On 2-dimensional local rings with Artin's approximation property

We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization. In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.

Let (R,m) be a local Noetherian Cohen-Macaulay ring with the approximation property. We show that R admits a canonical module.

Generating absolute prime ideals

Nested Artin strong approximation property

Let k be a field, R an associative k-algebra with identity, Δ a finite set of derivations of R, and R[Θ1, δ1] ··· [Θ n , δ n ] an iterated differential operator k-algebra over R such that δ j (Θ i ) ∈ R[Θ1, δ1] ··· [Θ i−1, δ i−1]; 1 ≤ i < j ≤ n. If R is Noetherian Δ-hypercentral, then every prime ideal P of A is classically localizable. The aim of this article is to show that under some additional hypotheses on the Δ-prime ideals of R, the local ring A P is regular in the sense of Robert Walker. We use this result to study the catenarity of A and to compute the numbers μ i of Bass. Let g be a nilpotent Lie algebra of finite dimension n acting on R by derivations and U(g) the enveloping algebra of g. Then the crossed product of R by U(g) is an iterated differential operator k-algebra as above. In this particular case, our results are known if k has characteristic zero.

Actions of Algebraic Groups on the Spectrum of Rational Ideals

Affine Chevalley algebras

Various composite polynomials have been studied as to the location of their zeros by a number of mathematicians, but mostly in relation to the complex plane. In more general spaces, however, this problem has so far been studied by Zervos, Marden, and Zaheer, but only for a special type of composite polynomial. In this paper, we introduce the class of composite polynomials Lu(z) (respectively Lv*(z)), derived from a given polynomial and its polar (respectively formal) derivatives in an algebraically closed field K of characteristic zero, and we investigate the location of the zeros of such polynomials. A comparative study of the two classes shows that they are essentially the same and differ only in form. From our main theorem we deduce as corollaries some known results due to Laguerre, Zervos, and Zaheer. It may be noted that although all our results are obviously valid in the complex plane, we mention only those applications which are interesting from a geometrical standpoint or which have physical interpretations.

A method is developed for finding all G-modules (where G is a connected and simply connected semisimple algebraic group over an algebraically closed field of characteristic zero) whose algebra of invariants has prescribed homological dimension. The main theorem says that the number of such G-modules, considered to within isomorphism and addition of a trivial direct summand, is finite. The same result is proved for finite groups G. All algebras of invariants of homological dimension ?10 of a single binary form are found, as well as all algebras of invariants of a system of binary forms that are hypersurfaces. It is shown that the exceptional simple groups have no irreducible modules with an algebra of invariants of small nonzero homological dimension. Bibliography: 46 titles.

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

In this paper, we establish a new coupled coincidence point results in partially ordered probabilistic metric spaces by utilizing the Gauge function. We use the compatibility condition between two mappings. We use monotone and mixed monotone properties of functions with respect to the ordering. Our main result has several corollaries. The main result is supported with an example which shows that the corollaries are actually contained in our main theorem. The methodology is a combination of analytic and order theoretic approaches.

We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with respect to continuous functions on $\mathbb{R}$. Notably, we explore several structural properties of Borel subsets of $\mathbb{R}$ that diverge from those of Polish spaces with dimension zero. Our first main result is on the existence of embeddings of several posets into the restriction of this quasi-order to any Borel class that is strictly above the classes of open and closed sets, for instance the linear order $\omega_1$, its reverse $\omega_1^\star$ and the poset $\mathcal{P}(\omega)/\mathsf{fin}$ of inclusion modulo finite error. As a consequence of its proof, it is shown that there are no complete sets for these classes. We further extend the previous theorem to targets that are reducible to $\mathbb{Q}$. These non-structure results motivate the study of further restrictions of the Wadge quasi-order. In our second main theorem, we introduce a combinatorial property that is shown to characterize those $F_\sigma$ sets that are reducible to $\mathbb{Q}$. This is applied to construct a minimal set below $\mathbb{Q}$ and prove its uniqueness up to Wadge equivalence. We finally prove several results concerning gaps and cardinal characteristics of the Wadge quasi-order and thereby answer questions of Brendle and Geschke.

We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$-separated types is again a reflective subuniverse, which we call $L'$. Furthermore, we prove results establishing that $L'$ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space $K(G,n)$ with $G$ abelian. We also include a partial converse to the main theorem.

We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having m = O(˜r/epsilon^2) rows. Here r˜ is the maximum stable rank, i.e., the squared ratio of Frobenius and operator norms, of the two matrices being multiplied. This is a quantitative improvement over previous work of [Magen and Zouzias, SODA, 2011] and [Kyrillidis et al., arXiv, 2014] and is also optimal for any oblivious dimensionality-reducing map. Furthermore, due to the black box reliance on the subspace embedding property in our proofs, our theorem can be applied to a much more general class of sketching matrices than what was known before, in addition to achieving better bounds. For example, one can apply our theorem to efficient subspace embeddings such as the Subsampled Randomized Hadamard Transform or sparse subspace embeddings, or even with subspace embedding constructions that may be developed in the future. Our main theorem, via connections with spectral error matrix multiplication proven in previous work, implies quantitative improvements for approximate least squares regression and low rank approximation, and implies faster low rank approximation for popular kernels in machine learning such as the gaussian and Sobolev kernels. Our main result has also already been applied to improve dimensionality reduction guarantees for k-means clustering, and also implies new results for nonparametric regression. Lastly, we point out that the proof of the BSS deterministic row-sampling result of [Batson et al., SICOMP, 2012] can be modified to obtain deterministic row-sampling for approximate matrix product in terms of the stable rank of the matrices. The original BSS proof was in terms of the rank rather than the stable rank.

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented.