Lifting elementary embeddings <Emphasis Type="Italic">j</Emphasis>: <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript> → <Emphasis Type="Italic">V</Emphasis><Subscript>λ</Subscript>

We describe a fairly general procedure for preserving I₃ embeddings j: V _λ → V _λ via λ-stage reverse Easton iterated forcings. We use this method to prove that, assuming the consistency of an I₃ embedding, V = HOD is consistent with the theory ZFC + WA where WA is an axiom schema in the language {∈, j} asserting a strong but not inconsistent form of “there is an elementary embedding V → V”. This improves upon an earlier result in which consistency was established assuming an I₁ embedding.      

Embeddings of the classesH ω

In this paper we study functions belonging to the classesV _ε and ΛBV, which are encountered in the theory of Fourier trigonometric series. Necessary and sufficient conditions for the embedding of the classesH _ω in the classesV _ϕ and ABV are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 713–719, November, 1998. This research was supported by the program “Leading Scientific Schools” under grant No. 96/97-15-96073.     

On Units of Group Algebras of 2-Groups of Maximal Class

Abstract

We determine the number of elements of order two in the group of normalized units V(𝔽₂ G) of the group algebra 𝔽₂ G of a 2-group of maximal class over the field 𝔽₂ of two elements. As a consequence for the 2-groups G and H of maximal class we have that V(𝔽₂ G) and V(𝔽₂ H) are isomorphic if and only if G and H are isomorphic.     

A process variability control chart

In this study a Shewhart type control chart namely the V _t chart, is proposed for improved monitoring of the process variability of a quality characteristic of interest Y. The proposed control chart is based on the ratio type estimator of the variance using a single auxiliary variable X. It is assumed that (Y, X) follows a bivariate normal distribution. The design structure of the V _t chart is developed for Phase-I quality control and its comparison is made with those of the S ² chart (a well-known Shewhart control chart) and the V _r chart (a Shewhart type control chart proposed by Riaz (Comput Stat, 2008a) used for the same purpose. It is observed that the proposed V _t chart outperforms the S ² and V _r charts, in terms of discriminatory power, for detecting moderate to large shifts in the process variability. It is observed that the performance of the V _t chart keeps improving with an increase in |ρ _yx | , where ρ _yx is the correlation between Y and X.     

A Construction of Uniquely C4-free colourable Graphs

Abstract

An F-free colouring of a graph G is a partition {V₁,V₂,…,V_n} of the vertex set V(G) of G such that F is not an induced subgraph of G[V_i] for each i. A graph is uniquely F-free colourable if any two .F-free colourings induce the same partition of V(G). We give a constructive proof that uniquely C₄-free colourable graphs exist.     

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑C ⁿ be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ|_D∖Z are either Hirzebruch surfaces (projective bundles overP ₁), Hopf surfaces (elliptic bundles overP ₁), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.     

On Relative K 2-Group of Split Radical Ideals

Let I be a split radical ideal of a ring R. In this article, the exact sequence 1 → K ₂(R, I) → U _R (I) → V(R, I) → 1 is given by using the method of extension of groups, where U _R (I) is determined by generators and relations. The results of Maazen and Stienstra on the presentation for relative K ₂ group of split radical pairs are extended and amplified.     

Twisted Bimodules and Hochschild Cohomology for Self-injective Algebras of Class A n , II

Up to derived equivalence, the representation-finite self-injective algebras of class A _n are divided into the wreath-like algebras (containing all Brauer tree algebras) and the Möbius algebras. In Part I (Forum Math. 11 (1999), 177–201), the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Möbius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Möbius algebras.     

A Random Set Extension of a Strong Law of Large Numbers of Cantrell and Rosalsky

Abstract

A strong law of large numbers under conditions irrespective of the joint distribution of the sequence is extended to random sets. The extension is such that the role of events of the form {||V _n || ≤ b _n } (where V _n is a random element of a separable Banach space) is played by events of the form {X _n  ? B _n } (where X _n is a random closed bounded set).     

Uniform vector bundles on quadrics

Summary In this paper we prove that a rankr uniform vector bundle on a nonsingular quadricQ with dimQ≥2r+2 is a direct sum of line bundles. We study also rank 2 uniform vector bundles onP ₁×P₁. We prove that they are not all homogeneous and that any rank 2 homogeneous vector bundles onP ₁×P₁ is decomposable.

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Riassunto In questo lavoro si dimostra che un fibrato uniforme di rangor su una quadrica non singolareQ con dimQ≥2r+2 è somma diretta di fibrati in rette. Si studiano poi i fibrati uniformi di rango 2 suP ₁×P₁. Si dimostra che non sono tutti omogenei e che ogni fibrato omogeneo di rango 2 suP ₁×P₁ è decomponibile.

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The author is member of G.N.S.A.G.A. of C.N.R.     

