Refined methods for the identifiability of tensors

We prove that the general tensor of size \(2^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.9997\frac{2^n}{n+1}\) (the constant 1 being the optimal value). Similarly, the general tensor of size \(3^n\) and rank \(k\) has a unique decomposition as the sum of decomposable tensors if \(k\le 0.998\frac{3^n}{2n+1}\) (the constant 1 being the optimal value). Some results of this flavor are obtained for tensors of any size, but the explicit bounds obtained are weaker.     

Generation of full cycles by a composition of NLFSRs

Non-linear feedback shift registers (NLFSRs) are a generalization of linear feedback shift registers in which a current state is a non-linear function of the previous state. The interest in NLFSRs is motivated by their ability to generate pseudo-random sequences which are typically hard to break with existing cryptanalytic methods. However, it is still not known how to construct large \(n\) -stage NLFSRs which generate full cycles of \(2^n\) possible states. This paper presents a method for generating full cycles by a composition of NLFSRs. First, we show that an \(n*k\) -stage register with period \(O(2^{2n})\) can be constructed from \(k\) NLFSRs with \(n\) -stages by adding to their feedback functions a logic block of size \(O(nk)\) , for \(k > 1\) . This logic block implements Boolean functions representing pairs of states whose successors have to be exchanged in order to join cycles. Then, we show how to join all cycles into one by using one more logic block of size \(O(nk)\) .     

Two-dimensional slices of nonpseudoconvex domains with rough boundary

For a domain \(D\subset {\mathbb C}^n,\; n\ge 3\) , the set \(E\) is defined as the set of all points \(z\in {\mathbb C}^n\) for which the intersection of \(D\) with every complex \(2\) -plane through \(z\) is pseudoconvex. For \(D\) nonpseudoconvex, it is shown that \(E\) is contained in an affine subspace of codimension \(2\) . This results solves a problem raised by Nikolov and Pflug.     

The ring of evenly weighted points on the line

Let \(M_w = ({\mathbb {P}}^1)^n /\!/\hbox {SL}_2\) denote the geometric invariant theory quotient of \(({\mathbb {P}}^1)^n\) by the diagonal action of \(\hbox {SL}_2\) using the line bundle \(\mathcal {O}(w_1,w_2,\ldots ,w_n)\) on \(({\mathbb {P}}^1)^n\) . Let \(R_w\) be the coordinate ring of \(M_w\) . We give a closed formula for the Hilbert function of \(R_w\) , which allows us to compute the degree of \(M_w\) . The graded parts of \(R_w\) are certain Kostka numbers, so this Hilbert function computes stretched Kostka numbers. If all the weights \(w_i\) are even, we find a presentation of \(R_w\) so that the ideal \(I_w\) of this presentation has a quadratic Gröbner basis. In particular, \(R_w\) is Koszul. We obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of \(M_w\) .     

Additive mappings satisfying algebraic conditions in rings

Let \(R\) be any \((n+1)!\) -torsion free ring and \(F,D: R\rightarrow R\) be additive mappings satisfying \(F(x^{n+1})=(\alpha (x))^nF(x)+\sum \nolimits _{i=1}^n (\alpha (x))^{n-i}(\beta (x))^iD(x)\) for all \(x\in R\) , where \(n\) is a fixed integer and \(\alpha \) , \(\beta \) are automorphisms of \(R\) . Then, \(D\) is Jordan left \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan left \((\alpha , \beta )\) -derivation on \(R\) and if additive mappings \(F\) and \(D\) satisfying \(F(x^{n+1})=F(x)(\alpha (x))^n+\sum \nolimits _{i=1}^n (\beta (x))^iD(x)(\alpha (x))^{n-i}\) for all \(x\in R\) . Then, \(D\) is Jordan \((\alpha , \beta )\) -derivation and \(F\) is generalized Jordan \((\alpha , \beta )\) -derivation on \(R\) . At last some immediate consequences of the above theorems have been given.     

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.     

