On macroscopic dimension of rationally inessential manifolds

We show that, for a rationally inessential orientable closed n-manifold M whose fundamental group is a duality group, the macroscopic dimension of its universal cover [(M)\tilde]\tilde M is strictly less than n: dim _MC [(M)\tilde] < n\tilde M < n. As a corollary, we obtain the following partial result towards Gromov’s conjecture     

A Practical Criterion For Some Submanifolds To Be Totally Geodesic

Our main theorem is a characterization of a totally geodesic K?hler immersion of a complex n-dimensional K?hler manifold M _n into an arbitrary complex (n + p)-dimensional K?hler manifold [(M)\tilde]_n+p\tilde{M}_{n+p} by observing the extrinsic shape of K?hler Frenet curves on the submanifold M _n . Those curves are closely related to the complex structure of M _n .     

Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVⁿ⁺¹,[(F_b)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVⁿ⁺¹{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(F_b)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||_[(a)\tilde], [(b)\tilde]^\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV³,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .     

On some Dirichlet series related to the Riemann zeta function,I

On the assumption of the truth of the Riemann hypothesis for the Riemann zeta function we construct a class of modified von-Mangoldt functions with slightly better mean value properties than the well known function L\Lambda . For every e ? (0,1/2)\varepsilon \in (0,1/2) there is a [(L)\tilde] : \Bbb N ? \Bbb C\tilde {\Lambda} : \Bbb N \to \Bbb C such that¶ i) [(L)\tilde] (n) = L (n) (1 + O(n^-1/4  logn))\tilde {\Lambda} (n) = \Lambda (n) (1 + O(n^{-1/4\,} \log n)) and¶ii) ?_{n \leqq x} [(L)\tilde] (n) (1- [(n)/(x)]) = [(x)/2] + O(x^1/4+e) (x \geqq 2).\sum \limits_{n \leqq x} \tilde {\Lambda} (n) \left(1- {{n}\over{x}}\right) = {{x}\over{2}} + O(x^{1/4+\varepsilon }) (x \geqq 2).¶Unfortunately, this does not lead to an improved error term estimation for the unweighted sum ?_{n \leqq x} [(L)\tilde] (n)\sum \limits_{n \leqq x} \tilde {\Lambda} (n), which would be of importance for the distance between consecutive primes.     

Envelope of holomorphy of a tube domain over a complex Lie group

According to S. Bochner [6, 7]: IfD =B +iℝ ⁿ is a tube domain in ℂ ⁿ , where B is a domain in ℝ ⁿ , and if [(B)\tilde]\tilde B is the convex envelope of B, then any holomorphic function on D extends to the tube domain [(D)\tilde] = [(B)\tilde] + i\mathbbRⁿ \tilde D = \tilde B + i\mathbb{R}^n , which is a univalent envelope of holomorphy of D. We give a generalization of this result to (nonunivalent) tube domains over a complex Lie group which admit a closed sub-group as a real form. Application: If (V, φ) is a tube domain over ℂ ⁿ and if B is the convex envelope of ϕ(V)∩ℝ ⁿ in ℝ ⁿ , then [(V)\tilde] = B + i\mathbbRⁿ \tilde V = B + i\mathbb{R}^n is an envelope of holomorphy of (V, φ).     

A Note on Aluthge Transforms of Complex Symmetric Operators and Applications

In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator [(T)\tilde] = |T|^\frac12 U|T|^\frac12\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}} is complex symmetric, (ii) if T is a complex symmetric operator, then ([(T)\tilde])^*(\tilde{T})^{*} and [(T^*)\tilde]\widetilde{T^{*}} are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then [((T^*))\tilde]_s,t\widetilde{(T^{*})}_{s,t} and ([(T)\tilde]_t,s)^*(\tilde{T}_{t,s})^{*} are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal.     

Growth of generalized Temperley–Lieb algebras connected with simple graphs

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , or [(E)\tilde]₇ {\tilde E_7} . An algebra TL_{G, t} T{L_{\Gamma, \tau }} has exponential growth if and only if the graph Γ coincides with none of the graphs A_n {A_n} , D_n {D_n} , E_n {E_n} , [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , and [(E)\tilde]₇ {\tilde E_7} .     

Multipliers of weighted semigroups and associated Beurling Banach algebras

Given a weighted discrete abelian semigroup (S, ω), the semigroup M _ω (S) of ω-bounded multipliers as well as the Rees quotient M _ω (S)/S together with their respective weights [(w)\tilde]\tilde{\omega} and [(w)\tilde]_q\tilde{\omega}_q induced by ω are studied; for a large class of weights ω, the quotient l¹(M_w(S),[(w)\tilde])/l¹(S,w)\ell^1(M_{\omega}(S),\tilde{\omega})/\ell^1(S,{\omega}) is realized as a Beurling algebra on the quotient semigroup M _ω (S)/S; the Gel’fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples.     

