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.     

On conditional permutability and factorized groups

Two subgroups \(A\) and \(B\) of a group \(G\) are said to be totally completely conditionally permutable (tcc-permutable) if \(X\) permutes with \(Y^g\) for some \(g\in \langle X,Y\rangle \) , for all \(X \le A\) and all \(Y\le B\) . In this paper, we study finite products of tcc-permutable subgroups, focussing mainly on structural properties of such products. As an application, new achievements in the context of formation theory are obtained.     

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.     

Noether–Lefschetz theory with base locus

Let \(Z\) be a closed subscheme of a smooth complex projective variety \(Y\subseteq \mathbb {P}^N\) , with \(\dim \,Y=2r+1\ge 3\) . We describe the intermediate Néron–Severi group (i.e. the image of the cycle map \(A_r(X)\rightarrow H_{2r}(X;\mathbb {Z})\) ) of a general smooth hypersurface \(X\subset Y\) of sufficiently large degree containing \(Z\) .     

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.     

Almost Ricci solitons and K-contact geometry

We give a short Lie-derivative theoretic proof of the following recent result of Barros et al. “A compact non-trivial almost Ricci soliton with constant scalar curvature is gradient, and isometric to a Euclidean sphere”. Next, we obtain the result: a complete almost Ricci soliton whose metric \(g\) is \(K\) -contact and flow vector field \(X\) is contact, becomes a Ricci soliton with constant scalar curvature. In particular, for \(X\) strict, \(g\) becomes compact Sasakian Einstein.     

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.     

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble

Let \(K\) be a global field and \(G\) a finite solvable \(K\) -group. Under certain hypotheses concerning the extension splitting \(G\) , we show that the homogeneous space \(V=G'/G\) with \(G'\) a semi-simple simply connected \(K\) -group has the weak approximation property. We use a more precise version of this result to prove the Hasse principle for homogeneous spaces \(X\) under a semi-simple simply connected \(K\) -group \(G'\) with finite solvable geometric stabilizer \({\bar{G}}\) , under certain hypotheses concerning the \(K\) -kernel (or \(K\) -lien) \(({\bar{G}},\kappa )\) defined by \(X\) .     

The V-filtration for tame unit F-crystals

Let \(X\) be a smooth variety over an algebraically closed field of characteristic \(p > 0, Z\) a smooth divisor, and \(j: U=X {\setminus } Z \rightarrow X\) the natural inclusion. We introduce in an axiomatic way the notion of a \(V\) -filtration on unit \(F\) -crystals and prove such axioms determine a unique filtration. It is shown that if \(\mathcal M \) is a tame unit \(F\) -crystal on \(U\) , then such a \(V\) -filtration along \(Z\) exists on \(j_*\mathcal M \) . The degree zero component of the associated graded module is proven to be the (unipotent) nearby cycles functor of Grothendieck and Deligne under the Emerton–Kisin Riemann–Hilbert correspondence. A few applications to \(\mathbb A ^1\) and gluing are then discussed.     

Pencils on separating (M-2)-curves

A separating ( \(M-2\) )-curve is a smooth geometrically irreducible real projective curve \(X\) such that \(X(\mathbb{R })\) has \(g-1\) connected components and \(X(\mathbb{C })\setminus X(\mathbb{R })\) is disconnected. Let \(T_g\) be a Teichmüller space of separating ( \(M-2\) )-curves of genus g. We consider two partitions of \(T_g\) , one by means of a concept of special type, the other one by means of the separating gonality. We show that those two partitions are very closely related to each other. As an application, we obtain the existence of real curves having isolated real linear systems \(g^1_{g-1}\) for all \(g\ge 4\) .     

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.     

Counting and Sampling Minimum Cuts in Genus g Graphs

Let \(G\) be a directed graph with \(n\) vertices embedded on an orientable surface of genus \(g\) with two designated vertices \(s\) and \(t\) . We show that computing the number of minimum \((s,t)\) -cuts in \(G\) is fixed-parameter tractable in \(g\) . Specifically, we give a \(2^{O(g)} n^2\) time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are no prior results that are fixed-parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count minimum cuts once, can sample an \((s,t)\) -minimum cut uniformly at random in \(O(n \log n)\) time per sample.     

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.     

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.     

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.     

Truncated Quillen complexes of p-groups

Let \(p\) be an odd prime and let \(P\) be a \(p\) -group. We examine the order complex of the poset of elementary abelian subgroups of \(P\) having order at least \(p^2\) . Bouc and Thévenaz showed that this complex has the homotopy type of a wedge of spheres. We show that, for each nonnegative integer \(l\) , the number of spheres of dimension \(l\) in this wedge is controlled by the number of extraspecial subgroups \(X\) of \(P\) having order \(p^{2l+3}\) and satisfying \(\Omega _1(C_P(X))=Z(X)\) . We go on to provide a negative answer to a question raised by Bouc and Thévenaz concerning restrictions on the homology groups of the given complex.     

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 \(A_4\) .     

On Gromov’s Method of Selecting Heavily Covered Points

A result of Boros and Füredi ( \(d=2\) ) and of Bárány (arbitrary \(d\) ) asserts that for every \(d\) there exists \(c_d>0\) such that for every \(n\) -point set \(P\subset {\mathbb {R}}^d\) , some point of \({\mathbb {R}}^d\) is covered by at least of the \(d\) -simplices spanned by the points of  \(P\) . The largest possible value of \(c_d\) has been the subject of ongoing research. Recently Gromov improved the existing lower bounds considerably by introducing a new, topological proof method. We provide an exposition of the combinatorial component of Gromov’s approach, in terms accessible to combinatorialists and discrete geometers, and we investigate the limits of his method. In particular, we give tighter bounds on the cofilling profiles for the \((n-1)\) -simplex. These bounds yield a minor improvement over Gromov’s lower bounds on \(c_d\) for large \(d\) , but they also show that the room for further improvement through the cofilling profiles alone is quite small. We also prove a slightly better lower bound for \(c_3\) by an approach using an additional structure besides the cofilling profiles. We formulate a combinatorial extremal problem whose solution might perhaps lead to a tight lower bound for  \(c_d\) .     

Severi inequality for varieties of maximal Albanese dimension

In this paper, we prove the general Severi inequality for varieties of maximal Albanese dimension. Suppose that \(X\) is an \(n\) -dimensional projective, normal, minimal and \(\mathbb {Q}\) -Gorenstein variety of general type in characteristic zero. If \(X\) is of maximal Albanese dimension, then \(K^n_X \ge 2 n! \chi (\omega _X)\) .