<Emphasis Type="Italic">∂</Emphasis>-operations and <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">V</Emphasis></Subscript>-fields

The hyperoperations, called theta-operations （δ）, are motivated from the usual property, which the derivative has on the derivation of a product of functions. Using any map on a set, one can define δ-operations. In this paper, we continue our study on the δ-operations on groupoids, rings, fields and vector spaces or on the corresponding hyperstructures. Using δ-operations one obtains, mainly, Hwstructures, which form the largest class of the hyperstructures. For representation theory of hyperstructures, by hypermatrices, one needs special Hv-rings or Hy-fields, so these hyperstructures can be used. Moreover, we study the relation of these δ-structures with other classes of hyperstructures, especially with the Hv-structures.     

Monitoring process variability using auxiliary information

In this study a Shewhart type control chart namely V _r chart is proposed for improved monitoring of process variability (targeting large shifts) of a quality characteristic of interest Y. The proposed control chart is based on regression type estimator of variance using a single auxiliary variable X. It is assumed that (Y, X) follow a bivariate normal distribution. The design structure of V _r chart is developed and its comparison is made with the well-known Shewhart control chart namely S ² chart used for the same purpose. Using power curves as a performance measure it is observed that V _r chart outperforms the S ² chart for detecting moderate to large shifts, which is main target of Shewhart type control charts, in process variability under certain conditions on ρ _yx . These efficiency conditions on ρ _yx are also obtained for V _r chart in this study.     

H2 optimal model order reduction on the Stiefel manifold for the MIMO discrete system by the cross Gramian

ABSTRACT

In this paper, the H₂ optimal model order reduction method for the large-scale multiple-input multiple-output (MIMO) discrete system is investigated. First, the MIMO discrete system is resolved into a number of single-input single-output (SISO) subsystems, and the H₂ norm of the original MIMO discrete system is expressed by the cross Gramian of each subsystem. Then, the retraction and the vector transport on the Stiefel manifold are introduced, and the geometric conjugate gradient model order reduction method is proposed. The reduced system of the original MIMO discrete system is generated by using the proposed method. Finally, two numerical examples show the efficiency of the proposed method.     

Orbits of the Actions of Odd-Order Solvable Groups

Suppose that V is a finite faithful irreducible G-module where G is a finite solvable group of odd order. We prove if the action is quasi-primitive, then either F(G) is abelian or G has at least 212 regular orbits on V. As an application, we prove that when V is a finite faithful completely reducible G-module for a solvable group G of odd order, then there exists v ∈ V such that C _G (v) ? F ₂(G) (where F ₂(G) is the 2nd ascending Fitting subgroup of G). We also generalize a result of Espuelas and Navarro. Let G be a group of odd order and let H be a Hall π-subgroup of G. Let V be a faithful G-module over a finite field of characteristic 2, then there exists v ∈ V such that C _H (v) ? O _π(G).     

Classi localmente filtrali

Riassunto Si dà la definizione di classe ?localmente filtrale?. Si diceche K è una classe localmente filtrale se per ogni n∈ω, per ogni A ₀,...,A _n−1, εK e per ogni famiglia di sotto-insiemi V_i di A_i (i∈n) con V_i finiti, la classe {B ₀,...,B _n−1 delle algebre generate da V₀, ..., V_n−1 è costituita da algebre finite ed è filtrale. Si dimostra che seK è localmente filtrale alloraV _L(K)=IR _L DS(K) e si dà un teorema di caratterizzazione per queste classi.

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Summary We define a ?classe localmente filtrale? as follows: LetK be a class of similar algebras;K is a ?classe localmente filtrale? if for andn ∈ ω and for anyA ₀,...,A _n−1 ink and for any family of finite subsetsV _i ofA _i(i∈n), the class {B ₀,...,B _n−1 of algebras generated byV _o, ...,V _n−1 consists of finite algebras and is ?filtrale?. We show that ifK is ?localmente filtrale? thenV _L (K)=IR _L DS(K) and we give a characterization theorem for these classes.

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Lavoro eseguito nell'ambito dei gruppi di ricerca matematici del C.N.R. per l'anno 1970–71.     

K 0 of Exchange Rings with All Idempotents Central

ABSTRACT

Let R be an exchange ring with all idempotents central, and let Max(R) ? Ξ(R) ? Spec(R). If ∩{P | P ∈ Ξ(R)} is nil, we prove that K ₀(R) ? {f : Ξ(R) → ?| f is continuous}.     

Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

The integrability of an m-component system of hydrodynamic type, u _t = V(u)u _x , by the generalized hodograph method requires the diagonalizability of the m ×  m matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains—infinite-component systems of hydrodynamic type for which the ∞ ×  ∞ matrix V(u) is ‘sufficiently sparse’. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.