Morse�CSmale theory for flows on covering spaces of compact manifolds

For a dynamical system X on a compact differentiable manifold M and for the dynamical system X(ρ) induced from X by a covering map r :  [(M)\tilde] ?  M{\rho \, : \, \widetilde{M}\, \rightarrow \, M}, we develop algebraic topology methods for estimating the lower bounds on the number of codimension-1 surfaces (i.e., on the number of index-1 equilibria of flows and their stable manifolds) on the boundary of regions of stability on [(M)\tilde]{\widetilde{M}}. We also develop methods for estimating the number of equilibria on the boundaries of stability regions of noncompact manifolds with very general assumptions. Our methods allow us to obtain results for noncompact manifolds in cases when Morse–Smale approach does not work.     

Entropy bounds and second cohomology

When X is a finite complex and p₁X\pi_{1}X acts on \mathbbR²{\mathbb{R}}^2 by translations we give criteria involving H²X for an equivariant map F : [(X)\tilde] ? \mathbbR²F : \tilde{X} \rightarrow {\mathbb{R}}^2 to be onto. Following work of Manning and Shub, this leads to entropy bounds related to Shub’s entropy conjecture.     

Generalized commutators for Marcinkiewicz type integrals with variable kernels

Let A be a function with derivatives of order m and D γ A ∈■β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ∞ (R n ) × L s (S n 1 ) (s ≥ n/(n β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ A Ω and its variation μ A Ω are bounded from L p (R n ) to L q (R n ), where 1 < p < n/β and 1/q = 1/p β/n. The authors also consider the boundedness of μ A Ω and its variation μ A Ω on Hardy spaces.     

A New Recursion in the Theory of Macdonald Polynomials

The bigraded Frobenius characteristic of the Garsia-Haiman module M _μ is known [7, 10] to be given by the modified Macdonald polynomial [(H)\tilde]_m[X; q, t]{\tilde{H}_{\mu}[X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?_p1 [(H)\tilde]_m[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]} is the bigraded Frobenius characteristic of the restriction of M _μ from S _n to S _n-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c _{μ v} occurring in the expansion ?_p1 [(H)\tilde]_m[X; q, t] = ?_{v ? m}c_mv [(H)\tilde]_v[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}. In particular, it follows from this formula that the bigraded Hilbert series F μ (q, t) of M _μ may be calculated from the recursion F_m (q, t) = ?_{v ? m}c_mv F_v (q, t){F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fm(q, t) ? N[q, t]{F\mu (q, t) \in \mathbf{N}[q, t]}. This difficulty arises from the fact that the c _{μ v} have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F _μ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].     

The mixing advantage is less than 2

Corresponding to n independent non-negative random variables X ₁,...,X _n , are values M ₁,...,M _n , where each M _i is the expected value of the maximum of n independent copies of X _i . We obtain an upper bound for the expected value of the maximum of X ₁,...,X _n in terms of M ₁,...,M _n . This inequality is sharp in the sense that the random variables can be chosen so that the bound is approached arbitrarily closely. We also present related comparison results.      

The 2-factoriality of the O’Grady moduli spaces

The aim of this work is to show that the moduli space M ₁₀ introduced by O’Grady is a 2-factorial variety. Namely, M ₁₀ is the moduli space of semistable sheaves with Mukai vector v: = (2, 0, −2) in H^ev(X,\mathbbZ){H^{ev}(X,\mathbb{Z})} on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between v^{^}{v^{\perp}} (sublattice of the Mukai lattice of X) and its image in H² ([(M)\tilde]₁₀, \mathbbZ){H^{2} (\widetilde{M}_{10}, \mathbb{Z})}, lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold [(M)\tilde]₁₀{\widetilde{M}_{10}}, obtained as symplectic resolution of M ₁₀. Similar results are shown for the moduli space M ₆ introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.     

On the Unital C*-Algebras Generated by Certain Subnormal Tuples

We consider an important class of subnormal operator m-tuples M _p (p = m,m + 1, . . .) that is associated with a class of reproducing kernel Hilbert spaces H_p{{\mathcal H}_p} (with M _m being the multiplication tuple on the Hardy space of the open unit ball \mathbb B^2m{{\mathbb B}^{2m}} in \mathbb C^m{{\mathbb C}^m} and M _m+1 being the multiplication tuple on the Bergman space of \mathbb B^2m{{\mathbb B}^{2m}}). Given any two C*-algebras A{\mathcal A} and B{\mathcal B} from the collection {C^*(M_p), C^*([(M)\tilde]_p): p 3 m}{\{C^*({M}_p), C^*({\tilde M}_p): p \geq m\}} , where C*(M _p ) is the unital C*-algebra generated by M _p and C^*([(M)\tilde]_p){C^*({\tilde M}_p)} the unital C*-algebra generated by the dual [(M)\tilde]_p{{\tilde M}_p} of M _p , we verify that A{\mathcal A} and B{\mathcal B} are either *-isomorphic or that there is no homotopy equivalence between A{\mathcal A} and B{\mathcal B} . For example, while C*(M _m ) and C*(M _m+1) are well-known to be *-isomorphic, we find that C^*([(M)\tilde]_m){C^*({\tilde M}_m)} and C^*([(M)\tilde]_m+1){C^*({\tilde M}_{m+1})} are not even homotopy equivalent; on the other hand, C*(M _m ) and C^*([(M)\tilde]_m){C^*({\tilde M}_{m})} are indeed *-isomorphic. Our arguments rely on the BDF-theory and K-theory.     

