Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

Homomorphisms from a finite group into wreath products

Let G be a finite group, A a finite abelian group. Each homomorphism \({\varphi:G\rightarrow A\wr S_n}\) induces a homomorphism \({\overline{\varphi}:G\rightarrow A}\) in a natural way. We show that as \({\varphi}\) is chosen randomly, then the distribution of \({\overline{\varphi}}\) is close to uniform. As application we prove a conjecture of T. Müller on the number of homomorphisms from a finite group into Weyl groups of type D _n .     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

Open book structures on semi-algebraic manifolds

Given a C ² semi-algebraic mapping \({F} : {\mathbb{R}^N \rightarrow \mathbb{R}^p}\), we consider its restriction to \({W \hookrightarrow \mathbb{R^{N}}}\) an embedded closed semi-algebraic manifold of dimension \({n-1 \geq p \geq 2}\) and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection \({\frac{F}{\Vert F \Vert}:W{\setminus} F^{-1}(0) \to S^{p-1}}\). Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering W as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of F with the canonical projection \({\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}}\) and prove that the fibers of \({\frac{F}{\Vert F \Vert}}\) and \({\frac{\pi \circ F}{\Vert \pi \circ F \Vert}}\) are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection \({\frac{F}{\Vert F \Vert}}\) and \({W \cap F^{-1}(0)}\). Similar formulae are proved for mappings obtained after composition of F with canonical projections.     

An inequality between Willmore functional and Weyl functional for submanifolds in space forms

Let \({\phi : M \to R^{n+p}(c)}\) be an n-dimensional submanifold in an (n + p)-dimensional space form R ^n+p(c) with the induced metric g. Willmore functional of \({\phi}\) is \({W(\phi) = \int_{M}(S - nH^{2})^{n/2}dv}\) , where \({S = \sum_{\alpha,i, j}(h^{\alpha}_{ij} )^2}\) is the square of the length of the second fundamental form, H is the mean curvature of M. The Weyl functional of (M, g) is \({\nu(g) = \int_{M}|W_{g}|^{n/2}dv}\) , where \({|W_{g}|^{2} = \sum_{i, j,k,l} W^{2}_{ijkl}}\) and W _ijkl are the components of the Weyl curvature tensor W _g of (M, g). In this paper, we discover an inequality relation between Willmore functional \({W(\phi)}\) and Weyl funtional ν(g).     

M-$${\mathcal{}}$$-Injective (Flat) and Strongly $${\mathcal{B}}$$B-Injective (Flat) Modules

Let M be a left R-module, \({\mathcal{A}}\)be a family of some submodules of M and \({\mathcal{B}}\)be a family of some left R-modules. In this article, we introduce and characterize \({\mathcal{A}}\)-coherent, \({P\mathcal{A}}\), \({F\mathcal{A}}\), M-\({\mathcal{A}}\)-injective (flat) and strongly \({\mathcal{B}}\)-injective (flat) modules, which are generalizations of coherent, PS, FS, M-injective (flat) and strongly M-injective modules, respectively. We extend some known results to this general structure.     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

Links of Faithful Partial Tilting Modules

For a basic, hereditary, finite dimensional algebra Λ over an algebraically closed field k we consider the quiver \({\overrightarrow{\mathcal K}\!_{\Lambda}}\) of tilting modules and the subquivers of \({\overrightarrow{\mathcal K}\!_{\Lambda}}\) which are links \({\overrightarrow{{\text{lk}}}(M)}\) to partial tilting modules M and show that \({\overrightarrow{{\text{lk}}}(M)}\) is connected if M is faithful.     

Codes with the identifiable parent property for multimedia fingerprinting

Let \({\mathcal {C}}\) be a q-ary code of length n and size M, and \({\mathcal {C}}(i) = \{\mathbf{c}(i) \ | \ \mathbf{c}=(\mathbf{c}(1), \mathbf{c}(2), \ldots , \mathbf{c}(n))^{T} \in {\mathcal {C}}\}\) be the set of ith coordinates of \({\mathcal {C}}\). The descendant code of a sub-code \({\mathcal {C}}^{'} \subseteq {\mathcal {C}}\) is defined to be \({\mathcal {C}}^{'}(1) \times {\mathcal {C}}^{'}(2) \times \cdots \times {\mathcal {C}}^{'}(n)\). In this paper, we introduce a multimedia analogue of codes with the identifiable parent property (IPP), called multimedia IPP codes or t-MIPPC(n, M, q), so that given the descendant code of any sub-code \({\mathcal {C}}^{'}\) of a multimedia t-IPP code \({\mathcal {C}}\), one can always identify, as IPP codes do in the generic digital scenario, at least one codeword in \({\mathcal {C}}^{'}\). We first derive a general upper bound on the size M of a multimedia t-IPP code, and then investigate multimedia 3-IPP codes in more detail. We characterize a multimedia 3-IPP code of length 2 in terms of a bipartite graph and a generalized packing, respectively. By means of these combinatorial characterizations, we further derive a tight upper bound on the size of a multimedia 3-IPP code of length 2, and construct several infinite families of (asymptotically) optimal multimedia 3-IPP codes of length 2.     

Tensor Characterizations of Summing Polynomials

Operators T that belong to some summing operator ideal can be characterized by means of the continuity of an associated tensor operator \({\overline{T}}\) that is defined between tensor products of sequences spaces. In this paper, we provide a unifying treatment of these tensor product characterizations of summing operators. We work in the more general frame provided by homogeneous polynomials, where an associated “tensor” polynomial—which plays the role of \({\overline{T}}\)—, needs to be determined first. Examples of applications are shown.     

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle \({\mathcal {E}}\) on X. Let \(\mathfrak {h}\) be the Lie algebra of H. Let \(\mathcal {S}(X,{\mathcal {E}})\) be the space of Schwartz sections of \({\mathcal {E}}\). We prove that \(\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})\) is a closed subspace of \(\mathcal {S}(X,{\mathcal {E}})\) of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let \(\pi \) be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants \(V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}\) is an isomorphism if and only if any linear \(\mathfrak {h}\)-invariant functional on V is continuous in the topology induced from \(\pi \). The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.     

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\), \({n \geq 2}\). First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\), \({\rho > \frac{3}{2}}\). If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\). Second, we study the class of potentials q(r) which can be split into q(r) = q ₁(r) + q ₂(r) such that q ₁(r) has compact support and \({q_2 (r) \in \mathcal{A}}\). If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\). The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\), \({\rho >1}\), we show that the Regge poles are confined in a vertical strip in the complex plane.     

Total Rainbow Connection Number and Complementary Graph

A vertex-colored graph G is rainbow vertex connected if any two distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. In this paper, we prove that for a connected graph G, if \({{\rm diam}(\overline{G}) \geq 3}\), then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. Next, we obtain that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 2}\), if G is connected, then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we prove that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 3}\), if G is connected, then trc\({(G) \leq 5}\), and this bound is tight. Next, a Nordhaus–Gaddum-type result for the total rainbow connection number is provided. We show that if G and \({\overline{G}}\) are both connected, then \({6 \leq {\rm trc} (G) + {\rm trc}(\overline{G}) \leq 4n - 6.}\) Examples are given to show that the lower bound is tight for \({n \geq 7}\) and n = 5. Tight lower bounds are also given for n = 4, 6.     

On countably {\mu}-paracompact spaces

A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {F_i} of non-empty closed sets with \({\bigcap_{i=1}^{\infty} F_{i} = \emptyset}\) there exists a sequence {G_i} of open sets such that \({\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}\) and \({F_{i} \subset G_{i}}\) for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a \({\mu}\)-normal generalized topological space satisfying the analogue of A which is not even countably \({\mu}\)-metacompact. Then we study the relationships between countably \({\mu}\)-paracompactness, countably \({\mu}\)-metacompactness and the condition corresponding to condition A in generalized topological spaces.     

