A Formal Chevalley Restriction Theorem for Kac–Moody Groups

Let G be a symmetrizable Kac–Moody group over a field of characteristic zero, let T be a split maximal torus of G. By using a completion of the algebra of strongly regular functions on G, and its restriction on T, we give a formal Chevalley restriction theorem. Specializing to the affine case, and to the field of complex numbers, we obtain a convergent Chevalley restriction theorem, by choosing the formal functions, which are convergent on the semi-groups of trace class elements G ^trG resp. T ^trT.     

Polynomial Rings over Non-noetherian Power Series Rings

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The Krull―Schmidt Theorem for Henselian Rings

A necessary and sufficient condition on a local ring over which all indecomposable finite-dimensional algebras are local is found. The KrullSchmidt theorem for a class of rings that includes both the Henselian valuation rings and the rings of integers of multidimensional fields is proved. Bibliography: 2 titles.     

An Infeasible Mizuno–Todd–Ye Type Algorithm for Convex Quadratic Programming with Polynomial Complexity

Infeasible interior point methods have been very popular and effective. In this paper, we propose a predictor–corrector infeasible interior point algorithm for convex quadratic programming, and we prove its convergence and analyze its complexity. The algorithm has the polynomial numerical complexity with O(nL)-iteration.     

On Polynomial Functions over Finite Commutative Rings

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained.     

On SZ°-Ideals in Polynomial Rings

The theory of z-ideals and z°-ideals, especially as pertaining to the ideal theory of C(X), the ring of continuous functions on a completely regular Hausdorff space X, has been attended to during the recent years; see Gillman and Jerison [9 Gillman , L. , Jerison , M. ( 1976 ). Rings of Continuous Functions . New York : Springer Verlag . [Google Scholar]], Mason [18 Mason , G. ( 1989 ). Prime ideals and quotient rings of reduced rings . Math. Japan 34 : 941 – 956 . [Google Scholar]], and Azarpanah et al. [4 Azarpanah , F. , Karamzadeh , O. A. S. , Rezaei Aliabad , A. ( 2000 ). On ideals consisting entirely of zerodivisor . Comm. Algebra 28 ( 2 ): 1061 – 1073 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. In this article we will consider the theory of z°-ideals as applied to the rings of polynomials over a commutative ring with identity. We introduce and study sz°-ideals (an ideal I of a ring is called sz°-ideal, if whenever S is a finite subset of I, then the intersection of all minimal prime ideals containing S is in I). In addition, we will pay attention to several annihilator conditions and find some new results. Finally, we use the two examples that appeared in Henriksen and Jerison [10 Henriksen , M. , Jerison , M. ( 1965 ). The space of minimal prime ideals of a commutative ring . Trans. Amer. Math. Soc. 115 : 110 – 130 .[Crossref], [Web of Science ®] , [Google Scholar]] and Huckaba [12 Huckaba , J. A. ( 1988 ). Commutative Rings with Zero Divisors . Marcel-Dekker Inc . [Google Scholar]], to answer some natural questions that might arise in the literature.     

A Krull―Schmidt Theorem for Artinian Modules over Local Rings

The paper contains a generalization of Fitting's lemma for Artinian modules of arbitrary length over some non-Noetherian local rings. A counterexample to a strong form of the KrullSchmidt theorem for Artinian modules over a local ring is constructed. Bibliography: 4 titles.     

Reduced Gröbner Bases in Polynomial Rings over a Polynomial Ring

We propose a new notion of reduced Gr?bner bases in polynomial rings over a polynomial ring and we show that every ideal has a unique reduced Gr?bner basis. We introduce an algorithm for computing them.     

Skew Pieri rules for Hall–Littlewood functions

We produce skew Pieri rules for Hall–Littlewood functions in the spirit of Assaf and McNamara (J. Comb. Theory Ser. A 118(1):277–290, 2011). The first two were conjectured by the first author (Konvalinka in J. Algebraic Comb. 35(4):519–545, 2012). The key ingredients in the proofs are a q-binomial identity for skew partitions and a Hopf algebraic identity that expands products of skew elements in terms of the coproduct and the antipode.     

Noetherian Skew Group Rings of F.C Groups

Let R *θG be the skew group ring with a F.C group G and the group homom-rphismθfrom G to Aut(R), the group of automorphisms of the ring R. In this paper,the necessary and sufficient condition such that R*θG will be Noetherian is given, which generalizes the results of I.G. connel.     

Cibils–Rosso's Theorem for Quantum Groupoids

Abstract

We generalize the Cibils–Rosso's theorem for categories of Sweedler's Hopf bimodules to the one for categories of weak entwined bimodules. We show that the weak entwined bimodules are modules over a certain algebra. Our best results are attained for categories of weak Hopf bimodules over quantum groupoids (weak Hopf algebras), as special cases of weak Doi–Hopf bimodules.     

Mackey–Ahrens Theorem for Spatial Scales

The Mackey–Ahrens theorem on topologies which are congruent in duality is extended to the duality of inductive scales of spaces.     

Skew Group Rings which are Semi-Hereditary Orders and Prüfer Orders in Simple Artinian Rings

Let R be a commutative ring, let G be a finite group acting on R as automorphisms of R and let R * G be the skew group ring. By using the decomposition subgroups of G, the inertial subgroups of G, the properties of the coefficient ring R and the properties of the fixed subring R ^G , some necessary and sufficient conditions for R * G to be a prime Goldie ring, a semi-hereditary order in a simple Artinian ring, or a Prüfer order in a simple Artinian ring are given.     

Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

In this article, we study the notion of radical perfectness in Prüfer and classical pullbacks issued from valuation domains. We answer positively a question by Erdogdu of whether a domain R such that every prime ideal of the polynomial ring R[X] is radically perfect is one-dimensional. Particularly, we prove that Prüfer and pseudo-valuation domains R over which every prime ideal of the polynomial ring R[X] is radically perfect are one-dimensional domains. Moreover, the class group of such a Prüfer domain is torsion.     

G ∞-Formality Theorem in Terms of Graphs and Associated Chevalley–Eilenberg–Harrison Cohomology #

In 1997, M. Kontsevich proved the L _∞-formality conjecture (which implies the existence of star-products for any Poisson manifold) using graphs. A year later, D. Tamarkin gave another proof of a more general conjecture (for G _∞-structures) using operads and cohomological methods. In this article, we show how Tamarkin's construction can be written using graphs. For that, we introduce a generalization of Kontsevich graphs on which we define a “Chevalley–Eilenberg–Harrison” complex. We show that this complex on graphs is related to the “Chevalley–Eilenberg–Harrison” complex for maps on polyvector fields, which is trivial and give Tamarkin's formality theorem as a consequence. This formality reduces to an L _∞-formality.     

Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians

We hypothesize the form of a transformation reducing the elliptic A _N Calogero–Moser operator to a differential operator with polynomial coefficients. We verify this hypothesis for N ≤ 3 and, moreover, give the corresponding polynomial operators explicitly.     

Andronov–Hopf Bifurcation Theorem for Relay Systems

Existence and stability problems for relay systems were examined in many research papers, but local methods that play an important role in deep understanding of dynamics have not yet been sufficiently developed. In this paper, we introduce the concept of the monodromy matrix for a cycle and state an analog of the Andronov–Hopf bifurcation theorem.