Units of compatible nearrings,IV

This is the fourth in a sequence of papers originating in a effort to study the units of a compatible nearring $R$ satisfying the descending chain condition on right ideals using a faithful compatible module $G$ of $R$ . A key point in this endeavor involves determining $1 + Ann_R(G/H)$ where $H$ is a direct sum of isomorphic minimal $R$ -ideals where success in doing so gives us not only information about the units of $R$ , but also information about $R$ and $J_2(R)$ . In the previous papers, $1 + Ann_R(G/H)$ has been determined whenever $G/H$ does not contain a minimal factor isomorphic to the minimal summands of $H$ . In this paper we determine $1 + Ann_R(G/H)$ when $G/H$ does contain a minimal factor isomorphic to the minimal summands of $H$ . With the completion of the determination of $1 + Ann_R(G/H)$ in all cases, we illustrate how things work in practice by considering the nearrings generated by the inner automorphisms of a finite dihedral group, special linear group, and general linear group and nearrings of congruence preserving functions on an expanded group.     

Quotient Algebras of Toeplitz-Composition C^{*}-Algebras for Finite Blaschke Products

Let $R$ be a finite Blaschke product. We study the $C^*$ -algebra $\mathcal TC _R$ generated by both the composition operator $C_R$ and the Toeplitz operator $T_z$ on the Hardy space. We show that the simplicity of the quotient algebra $\mathcal OC _R$ by the ideal of the compact operators can be characterized by the dynamics near the Denjoy–Wolff point of $R$ if the degree of $R$ is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of $\mathcal OC _R$ such that $R$ is a finite Blaschke product of degree at least two and the Julia set of $R$ is the unit circle, using the Kirchberg–Phillips classification theorem.     

Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Let $R$ be a non-commutative prime ring, with center $Z(R)$ , extended centroid $C$ and let $F$ be a non-zero generalized derivation of $R$ . Denote by $L$ a non-central Lie ideal of $R$ . If there exists $0\ne a\in R$ such that $a[F(x),x]_k\in Z(R)$ for all $x\in L$ , where $k$ is a fixed integer, then one of the followings holds: (1) either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$ , (2) or $R$ satisfies $s_4$ , the standard identity in $4$ variables, and $char(R)=2$ ; (3) or $R$ satisfies $s_4$ and there exist $q\in U, \gamma \in C$ such that $F(x)=qx+xq+\gamma x$ .     

Syzygies and tensor product of modules

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$ -modules. Assume $n$ is a nonnegative integer and that the tensor product $M\otimes _{R}N$ is an $(n+c)$ th syzygy of some finitely generated $R$ -module. If ${{\mathrm{Tor}}}^{R}_{>0}(M,N)=0$ , then both $M$ and $N$ are $n$ th syzygies of some finitely generated $R$ -modules.     

C^*-Algebras Associated with Complex Dynamical Systems and Backward Orbit Structure

Let $R$ be a rational function. The iterations $(R^n)_n$ of $R$ gives a complex dynamical system on the Riemann sphere. We associate a $C^*$ -algebra and study a relation between the $C^*$ -algebra and the original complex dynamical system. In this short note, we recover the number of $n$ th backward orbits counted without multiplicity starting at branched points in terms of associated $C^*$ -algebras with gauge actions. In particular, we can partially imagine how a branched point is moved to another branched point under the iteration of $R$ . We use KMS states and a Perron–Frobenius type operator on the space of traces to show it.     

Triplets of Closely Embedded Dirichlet Type Spaces on the Unit Polydisc

We propose a general concept of triplet of Hilbert spaces with closed embeddings, instead of continuous ones, and we show how rather general weighted $L^2$ spaces yield this kind of generalized triplets of Hilbert spaces for which the underlying spaces and operators can be explicitly calculated. Then we show that generalized triplets of Hilbert spaces with closed embeddings can be naturally associated to any pair of Dirichlet type spaces $\mathcal{D }_\alpha (\mathbb{D }^N)$ of holomorphic functions on the unit polydisc $\mathbb{D }^N$ and we explicitly calculate the associated operators in terms of reproducing kernels and radial derivative operators. We also point out a rigging of the Hardy space $H^2(\mathbb{D }^N)$ through a scale of Dirichlet type spaces and Bergman type spaces.     