Maximum Principle for H-Surfaces in the Unit Cone and Dirichlet’s Problem for their Equation in Central Projection

In the unit cone\({\mathcal{C} := \{(x, y, z)} \in {\mathbb R}^{3} : {x}^{2} + {y}^{2} < {z}^{2}, {z} > {0}\}\) we establish a geometric maximum principle for H-surfaces, where its mean curvature \({H = H(x, y, z)}\) is optimally bounded. Consequently, these surfaces cannot touch the conical boundary \({\partial \mathcal{C}}\) at interior points and have to approach \({\partial \mathcal{C}}\) transversally. By a nonlinear continuity method, we then solve the Dirichlet problem of the H-surface equation in central projection for Jordan-domains \({\Omega}\) which are strictly convex in the following sense: On its whole boundary \({\partial \mathcal{C}(\Omega)}\) their associate cone \({\mathcal{C}(\Omega) := \{(rx, ry, r) \in {\mathbb R}^{3} : (x, y) \in \Omega, r \in (0,+\infty)}\}\) admits rotated unit cones \({O \circ \mathcal{C}}\) as solids of support, where \({O \in {\mathbb R}^{3\times3}}\) represents a rotation in the Euclidean space. Thus we construct the unique H-surface with one-to-one central projection onto these domains \({\Omega}\) bounding a given Jordan-contour \({\Gamma \subset \mathcal{C} \backslash \{0\}}\) with one-toone central projection.     

Universal Padé approximants and their behaviour on the boundary

There are several kinds of universal Taylor series. In one such kind the universal approximation is required at every boundary point of the domain of definition \({\varOmega }\) of the universal function f. In another kind the universal approximation is not required at any point of \(\partial {\varOmega }\) but in this case the universal function f can be taken smooth on \(\overline{\varOmega }\) and, moreover, it can be approximated by its Taylor partial sums on every compact subset of \(\overline{\varOmega }\). Similar generic phenomena hold when the partial sums of the Taylor expansion of the universal function are replaced by some Padé approximants of it. In the present paper we show that in the case of Padé approximants, if \({\varOmega }\) is an open set and S, T are two subsets of \(\partial {\varOmega }\) that satisfy some conditions, then there exists a universal function \(f\in H({\varOmega })\) which is smooth on \({\varOmega }\cup S\) and has some Padé approximants that approximate f on each compact subset of \({\varOmega }\cup S\) and simultaneously obtain universal approximation on each compact subset of \((\mathbb {C}{\backslash }\overline{\varOmega })\cup T\). A sufficient condition for the above to happen is \(\overline{S}\cap \overline{T}=\emptyset \), while a necessary and sufficient condition is not known.     

A remark on Berger’s conjecture,Kolchin’s theorem,and arc schemes

Let k be a field of characteristic zero. Let V be a k-scheme of finite type, i.e., a k-variety, which is integral. We prove that if the associated arc scheme \({\mathcal{L}_{\infty}(V)}\) is reduced, then the \({\mathcal{O}_{V}}\)-Module \({\Omega_{V/k}^{1}}\) is torsion-free. Then if the k-variety V is assumed to be locally a complete intersection (lci), we deduce that the k-variety V is normal. We also obtain the following consequence: for every class \({\mathfrak{C}}\) of integral k-curves which satisfies the Berger conjecture, and for every \({\mathscr{C} \in \mathfrak{C}}\), the k-curve \({\mathscr{C}}\) is smooth if and only if \({\mathcal{L}(\mathscr{C})}\) is reduced.     

Isoperimetric inequality for the polydisk

Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E ^r (Ω ⁿ ) be the holomorphic Smirnov class on Ω ⁿ with index r. We show that the generalized isoperimetric inequality

$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $

holds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.