A Second Main Theorem of Nevanlinna Theory for Closed Subschemes in Subgeneral Position

In this paper, by using Seshadri constants for subschemes, the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in (weak) subgeneral position. As an application of his second main theorem,he obtain a Brody hyperbolicity result for the complement of nef effective divisors. He also give the corresponding Schmidt’s subspace theorem and arithmetic hyperbolicity result in Diophantine approximation.     

The Equivalence between Property (ω) and Weyl’s Theorem

We call T ∈ B(H) consistent in Fredholm and index (briefly a CFI operator) if for each B ∈ B(H),T B and BT are Fredholm together and the same index of B,or not Fredholm together.Using a new spectrum defined in view of the CFI operator,we give the equivalence of Weyl’s theorem and property (ω) for T and its conjugate operator T* .In addition,the property (ω) for operator matrices is considered.     

Holomorphic Curves into Projective Varieties Intersecting Closed Subschemes in Subgeneral Position*

In this paper, the authors introduce the index of subgeneral position for closed subschemes and obtain a second main theorems based on this notion. They also give the corresponding Schmidt’s subspace type theorem via the analogue between Nevanlinna theory and Diophantine approximation.     

ON THE VALIRON’S THEOREM IN THE POLYDISK

In this paper, we discuss the Valiron's theorem in the unit polydisk DN. We prove that for a holomorphic map φ : DN→ DNsatisfying some regular conditions, there exists a holomorphic map θ : DN→ H and a constant α 0 such that θoφ =1/aθ.It is based on the extension of Julia-Wolff-Carath′eodory(JWC) theorem of D in the polydisk.     

A general vectorial Ekeland’s variational principle with a P-distance

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.     

VECTORIAL EKELAND＇S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS

By using the properties of w-distances and Gerstewitz’s functions,we first give a vectorial Takahashi’s nonconvex minimization theorem with a w-distance.From this,we deduce a general vectorial Ekeland’s variational principle,where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value.From the general vectorial variational principle,we deduce a vectorial Caristi’s fixed point theorem with a w-distance.Finally we show that the above three theorems are equivalent to each other.The related known results are generalized and improved.In particular,some conditions in the theorems of [Y.Araya,Ekeland’s variational principle and its equivalent theorems in vector optimization,J.Math.Anal.Appl.346(2008),9-16] are weakened or even completely relieved.     

Bolzano's Theorems for Holomorphic Mappings

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions.A more complicated proof,using Cauchy's argument principle,provides uniqueness of the zero,when the sign conditions on the boundary are strict.Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions.Extensions to holomorphic mappings from Cn to Cn are obtained using Brouwer degree.     

Picard’s theorem and the Rickman construction

Nevanlinna theory (value-distribution theory) has its genesis in Picard’s discovery that a function analytic in the plane which omits two values is constant. Nearly a century later, attention turned to the analogous situation in Rn, n≥3, where entire functions are necesarily replaced by entire quasiregular mappings. This expository article centers on one of Seppo Rickman’s main contributions to this issue, including an outline of his famous example showing that the omitted set in R3, while finite, can be much larger than possible in the plane.     

EXISTENCE AND NONEXISTENCE OF POSITIVE SOLUTIONS TO A THREE-POINT BOUNDARY VALUE PROBLEM

In this paper, we are concerned with the existence and nonexistence of positive solutions to a three-point boundary value problems. By Krasnoselskii’s fixed point theorem in Banach space, we obtain sufficient conditions for the existence and non-existence of positive solutions to the above three-point boundary value problems.     