The extremal solution for the fractional Laplacian

We study the extremal solution for the problem \((-\Delta )^s u=\lambda f(u)\) in \(\Omega \) , \(u\equiv 0\) in \(\mathbb R ^n\setminus \Omega \) , where \(\lambda >0\) is a parameter and \(s\in (0,1)\) . We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions \(n<4s\) . We also show that, for exponential and power-like nonlinearities, the extremal solution is bounded whenever \(n<10s\) . In the limit \(s\uparrow 1\) , \(n<10\) is optimal. In addition, we show that the extremal solution is \(H^s(\mathbb R ^n)\) in any dimension whenever the domain is convex. To obtain some of these results we need \(L^q\) estimates for solutions to the linear Dirichlet problem for the fractional Laplacian with \(L^p\) data. We prove optimal \(L^q\) and \(C^\beta \) estimates, depending on the value of \(p\) . These estimates follow from classical embedding results for the Riesz potential in \(\mathbb R ^n\) . Finally, to prove the \(H^s\) regularity of the extremal solution we need an \(L^\infty \) estimate near the boundary of convex domains, which we obtain via the moving planes method. For it, we use a maximum principle in small domains for integro-differential operators with decreasing kernels.     

A nonlinear Stein theorem

For vector valued solutions \(u\) to the \(p\) -Laplacian system \(-\triangle _p u=F\) in a domain of \({\mathbb {R}}^n,\,p>1,\,n \ge 2,\) if \(F\) belongs to the limiting Lorentz space \(L(n,1),\) then \(Du\) is continuous.     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

On matching cover of graphs

A k-matching cover of a graph \(G\) is a union of \(k\) matchings of \(G\) which covers \(V(G)\) . The matching cover number of \(G\) , denoted by \(mc(G)\) , is the minimum number \(k\) such that \(G\) has a \(k\) -matching cover. A matching cover of \(G\) is optimal if it consists of \(mc(G)\) matchings of \(G\) . In this paper, we present an algorithm for finding an optimal matching cover of a graph on \(n\) vertices in \(O(n^3)\) time (if use a faster maximum matching algorithm, the time complexity can be reduced to \(O(nm)\) , where \(m=|E(G)|\) ), and give an upper bound on matching cover number of graphs. In particular, for trees, a linear-time algorithm is given, and as a by-product, the matching cover number of trees is determined.     

Bounding the exponent of a verbal subgroup

We deal with the following conjecture. If \(w\) is a group word and \(G\) is a finite group in which any nilpotent subgroup generated by \(w\) -values has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only. We show that this is true in the case where \(w\) is either the \(n\text{ th }\) Engel word or the word \([x^n,y_1,y_2,\ldots ,y_k]\) (Theorem A). Further, we show that for any positive integer \(e\) there exists a number \(k=k(e)\) such that if \(w\) is a word and \(G\) is a finite group in which any nilpotent subgroup generated by products of \(k\) values of the word \(w\) has exponent dividing \(e\) , then the exponent of the verbal subgroup \(w(G)\) is bounded in terms of \(e\) and \(w\) only (Theorem B).     

Weak lower semicontinuity for polyconvex integrals in the limit case

We prove a lower semicontinuity result for polyconvex functionals of the Calculus of Variations along sequences of maps \(u:\Omega \subset \mathbb{R }^n\rightarrow \mathbb{R }^m\) in \(W^{1,m}\) , \(2\le m\le n\) , bounded in \(W^{1,m-1}\) and convergent in \(L^1\) under mild technical conditions but without any extra coercivity assumption on the integrand.     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

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 \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

L^p-Integrability,Dimensions of Supports of Fourier Transforms and Applications

It is proved that there does not exist any non zero function in \(L^p({\mathbb R}^n)\) with \(1\le p\le 2n/\alpha \) if its Fourier transform is supported by a set of finite packing \(\alpha \) -measure where \(0<\alpha <n\) . It is shown that the assertion fails for \(p>2n/\alpha \) . The result is applied to prove \(L^p\) Wiener Tauberian theorems for \({\mathbb R}^n\) and \(M(2)\) .     