The convex hull of the lattice points inside a curve

Let $C$ be a smooth convex closed plane curve. The $C$ -ovals $C(R,u,v)$ are formed by expanding by a factor  $R$ , then translating by  $(u,v)$ . The number of vertices $V(R,u,v)$ of the convex hull of the integer points within or on  $C(R,u,v)$ has order  $R^{2/3}$ (Balog and Bárány) and has average size $BR^{2/3}$ as $R$ varies (Balog and Deshouillers). We find the space average of  $V(R,u,v)$ over vectors  $(u,v)$ to be  $BR^{2/3}$ with an explicit coefficient  $B$ , and show that the average over  $R$ has the same  $B$ . The proof involves counting edges and finding the domain $D(q,a)$ of an integer vector, the set of  $(u,v)$ for which the convex hull has a directed edge parallel to  $(q,a)$ . The resulting sum over bases of the integer lattice is approximated by a triple integral.     

Reorthogonalized block classical Gram–Schmidt

A reorthogonalized block classical Gram–Schmidt algorithm is proposed that factors a full column rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular and nonsingular. This block Gram–Schmidt algorithm can be implemented using matrix–matrix operations making it more efficient on modern architectures than orthogonal factorization algorithms based upon matrix-vector operations and purely vector operations. Gram–Schmidt orthogonal factorizations are important in the stable implementation of Krylov space methods such as GMRES and in approaches to modifying orthogonal factorizations when columns and rows are added or deleted from a matrix. With appropriate assumptions about the diagonal blocks of $R$ , the algorithm, when implemented in floating point arithmetic with machine unit $\varepsilon _M$ , produces $Q$ and $R$ such that $\Vert I- Q ^T\!~ Q \Vert =O(\varepsilon _M)$ and $\Vert A-QR \Vert =O(\varepsilon _M\Vert A \Vert )$ . The first of these bounds has not been shown for a block Gram–Schmidt procedure before. As consequence of these results, we provide a different analysis, with a slightly different assumption, that re-establishes a bound of Giraud et al. (Num Math, 101(1):87–100, 2005) for the CGS2 algorithm.     

C^*-algebras of Toeplitz type associated with algebraic number fields

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra ${\mathfrak T }[R]$ . It is an extension of the ring C*-algebra ${\mathfrak A }[R]$ studied previously by the first named author in collaboration with X. Li. In contrast to ${\mathfrak A }[R]$ , it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$ -semigroup $R\rtimes R^\times $ on $\ell ^2 (R\rtimes R^\times )$ . The algebra ${\mathfrak T }[R]$ carries a natural one-parameter automorphism group $(\sigma _t)_{t\in {\mathbb R }}$ . We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The “partition functions” are partial Dedekind $\zeta $ -functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta $ -functions, which we then use to show uniqueness of the $\beta $ -KMS state for each inverse temperature $\beta \in (1,2]$ .     

Accurate higher-order likelihood inference on P(Y<X)

The stress-strength reliability $R=P(Y<X)$ , where $X$ and $Y$ are independent continuous random variables, has obtained wide attention in many areas of application, such as in engineering statistics and biostatistics. Classical likelihood-based inference about $R$ has been widely examined under various assumptions on $X$ and $Y$ . However, it is well-known that first order inference can be inaccurate, in particular when the sample size is small or in presence of unknown parameters. The aim of this paper is to illustrate higher-order likelihood-based procedures for parametric inference in small samples, which provide accurate point estimators and confidence intervals for $R$ . The proposed procedures are illustrated under the assumptions of Gaussian and exponential models for $(X,Y)$ . Moreover, simulation studies are performed in order to study the accuracy of the proposed methodology, and an application to real data is discussed. An implementation of the proposed method in the R software is provided.     

Geometric complexity of embeddings in {\mathbb{R}^{d}}

Given a simplicial complex K, we consider several notions of geometric complexity of embeddings of K in a Euclidean space \({\mathbb{R}^d}\) : thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any n-complex with N simplices which topologically embeds in \({\mathbb{R}^{2n}, n > 2}\) , can be PL embedded in \({\mathbb{R}^{2n}}\) with refinement complexity \({O(e^{N^{4+{\epsilon}}})}\) . Families of simplicial n-complexes K are constructed such that any embedding of K into \({\mathbb{R}^{2n}}\) has an exponential lower bound on thickness and refinement complexity as a function of the number of simplices of K. This contrasts embeddings in the stable range, \({K\subset \mathbb{R}^{2n+k}, k > 0}\) , where all known bounds on geometric complexity functions are polynomial. In addition, we give a geometric argument for a bound on distortion of expander graphs in Euclidean spaces. Several related open problems are discussed, including questions about the growth rate of complexity functions of embeddings, and about the crossing number and the ropelength of classical links.     