Large deviations for functionals of stationary processes

Summary Let (S _n ) be a sequence ofR ^d -valued random variables adapted to the internal history of a stationary sequence of random elements (X _n ). We formulate conditions under which the principle of large deviations holds true for the sequence (S _n ).     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H _{\mathfrakg\mathfrak{g} ) of the tangent Lie algebra \mathfrakg\mathfrak{g} of the group G with coefficients in the one-dimensional representation \mathfrakg\mathfrak{g} \mathbbK\mathbb{K} defined by [(W)\tilde] \mathfrakg \tilde \Omega _\mathfrak{g} of H ¹(G/ \mathfrakg\mathfrak{g} .}     

On the 3-Torsion Part of the Homology of the Chessboard Complex

Let 1 ≤ m ≤ n. We prove various results about the chessboard complex M _m,n , which is the simplicial complex of matchings in the complete bipartite graph K _m,n . First, we demonstrate that there is nonvanishing 3-torsion in [(H)\tilde]_d(\sf M_m,n; \mathbb Z){{\tilde{H}_d({\sf M}_{m,n}; {\mathbb Z})}} whenever \fracm+n-43 ￡ d ￡ m-4{{\frac{m+n-4}{3}\leq d \leq m-4}} and whenever 6 ≤ m < n and d = m − 3. Combining this result with theorems due to Friedman and Hanlon and to Shareshian and Wachs, we characterize all triples (m, n, d ) satisfying [(H)\tilde]_d (\sf M_m,n; \mathbb Z) 1 0{{\tilde{H}_d \left({\sf M}_{m,n}; {\mathbb Z}\right) \neq 0}}. Second, for each k ≥ 0, we show that there is a polynomial f _k (a, b) of degree 3k such that the dimension of [(H)\tilde]_k+a+2b-2 (\sf M_{k+a+3b-1,k+2a+3b-1}; \mathbb Z₃){{\tilde{H}_{k+a+2b-2}}\,\left({{\sf M}_{k+a+3b-1,k+2a+3b-1}}; \mathbb Z_{3}\right)}, viewed as a vector space over \mathbbZ₃{\mathbb{Z}_3}, is at most f _k (a, b) for all a ≥ 0 and b ≥ k + 2. Third, we give a computer-free proof that [(H)\tilde]₂ (\sf M_5,5; \mathbb Z) @ \mathbb Z₃{{\tilde{H}_2 ({\sf M}_{5,5}; \mathbb {Z})\cong \mathbb Z_{3}}}. Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of M _m,n to the homology of M _m-2,n-1 and M _m-2,n-3.     

Large deviation results for generalized compound negative binomial risk models

In this paper we extend and improve some results of the large deviation for random sums of random variables. Let {Xn;n 〉 1} be a sequence of non-negative, independent and identically distributed random variables with common heavy-tailed distribution function F and finite mean μ ∈R^＋, {N（n）; n ≥0} be a sequence of negative binomial distributed random variables with a parameter p C （0, 1）, n ≥ 0, let {M（n）; n ≥ 0} be a Poisson process with intensity λ 〉 0. Suppose {N（n）; n ≥ 0}, {Xn; n≥1} and {M（n）; n ≥ 0} are mutually independent. Write S（n） =N（n）∑i=1 Xi-cM（n）.Under the assumption F ∈ C, we prove some large deviation results. These results can be applied to certain problems in insurance and finance.     

Gromov hyperbolicity of negatively curved Finsler manifolds

Let (M, F) be a closed C ^∞ Finsler manifold. The lift of the Finsler metric F to the universal covering space defines an asymmetric distance [(d)\tilde]{\widetilde d} on [(M)\tilde]{\widetilde M}. It is well-known that the classical comparison theorem of Aleksandrov does not exist in the Finsler setting. Therefore, it is necessary to introduce new Finsler tools for the study of the asymmetric metric space ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)}. In this paper, by using the geometric flip map and the unstable-stable angle introduced in [2], we prove that if (M, F) is a closed Finsler manifold of negative flag curvature, then ([(M)\tilde], [(d)\tilde]){(\widetilde M, \widetilde d)} is an asymmetric δ-hyperbolic space in the sense of Gromov.