4,870 results
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2. Counterexample to the paper 'On the Gorenstein injective dimension and Bass formula'
- Author
-
Moharram Aghapournahr
- Subjects
Discrete mathematics ,Bass (sound) ,Class (set theory) ,Pure mathematics ,Generalization ,General Mathematics ,Dimension (graph theory) ,Finitely-generated abelian group ,Injective function ,Counterexample ,Mathematics - Abstract
In this note, we give a counterexample for Theorem 2.3 of the above mentioned paper that is a generalization of the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules.
- Published
- 2009
3. Correction to the paper ‘Copies of $\ell _{\infty }$ in the space of Pettis integrable functions with integrals of finite variation’ (Studia Math. 210 (2012), 93–98)
- Author
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Juan Carlos Ferrando
- Subjects
Discrete mathematics ,Pure mathematics ,Finite variation ,Integrable system ,General Mathematics ,Space (mathematics) ,Mathematics - Published
- 2016
4. A Note on A Paper of Cellina Concerning Differential Equations in Banach Spaces
- Author
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M. Kunze
- Subjects
Discrete mathematics ,Pure mathematics ,Approximation property ,General Mathematics ,Eberlein–Šmulian theorem ,Infinite-dimensional vector function ,Banach space ,Interpolation space ,Banach manifold ,Lp space ,C0-semigroup ,Mathematics - Published
- 1996
5. A remark on V.A. Iskovskikh's paper 'A simple proof of the non-rationality of a three-dimensional quartic'
- Author
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M M Grinenko
- Subjects
Discrete mathematics ,Pure mathematics ,Simple (abstract algebra) ,General Mathematics ,Quartic function ,Rationality ,Quartic surface ,Mathematics - Published
- 2002
6. Remark on Belyĭ's paper concerning Galois extensions of the maximal cyclotomic field with certain linear groups as Galois groups
- Author
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Hisashi Kojima
- Subjects
Discrete mathematics ,Pure mathematics ,12F10 ,Galois cohomology ,12F12 ,General Mathematics ,Fundamental theorem of Galois theory ,Galois group ,Abelian extension ,Galois module ,11R32 ,Differential Galois theory ,Embedding problem ,symbols.namesake ,symbols ,Galois extension ,Mathematics - Published
- 1991
7. Relations Between the Divisors of the First n Natural Numbers: (Second Paper.)
- Author
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J. W. L. Glaisher
- Subjects
Discrete mathematics ,Pure mathematics ,Practical number ,Amicable numbers ,General Mathematics ,Natural number ,Quasiperfect number ,Table of divisors ,Refactorable number ,Mathematics - Abstract
n/a
- Published
- 1908
8. A note of Shimura's paper ?discontinuous groups and abelian varieties?
- Author
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David Mumford
- Subjects
Shimura variety ,Discrete mathematics ,Pure mathematics ,Abelian variety of CM-type ,General Mathematics ,Schottky problem ,Elementary abelian group ,Abelian category ,Hilbert's twelfth problem ,Abelian group ,Mathematics ,Arithmetic of abelian varieties - Published
- 1969
9. Geometric analogue of the Mumford-Tate conjecture for stably nondegenerate abelian varieties (a note on Mustafin's paper)
- Author
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Fumio Hazama
- Subjects
Discrete mathematics ,Pure mathematics ,11G10 ,General Mathematics ,Degenerate energy levels ,32G20 ,Elementary abelian group ,Rank of an abelian group ,Abelian variety of CM-type ,Abelian category ,Abelian group ,14K05 ,Tate conjecture ,Mathematics ,Arithmetic of abelian varieties - Published
- 1988
10. Remark on Lehto's paper 'a generalization of Picard's theorem'
- Author
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Kikuji Matsumoto
- Subjects
Discrete mathematics ,Pure mathematics ,Picard–Lindelöf theorem ,Generalization ,General Mathematics ,Picard theorem ,Mathematics - Published
- 1962
11. Correction to the paper 'On weighted $H^{p}$ spaces' (Studia Math. 40 (1971), pp. 109-159)
- Author
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T. Walsh
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Mathematics - Published
- 1972
12. Generalization of Certain Theorems of Bohl, [Second Paper]
- Author
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F. H. Murray
- Subjects
Discrete mathematics ,Pure mathematics ,Generalization ,General Mathematics ,Point (geometry) ,Extension (predicate logic) ,Mathematical proof ,Algebraic method ,Mathematics - Abstract
near a point solution xi ai (i 1, 1 *, n) when certain conditions are satisfied. In the present paper these conditions are replaced by less stringent ones; the methods of proof of certain existence theorems are very similar to those employed in the first paper, and these proofs are given here in an abbreviated forin. In addition, the asymptotic properties of certain trajectories are discussed by an extension of the methods of Bohl. On account of the more complicated form of certain quadratic forms which occur here, it has been convenient to leave undetermined certain constants which are determined explicitly in the first paper; this procedure, together with the algebraic method of transforming the canonical equations, reduces to a small amount the results common to both papers.
