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.     

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.     

Endomorphism rings of bimodules

Let \(M\) be an \(R\) - \(R\) -bimodule over a semi-prime right and left Goldie ring \(R\) . We investigate how non-singularity conditions on \(M_R\) are related to such conditions on \(_RM\) . In particular, we say an \(R\) - \(R\) -bimodule \(M\) such that \(_RM\) and \(M_R\) are non-singular has the right essentiality property if \(IM_R\) is essential in \(M_R\) for all essential right ideals \(I\) of \(R\) , and investigate several questions related to this property.     

On the commuting probability and supersolvability of finite groups

For a finite group \(G\) , let \(d(G)\) denote the probability that a randomly chosen pair of elements of \(G\) commute. We prove that if \(d(G)>1/s\) for some integer \(s>1\) and \(G\) splits over an abelian normal nontrivial subgroup \(N\) , then \(G\) has a nontrivial conjugacy class inside \(N\) of size at most \(s-1\) . We also extend two results of Barry, MacHale, and Ní Shé on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if \(d(G)>5/16\) then either \(G\) is supersolvable, or \(G\) isoclinic to \(A_4\) , or \(G/\mathbf{Z}(G)\) is isoclinic to \(A_4\) .     

Optimal mass transport and symmetric representations of their cost functions

We consider Monge–Kantorovich problems corresponding to general cost functions \(c(x,y)\) but with symmetry constraints on a Polish space \(X\times X\) . Such couplings naturally generate anti-symmetric Hamiltonians on \(X\times X\) that are \(c\) -convex with respect to one of the variables. In particular, if \(c\) is differentiable with respect to the first variable on an open subset \(X\) in \( \mathbb {R}^d\) , we show that for every probability measure \(\mu \) on \(X\) , there exists a symmetric probability measure \(\pi _0\) on \(X\times X\) with marginals \(\mu \) , and an anti-symmetric Hamiltonian \(H\) such that \(\nabla _2H(y, x)=\nabla _1c(x,y)\) for \( \pi _0\) -almost all \((x,y) \in X \times X.\) If \(\pi _0\) is supported on a graph \((x, Sx)\) , then \(S\) is necessarily a \(\mu \) -measure preserving involution (i.e., \(S^2=I\) ) and \(\nabla _2H(x, Sx)=\nabla _1c(Sx,x)\) for \(\mu \) -almost all \(x \in X.\) For monotone cost functions such as those given by \(c(x,y)=\langle x, u(y)\rangle \) or \(c(x,y)=-|x-u(y)|^2\) where \(u\) is a monotone operator, \(S\) is necessarily the identity yielding a classical result by Krause, namely that \(u(x)=\nabla _2H(x, x)\) where \(H\) is anti-symmetric and concave-convex.     

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 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 \(U\) .     

Characterization of ellipses as uniformly dense sets with respect to a family of convex bodies

Let \(K\subset \mathbb R ^N\) be a convex body containing the origin. A measurable set \(G\subset \mathbb R ^N\) with positive Lebesgue measure is said to be uniformly \(K\) -dense if, for any fixed \(r>0\) , the measure of \(G\cap (x+r K)\) is constant when \(x\) varies on the boundary of \(G\) (here, \(x+r K\) denotes a translation of a dilation of \(K\) ). We first prove that \(G\) must always be strictly convex and at least \(C^{1,1}\) -regular; also, if \(K\) is centrally symmetric, \(K\) must be strictly convex, \(C^{1,1}\) -regular and such that \(K=G-G\) up to homotheties; this implies in turn that \(G\) must be \(C^{2,1}\) -regular. Then for \(N=2\) , we prove that \(G\) is uniformly \(K\) -dense if and only if \(K\) and \(G\) are homothetic to the same ellipse. This result was already proven by Amar et al. in 2008 . However, our proof removes their regularity assumptions on \(K\) and \(G\) , and more importantly, it is susceptible to be generalized to higher dimension since, by the use of Minkowski’s inequality and an affine inequality, avoids the delicate computations of the higher-order terms in the Taylor expansion near \(r=0\) for the measure of \(G\cap (x+r\,K)\) (needed in 2008).     

The Ubiquity of the Orders of Fractional Semifields of Even Characteristic

To each non-square integer \(2^{2N+1}\ge 2^5\) there correspond semifields \(D\) of order of \(2^{2N+1}\) that contain \(\text{ GF}(4)\) . Hence there exist affine planes for each non-square order \(2^{2N+1}\ge 2^{5}\) that contain subaffine planes of order \(2^2\) . Moreover, there also exists semifields \(D_1\) and \(D_2\) , with \(|D_1|= |D_2| =|D|\) such that \(D_1\) is commutative and \(D_2\) is non-commutative but neither \(D_1\) nor \(D_2\) contains \(\text{ GF}(4)\) .     

