A rigidity theorem for the pair ${\cal q}{\Bbb C} P^n$ (complex hyperquadric, complex projective space)

Given a compact Kähler manifold M of real dimension 2n, let P be either a compact complex hypersurface of M or a compact totally real submanifold of dimension n. Let q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n) be the complex hyperquadric (resp. the totally geodesic real projective space) in the complex projective space \Bbb C Pⁿ{\Bbb C} P^n of constant holomorphic sectional curvature 4l \lambda . We prove that if the Ricci and some (n-1)-Ricci curvatures of M (and, when P is complex, the mean absolute curvature of P) are bounded from below by some special constants and volume (P) / volume (M) ￡\leq volume (q\cal q)/ volume (\Bbb C Pⁿ)({\Bbb C} P^n) (resp. ￡\leq volume (\Bbb R Pⁿ)({\Bbb R} P^n) / volume (\Bbb C Pⁿ)({\Bbb C} P^n)), then there is a holomorphic isometry between M and \Bbb C Pⁿ{\Bbb C} P^n taking P isometrically onto q\cal q (resp. \Bbb R Pⁿ{\Bbb R} P^n). We also classify the Kähler manifolds with boundary which are tubes of radius r around totally real and totally geodesic submanifolds of half dimension, have the holomorphic sectional and some (n-1)-Ricci curvatures bounded from below by those of the tube \Bbb R Pⁿ_r{\Bbb R} P^n_r of radius r around \Bbb R Pⁿ{\Bbb R} P^n in \Bbb C Pⁿ{\Bbb C} P^n and have the first Dirichlet eigenvalue not lower than that of \Bbb R Pⁿ_r{\Bbb R} P^n_r.     

Circular Favard Length of the Four-Corner Cantor Set

Let C_n\mathcal{C}_{n} be the n-th generation in the construction of the middle-half Cantor set. The Cartesian square K_n\mathcal{K}_{n} of C_n\mathcal{C}_{n} consists of 4 ⁿ squares of side-length 4⁻ⁿ . We drop a circle of radius r on the plane and try to estimate from below the conditional probability of this circle to intersect K_n\mathcal{K}_{n} if it already intersects a disc containing K_n\mathcal{K}_{n}. If the radius is very large  ≈4 ⁿ then clearly this should not differ too much from the usual Buffon needle probability. But it turns out that the best known lower bound (Bateman and Volberg in , 2008) persists even when the radius is much smaller than this—r>Cn ^ε suffices—and the intersection probability is at least \fracC_elognn\frac{C_{\varepsilon}\log n}{n}. This suggests that the method of Bateman and Volberg (, 2008) may be of use in proving a certain estimate for the lacunary circular maximal function from Seeger et al. (Preprint, 2005).     

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.     

Calderón-Zygmund Capacities and Wolff Potentials on Cantor Sets

We show that, for some Cantor sets in ℝ ^d , the capacity γ _s associated with the s-dimensional Riesz kernel x/|x| ^s+1 is comparable to the capacity [(C)\dot]_{\frac23(d-s),\frac32}\dot{C}_{\frac{2}{3}(d-s),\frac{3}{2}} from non-linear potential theory. It is an open problem to show that, when s is positive and non-integer, they are comparable for all compact sets in ℝ ^d . We also discuss other open questions in the area.     

Projective normality of Weyl group quotients

In this note, we prove that for the standard representation Vof the Weyl group W of a semi-simple algebraic group of type A _n , B _n , C _n , D _n , F ₄ and G ₂ over \mathbbC\mathbb{C}, the projective variety \mathbbP(V^m)/W\mathbb{P}(V^m)/W is projectively normal with respect to the descent of O(1)^?|W|\mathcal{O}(1)^{\otimes |W|}, where V ^m denote the direct sum of m copies of V.     

On matrix near-rings

Let R be a right near-ring with identity and M_n(R) be the near-ring of n 2 n matrices over R in the sense of Meldrum and Van der Walt. In this paper, M_n(R) is said to be s\sigma-generated if every n 2 n matrix A over R can be expressed as a sum of elements of X_n(R), where X_n(R)={f_ij^r | 1\leqq i, j\leqq n, r ? R}X_n(R)=\{f_{ij}^r\,|\,1\leqq i, j\leqq n, r\in R\}, is the generating set of M_n(R). We say that R is s\sigma-generated if M_n(R) is s\sigma-generated for every natural number n. The class of s\sigma-generated near-rings contains distributively generated and abstract affine near-rings. It is shown that this class admits homomorphic images. For abelian near-rings R, we prove that the zerosymmetric part of R is a ring, so the class of zerosymmetric abelian s\sigma-generated near-rings coincides with the class of rings. Further, for every n, there is a bijection between the two-sided subgroups of R and those of M_n(R).     