- Published
- 1927
13. A formula for generating weakly modular forms with weight 12
- Author
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Aykut Ahmet Aygunes
- Subjects
Discrete mathematics ,symbols.namesake ,Pure mathematics ,Special solution ,General Mathematics ,Short paper ,Modular form ,Eisenstein series ,symbols ,Derivative ,Function (mathematics) ,Mathematics ,Möbius transformation - Abstract
In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.
- Published
- 2016
14. Addendum to My Paper 'Foundations of a General Theory of Equisingularity on r-Dimensional Algebroid and Algebraic Varieties, of Embedding Dimension r + 1'
- Author
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Oscar Zariski
- Subjects
Discrete mathematics ,Pure mathematics ,General theory ,General Mathematics ,Dimension (graph theory) ,Embedding ,Addendum ,Algebraic variety ,Mathematics - Published
- 1980
15. A Note on Quillen's Paper 'Projective Modules Over Polynomial Rings'
- Author
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Moshe Roitman
- Subjects
Discrete mathematics ,Pure mathematics ,Collineation ,Applied Mathematics ,General Mathematics ,Polynomial ring ,Complex projective space ,Projective cover ,Projective line over a ring ,Projective space ,Projective module ,Quaternionic projective space ,Mathematics - Published
- 1977
16. Correction to the paper ' On functions and equations in distributive lattices '
- Author
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Sergiu Rudeanu
- Subjects
Discrete mathematics ,Pure mathematics ,Distributive property ,General Mathematics ,Distributive lattice ,Birkhoff's representation theorem ,Congruence lattice problem ,Map of lattices ,Complemented lattice ,Mathematics - Published
- 1970
17. Homological stability of non-orientable mapping class groups with marked points
- Author
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Elizabeth Hanbury
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Algebraic Geometry ,Applied Mathematics ,General Mathematics ,Short paper ,Homology (mathematics) ,Mathematics::Geometric Topology ,Mapping class group ,Mathematics - Abstract
Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short paper we analyse the situation where the underlying non-orientable surfaces have marked points.
- Published
- 2008
18. On local observability for invariant systems on Lie groups and coset spaces
- Author
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N. Apostolou and D. Kazakos
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Simple Lie group ,Short paper ,General Engineering ,General Physics and Astronomy ,Coset ,Lie group ,Field (mathematics) ,Observability ,Invariant (mathematics) ,Mathematics - Abstract
The purpose of this short paper is to provide a proof of a local observability criterion for invariant systems on Lie groups and coset spaces. The criterion was originally presented with an ad hoc proof based on Lie theoretic arguments. The alternative proof presented here shows that the criterion essentially comes from the classical observability results involving the observability codistribution and its properties. In this way it unifies the two approaches and extends the application field of the classical results.
- Published
- 1996
19. Remarks on blowing-up divisorial ideals
- Author
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Lorenzo Robbiano
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Algebraic Geometry ,Number theory ,Mathematics::Commutative Algebra ,General Mathematics ,Prime ideal ,Short paper ,Local ring ,Topological group ,Algebraic geometry ,Blowing up ,Mathematics - Abstract
In this short paper I describe special situations where a local ring is normally flat along a divisorial prime ideal.
- Published
- 1979
20. Some results on complex valued metric spaces employing an implicit relation with complex coefficients and its applications
- Author
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Fayyaz Rouzkard
- Subjects
Discrete mathematics ,Pure mathematics ,Weakly compatible ,Relation (database) ,Complex valued metric space ,General Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Complex valued ,Product metric ,010103 numerical & computational mathematics ,lcsh:QA1-939 ,01 natural sciences ,Common fixed point ,Metric space ,Contractive type mapping ,Complex Coefficient ,0101 mathematics ,Contraction (operator theory) ,Coincidence point ,Mathematics - Abstract
In this paper, we establish coincidence point and common fixed point theorems involving two pairs of weakly compatible mapping satisfying contraction condition with complex coefficient are proved in complex valued metric space. The presented theorems generalize, extend and improve many existing results in the literature. An example is given at the end of the paper.
