The k-error linear complexity distribution for 2^n-periodic binary sequences

The linear complexity and the \(k\) -error linear complexity of a sequence have been used as important security measures for key stream sequence strength in linear feedback shift register design. By using the sieve method of combinatorics, we investigate the \(k\) -error linear complexity distribution of \(2^n\) -periodic binary sequences in this paper based on Games–Chan algorithm. First, for \(k=2,3\) , the complete counting functions for the \(k\) -error linear complexity of \(2^n\) -periodic binary sequences (with linear complexity less than \(2^n\) ) are characterized. Second, for \(k=3,4\) , the complete counting functions for the \(k\) -error linear complexity of \(2^n\) -periodic binary sequences with linear complexity \(2^n\) are presented. Third, as a consequence of these results, the counting functions for the number of \(2^n\) -periodic binary sequences with the \(k\) -error linear complexity for \(k = 2\) and \(3\) are obtained.     

Non-Self-Adjoint Resolutions of the Identity and Associated Operators

Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity \(\{X(\lambda )\}_{\lambda \in {\mathbb R}}\) , whose adjoints constitute also a resolution of the identity, are studied. In particular, it is shown that a closed operator \(B\) has a spectral representation analogous to the familiar one for self-adjoint operators if and only if \(B=\textit{TAT}^{-1}\) where \(A\) is self-adjoint and \(T\) is a bounded inverse.     

On weakly closed subgroups of finite groups

Suppose that \(G\) is a finite group and \(H\) , \(K\) are subgroups of \(G\) . We say that \(H\) is weakly closed in \(K\) with respect to \(G\) if, for any \(g \in G\) such that \(H^{g}\le K\) , we have \(H^{g}=H\) . In particular, when \(H\) is a subgroup of prime-power order and \(K\) is a Sylow subgroup containing it, \(H\) is simply said to be a weakly closed subgroup of \(G\) or weakly closed in \(G\) . In the paper, we investigate the structure of finite groups by means of weakly closed subgroups.     

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).     

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

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 \(p\) .     

Weighted Hardy inequality with higher dimensional singularity on the boundary

Let \(\Omega \) be a smooth bounded domain in \(\mathbb R ^N\) with \(N\ge 3\) and let \(\Sigma _k\) be a closed smooth submanifold of \(\partial \Omega \) of dimension \(1\le k\le N-2\) . In this paper we study the weighted Hardy inequality with weight function singular on \(\Sigma _k\) . In particular we provide necessary and sufficient conditions for existence of minimizers.     

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).     

Regularity of minimal hypersurfaces with a common free boundary

Let \(N\) be a Riemannian manifold and consider a stationary union of three or more \(C^{1,\mu }\) hypersurfaces-with-boundary \(M_k \subset N\) with a common boundary \(\Gamma \) . We show that if \(N\) is smooth, then \(\Gamma \) is smooth and each \(M_k\) is smooth up to \(\Gamma \) (real analytic in the case \(N\) is real analytic). Consequently we strengthen a result of Wickramasekera for stable codimension 1 integral varifolds regularity to conclude that under the stronger hypothesis that \(V\) is a stationary, stable, integral \(n\) -varifold in an \((n+1)\) -dimensional, smooth (real analytic) Riemannian manifold such that the support of \(\Vert V\Vert \) is nowhere locally the union of three or more smooth (real analytic) hypersurfaces-with-boundary meeting along a common boundary, the singular set of \(V\) is empty if \(n \le 6\) , is discrete if \(n = 7\) , and has Hausdorff dimension at most \(n-7\) if \(n \ge 8\) .     

Compactness and sequential completeness in some spaces of operators

Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the Banach lattice of all real-valued bounded continuous functions on \(X\) , endowed with the strict topologies \(\beta _\sigma ,\) \(\beta _\tau \) and \(\beta _t\) . Let \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) \((z=\sigma ,\tau ,t)\) stand for the space of all \((\beta _z,\xi )\) -continuous linear operators from \(C_b(X)\) to a locally convex Hausdorff space \((E,\xi ),\) provided with the topology \(\mathcal{T}_s\) of simple convergence. We characterize relative \(\mathcal{T}_s\) -compactness in \(\mathcal{L}_{\beta _z,\xi }(C_b(X),E)\) in terms of the representing Baire vector measures. It is shown that if \((E,\xi )\) is sequentially complete, then the spaces \((\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)\) are sequentially complete whenever \(z=\sigma \) ; \(z=\tau \) and \(X\) is paracompact; \(z=t\) and \(X\) is paracompact and ?ech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on \(C_b(X)\) is given.     