INJECTIVE RESOLVENTS AND PREENVELOPES

Abstract

Let R be a noetherian ring, and denote the full subcategories of R-modules L such that Extⁱ(E,L)=0 for all injective R-modules E for 1?i?n and O?i?n by C_n, and C′_n respectively. Then LεC_n, if and only if every injective resolution of L is an injective resolvent of the nth cosyzygy. In this case, L is not injective if and only if its injective dimension is greater than n. If LεC′_n and idN?n. then Hom(N,L)=0 for all R-modules N. As an application, let K_n be the nth syzygy of an injective resolvent of the nth cosyzygy of an R-module N, then there exists a homomorphism φ:N → K such that ((φ,i_N), K_n ? E(N)) and (φ,K_n) are preenvelopes of N for C_s and C′_s respectively, for s≥n. If the global dimension of R is at most 2, then C′₁ is reflective in the category of R-modules.     

The fundamental theory of abstract majorization inequalities

Using the axiomatic method,abstract concepts such as abstract mean, abstract convex function and abstract majorization are proposed. They are the generalizations of concepts of mean, convex function and majorization, respectively. Through the logical deduction, the fundamental theorems about abstract majorization inequalities are established as follows: for arbitrary abstract mean Σ and Σ , and abstract Σ → Σ strict convex function f(x) on the interval I, if xi, yi ∈ I (i = 1, 2, . . . , n) satisfy that (x1...     

On the mean square of Lerch zeta-functions

Using the approximate functional equation for L(l,a, s) = ?_n=0^￥ [(e(ln))/((n+a)^s)] L(\lambda,\alpha, s) = \sum\limits_{n=0}^{\infty} {e(\lambda n)\over (n+\alpha)^s} , we prove for fixed parameters $ 0<\lambda,\alpha\leq 1 $ 0<\lambda,\alpha\leq 1 asymptotic formulas for the mean square of L(l,a,s) L(\lambda,\alpha,s) inside the critical strip. This improves earlier results of D. Klusch and of A. Laurin)ikas.     

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

A criterion for the splitting of a principal bundle over a projective space

Let E_G be an algebraic principal G-bundle over \mathbbC\mathbbPⁿ ,\mathbb{C}\mathbb{P}^n , n  \mathbbC.\mathbb{C}. We prove that E_G admits a reduction of structure group to a one-parameter subgroup of G if and only if

