Abstract: Masuoka proved (Proc Am Math Soc 137(6):1925–1932, 2009) that a finite-dimensional irreducible Hopf algebra H in positive characteristic is semisimple if and only if it is commutative semisimple if and only if the Hopf subalgebra generated by all primitives is semisimple. In this note, we give another proof of this result by using Hochschild cohomology of coalgebras. PubDate: 2019-03-19

Abstract: In 1941, Brauer-Nesbitt established a characterization of a block with trivial defect group as a block B with \(k(B) = 1\) where k(B) is the number of irreducible ordinary characters of B. In 1982, Brandt established a characterization of a block with defect group of order two as a block B with \(k(B) = 2\) . These correspond to the cases when the block is Morita equivalent to the one-dimensional algebra and to the non-semisimple two-dimensional algebra, respectively. In this paper, we redefine k(A) to be the codimension of the commutator subspace K(A) of a finite-dimensional algebra A and prove analogous statements for arbitrary (not necessarily symmetric) finite-dimensional algebras. This is achieved by extending the Okuyama refinement of the Brandt result to this setting. To this end, we study the codimension of the sum of the commutator subspace K(A) and the nth power \({\text {Rad}}^n(A)\) of the Jacobson radical \({\text {Rad}}(A)\) . We prove that this is Morita invariant and give an upper bound for the codimension as well. PubDate: 2019-03-19

Abstract: We prove that the three-dimensional periodic Burgers’ equation has a unique global in time solution in a critical Gevrey–Sobolev space. Comparatively to Navier–Stokes equations, the main difficulty is the lack of an incompressibility condition. In our proof of existence, we overcome the bootstrapping argument, which was a technical step in a precedent proof in Sololev spaces. This makes our proof shorter and gives sense of considering the Gevrey class for a mathematical study to Burgers’ equation. To prove that the unique solution is global in time, we use the maximum principle. Energy methods, Sobolev product laws, compactness methods, and Fourier analysis are the main tools. PubDate: 2019-03-19

Abstract: We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules. PubDate: 2019-03-12

Abstract: We study the class \(\mathcal {M}_p\) of Schur multipliers on the Schatten-von Neumann class \(\mathcal {S}_p\) with \(1 \le p \le \infty \) as well as the class of completely bounded Schur multipliers \(\mathcal {M}^{cb}_p\) . We first show that for \(2 \le p < q \le \infty \) there exists \(m \in \mathcal {M}^{cb}_p\) with \(m \not \in \mathcal {M}_q\) , so in particular the following inclusions that follow from interpolation are strict: \(\mathcal {M}_q \subsetneq \mathcal {M}_p\) and \(\mathcal {M}^{cb}_q \subsetneq \mathcal {M}^{cb}_p\) . In the remainder of the paper we collect computational evidence that for \(p\not = 1,2, \infty \) we have \(\mathcal {M}_p = \mathcal {M}^{cb}_p\) , moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Astérisque 247:vi+131, 1998) is false. PubDate: 2019-03-12

Abstract: Let \(\mathcal {\scriptstyle {O}}_K\) be the ring of integers of an imaginary quadratic number field K. In this paper we give a new description of the maximal discrete extension of the group \(SL_2(\mathcal {\scriptstyle {O}}_K)\) inside \(SL_2(\mathbb {C})\) , which uses generalized Atkin–Lehner involutions. Moreover we find a natural characterization of this group in SO(1, 3). PubDate: 2019-03-09

Abstract: The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces. PubDate: 2019-03-08

Abstract: We prove a formula for the number of representations of an element in a finite basic ring as a sum of k exceptional units and find bounds for this number in an arbitrary finite ring with identity. PubDate: 2019-03-07

Abstract: The space \({{\mathcal {L}}}(X, Y)\) stands for the Banach space of all bounded linear operators from X to Y endowed with the operator norm. It is shown that \(c_{0}(\Gamma )\) embeds into \({{\mathcal {L}}}(X, Y)\) if and only if \(l_{\infty }(\Gamma )\) embeds into \({{\mathcal {L}}}(X, Y)\) or \(c_{0}(\Gamma )\) embeds into Y. As a consequence, we extend a classical Kalton theorem by proving that if \(c_{0}(\Gamma )\) embeds into \({{\mathcal {L}}}(X, Y)\) and X has the \( \Gamma \) -Josefson–Nissenzweig property, then \(l_{\infty }(\Gamma )\) also embeds into \({{\mathcal {L}}}(X, Y)\) . We also show that, for certain Banach spaces X and Y, \(c_{0}(\Gamma )\) embeds complementably into \({{\mathcal {L}}}(X, Y)\) if and only if \(c_{0}(\Gamma )\) embeds into Y. PubDate: 2019-03-04