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.     

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.     

Hodge decomposition and solution formulas for some first-order time-dependent parabolic operators with non-constant coefficients

In this paper, we present a Hodge decomposition for the \(L_p\) -space of some parabolic first-order partial differential operators with non-constant coefficients. This is done over different types of domains in Euclidean space \(\mathbb{R }^n\) and on some conformally flat cylinders and the \(n\) -torus associated with different spinor bundles. Initially, we apply a regularization procedure in order to control the non-removable singularities over the hyperplane \(t=0\) . Using the setting of Clifford algebras combined with a Witt basis, we introduce some specific integral and projection operators. We present an \(L_p\) -decomposition where one of the components is the kernel of the regularized parabolic Dirac operator with non-constant coefficients. After that, we study the behavior of the solutions and the validity of our results when the regularization parameter tends to zero. To round off, we give some analytic solution formulas for the special context of domains on cylinders and \(n\) -tori.     

Measurable bundles of C^*-dynamical systems and its applications

In the present paper we investigate \(L_0\) -valued states and Markov operators on \( C^*\) -algebras over \(L_0\) . Here, \(L_0=L_0(\Omega )\) is the algebra of equivalence classes of complex measurable functions on \((\Omega ,\Sigma ,\mu )\) . In particular, we give representations for \(L_0\) -valued states and Markov operators on \(C^*\) -algebras over \(L_0\) , respectively, as measurable bundles of states and Markov operators. Moreover, we apply the obtained representations to study certain ergodic properties of \( C^*\) -dynamical systems over \(L_0\) .     

Independence Under the G-Expectation Framework

We show that, for two non-trivial random variables \(X\) and \(Y\) under a sublinear expectation space, if \(X\) is independent from \(Y\) and \(Y\) is independent from \(X\) , then \(X\) and \(Y\) must be maximally distributed.     

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 \(\mathfrak{X }_p\) .     

Pseudo-effective cone of Grassmann bundles over a curve

Let \(E\) be a vector bundle over a smooth projective curve \(X\) defined over an algebraically closed field \(k\) . For any integer \(1\,\le \, r\, <\, \mathrm{rank}(E)\) , let \(\mathrm{Gr}_r(E)\,\longrightarrow \, X\) be a Grassmann bundle parametrizing all \(r\) dimensional quotients of the fibers of \(E\) . We compute the pseudo-effective cone in the real Néron–Severi group \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) . We prove that this cone coincides with the nef cone in \(\mathrm{NS}(\mathrm{Gr}_r(E))_\mathbb{R }\) if and only if the vector bundle \(E\) is semistable (respectively, strongly semistable) when the characteristic of \(k\) is zero (respectively, positive). Examples are given to show that this characterization of (strong) semistability is not true for vector bundles on higher dimensional projective varieties.     

Michael–Simon inequalities for k-th mean curvatures

This paper continues the study of Alexandrov–Fenchel inequalities for quermassintegrals for \(k\) -convex domains. It focuses on the application to the Michael–Simon type inequalities for \(k\) -curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a domain.     

Intrinsically p-biharmonic maps

For a compact Riemannian manifold \(N\) , a domain \(\Omega \subset \mathbb {R}^m\) and for \(p\in (1, \infty )\) , we introduce an intrinsic version \(E_p\) of the \(p\) -biharmonic energy functional for maps \(u : \Omega \rightarrow N\) . This requires finding a definition for the intrinsic Hessian of maps \(u : \Omega \rightarrow N\) whose first derivatives are merely \(p\) -integrable. We prove, by means of the direct method, existence of minimizers of \(E_p\) within the corresponding intrinsic Sobolev space, and we derive a monotonicity formula. Finally, we also consider more general functionals defined in terms of polyconvex functions.     

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.