Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

There is exactly one $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-cyclic 1-perfect code

Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect code. We also prove that such a code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic.     

On the additivity of block designs

We show that symmetric block designs \({\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})\) can be embedded in a suitable commutative group \({\mathfrak {G}}_{\mathcal {D}}\) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of \({\mathrm {PG}}(d,2)\) and \({\mathrm {AG}}(d,3)\). In both cases, the blocks can be characterized as the only k-subsets of \(\mathcal {P}\) whose elements sum to zero. It follows that the group of automorphisms of any such design \(\mathcal {D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal {D}\) that leave \(\mathcal {P}\) invariant. In some special cases, the group \({\mathfrak {G}}_\mathcal {D}\) can be determined uniquely by the parameters of \(\mathcal {D}\). For instance, if \(\mathcal {D}\) is a 2-\((v,k,\lambda )\) symmetric design of prime order p not dividing k, then \({\mathfrak {G}}_\mathcal {D}\) is (essentially) isomorphic to \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of \(\mathcal {B}\) can be characterized also as the v intersections of \(\mathcal {P}\) with v suitable hyperplanes of \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\).     

Derivations,Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

For a commutative C*-algebra \({\mathcal {A}}\) with unit e and a Hilbert \({\mathcal {A}}\)-module \({\mathcal {M}}\), denote by End\(_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all bounded \({\mathcal {A}}\)-linear mappings on \({\mathcal {M}}\), and by End\(^*_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all adjointable mappings on \({\mathcal {M}}\). We prove that if \({\mathcal {M}}\) is full, then each derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is \({\mathcal {A}}\)-linear, continuous, and inner, and each 2-local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) or End\(^{*}_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation. If there exist \(x_0\) in \({\mathcal {M}}\) and \(f_0\) in \({\mathcal {M}}^{'}\), such that \(f_0(x_0)=e\), where \({\mathcal {M}}^{'}\) denotes the set of all bounded \({\mathcal {A}}\)-linear mappings from \({\mathcal {M}}\) to \({\mathcal {A}}\), then each \({\mathcal {A}}\)-linear local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation.     

On MDS convolutional codes over $${\mathbb {Z}}_{p^{r}}$$

Maximum distance separable (MDS) convolutional codes are characterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) was recently discovered in Oued and Sole (IEEE Trans Inf Theory 59(11):7305–7313, 2013) via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over \({\mathbb {Z}}_{p^{r}}\) from a new perspective. We introduce the notions of p-standard form and r-optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over \({\mathbb {Z}}_{p^{r}}\) for any given set of parameters.     

$${{\mathbb {Z}}}_2$$-double cyclic codes

A binary linear code C is a \({\mathbb {Z}}_2\)-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the \({\mathbb {Z}}_2[x]\)-module \({\mathbb {Z}}_2[x]/(x^r-1)\times {\mathbb {Z}}_2[x]/(x^s-1).\) We determine the structure of \({\mathbb {Z}}_2\)-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of \({\mathbb {Z}}_2\)-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between \({{\mathbb {Z}}}_2\)-double cyclic and other families of cyclic codes, and show some examples of distance optimal \({\mathbb {Z}}_2\)-double cyclic codes.     

Integer Decomposition Property of Free Sums of Convex Polytopes

Let \({\mathcal{P} \subset \mathbb{R}^{d}}\) and \({\mathcal{Q} \subset \mathbb{R}^{e}}\) be integral convex polytopes of dimension d and e which contain the origin of \({\mathbb{R}^{d}}\) and \({\mathbb{R}^{e}}\), respectively. We say that an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^{d}}\) possesses the integer decomposition property if, for each \({n\geq1}\) and for each \({\gamma \in n\mathcal{P}\cap\mathbb{Z}^{d}}\), there exist \({\gamma^{(1)}, . . . , \gamma^{(n)}}\) belonging to \({\mathcal{P}\cap\mathbb{Z}^{d}}\) such that \({\gamma = \gamma^{(1)} +. . .+\gamma^{(n)}}\). In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of \({\mathcal{P}}\) and \({\mathcal{Q}}\) to possess the integer decomposition property will be presented.     