Abstract: We extend a well-known theorem of Burnside in the setting of general fields as follows: for a general field F the matrix algebra \(M_n(F)\) is the only algebra in \(M_n(F)\) which is spanned by an irreducible semigroup of triangularizable matrices. In other words, for a semigroup of triangularizable matrices with entries from a general field irreducibility is equivalent to absolute irreducibility. As a consequence of our result we prove a stronger version of a theorem of Janez Bernik. PubDate: 2019-03-02

Abstract: We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of \({\mathbb {C}}^n\) . Namely, two non-pluripolar, polynomially closed, compact subsets of \({\mathbb {C}}^n\) are interpolated as level sets \(L_t=\{z: u_t(z)=-1\}\) for the geodesic \(u_t\) between their relative extremal functions with respect to any ambient bounded domain. The sets \(L_t\) are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge–Ampère capacities of \(L_t\) satisfy a dual Brunn–Minkowski inequality. PubDate: 2019-03-02

Abstract: Recently S. Patrikis, J.F. Voloch, and Y. Zarhin have proven, assuming several well-known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for K3 surfaces, under some technical restrictions on the Picard rank. This is possible since abelian varieties and K3s are quite well described by ‘Hodge-theoretical’ results. In particular the theorem we present can be interpreted as follows: a family of \(\ell \) -adic representations that looks like the one induced by the transcendental part of the \(\ell \) -adic cohomology of a K3 surface (defined over a number field) determines a Hodge structure which in turn determines a K3 surface (which may be defined over a number field). PubDate: 2019-03-02

Abstract: Let \(I_n(G)\) denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that \(I_2(G)=I_2(S)\) and \(I_p(G) =I_p(S)\) for some odd prime divisor p, then \( G = S \) ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture. PubDate: 2019-03-01

Abstract: In this article, we study simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different non-parallel weights. We also study simultaneous non-vanishing of Fourier coefficients of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights at powers of prime ideals. PubDate: 2019-03-01

Abstract: We show that the Faber–Krahn inequality implies Pólya’s conjecture for eigenvalues \(\lambda _{k}\) of the Dirichlet Laplacian in \(\mathbb {R}^n\) up to \(k = \lfloor b(n) \rfloor \) , where b(n) is a function with exponential growth on the dimension. This function also appears in Pleijel’s bound for the number of nodal domains of the sequence of eigenfunctions of the ball and we improve on previous estimates by providing precise upper and lower bounds for b which coincide up to the first four terms in the expansion of \(\log (b(n))\) for large n. PubDate: 2019-03-01

Abstract: Let G be a finite p-group, \(g\in G\) , \(A\subseteq G\) , and \(q:=p^k\ge G \) . We determine, modulo p, the number of solutions \((x_1,\ldots ,x_q)\in A^q\) of the equation \(x_1\cdots x_q=g\) . More generally, we solve the analogous problem for \((x_1,\ldots ,x_q)\in B_1\times \ldots \times B_q\) with given \(B_i\in \{A,G\setminus A\}\) . PubDate: 2019-03-01

Abstract: In this paper, we obtain bounds on the \(L^1\) -norm of the sum \(\sum _{n\le x}\tau (n)e(\alpha n)\) where \(\tau (n)\) is the divisor function. PubDate: 2019-03-01

Abstract: Let \(m\in \mathbb {N}\) and \(\vec {b}=(b_{1}, \ldots ,b_{m})\) be a collection of locally integrable functions. It is proved that \(b_{1},b_{2}, \ldots , b_{m}\in BMO\) if and only if $$\begin{aligned} \sup _{Q}\frac{1}{ Q ^{m}}\mathop {\int }\limits _{Q^{m}}\Big \sum _{i=1}^{m} \big (b_{i}(x_{i})-(b_{i})_{Q}\big )\Big d\vec {x}<\infty , \end{aligned}$$ where \((b_{i})_{Q}=\frac{1}{ Q }\int _{Q}b_{i}(x)dx\) . As an application, we show that if the linear commutator of a certain multilinear Calderón–Zygmund operator \([\Sigma \vec {b},T]\) is bounded from \(L^{p_{1}}\times \cdots \times L^{p_{m}}\) to \(L^{p}\) with \(\sum _{i=1}^{m}1/p_{i}=1/p\) and \(1<p,p_{1}, \ldots ,p_{m}<\infty \) , then \(b_{1}, \ldots ,b_{m}\in BMO\) . Therefore, the result of Chaffee (Proc R Soc Edinb Sect A 146(6):1159–1166, 2016) or Li and Wick (Can Math Bull 60(3):571–585, 2017) is extended to the general case. PubDate: 2019-03-01