RIESZ IDEMPOTENT AND BROWDER＇S THEOREM FOR ABSOLUTE-（p,r ）-PARANORMAL OPERATORS

An operator T is said to be paranormal if ||T 2x|| ≥ ||T x||2 holds for every unit vector x.Several extensions of paranormal operators are considered until now,for example absolute-k-paranormal and p-paranormal introduced in [10],[14],respectively.Yamazaki and Yanagida [38] introduced the class of absolute-(p,r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators.An operator T ∈ B(H) is called absolute-(p,r)-paranormal operator if |||T |p|T |r x||r ≥ |||T |rx||p+r for every unit vector x ∈ H and for positive real numbers p > 0 and r > 0.The famous result of Browder,that self adjoint operators satisfy Browder’s theorem,is extended to several classes of operators.In this paper we show that for any absolute-(p,r)paranormal operator T,T satisfies Browder’s theorem and a-Browder’s theorem.It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p,r)-paranormal operator T,then E is self-adjoint if and only if the null space of T μ,N(T μ) N(T μ).     

Second Main Theorem for Meromorphic Maps into Algebraic Varieties Intersecting Moving Hypersurfaces Targets*

Since the great work on holomorphic curves into algebraic varieties intersecting hypersurfaces in general position established by Ru in 2009, recently there has been some developments on the second main theorem into algebraic varieties intersecting moving hypersurfaces targets. The main purpose of this paper is to give some interesting improvements of Ru’s second main theorem for moving hypersurfaces targets located in subgeneral position with index.     

Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank≥2

We obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥2 into any complex space Z. After lifting to the normalization of the subvariety F (Ω) Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou’s theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f : Ω→ D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described.     

ON THE LOWER BOUND FOR A CLASS OF HARMONIC FUNCTIONS IN THE HALF SPACE

The main objective is to derive a lower bound from an upper one for harmonic functions in the half space, which extends a result of B. Y. Levin from dimension 2 to dimension n ≥ 2. To this end, we first generalize the Carleman’s formula for harmonic functions in the half plane to higher dimensional half space, and then establish a Nevanlinna’s representation for harmonic functions in the half sphere by using Hrmander’s theorem.     

性质$(\omega)$的注解

In this note we study the property （ω）, a variant of Weyl＇s theorem introduced by Rakocevic, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a necessary and sufficient condition for which both property （ω） and approximate Weyl＇s theorem hold. As a consequence of the main result, we study the property （ω） and approximate Weyl＇s theorem for a class of operators which we call the λ-weak-H（p） operators.     

Property (ω′) and Its Perturbation

In this note we define the property (ω′), a variant of Weyl’s theorem, and establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for which property (ω′) holds by means of the variant of the essential approximate point spectrum σ1(·) and the spectrum defined in view of the property of consistency in Fredholm and index. In addition, the perturbation of property (ω′) is discussed.     

A Remark on Chen’s Theorem (Ⅱ)

Let p denote a prime and P2 denote an almost prime with at most two prime factors. The author proves that for suffciently large x,sum from p≤x p 2=P2 1>(1.13Cx)/(log~2x), where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.     

The truncated Second Main Theorem and uniqueness theorems

In this paper, we first establish a truncated Second Main Theorem for algebraically nondegenerate holomorphic mappings from the complex plane into a complex projective variety V intersecting hypersurfaces. We then prove some uniqueness results for meromorphic mappings. The result of Demailly about a partial solution to the Fujita’s conjecture is used.     

$(\omega')$性质及其摄动

In this note we define the property （ω′）, a variant of Weyl’s theorem, and establish for a bounded linear operator defined on a Hilbert space the necessary and sufficient conditions for which property （ω′） holds by means of the variant of the essential approximate point spectrum σ1（·） and the spectrum defined in view of the property of consistency in Fredholm and index. In addition, the perturbation of property （ω′） is discussed.     

Monomorphism categories associated with symmetric groups and parity in finite groups

Monomorphism categories of the symmetric and alternating groups are studied via Cayley’s Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley’s theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.     

ON THE GENERALIZATIONS OF DEN JOY-WOLFF THEOREM

We consider generalizations of Denjoy-Wolff theorem on strongly pseudo-convex domains. Beardon [3] gave a general Denjoy-Wolff theorem in hyperbolic domain in the complex plane. Our main results are generalizations of Beardon’s result in a higher dimensional setting.