- Published
- 2018
21. Redheffer type bounds for Bessel and modified Bessel functions of the first kind
- Author
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Árpád Baricz and Khaled Mehrez
- Subjects
Discrete mathematics ,Pure mathematics ,Hankel transform ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dirichlet eta function ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Bessel polynomials ,Struve function ,symbols ,Discrete Mathematics and Combinatorics ,Bessel's inequality ,0101 mathematics ,Bessel function ,Mathematics - Abstract
In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.
- Published
- 2018
22. The lattices of invariant subspaces of a class of operators on the Hardy space
- Author
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Zeljko Cuckovic and Bhupendra Paudyal
- Subjects
Discrete mathematics ,Pure mathematics ,Volterra operator ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,Hardy space ,Reflexive operator algebra ,01 natural sciences ,Linear subspace ,symbols.namesake ,Operator (computer programming) ,Lattice (order) ,FOS: Mathematics ,symbols ,Complex Variables (math.CV) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**
- Published
- 2018
23. Some weak specification properties and strongly mixing
- Author
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Jiandong Yin, Tao Wang, and Qi Yan
- Subjects
010101 applied mathematics ,Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Equivalence (formal languages) ,01 natural sciences ,Mathematics - Abstract
In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.
- Published
- 2017
24. Bounds for Calderón–Zygmund operators with matrix A2 weights
- Author
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Sandra Pott and Andrei Stoica
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Pure mathematics ,Representation theorem ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Mathematics::Classical Analysis and ODEs ,Haar ,01 natural sciences ,Operator (computer programming) ,0103 physical sciences ,Embedding ,010307 mathematical physics ,0101 mathematics ,Special case ,Martingale (probability theory) ,Singular integral operators ,Mathematics - Abstract
It is well-known that dyadic martingale transforms are a good model for Calderon–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A 2 matrix weight, then the weighted L 2 -norm of a Calderon–Zygmund operator with cancellation has the same dependence on the A 2 characteristic of W as the weighted L 2 -norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderon–Zygmund operators on the A 2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderon–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytonen to extend the result to general Calderon–Zygmund operators.
- Published
- 2017
25. On the Auslander–Reiten conjecture for Cohen–Macaulay local rings
- Author
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Ryo Takahashi and Shiro Goto
- Subjects
Discrete mathematics ,Pure mathematics ,Conjecture ,Cohen–Macaulay ring ,Applied Mathematics ,General Mathematics ,Gorenstein ring ,Local ring ,Mathematics - Abstract
This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.
- Published
- 2017
26. Fixed Point Theorems for Mappings Satisfying Weak Nonexpansivity Condition (Weak Contractivity Condition) into (from) Cartesian Products Normed Spaces
- Author
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Sahar Mohamed Ali Abou Bakr
- Subjects
Statistics and Probability ,Discrete mathematics ,Pure mathematics ,General Mathematics ,Banach space ,Fixed-point theorem ,Cartesian product ,Type (model theory) ,Fixed point ,Space (mathematics) ,symbols.namesake ,Monotone polygon ,symbols ,Mathematics ,Normed vector space - Abstract
This paper suggests new types of weak nonexpansive mappings defined from normed space X into its Cartesian product X × X, studies the main features of the fixed points for those mappings and extends the concept of (C)-contractivity condition introduced in some previous research papers. On other side, it introduces new types of contraction mappings with a mixed monotone property; the {a, b, c} M-first type and the {a, b, c} M-second type contractions, these types are defined from the Cartesian product space X × X into X, where X is a sequentially ordered Banach space, proves the existence of first-anti-second and second-anti-first couple fixed points of such types and generalizes some of the results given before.
- Published
- 2017
27. A class of generalized operator equilibrium problems
- Author
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Jong Kyu Kim and Abdul Raouf
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Pure mathematics ,021103 operations research ,General Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we consider a class of generalized operator equilibrium problems and derive a Minty type lemma for this class of problems. Further, we establish some existence theorems for the generalized operator equilibrium problems. The theorems presented in this paper generalize and unify many well-known results in the literature.
- Published
- 2017
28. Degenerate abstract Volterra equations in locally convex spaces
- Author
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Marko Kostić
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Degenerate energy levels ,Volterra equations ,Equicontinuity ,01 natural sciences ,Volterra integral equation ,010101 applied mathematics ,symbols.namesake ,Locally convex topological vector space ,Resolvent operator ,symbols ,0101 mathematics ,Well posedness ,Mathematics - Abstract
In the paper under review, we analyze various types of degenerate abstract Volterra integrodifferential equations in sequentially complete locally convex spaces. From the theory of non-degenerate equations, it is well known that the class of (a,k)-regularized C-resolvent families provides an efficient tool for dealing with abstract Volterra integro-differential equations of scalar type. Following the approach of T.-J. Xiao and J. Liang [41]-[43], we introduce the class of degenerate exponentially equicontinuous (a,k)- regularized C-resolvent families and discuss its basic structural properties. In the final section of paper, we will look at generation of degenerate fractional resolvent operator families associated with abstract differential operators.
