Orthogonality in \ell _p-spaces and its bearing on ordered Banach spaces

We introduce a notion of \(p\) -orthogonality in a general Banach space for \(1 \le p \le \infty \) . We use this concept to characterize \(\ell _p\) -spaces among Banach spaces and also among complete order smooth \(p\) -normed spaces as (ordered) Banach spaces with a total \(p\) -orthonormal set (in the positive cone). We further introduce a notion of \(p\) -orthogonal decomposition in order smooth \(p\) -normed spaces. We prove that if the \(\infty \) -orthogonal decomposition holds in an order smooth \(\infty \) -normed space, then the \(1\) -orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.     

Hyperreflexivity of bounded N-cocycle spaces of banach algebras

Let \(X\) and \(Y\) be Banach spaces, \(n\in \mathbb {N}\) , and \(B^n(X,Y)\) the space of bounded \(n\) -linear maps from \(X\times \ldots \times X\) ( \(n\) -times) into \(Y\) . The concept of hyperreflexivity has already been defined for subspaces of \(B(X,Y)\) , where \(X\) and \(Y\) are Banach spaces. We extend this concept to the subspaces of \(B^n(X,Y)\) , taking into account its \(n\) -linear structure. We then investigate when \(\mathcal {Z}^n(A,X)\) , the space of all bounded \(n\) -cocycles from a Banach algebra \(A\) into a Banach \(A\) -bimodule \(X\) , is hyperreflexive. Our approach is based on defining two notions related to a Banach algebra, namely the strong property \((\mathbb {B})\) and bounded local units, and then applying them to find uniform criterions under which \(\mathcal {Z}^n(A,X)\) is hyperreflexive. We also demonstrate that these criterions are satisfied in variety of examples including large classes of C \(^*\) -algebras and group algebras and thereby providing various examples of hyperreflexive \(n\) -cocyle spaces. One advantage of our approach is that not only we obtain the hyperreflexivity for bounded \(n\) -cocycle spaces in different cases but also our results generalize the earlier ones on the hyperreflexivity of bounded derivation spaces, i.e. when \(n=1\) , in the literature. Finally, we investigate the hereditary properties of the strong property \((\mathbb {B})\) and b.l.u. This allows us to come with more examples of bounded \(n\) -cocycle spaces which are hyperreflexive.     

Supercyclic abelian semigroups of matrices on \mathbb {C}^{n}

We give a complete characterization of a supercyclic abelian semigroup of matrices on \(\mathbb {C}^{n}\) . For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over \(\mathbb {C}\) that form a supercyclic abelian semigroup on \({\mathbb {C}}^{n}\) . In particular, no abelian semigroup generated by \(n-1\) matrices on \(\mathbb {C}^{n}\) can be supercyclic.     

Classification of conformal minimal immersions of constant curvature from S^2 to { HP}^2

In this paper, we study geometry of conformal minimal two-spheres immersed in quaternionic projective spaces. We firstly use Bahy-El-Dien and Wood’s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from \(S^2\) to the quaternionic projective space \({ HP}^2\) . Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from \(S^2\) to the quaternionic projective space \({ HP}^2\) .     

On the Chromatic Number of \mathbb {R}^4

The lower bound for the chromatic number of \(\mathbb {R}^4\) is improved from \(7\) to \(9\) . Three graphs with unit distance embeddings in \(\mathbb {R}^4\) are described. The first is a \(7\) -chromatic graph of order \(14\) whose chromatic number can be verified by inspection. The second is an \(8\) -chromatic graph of order \(26\) . In this case the chromatic number can be verified quickly by a simple computer program. The third graph is a \(9\) -chromatic graph of order \(65\) for which computer verification takes about one minute.     

A class of \ell ^p saturated Banach spaces

We present a new class of reflexive \(\ell ^p\) saturated Banach spaces \(\mathfrak{X }_p\) for \(1<p<\infty \) with rather tight structure. The norms of these spaces are defined with the use of a modification of the standard method yielding hereditarily indecomposable Banach spaces. The space \(\mathfrak{X }_p\) does not embed into a space with an unconditional basis and for any analytic decomposition into two subspaces, it is proved that one of them embeds isomorphically into the \(\ell ^p\) -sum of a sequence of finite dimensional normed spaces. We also study the space of operators of \(\mathfrak{X }_p\) .     

Square functions in the Hermite setting for functions with values in UMD spaces

In this paper, we characterize the Lebesgue Bochner spaces \(L^p({\mathbb{R }}^{n},B),\, 1 , by using Littlewood–Paley \(g\) -functions in the Hermite setting, provided that \(B\) is a UMD Banach space. We use \(\gamma \) -radonifying operators \(\gamma (H,B)\) where \(H=L^2((0,\infty ),\frac{\mathrm{d}t}{t})\) . We also characterize the UMD Banach spaces in terms of \(L^p({\mathbb{R }}^{n},B)-L^p({\mathbb{R }}^{n},\gamma (H,B))\) boundedness of Hermite Littlewood–Paley \(g\) -functions.     