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      14. Smoothed analysis of condition numbers and complexity implications for linear programming      John Dunagan  Daniel A. Spielman  Shang-Hua Teng 《Mathematical Programming》2011,126(2):315-350 We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every n-by-d matrix Ā, n-vector [`(\varvec b)]{\bar{\varvec b}}, and d-vector [`(\varvec c)]{\bar{\varvec c}} satisfying ||[`(A)], [`(\varvec b)], [`(\varvec c)]||F ￡ 1{{||\bar{\bf A}, \bar{\varvec b}, \bar{\varvec c}||_F \leq 1}} and every σ ≤ 1, EA,\varvec b,\varvec c [logC (A,\varvec b,\varvec c) = O (log(nd/s)),\mathop{\bf E}\limits_{\bf A,\varvec b,\varvec c }{{[\log C (\bf A,\varvec b,\varvec c)} = O (\log (nd/\sigma)),}      15. On Regular Vertices of the Union of Planar Convex Objects      Esther Ezra  János Pach  Micha Sharir 《Discrete and Computational Geometry》2009,41(2):216-231 Let be a collection of n compact convex sets in the plane such that the boundaries of any pair of sets in intersect in at most s points for some constant s≥4. We show that the maximum number of regular vertices (intersection points of two boundaries that intersect twice) on the boundary of the union U of is O *(n 4/3), which improves earlier bounds due to Aronov et al. (Discrete Comput. Geom. 25, 203–220, 2001). The bound is nearly tight in the worst case. In this paper, a bound of the form O *(f(n)) means that the actual bound is C ε f(n)⋅n ε for any ε>0, where C ε is a constant that depends on ε (and generally tends to ∞ as ε decreases to 0). Work by János Pach and Micha Sharir was supported by NSF Grant CCF-05-14079, and by a grant from the U.S.–Israeli Binational Science Foundation. Work by Esther Ezra and Micha Sharir was supported by grant 155/05 from the Israel Science Fund and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work on this paper by the first author has also been supported by an IBM Doctoral Fellowship. A preliminary version of this paper has been presented in Proc. 23nd Annu. ACM Sympos. Comput. Geom., 2007, pp. 220–226. E. Ezra’s current address: Department of Computer Science, Duke University, Durham, NC 27708-0129, USA. E-mail: esther@cs.duke.edu      16. The Probability that a Convex Body Intersects the Integer Lattice in a <Emphasis Type="Italic">k</Emphasis>-dimensional Set      Edgardo Roldán-Pensado 《Discrete and Computational Geometry》2012,47(2):288-300 Let K be a convex body in ℝ d . It is known that there is a constant C 0 depending only on d such that the probability that a random copy ρ(K) of K does not intersect ℤ d is smaller than \fracC0|K|\frac{C_{0}}{|K|} and this is best possible. We show that for every k<d there is a constant C such that the probability that ρ(K) contains a subset of dimension k is smaller than \fracC|K|\frac{C}{|K|}. This is best possible if k=d−1. We conjecture that this is not best possible in the rest of the cases; if d=2 and k=0 then we can obtain better bounds. For d=2, we find the best possible value of C 0 in the limit case when width(K)→0 and |K|→∞.      17. Complete hypersurfaces with constant scalar curvature in spheres      Aldir Brasil  A. Gervasio Colares  Oscar Palmas 《Monatshefte für Mathematik》2010,24(3):369-380 To a given immersion i:Mn? \mathbb Sn+1{i:M^n\to \mathbb S^{n+1}} with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|2. We prove the existence of a constant C n (R) depending on R and n so that R ≥ 1 and sup |A|2 = C n (R) imply that the hypersurface is a H(r)-torus \mathbb S1(?{1-r2})×\mathbb Sn-1 (r){\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}. For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > C n (R) there is a complete hypersurface in \mathbb Sn+1{\mathbb S^{n+1}} with constant scalar curvature R and sup |A|2 = C, answering questions raised by Q. M. Cheng.      18. Jordan-Ketten und realisierungen rationaler matrizen      Harald K. Wimmer 《Linear algebra and its applications》1978,20(2):101-110 If the inverse of a square polynomial matrix L(s) is proper rational, then L(s)-1 can be written as C(sI–J)-1B. The result of this note states that if J is an nXn Jordan matrix, with n=degreedetL(s), then C consists of Jordan chains of L(s), and BT of Jordan chains of L(s)T. This is a generalization of the fact that each matrix which transforms a complex matrix A into Jordan form is made up of eigenvectors and generalized eigenvectors of A. The proof of our result relies on the realization theory of rational matrices.      19. The corona theorem on the complement of certain square cantor sets      Jon Handy 《Journal d'Analyse Mathématique》2009,108(1):1-18 Let K be a square Cantor set, i.e., the Cartesian product K = E × E of two linear Cantor sets. Let δ n denote the proportion of the intervals removed in the nth stage of the construction of E. It is shown that if $ \delta _n = o(\frac{1} {{\log \log n}}) $ \delta _n = o(\frac{1} {{\log \log n}}) , then the corona theorem holds on the domain Ω = ℂ* \ K.      20. The best extension of algebraic polynomials from the unit circle      A. V. Parfenenkov 《Proceedings of the Steklov Institute of Mathematics》2009,266(Z1):194-204 We study the class $ \mathfrak{P}_n $ \mathfrak{P}_n of algebraic polynomials P n (x, y) in two variables of total degree n whose uniform norm on the unit circle Γ1 centered at the origin is at most 1: $ \left\| {P_n } \right\|_{C(\Gamma _1 )} $ \left\| {P_n } \right\|_{C(\Gamma _1 )} ≤ 1. The extension of polynomials from the class $ \mathfrak{P}_n $ \mathfrak{P}_n to the plane with the least uniform norm on the concentric circle Γ r of radius r is investigated. It is proved that the values θ n (r) of the best extension of the class $ \mathfrak{P}_n $ \mathfrak{P}_n satisfy the equalities θ n (r) = r n for r > 1 and θ n (r) = r n−1 for 0 < r < 1.

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Smoothed analysis of condition numbers and complexity implications for linear programming

We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every n-by-d matrix Ā, n-vector [`(\varvec b)]{\bar{\varvec b}}, and d-vector [`(\varvec c)]{\bar{\varvec c}} satisfying ||[`(A)], [`(\varvec b)], [`(\varvec c)]||_F ￡ 1{{||\bar{\bf A}, \bar{\varvec b}, \bar{\varvec c}||_F \leq 1}} and every σ ≤ 1,

