{2} $
which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces.
In this paper, we study the properties of
$k$
-plurisubharmonic functions defined on domains in
${ \mathbb{C} }^{n} $
. By the monotonicity formula, we give an alternative proof of the weak continuity of complex
$k$
-Hessian operators with respect to local uniform convergence.
For a Lie algebra
$L$
over an algebraically closed field
$F$
of nonzero characteristic, every finite dimensional
$L$
-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra
$L$
is in the saturated formation
$\mathfrak{F}$
and if
$V, W$
are irreducible
$L$
-modules with the same cluster and the
$p$
-operation vanishes on the centre of the
$p$
-envelope used, then
$V, W$
are either both
$\mathfrak{F}$
-central or both
$\mathfrak{F}$
-eccentric. Clusters are used to generalise the construction of induced modules.
Let
$\Omega $
be a bounded open interval, and let
$p\gt 1$
and
$q\in (0, p- 1)$
. Let
$m\in {L}^{{p}^{\prime } } (\Omega )$
and
$0\leq c\in {L}^{\infty } (\Omega )$
. We study the existence of strictly positive solutions for elliptic problems of the form
$- (\vert {u}^{\prime } \mathop{\vert }\nolimits ^{p- 2} {u}^{\prime } ){\text{} }^{\prime } + c(x){u}^{p- 1} = m(x){u}^{q} $
in
$\Omega $
,
$u= 0$
on
$\partial \Omega $
. We mention that our results are new even in the case
$c\equiv 0$
.
Let
$G$
be a finite group. We show that the order of the subgroup generated by coprime
${\gamma }_{k} $
-commutators (respectively,
${\delta }_{k} $
-commutators) is bounded in terms of the size of the set of coprime
${\gamma }_{k} $
-commutators (respectively,
${\delta }_{k} $
-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words
${\gamma }_{k} $
and
${\delta }_{k} $
are concise.
It is known that
$\zeta (1+ it)\ll \mathop{(\log t)}\nolimits ^{2/ 3} $
when
$t\gg 1$
. This paper provides a new explicit estimate
$\vert \zeta (1+ it)\vert \leq \frac{3}{4} \log t$
, for
$t\geq 3$
. This gives the best upper bound on
$\vert \zeta (1+ it)\vert $
for
$t\leq 1{0}^{2\cdot 1{0}^{5} } $
.
In this paper, explicit auxiliary functions are used to get upper and lower bounds for the Mahler measure of monic irreducible totally positive polynomials with integer coefficients. These bounds involve the length and the trace of the polynomial.
In 1977 Hartwig and Luh asked whether an element
$a$
in a Dedekind-finite ring
$R$
satisfying
$aR= {a}^{2} R$
also satisfies
$Ra= R{a}^{2} $
. In this paper, we answer this question in the negative. We also prove that if
$a$
is an element of a Dedekind-finite exchange ring
$R$
and
$aR= {a}^{2} R$
, then
$Ra= R{a}^{2} $
. This gives an easier proof of Dischinger’s theorem that left strongly
$\pi $
-regular rings are right strongly
$\pi $
-regular, when it is already known that
$R$
is an exchange ring.
For a semigroup
$S$
, let
${S}^{1} $
be the semigroup obtained from
$S$
by adding a new symbol 1 as its identity if
$S$
has no identity; otherwise let
${S}^{1} = S$
. Mitsch defined the natural partial order
$\leqslant $
on a semigroup
$S$
as follows: for
$a, b\in S$
,
$a\leqslant b$
if and only if
$a= xb= by$
and
$a= ay$
for some
$x, y\in {S}^{1} $
. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.
We use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime
$p$
. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below
${p}^{1/ 2} $
, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.
Consider a map of class
${C}^{3} $
with nonflat critical points and with all periodic points hyperbolic repelling. We show that the ‘backward contracting condition’ implies the summability condition. This result is the converse of Theorem 3 of Bruin et al. [‘Large derivatives, backward contraction and invariant densities for interval maps’, Invent. Math.172 (2008), 509–533].
Let
$(a, b, c)$
be a primitive Pythagorean triple satisfying
${a}^{2} + {b}^{2} = {c}^{2} . $
In 1956, Jeśmanowicz conjectured that for any given positive integer
$n$
the only solution of
$\mathop{(an)}\nolimits ^{x} + \mathop{(bn)}\nolimits ^{y} = \mathop{(cn)}\nolimits ^{z} $
in positive integers is
$x= y= z= 2. $
In this paper, for the primitive Pythagorean triple
$(a, b, c)= (4{k}^{2} - 1, 4k, 4{k}^{2} + 1)$
with
$k= {2}^{s} $
for some positive integer
$s\geq 0$
, we prove the conjecture when
$n\gt 1$
and certain divisibility conditions are satisfied.
We show that the permutation module over
$ \mathbb{C} $
afforded by the action of
${\mathrm{Sp} }_{2m} ({2}^{f} )$
on its natural module is isomorphic to the permutation module over
$ \mathbb{C} $
afforded by the action of
${\mathrm{Sp} }_{2m} ({2}^{f} )$
on the union of the right cosets of
${ \mathrm{O} }_{2m}^{+ } ({2}^{f} )$
and
${ \mathrm{O} }_{2m}^{- } ({2}^{f} )$
.
We prove that the hypotheses in the Pigola–Rigoli–Setti version of the Omori–Yau maximum principle are logically equivalent to the assumption that the manifold carries a
${C}^{2} $
proper function whose gradient and Hessian (Laplacian) are bounded. In particular, this result extends the scope of the original Omori–Yau principle, formulated in terms of lower bounds for curvature.