{l_p}-Recovery of the Most Significant Subspace Among Multiple Subspaces with Outliers

We assume data sampled from a mixture of \(d\) -dimensional linear subspaces with spherically symmetric distributions within each subspace and an additional outlier component with spherically symmetric distribution within the ambient space (for simplicity, we may assume that all distributions are uniform on their corresponding unit spheres). We also assume mixture weights for the different components. We say that one of the underlying subspaces of the model is most significant if its mixture weight is higher than the sum of the mixture weights of all other subspaces. We study the recovery of the most significant subspace by minimizing the \(l_p\) -averaged distances of data points from \(d\) -dimensional subspaces of \(\mathbb R^D\) , where \(0 < p \in \mathbb R\) . Unlike other \(l_p\) minimization problems, this minimization is nonconvex for all \(p>0\) and thus requires different methods for its analysis. We show that if \(0 , then for any fraction of outliers, the most significant subspace can be recovered by \(l_p\) minimization with overwhelming probability (which depends on the generating distribution and its parameters). We show that when adding small noise around the underlying subspaces, the most significant subspace can be nearly recovered by \(l_p\) minimization for any \(0 with an error proportional to the noise level. On the other hand, if \(p>1\) and there is more than one underlying subspace, then with overwhelming probability the most significant subspace cannot be recovered or nearly recovered. This last result does not require spherically symmetric outliers.     

Extremal effective divisors on \overline{\mathcal M}_{1,n}

For every \(n\ge 3\) , we exhibit infinitely many extremal effective divisors on \(\overline{\mathcal M}_{1,n}\) , the Deligne-Mumford moduli space parameterizing stable genus one curves with \(n\) ordered marked points.     

C^{1,\alpha }-regularity for surfaces with H\in L^p

In this paper, we prove several results on the geometry of surfaces immersed in \(\mathbb {R}^3\) with small or bounded \(L^2\) norm of \(|A|\) . For instance, we prove that if the \(L^2\) norm of \(|A|\) and the \(L^p\) norm of \(H\) , \(p>2\) , are sufficiently small, then such a surface is graphical away from its boundary. We also prove that given an embedded disk with bounded \(L^2\) norm of \(|A|\) , not necessarily small, then such a disk is graphical away from its boundary, provided that the \(L^p\) norm of \(H\) is sufficiently small, \(p>2\) . These results are related to previous work of Schoen–Simon (Surfaces with quasiconformal Gauss map. Princeton University Press, Princeton, vol 103, pp 127–146, 1983) and Colding–Minicozzi (Ann Math 160:69–92, 2004).     

Pro-\mathcal {C} completions of orientable \text{ PD }^3-pairs

Let \({\mathcal {C}}\) be a class of finite groups. We study some sufficient conditions for the pro- \({\mathcal {C}}\) completion of an orientable \(\text{ PD }^3\) -pair over \(\mathbb {Z}\) to be an orientable profinite \(\text{ PD }^3\) -pair over \(\mathbb {F}_p\) . More results are proven for the pro- \(p\) completion of \(\text{ PD }^3\) -pairs.     

On an extension of the H^{k} mean curvature flow of closed convex hypersurfaces

In this paper we prove that the \(H^{k}\) ( \(k\) is odd and larger than \(2\) ) mean curvature flow of a closed convex hypersurface can be extended over the maximal time provided that the total \(L^{p}\) integral of the mean curvature is finite for some \(p\) .     

Growth in solvable subgroups of {{\mathrm{GL}}}_r({\mathbb {Z}}/p{\mathbb {Z}})

Let \(K={\mathbb {Z}}/p{\mathbb {Z}}\) and let \(A\) be a subset of \({{\mathrm{GL}}}_r(K)\) such that \(\langle A \rangle \) is solvable. We reduce the study of the growth of \(A\) under the group operation to the nilpotent setting. Fix a positive number \(C\ge 1\) ; we prove that either \(A\) grows (meaning \(|A_3|\ge C|A|\) ), or else there are groups \(U_R\) and \(S\) , with \(U_R\unlhd S \unlhd \langle A\rangle \) , such that \(S/U_R\) is nilpotent, \(A_k\cap S\) is large and \(U_R\subseteq A_k\) , where \(k\) depends only on the rank \(r\) of \({{\mathrm{GL}}}_r(K)\) . Here \(A_k = \{x_1 x_2 \cdots x_k : x_i \in A \cup A^{-1} \cup \{1\}\}\) . When combined with recent work by Pyber and Szabó, the main result of this paper implies that it is possible to draw the same conclusions without supposing that \(\langle A \rangle \) is solvable.     