Dirac–Krein Systems on Star Graphs

Suppose that \({\mathcal {M}}\) is a countably decomposable type II\({_1}\) von Neumann algebra and \({\mathcal {A}}\) is a separable, non-nuclear, unital C\({^*}\)-algebra. We show that, if \({\mathcal {M}}\) has Property \({\Gamma}\), then the similarity degree of \({\mathcal {M}}\) is less than or equal to 5. If \({\mathcal {A}}\) has Property c\({^*}\)-\({\Gamma}\), then the similarity degree of \({\mathcal {A}}\) is equal to 3. In particular, the similarity degree of a \({\mathcal {Z}}\)-stable, separable, non-nuclear, unital C\({^*}\)-algebra is equal to 3.     

Joins and subdirect products of varieties

We generalise in three different directions two well-known results in universal algebra. Grätzer, Lakser and P?onka proved that independent subvarieties \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) of a variety \({\mathcal{V}}\) are disjoint and such that their join \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) (in the lattice of subvarieties of \({\mathcal{V}}\)) is their direct product \({\mathcal{V}_{1} \times \mathcal{V}_{2}}\) . Jónsson and Tsinakis provided a partial converse to this result: if \({\mathcal{V}}\) is congruence permutable and \({\mathcal{V}_{1}, \mathcal{V}_{2}}\) are disjoint, then they are independent (and so \({\mathcal{V}_{1} \vee \mathcal{V}_{2} = \mathcal{V}_{1} \times \mathcal{V}_{2}}\)). We show that (i) if \({\mathcal{V}}\) is subtractive, then Jónsson’s and Tsinakis’ result holds under some minimal assumptions; (ii) if \({\mathcal{V}}\) satisfies some weakened permutability conditions, then disjointness implies a generalised notion of independence and \({\mathcal{V}_{1} \vee \mathcal{V}_{2}}\) is the subdirect product of \({\mathcal{V}_{1}}\) and \({\mathcal{V}_2}\) ; (iii) the same holds if \({\mathcal{V}}\) is congruence 3-permutable.     

Normes cyclotomiques naïves et unités logarithmiques

We compute the \({\mathbb {Z}}\)-rank of the subgroup \(\widetilde{E}_K =\bigcap _{n\in {\mathbb {N}}} N_{K_n/K}(K_n^\times )\) of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic \({\mathbb {Z}}_\ell \)-extension \(K^c\) of K. Thus we compare its \(\ell \)-adification \({\mathbb {Z}}_\ell \otimes _{\mathbb {Z}}\widetilde{E}_K\) with the group of logarithmic units \(\widetilde{\varepsilon }_K\). By the way we point out an easy proof of the Gross–Kuz’min conjecture for \(\ell \)-undecomposed extensions of abelian fields.     

The Eight Cayley–Dickson Doubling Products

The purpose of this paper is to identify all eight of the basic Cayley–Dickson doubling products. A Cayley–Dickson algebra \({\mathbb{A}_{N+1}}\) of dimension \({2^{N+1}}\) consists of all ordered pairs of elements of a Cayley–Dickson algebra \({\mathbb{A}_{N}}\) of dimension \({2^N}\) where the product \({(a, b)(c, d)}\) of elements of \({\mathbb{A}_{N+1}}\) is defined in terms of a pair of second degree binomials \({(f(a, b, c, d), g(a, b, c,d))}\) satisfying certain properties. The polynomial pair\({(f, g)}\) is called a ‘doubling product.’ While \({\mathbb{A}_{0}}\) may denote any ring, here it is taken to be the set \({\mathbb{R}}\) of real numbers. The binomials \({f}\) and \({g}\) should be devised such that \({\mathbb{A}_{1} = \mathbb{C}}\) the complex numbers, \({\mathbb{A}_{2} = \mathbb{H}}\) the quaternions, and \({\mathbb{A}_{3} = \mathbb{O}}\) the octonions. Historically, various researchers have used different yet equivalent doubling products.     