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.     

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.     

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.     

Scattered linear sets generated by collineations between pencils of lines

Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\) , and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\) . In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\) -linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n\) with \(A\) and \(B\) . We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\) -linear set of pseudoregulus type with transversal points \(A\) and \(B\) .     

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.     

Salem numbers in dynamics on Kähler threefolds and complex tori

Let \(X\) be a compact Kähler manifold of dimension \(k\!\le \! 4\) and \(f{:}X\!\rightarrow \! X\) a pseudo-automorphism. If the first dynamical degree \(\lambda _1(f)\) is a Salem number, we show that either \(\lambda _1(f)=\lambda _{k-1}(f)\) or \(\lambda _1(f)^2=\lambda _{k-2}(f)\) . In particular, if \({\dim }(X)=3\) then \(\lambda _1(f)=\lambda _2(f)\) . We use this to show that if \(X\) is a complex 3-torus and \(f\) is an automorphism of \(X\) with \(\lambda _1(f)>1\) , then \(f\) has a non-trivial equivariant holomorphic fibration if and only if \(\lambda _1(f)\) is a Salem number. If \(X\) is a complex 3-torus having an automorphism \(f\) with \(\lambda _1(f)=\lambda _2(f)>1\) but is not a Salem number, then the Picard number of \(X\) must be 0, 3 or 9, and all these cases can be realized.     

Fully inert subgroups of free Abelian groups

A subgroup \(H\) of an Abelian group \(G\) is called fully inert if \((\phi H + H)/H\) is finite for every \(\phi \in \mathrm{End}(G)\) . Fully inert subgroups of free Abelian groups are characterized. It is proved that \(H\) is fully inert in the free group \(G\) if and only if it is commensurable with \(n G\) for some \(n \ge 0\) , that is, \((H + nG)/H\) and \((H + nG)/nG\) are both finite. From this fact we derive a more structural characterization of fully inert subgroups \(H\) of free groups \(G\) , in terms of the Ulm–Kaplansky invariants of \(G/H\) and the Hill–Megibben invariants of the exact sequence \(0 \rightarrow H \rightarrow G \rightarrow G/H \rightarrow 0\) .     

Proper circulant weighing matrices of weight p^2

A circulant weighing matrix \(CW(v,n)\) is a circulant matrix \(M\) of order \(v\) with \(0,\pm 1\) entries such that \(MM^T=nI_v\) . In this paper, we study proper circulant matrices with \(n=p^2\) where \(p\) is an odd prime divisor of \(v\) . For \(p\ge 5\) , it turns out that to search for such circulant matrices leads us to two group ring equations and by studying these two equations, we manage to prove that no proper \(CW(pw,p^2)\) exists when \(p\equiv 3\pmod {4}\) or \(p=5\) .     

Prime graphs of finite groups with a small number of triangles

The prime graph \(\Delta (G)\) of a finite group \(G\) is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of \(G\) and two distinct primes \(p\) and \(q\) are joined by an edge if the product \(pq\) divides some character degree of \(G\) . In this paper, we determine the upper bounds for the numbers of vertices of the prime graphs of finite groups which possess a small number of triangles. In some cases, we study the structure of such finite groups and their prime graphs in detail.     

Generalization of Sabitov’s Theorem to Polyhedra of Arbitrary Dimensions

In 1996 Sabitov proved that the volume \(V\) of an arbitrary simplicial polyhedron \(P\) in the \(3\) -dimensional Euclidean space \(\mathbb {R}^3\) satisfies a monic (with respect to \(V\) ) polynomial relation \(F(V,\ell )=0\) , where \(\ell \) denotes the set of the squares of edge lengths of \(P\) . In 2011 the author proved the same assertion for polyhedra in \(\mathbb {R}^4\) . In this paper, we prove that the same result is true in arbitrary dimension \(n\ge 3\) . Moreover, we show that this is true not only for simplicial polyhedra, but for all polyhedra with triangular \(2\) -faces. As a corollary, we obtain the proof in arbitrary dimension of the well-known Bellows Conjecture posed by Connelly in 1978. This conjecture claims that the volume of any flexible polyhedron is constant. Moreover, we obtain the following stronger result. If \(P_t, t\in [0,1]\) , is a continuous deformation of a polyhedron such that the combinatorial type of \(P_t\) does not change and every \(2\) -face of \(P_t\) remains congruent to the corresponding face of \(P_0\) , then the volume of \(P_t\) is constant. We also obtain non-trivial estimates for the oriented volumes of complex simplicial polyhedra in \(\mathbb {C}^n\) from their orthogonal edge lengths.