Note on prehomogeneous vector spaces

Let \(V\) be a complex prehomogeneous vector space under the action of a linear algebraic group \(G\) . Assume the poset of orbit closures in the Zariski topology \(\{\overline{Gx}:x\in V\}\) coincides with a (partial) flag \(V_0=0<V_1<\dots <V_k=V\) in \(V\) . Then for any Borel subgroup \(B\) of \(G\) , the poset \(\{\overline{B x}:x\in V\}\) coincides with a full flag in \(V\) .     

Universal geometric cluster algebras

We consider, for each exchange matrix $B$ , a category of geometric cluster algebras over $B$ and coefficient specializations between the cluster algebras. The category also depends on an underlying ring $R$ , usually $\mathbb {Z},\,\mathbb {Q}$ , or $\mathbb {R}$ . We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over $B$ with universal geometric coefficients, or the universal geometric cluster algebra over $B$ . Constructing universal geometric coefficients is equivalent to finding an $R$ -basis for $B$ (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan ${\mathcal {F}}_B$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between ${\mathcal {F}}_B$ and $\mathbf{g}$ -vectors. We construct universal geometric coefficients in rank $2$ and in finite type and discuss the construction in affine type.     

New results about normal pairs of rings with zero-divisors

Let $R\subset S$ be a (unital) extension of (commutative) rings. It is proved in Theorem 1, that $(R, S)$ is a normal pair (i.e. $T$ is integrally closed in $S$ for each ring $T$ such that $R \subseteq T \subseteq S$ ) if and only if $R\subset S$ is a $P$ -extension and $R$ is integrally closed in $S$ . Theorem 2 states that for rings $R\subseteq T \subseteq S, R\subseteq S$ is a $P$ -extension if and only if $R\subseteq T$ and $T\subseteq S$ are $P$ -extensions. As a consequence, we prove that if $R\subseteq T \subseteq B$ are rings and if $\overline{R}_T$ (respectively, $\overline{R}_B$ ) is the integral closure of $R$ in $T$ (respectively, in $B$ ), then $(\overline{R}_T, T)$ is a normal pair if and only if $(\overline{R}_B, \overline{R}_BT)$ is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of arbitrary rings.     

The Euler class group of a polynomial algebra with coefficients in a line bundle

Let $A$ be a commutative Noetherian ring and $P$ be a projective $A$ -module of rank $=(\text {dim}(A)-1)$ . An intriguing open question is to find the precise obstruction for $P$ to split as: $P\simeq Q\oplus A$ for some $A$ -module $Q$ . In this paper we settle this question when $A=R[T]$ for some ring $R$ containing the field of rationals and $P$ is a projective $A$ -module of rank $=\text {dim}(R)$ .     

Volume in General Metric Spaces

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be “faithfully preserved” by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of \(k\) points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set \(S = \{v_1, \ldots , v_k \}\) of points in a general metric space: the product of the distances from \(v_i\) to \(\{ v_1, \dots , v_{i-1} \}\) , normalized by \(\tfrac{1}{(k-1)!}\) , where the ordering of the points is that given by Prim’s minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of \(S\) into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige’s definition: the use of any other order in the product affects volume \(^{1/(k-1)}\) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

Shellability of the higher pinched Veronese posets

The pinched Veronese poset \({\mathcal {V}}^{\bullet }_n\) is the poset with ground set consisting of all nonnegative integer vectors of length \(n\) such that the sum of their coordinates is divisible by \(n\) with exception of the vector \((1,\ldots ,1)\) . For two vectors \(\mathbf {a}\) and \(\mathbf {b}\) in \({\mathcal {V}}^{\bullet }_n\) , we have \(\mathbf {a}\preceq \mathbf {b}\) if and only if \(\mathbf {b}- \mathbf {a}\) belongs to the ground set of \({\mathcal {V}}^{\bullet }_n\) . We show that every interval in \({\mathcal {V}}^{\bullet }_n\) is shellable for \(n \ge 4\) . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in \({\mathcal {V}}^{\bullet }_n\) has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for \(n \ge 4\) . (This also follows from a result by Conca, Herzog, Trung, and Valla.)