A characterization of $$\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$$-linear codes

We prove that the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is exactly the class of \(\mathbb {Z}_2\)-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which have a nontrivial \(\mathbb {Z}_2\mathbb {Z}_2[u]\) structure. Moreover, we exhibit some examples of \(\mathbb {Z}_2\)-linear codes which are not \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear. Also, we state that the duality of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is the same as the duality of \(\mathbb {Z}_2\)-linear codes. Finally, we prove that the class of \(\mathbb {Z}_2\mathbb {Z}_4\)-linear codes which are also \(\mathbb {Z}_2\)-linear is strictly contained in the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes.     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

On the Convergence to Equilibrium of Unbounded Observables Under a Family of Intermittent Interval Maps

We consider a family \({\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}\) of Markov interval maps interpolating between the tent map \({T_{0}}\) and the Farey map \({T_{1}}\). Letting \({\mathcal{P}_{r}}\) denote the Perron–Frobenius operator of \({T_{r}}\), we show, for \({\beta \in [0, 1]}\) and \({\alpha \in (0, 1)}\), that the asymptotic behaviour of the iterates of \({\mathcal{P}_{r}}\) applied to observables with a singularity at \({\beta}\) of order \({\alpha}\) is dependent on the structure of the \({\omega}\)-limit set of \({\beta}\) with respect to \({T_{r}}\). The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.     

Convergence in p-mean for arrays of row-wise extended negatively dependent random variables

It is known that the maximal operator \({\sigma^{\kappa,*}(f)} := sup_{n \in \mathbf{P}}{|{\sigma}_{n}^{\kappa} (f)|}\) is bounded from the dyadic Hardy space \({H_{p}}\) into the space \({L_{p}}\) for \({p > 2/3}\) [6]. Moreover, Goginava and Nagy showed that \({\sigma^{\kappa,*}}\) is not bounded from the Hardy space \({H_{2/3}}\) to the space \({L_{2/3}}\) [9]. The main aim of this paper is to investigate the case \({0 < p < 2/3}\). We show that the weighted maximal operator \({\tilde{\sigma}^{\kappa,*,p}(f) :=sup_{n\in \mathbf{P}} \frac{|{\sigma}_{n}^\kappa (f)|}{n^{2/p-3}}}\), is bounded from the Hardy space \({H_{p}}\) into the space \({L_{p}}\) for any \({0 < p < 2/3}\). With its aid we provide a necessary and sufficient condition for the convergence of Walsh–Kaczmarz–Marcinkiewicz means in terms of modulus of continuity on the Hardy space \({H_p}\), and prove a strong convergence theorem for this means.     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S ^s (M).     

Remarks on Complex Symmetric Operators

An operator \({T\in{\mathcal{L}}({\mathcal{H}})}\) is said to be complex symmetric if there exists a conjugation C on \({{\mathcal H}}\) such that \({T= CT^{\ast}C}\). In this paper, we study the spectral radius algebras for complex symmetric operators. In particular, we prove that if A is a complex symmetric operator, then the spectral radius algebra \({{\mathcal B}_{A}}\) associated with A has a nontrivial invariant subspace under some conditions. Finally, we give some relations between \({P_{\tilde{A}}}\) and \({P_{\widetilde{A^{\ast}}}}\) (defined below) when A is complex symmetric.     

Corona problem with data in ideal spaces of sequences

Let E be a Banach lattice on \({\mathbb {Z}}\) with order continuous norm. We show that for any function \(f = \{f_j\}_{j \in {\mathbb {Z}}}\) from the Hardy space \(\mathrm H_{\infty }\left( E \right) \) such that \(\delta \leqslant \Vert f (z)\Vert _E \leqslant 1\) for all z from the unit disk \({\mathbb {D}}\) there exists some solution \(g = \{g_j\}_{j \in {\mathbb {Z}}} \in \mathrm H_{\infty }\left( E' \right) \), \(\Vert g\Vert _{\mathrm H_{\infty }\left( E' \right) } \leqslant C_\delta \) of the Bézout equation \(\sum _j f_j g_j = 1\), also known as the vector-valued corona problem with data in \(\mathrm H_{\infty }\left( E \right) \).