On an extension of the Frobenius^{\prime } theorem about p-nilpotency of a finite group

Suppose that \(G\) is a finite group and \(H\) is a subgroup of \(G\) . \(H\) is said to be \(s\) -quasinormally embedded in \(G\) if for each prime \(p\) dividing the order of \(H\) , a Sylow \(p\) -subgroup of \(H\) is also a Sylow \(p\) -subgroup of some \(s\) -quasinormal subgroup of \(G\) . We fix in every non-cyclic Sylow subgroup \(P\) of \(G\) some subgroup \(D\) satisfying \(1<|D|<|P|\) and study the \(p\) -nilpotency of \(G\) under the assumption that every subgroup \(H\) of \(P\) with \(|H|=|D|\) is \(s\) -quasinormally embedded in \(G\) . Some recent results and the Frobenius \(^{\prime }\) theorem are generalized.     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

Representations of set-valued risk measures defined on the l-tensor product of Banach lattices

We obtain a representation for set-valued risk measures which are defined on the completed \(l\) -tensor product \(E\widetilde{\otimes }_l G\) of Banach lattices \(E\) and \(G\) . This representation extends known representations for set-valued risk measures defined on Bochner spaces \(L^p(\mathbb {P}, \mathbb {R}^d)\) of \(p\) -integrable functions with values in \(\mathbb {R}^d\) .     

Polynomials for \mathrm{GL}_p\times \mathrm{GL}_q orbit closures in the flag variety

The subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\) -orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\) -polynomials and the Kazhdan–Lusztig–Vogan polynomials.     

On extensions of completely simple semigroups by groups

An example of an extension of a completely simple semigroup \(U\) by a group \(H\) is given which cannot be embedded into the wreath product of \(U\) by \(H\) . On the other hand, every central extension of \(U\) by \(H\) is shown to be embeddable in the wreath product of \(U\) by \(H\) , and any extension of \(U\) by \(H\) is proved to be embeddable in a semidirect product of a completely simple semigroup \(V\) by \(H\) where the maximal subgroups of \(V\) are direct powers of those of \(U\) .     

The {\mathcal {A}}-Truncated K-Moment Problem

Let \({\mathcal {A}}\subseteq {\mathbb {N}}^n\) be a finite set, and \(K\subseteq {\mathbb {R}}^n\) be a compact semialgebraic set. An \({\mathcal {A}}\) -truncated multisequence ( \({\mathcal {A}}\) -tms) is a vector \(y=(y_{\alpha })\) indexed by elements in \({\mathcal {A}}\) . The \({\mathcal {A}}\) -truncated \(K\) -moment problem ( \({\mathcal {A}}\) -TKMP) concerns whether or not a given \({\mathcal {A}}\) -tms \(y\) admits a \(K\) -measure \(\mu \) , i.e., \(\mu \) is a nonnegative Borel measure supported in \(K\) such that \(y_\alpha = \int _K x^\alpha \mathtt {d}\mu \) for all \(\alpha \in {\mathcal {A}}\) . This paper proposes a numerical algorithm for solving \({\mathcal {A}}\) -TKMPs. It aims at finding a flat extension of \(y\) by solving a hierarchy of semidefinite relaxations \(\{(\mathtt {SDR})_k\}_{k=1}^\infty \) for a moment optimization problem, whose objective \(R\) is generated in a certain randomized way. If \(y\) admits no \(K\) -measures and \({\mathbb {R}}[x]_{{\mathcal {A}}}\) is \(K\) -full (there exists \(p \in {\mathbb {R}}[x]_{{\mathcal {A}}}\) that is positive on \(K\) ), then \((\mathtt {SDR})_k\) is infeasible for all \(k\) big enough, which gives a certificate for the nonexistence of representing measures. If \(y\) admits a \(K\) -measure, then for almost all generated \(R\) , this algorithm has the following properties: i) we can asymptotically get a flat extension of \(y\) by solving the hierarchy \(\{(\mathtt {SDR})_k\}_{k=1}^\infty \) ; ii) under a general condition that is almost sufficient and necessary, we can get a flat extension of \(y\) by solving \((\mathtt {SDR})_k\) for some \(k\) ; iii) the obtained flat extensions admit a \(r\) -atomic \(K\) -measure with \(r\le |{\mathcal {A}}|\) . The decomposition problems for completely positive matrices and sums of even powers of real linear forms, and the standard truncated \(K\) -moment problems, are special cases of \({\mathcal {A}}\) -TKMPs. They can be solved numerically by this algorithm.     

Computing Cup Products in \mathbb {Z}_2-Cohomology of 3D Polyhedral Complexes

Let \(I=(\mathbb {Z}^3,26,6,B)\) be a three-dimensional (3D) digital image, let \(Q(I)\) be an associated cubical complex, and let \(\partial Q(I)\) be a subcomplex of \(Q(I)\) whose maximal cells are the quadrangles of \(Q(I)\) shared by a voxel of \(B\) in the foreground—the object under study—and by a voxel of \(\mathbb {Z}^3\backslash B\) in the background—the ambient space. We show how to simplify the combinatorial structure of \(\partial Q(I)\) and obtain a 3D polyhedral complex \(P(I)\) homeomorphic to \(\partial Q(I)\) but with fewer cells. We introduce an algorithm that computes cup products in \(H^*(P(I);\mathbb {Z}_2)\) directly from the combinatorics. The computational method introduced here can be effectively applied to any polyhedral complex embedded in \(\mathbb {R}^3\) .