\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.     

Shellability of the higher pinched Veronese posets

The pinched Veronese poset \({\mathcal {V}}^{\bullet }_n\) is the poset with ground set consisting of all nonnegative integer vectors of length \(n\) such that the sum of their coordinates is divisible by \(n\) with exception of the vector \((1,\ldots ,1)\) . For two vectors \(\mathbf {a}\) and \(\mathbf {b}\) in \({\mathcal {V}}^{\bullet }_n\) , we have \(\mathbf {a}\preceq \mathbf {b}\) if and only if \(\mathbf {b}- \mathbf {a}\) belongs to the ground set of \({\mathcal {V}}^{\bullet }_n\) . We show that every interval in \({\mathcal {V}}^{\bullet }_n\) is shellable for \(n \ge 4\) . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in \({\mathcal {V}}^{\bullet }_n\) has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for \(n \ge 4\) . (This also follows from a result by Conca, Herzog, Trung, and Valla.)     

{\mathfrak {sl}}_3-Web bases,intermediate crystal bases and categorification

We give an explicit graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra \(K_S\) . In order to do this, we identify Kuperberg’s basis for the \({\mathfrak {sl}}_3\) -web space \(W_S\) with a version of Leclerc–Toffin’s intermediate crystal basis and we identify Brundan, Kleshchev and Wang’s degree of tableaux with the weight of flows on webs and the \(q\) -degree of foams. We use these observations to give a “foamy” version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra. We restrict ourselves to the \({\mathfrak {sl}}_3\) case over \(\mathbb {C}\) here, but our approach should, up to the combinatorics of \({\mathfrak {sl}}_N\) -webs, work for all \(N>1\) or over \(\mathbb {Z}\) .     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .     

On the Largest Graph-Lagrangian of 3-Graphs with Fixed Number of Edges

The Graph-Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Graph-Lagrangian of a hypergraph. Frankl and Füredi conjectured that the \({r}\) -graph with \(m\) edges formed by taking the first \(\textit{m}\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({r}\) has the largest Graph-Lagrangian of all \(r\) -graphs with \(m\) edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed \(3\) -graphs with \(m\) edges is at most the Graph-Lagrangian of the \(\mathrm 3 \) -graph with \(m\) edges formed by taking the first \(m\) sets in the colex ordering of the collection of all subsets of \({\mathbb N}\) of size \({3}\) .     

Trace Formulae for Curvature of Jet Bundles over Planar Domains

For a domain \(\varOmega \) in \(\mathbb {C}\) and an operator \(T\) in \({\mathcal {B}}_n(\varOmega )\) , Cowen and Douglas construct a Hermitian holomorphic vector bundle \(E_T\) over \(\varOmega \) corresponding to \(T\) . The Hermitian holomorphic vector bundle \(E_T\) is obtained as a pull-back of the tautological bundle \(S(n,{\mathcal {H}})\) defined over \({\mathcal {G}}r(n,{\mathcal {H}})\) by a nondegenerate holomorphic map \(z\mapsto {\mathrm{ker}}(T-z),\;z\in \varOmega \) . To find the answer to the converse, Cowen and Douglas studied the jet bundle in their foundational paper. The computations in this paper for the curvature of the jet bundle are rather intricate. They have given a set of invariants to determine if two rank \(n\) Hermitian holomorphic vector bundle are equivalent. These invariants are complicated and not easy to compute. It is natural to expect that the equivalence of Hermitian holomorphic jet bundles should be easier to characterize. In fact, in the case of the Hermitian holomorphic jet bundle \({\mathcal {J}}_k({\mathcal {L}}_f)\) , we have shown that the curvature of the line bundle \({\mathcal {L}}_f\) completely determines the class of \({\mathcal {J}}_k({\mathcal {L}}_f)\) . In case of rank \(n\) Hermitian holomorphic vector bundle \(E_f\) , We have calculated the curvature of jet bundle \({\mathcal {J}}_k(E_f)\) and also obtained a trace formula for jet bundle \({\mathcal {J}}_k(E_f)\) .     

Non-Lipschitz points and the {\textit{SBV}} regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function \(T\) under controllability conditions which do not imply the Lipschitz continuity of \(T\) . We consider first the case of normal linear control systems with constant coefficients in \({\mathbb {R}}^N\) . We characterize points around which \(T\) is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call \(\mathcal {S}\) . Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\) -rectifiable with positive \(\mathcal {H}^{N-1}\) -measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\) -rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\) . In both cases, \(T\) is known to have epigraph with positive reach, hence to be a locally \(BV\) function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of \(DT\) must be concentrated in \(\mathcal {S}\) , our analysis yields that \(T\) is locally \(SBV\) , i.e., the Cantor part of \(DT\) vanishes. Our results imply also that \(T\) is differentiable outside a \(\mathcal {H}^{N-1}\) -rectifiable set. With small changes, our results are valid also in the case of multiple control input.     

