Distinguishing embedded curves in rational complex surfaces

We construct many pairs of smoothly embedded complex curves with the same genus and self-intersection number in the rational complex surfaces with the property that no self-diffeomorphism of sends one to the other. In particular, as a special case we answer a question originally posed by R. Gompf (1995) concerning genus two curves of self-intersection number 0 in .

     

Bases for some reciprocity algebras I

For a complex vector space , let be the algebra of polynomial functions on . In this paper, we construct bases for the algebra of all highest weight vectors in , where and for all , and the algebra of highest weight vectors in .

     

Some iterative algorithms for positive definite solution to nonlinear matrix equations

This paper is concerned with the unique positive definite solution to a system of nonlinear matrix equations $X-A^*\bar{Y}^{-1}A=I_n$ and $Y-B^*\bar{X}^{-1}B=I_n$, where $A,B\in\mathbb{C}^{n\times n}$ are given matrices. Based on the special structure of the system of nonlinear matrix equations, the system can be equivalently reformulated as $V-C^*\bar{V}^{-1}C=I_{2n}$. Moreover, by means of Sherman-Moorison-Woodbury formula, we derive the relationship between the solutions of $V-C^*\bar{V}^{-1}C =I_{2n}$ and the well studied standard nonlinear matrix equation $Z+D^*Z^{-1}D=Q$, where $D$, $Q$ are uniquely determined by $C$. Then, we present a structure-preserving doubling algorithm and two modified structure-preserving doubling algorithms to compute the positive definite solution of the system. Furthermore, cyclic reduction algorithm and two modified cyclic reduction algorithms for the positive definite solution of the system are proposed. Finally, some numerical examples are presented to illustrate the efficiency of the theoretical results and the behavior of the considered algorithms.     

Boundedness of the Cesàro Operator in Hardy Spaces

The Ces\aro operator $\mathcal{C}_{\alpha}$ is defined by \begin{equation*} (\mathcal{C}_{\alpha}f)(x) = \int_{0}^{1}t^{-1}f\left( t^{-1}x \right)\alpha (1-t)^{\alpha -1}\,dt~, \end{equation*} where $f$ denotes a function on $\mathbb{R}$. We prove that $\mathcal{C}_{\alpha}$, $\alpha >0$, is a bounded operator in the Hardy space $H^{p}$ for every $0 < p \leqq 1$.     

Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns of non-compact Riemannian symmetric spaces . The basic result is that affine symmetric spaces of can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

     

Simple birational extensions of the polynomial algebra

The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .

     

Clones from creatures

We show that (consistently) there is a clone on a countable set such that the interval of clones above is linearly ordered and has no coatoms.

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

Equlibrium measure of a product subset of

In this note we show that an equilibrium measure of a product of two subsets of and , respectively, is a product of their equilibrium measures. We also obtain a formula for , where are locally bounded plurisubharmonic functions and .

     

Projective equivalence for the roots of unity

Journal of Algebraic Combinatorics - Let $$\mu _{\infty }\subseteq \mathbb {C}$$ be the collection of roots of unity and $$\mathcal {C}_{n}:=\{(s_{1},\ldots ,s_{n})\in \mu _{\infty }^{n}:s_{i}\ne...     

The polydisk nullstellensatz

The Nullstellensatz for zeros in a closed polydisk of is proved constructively with no appeal to choice axioms.

     

Relative Tor Functors for Level Modules with Respect to a Semidualizing Bimodule

Let R and S be rings and _S C _R a semidualizing bimodule. We investigate the relative Tor functors \(\text {Tor}_{i}^{\mathcal {M}\mathcal {L}_{C}}(-,-)\) defined via C-level resolutions, and these functors are exactly the relative Tor functors \(\text {Tor}_{i}^{\mathcal {M}\mathcal {F}_{C}}(-,-)\) defined by Salimi, Sather-Wagstaff, Tavasoli and Yassemi provided that S = R is a commutative Noetherian ring. Vanishing of these functors characterizes the finiteness of \(\mathcal {L}_{C}(S)\)-projective dimension. Applications go in two directions. The first is to characterize when every S-module has a monic (or epic) C-level precover (or preenvelope). The second is to give some criteria for the isomorphism \(\text {Tor}_{i}^{\mathcal {M}\mathcal {L}_{C}}(-,-)\cong \text {Tor}_{i}^{\mathcal {M}\mathcal {F}_{C}}(-,-)\) between the bifunctors.     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Note on arithmetic convolution equations

We investigate the solvability of polynomial equations on the -algebra of arithmetic functions .

     

Stein fillability and the realization of contact manifolds

There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, its germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact contact -manifold has a geometric realization in via an embedding, or in via an immersion.

     

On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .

     

Manifolds with an -action on the tangent bundle

We study manifolds arising as spaces of sections of complex manifolds fibering over with the normal bundle of each section isomorphic to .

     

-saddle method and Beukers' integral

We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers' double integral. The two-dimensional version of the saddle method, which we call -saddle method, is applied.

     

Entire solutions of certain partial differential equations and factorization of partial derivatives

We show that the problem of characterizing entire solutions of certain partial differential equations and the problem of characterizing common right factors of partial derivatives of meromorphic functions in are closely related, and characterizations will be given using their relations.