Conical stochastic maximal L^p-regularity for 1\leqslant p<\infty

Let \(A = -\mathrm{div} \,a(\cdot ) \nabla \) be a second order divergence form elliptic operator on \({\mathbb R}^n\) with bounded measurable real-valued coefficients and let \(W\) be a cylindrical Brownian motion in a Hilbert space \(H\) . Our main result implies that the stochastic convolution process $$\begin{aligned} u(t) = \int _0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\geqslant 0, \end{aligned}$$ satisfies, for all \(1\leqslant p<\infty \) , a conical maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)}^p \leqslant C_p^p {\mathbb E}\Vert g \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)}^p. \end{aligned}$$ Here, \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)\) and \(T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)\) are the parabolic tent spaces of real-valued and \(H\) -valued functions, respectively. This contrasts with Krylov’s maximal \(L^p\) -regularity estimate $$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;{\mathbb R}^n))}^p \leqslant C^p {\mathbb E}\Vert g \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;H))}^p \end{aligned}$$ which is known to hold only for \(2\leqslant p<\infty \) , even when \(A = -\Delta \) and \(H = {\mathbb R}\) . The proof is based on an \(L^2\) -estimate and extrapolation arguments which use the fact that \(A\) satisfies suitable off-diagonal bounds. Our results are applied to obtain conical stochastic maximal \(L^p\) -regularity for a class of nonlinear SPDEs with rough initial data.     

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.     

New M-ary sequences with low autocorrelation from interleaved technique

Let \(p\) and \(q\) be two odd primes with \(p=Mf+1\) and \(M\) is even. A new construction of \(M\) -ary sequences of period \(pq\) with low periodic autocorrelation is presented in this paper based on interleaving the \(M\) -ary power residue sequence of period \(p\) according to the quadratic residue with respect to \(q\) . This construction can generate the well-known twin-prime sequence and generalized cyclotomy sequence of order two if \(M=2\) . For \(M=4\) , a new class of quaternary sequences of period \(pq\) with maximal nontrivial autocorrelation value being either \(\sqrt{5}\) or \(3\) is obtained. This achieves the best known results for such kind of quaternary sequences.     

Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

We consider the conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian \({{\mathrm{\mathrm {Gr}}}}\) of \(k\) -planes in \({{\mathrm{\mathbb {C}}}}^n\) . This conormal bundle has a fundamental class \({\kappa _I}\) in the equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . Here \({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\) . The torus \(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor \({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak {gl}_2)\) weight functions \((W_J)_J\) . We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\) . A modified version of the \(\kappa _I\) classes, named \(\kappa '_I\) , satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) . This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \({{\mathrm{\mathrm {Gr}}}}\) . The classes \((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\) . This basis depends on the choice of the coordinate flag in \({{\mathrm{\mathbb {C}}}}^n\) . We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.     

Mean Squared Error Minimization for Inverse Moment Problems

We consider the problem of approximating the unknown density \(u\in L^2(\Omega ,\lambda )\) of a measure \(\mu \) on \(\Omega \subset \mathbb {R}^n\) , absolutely continuous with respect to some given reference measure \(\lambda \) , only from the knowledge of finitely many moments of \(\mu \) . Given \(d\in \mathbb {N}\) and moments of order \(d\) , we provide a polynomial \(p_d\) which minimizes the mean square error \(\int (u-p)^2d\lambda \) over all polynomials \(p\) of degree at most \(d\) . If there is no additional requirement, \(p_d\) is obtained as solution of a linear system. In addition, if \(p_d\) is expressed in the basis of polynomials that are orthonormal with respect to \(\lambda \) , its vector of coefficients is just the vector of given moments and no computation is needed. Moreover \(p_d\rightarrow u\) in \(L^2(\Omega ,\lambda )\) as \(d\rightarrow \infty \) . In general nonnegativity of \(p_d\) is not guaranteed even though \(u\) is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing \(p_d\ge 0\) that minimizes \(\int (u-p)^2d\lambda \) now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating \(u\) .