Proceedings of the Edinburgh Mathematical Society最新文献

We show that if an open set in $mathbb{R}^d$ can be fibered by unit n-spheres, then $d geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n in left{0, 1, 3, 7 right}$. For these values of n, we construct unit n-sphere fibrations in $mathbb{R}^{2n+1}$.

We consider the broadest concept of the gradient Yamabe soliton, the conformal gradient soliton. In this paper, we elucidate the structure of complete gradient conformal solitons under some assumption, and provide some applications to gradient Yamabe solitons. These results enhance the understanding gained from previous research. Furthermore, we give an affirmative partial answer to the Yamabe soliton version of Perelman’s conjecture.

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Thenbegin{align*}&A_{2p}equiv A_2-frac{1648}3p^3B_{p-3} ({rm{mod}} p^4),&A_{2p-1}equiv A_1+frac{16p^3}3B_{p-3} ({rm{mod}} p^4),&A_{3p}equiv A_3-36738p^3B_{p-3} ({rm{mod}} p^4),end{align*}

where $A_n=sum_{k=0}^nbinom{n}k^2binom{n+k}{k}^2$ is the nth Apéry number, and Bn is the nth Bernoulli number.

Consider the multiplication operator MB in $L^2(mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

We establish a new improvement of the classical Lp-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one-dimensional Hardy inequality. Using some radialisation techniques of functions and then exploiting symmetric decreasing rearrangement arguments on the real line, the new multidimensional version of the Hardy inequality is given. Some consequences are also discussed.

In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

Li, Ma and Wang have provided in [13] a partial classification of the so-called Moebius deformable hypersurfaces, that is, the umbilic-free Euclidean hypersurfaces $fcolon M^nto mathbb{R}^{n+1}$ that admit non-trivial deformations preserving the Moebius metric. For $ngeq 5$, the classification was completed by the authors in [12]. In this article we obtain an infinitesimal version of that classification. Namely, we introduce the notion of an infinitesimal Moebius variation of an umbilic-free immersion $fcolon M^nto mathbb{R}^m$ into Euclidean space as a one-parameter family of immersions $f_tcolon M^nto mathbb{R}^m$, with $tin (-epsilon, epsilon)$ and $f_0=f$, such that the Moebius metrics determined by ft coincide up to the first order. Then we characterize isometric immersions $fcolon M^nto mathbb{R}^m$ of arbitrary codimension that admit a non-trivial infinitesimal Moebius variation among those that admit a non-trivial conformal infinitesimal variation, and use such characterization to classify the u

Let Σ be a σ-algebra of subsets of a set Ω and $B(Sigma)$ be the Banach space of all bounded Σ-measurable scalar functions on Ω. Let $tau(B(Sigma),ca(Sigma))$ denote the natural Mackey topology on $B(Sigma)$. It is shown that a linear operator T from $B(Sigma)$ to a Banach space E is Bochner representable if and only if T is a nuclear operator between the locally convex space $(B(Sigma),tau(B(Sigma),ca(Sigma)))$ and the Banach space E. We derive a formula for the trace of a Bochner representable operator $T:B({cal B} o)rightarrow B({cal B} o)$ generated by a function $fin L^1({cal B} o, C(Omega))$, where Ω is a compact Hausdorff space.