On Regular Vertices of the Union of Planar Convex Objects

Let be a collection of n compact convex sets in the plane such that the boundaries of any pair of sets in intersect in at most s points for some constant s≥4. We show that the maximum number of regular vertices (intersection points of two boundaries that intersect twice) on the boundary of the union U of is O ^*(n ^4/3), which improves earlier bounds due to Aronov et al. (Discrete Comput. Geom. 25, 203–220, 2001). The bound is nearly tight in the worst case. In this paper, a bound of the form O ^*(f(n)) means that the actual bound is C _ε f(n)⋅n ^ε for any ε>0, where C _ε is a constant that depends on ε (and generally tends to ∞ as ε decreases to 0). Work by János Pach and Micha Sharir was supported by NSF Grant CCF-05-14079, and by a grant from the U.S.–Israeli Binational Science Foundation. Work by Esther Ezra and Micha Sharir was supported by grant 155/05 from the Israel Science Fund and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Work on this paper by the first author has also been supported by an IBM Doctoral Fellowship. A preliminary version of this paper has been presented in Proc. 23nd Annu. ACM Sympos. Comput. Geom., 2007, pp. 220–226. E. Ezra’s current address: Department of Computer Science, Duke University, Durham, NC 27708-0129, USA. E-mail: esther@cs.duke.edu     

The Probability that a Convex Body Intersects the Integer Lattice in a <Emphasis Type="Italic">k</Emphasis>-dimensional Set

Let K be a convex body in ℝ ^d . It is known that there is a constant C ₀ depending only on d such that the probability that a random copy ρ(K) of K does not intersect ℤ ^d is smaller than \fracC₀|K|\frac{C_{0}}{|K|} and this is best possible. We show that for every k<d there is a constant C such that the probability that ρ(K) contains a subset of dimension k is smaller than \fracC|K|\frac{C}{|K|}. This is best possible if k=d−1. We conjecture that this is not best possible in the rest of the cases; if d=2 and k=0 then we can obtain better bounds. For d=2, we find the best possible value of C ₀ in the limit case when width(K)→0 and |K|→∞.     

Complete hypersurfaces with constant scalar curvature in spheres

To a given immersion i:Mⁿ? \mathbb Sⁿ⁺¹{i:M^n\to \mathbb S^{n+1}} with constant scalar curvature R, we associate the supremum of the squared norm of the second fundamental form sup |A|². We prove the existence of a constant C _n (R) depending on R and n so that R ≥ 1 and sup |A|² = C _n (R) imply that the hypersurface is a H(r)-torus \mathbb S¹(?{1-r²})×\mathbb S^n-1 (r){\mathbb S^1(\sqrt{1-r^2})\times\mathbb S^{n-1} (r)}. For R > (n − 2)/n we use rotation hypersurfaces to show that for each value C > C _n (R) there is a complete hypersurface in \mathbb Sⁿ⁺¹{\mathbb S^{n+1}} with constant scalar curvature R and sup |A|² = C, answering questions raised by Q. M. Cheng.     

Jordan-Ketten und realisierungen rationaler matrizen

If the inverse of a square polynomial matrix L(s) is proper rational, then L(s)^-1 can be written as C(sI–J)^-1B. The result of this note states that if J is an nXn Jordan matrix, with n=degreedetL(s), then C consists of Jordan chains of L(s), and B^T of Jordan chains of L(s)^T. This is a generalization of the fact that each matrix which transforms a complex matrix A into Jordan form is made up of eigenvectors and generalized eigenvectors of A. The proof of our result relies on the realization theory of rational matrices.     

The corona theorem on the complement of certain square cantor sets

Let K be a square Cantor set, i.e., the Cartesian product K = E × E of two linear Cantor sets. Let δ _n denote the proportion of the intervals removed in the nth stage of the construction of E. It is shown that if $ \delta _n = o(\frac{1} {{\log \log n}}) $ \delta _n = o(\frac{1} {{\log \log n}}) , then the corona theorem holds on the domain Ω = ℂ^* \ K.     

The best extension of algebraic polynomials from the unit circle

We study the class $ \mathfrak{P}_n $ \mathfrak{P}_n of algebraic polynomials P _n (x, y) in two variables of total degree n whose uniform norm on the unit circle Γ₁ centered at the origin is at most 1: $ \left\| {P_n } \right\|_{C(\Gamma _1 )} $ \left\| {P_n } \right\|_{C(\Gamma _1 )} ≤ 1. The extension of polynomials from the class $ \mathfrak{P}_n $ \mathfrak{P}_n to the plane with the least uniform norm on the concentric circle Γ _r of radius r is investigated. It is proved that the values θ _n (r) of the best extension of the class $ \mathfrak{P}_n $ \mathfrak{P}_n satisfy the equalities θ _n (r) = r ⁿ for r > 1 and θ _n (r) = r ⁿ⁻¹ for 0 < r < 1.