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.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

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.     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

On sharp local turns of planar polynomials

We show that for a real polynomial of degree \(n\) in two variables \(x\) and \(y\) , any local “sharp turn” must have its “size” \({\gtrsim }e^{-Cn^{2}}\) . We also show that there is indeed an example that has a sharp turn of size \({\lesssim }e^{-Cn}\) . This gives a quite satisfactory answer to a problem raised by Guth. The formulation of the problem was inspired by applications of the polynomial method in the study of Kakeya conjecture.     

A remark on the generalized Hodge conjecture

It is well known that for a smooth, projective variety \(X\) over \({\mathbb {C}}\) we have \(N^{p}H^{i}(X,{\mathbb {Q}})\subset F^{p} H^{i}(X,{\mathbb {C}})\cap H^{i}(X,{\mathbb {Q}})\) , where \(N^{\bullet }\) and \(F^{\bullet }\) are respectively the coniveau and Hodge filtrations. In general this inclusion is strict. We introduce a natural subspace \(S^{p,i}\subset F^{p}H^{i}(X,{\mathbb {C}})\) such that \(N^{p}H^{i}(X,{\mathbb {Q}})= S^{p,i}\cap H^{i}(X,{\mathbb {Q}})\) holds true for any \(i,p\) . The main technical tool is the use of semi-algebraic sets, which are available by the triangulation of complex projective varieties.     

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.     

Iterative reweighted minimization methods for l_p regularized unconstrained nonlinear programming

In this paper we study general \(l_p\) regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of the first- and second-order stationary points and hence also of local minimizers of the \(l_p\) minimization problems. We extend some existing iterative reweighted \(l_1\) ( \(\mathrm{IRL}_1\) ) and \(l_2\) ( \(\mathrm{IRL}_2\) ) minimization methods to solve these problems and propose new variants for them in which each subproblem has a closed-form solution. Also, we provide a unified convergence analysis for these methods. In addition, we propose a novel Lipschitz continuous \({\epsilon }\) -approximation to \(\Vert x\Vert ^p_p\) . Using this result, we develop new \(\mathrm{IRL}_1\) methods for the \(l_p\) minimization problems and show that any accumulation point of the sequence generated by these methods is a first-order stationary point, provided that the approximation parameter \({\epsilon }\) is below a computable threshold value. This is a remarkable result since all existing iterative reweighted minimization methods require that \({\epsilon }\) be dynamically updated and approach zero. Our computational results demonstrate that the new \(\mathrm{IRL}_1\) method and the new variants generally outperform the existing \(\mathrm{IRL}_1\) methods (Chen and Zhou in 2012; Foucart and Lai in Appl Comput Harmon Anal 26:395–407, 2009).     

Some Integral Representations and Singular Integral over Plane in Clifford Analysis

In this paper, Cauchy type integral and singular integral over hyper-complex plane \({\prod}\) are considered. By using a special Möbius transform, an equivalent relation between \({\widehat{H}^\mu}\) class functions over \({\prod}\) and \({H^\mu}\) class functions over the unit sphere is shown. For \({\widehat{H}^\mu}\) class functions over \({\prod}\) , we prove the existence of Cauchy type integral and singular integral over \({\prod}\) . Cauchy integral formulas as well as Poisson integral formulas for monogenic functions in upper-half and lower-half space are given respectively. By using Möbius transform again, the relation between the Cauchy type integrals and the singular integrals over \({\prod}\) and unit sphere is built.     

A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold

It is a classical fact that the cotangent bundle \(T^* {\mathcal {M}}\) of a differentiable manifold \({\mathcal {M}}\) enjoys a canonical symplectic form \(\Omega ^*\) . If \(({\mathcal {M}},\mathrm{J} ,g,\omega )\) is a pseudo-Kähler or para-Kähler \(2n\) -dimensional manifold, we prove that the tangent bundle \(T{\mathcal {M}}\) also enjoys a natural pseudo-Kähler or para-Kähler structure \(({\tilde{\hbox {J}}},\tilde{g},\Omega )\) , where \(\Omega \) is the pull-back by \(g\) of \(\Omega ^*\) and \(\tilde{g}\) is a pseudo-Riemannian metric with neutral signature \((2n,2n)\) . We investigate the curvature properties of the pair \(({\tilde{\hbox {J}}},\tilde{g})\) and prove that: \(\tilde{g}\) is scalar-flat, is not Einstein unless \(g\) is flat, has nonpositive (resp. nonnegative) Ricci curvature if and only if \(g\) has nonpositive (resp. nonnegative) Ricci curvature as well, and is locally conformally flat if and only if \(n=1\) and \(g\) has constant curvature, or \(n>2\) and \(g\) is flat. We also check that (i) the holomorphic sectional curvature of \(({\tilde{\hbox {J}}},\tilde{g})\) is not constant unless \(g\) is flat, and (ii) in \(n=1\) case, that \(\tilde{g}\) is never anti-self-dual, unless conformally flat.