- Published
- 2017
29. Rigidity theory for $C^*$-dynamical systems and the 'Pedersen Rigidity Problem', II
- Author
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Tron Omland, John Quigg, and Steven Kaliszewski
- Subjects
Discrete mathematics ,Exterior equivalences ,Pure mathematics ,Dynamical systems theory ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Outer conjugacy ,Generalized fixed point algebra ,01 natural sciences ,Rigidity (electromagnetism) ,Crossed product ,Primary 46L55 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,Abelian group ,Rigidity theory ,Operator Algebras (math.OA) ,Crossed products ,Mathematics ,Conjugate - Abstract
This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group $G$, saying that two actions $(A,\alpha)$ and $(B,\beta)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_{\alpha}G,\widehat\alpha)$ and $(B\rtimes_{\beta}G,\widehat\beta)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images, and we have decided to use the term "Pedersen rigid" for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call "fixed-point rigidity". In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known., Comment: Minor revision. To appear in Internat. J. Math
- Published
- 2018
30. On the Langlands correspondence for symplectic motives
- Author
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Benedict H. Gross
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Langlands dual group ,01 natural sciences ,Cohomology ,Langlands program ,Elliptic curve ,Local Langlands conjectures ,Orthogonal group ,0101 mathematics ,Weil group ,Symplectic geometry ,Mathematics - Abstract
In this paper, we present a refinement of the global Langlands correspondence for discrete symplectic motives of rank 2n over Q. To such a motive Langlands conjecturally associates a generic, automorphic representation of the split orthogonal group SO2n+1 over Q, which appears with multiplicity one in the cuspidal spectrum. Using the local theory of generic representations of odd orthogonal groups, we define a new vector F in this representation, which is the tensor product of local test vectors for the Whittaker functionals [9]. I hope that the defining properties ofF will make it easier to investigate the Langlands correspondence computationally, especially for the cohomology of algebraic curves. Our refinement is similar to the refinement that Weil [24] proposed for the conjecture that elliptic curves over Q are modular. Namely, Weil proposed that such a curve should be associated with a homomorphic newform F = P anq n of weight 2 on 0(N), where N is equal to the conductor of the curve. This paper expands on a letter that I wrote to Serre in 2010. It was motivated by a question Serre posed at my 60th birthday conference, and a suggestion Brumer made of a family of discrete subgroups generalizing 0(N). I would like to thank them, and to thank Deligne for his comments.
- Published
- 2016
31. L p estimates of rough maximal functions along surfaces with applications
- Author
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Abdulla M. Jarrah and Ahmad Al-Salman
- Subjects
Discrete mathematics ,Class (set theory) ,Pure mathematics ,General theorem ,Applied Mathematics ,General Mathematics ,Block (permutation group theory) ,Maximal function ,Singular integral ,Space (mathematics) ,Singular integral operators ,Mathematics - Abstract
In this paper, we study the Lp mapping properties of certain class of maximal oscillatory singular integral operators. We prove a general theorem for a class of maximal functions along surfaces. As a consequence of such theorem, we establish the Lp boundedness of various maximal oscillatory singular integrals provided that their kernels belong to the natural space Llog L(Sn−1). Moreover, we highlight some additional results concerning operators with kernels in certain block spaces. The results in this paper substantially improve previously known results.
- Published
- 2016
32. $k^{th}$ root transformations for some subclasses of alpha convex functions defined through convolution
- Author
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R. B. Sharma, T. Ram Reddy, and M. Haripriya
- Subjects
Discrete mathematics ,Pure mathematics ,Transformation (function) ,General Mathematics ,Proper convex function ,Function (mathematics) ,Inverse function ,Convolution power ,Convex function ,Analytic function ,Convolution ,Mathematics - Abstract
In this paper we introduce a new subclass of analytic functions defined through Convolution. We obtain the sharp upper bounds for the coefficient functional corresponding to the k th root transformation for the function in this class. Similar problems are investigated for the Inverse function and .The results of this paper generalise the work of earlier researchers in this direction.
- Published
- 2016
33. Étale extensions with finitely many subextensions
- Author
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Martine Picavet-L'Hermitte and Gabriel Picavet
- Subjects
Discrete mathematics ,Pure mathematics ,Ring (mathematics) ,Canonical decomposition ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Diagonal ,Support of a module ,Artinian ring ,010103 numerical & computational mathematics ,Extension (predicate logic) ,Type (model theory) ,Characterization (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
We study etale extensions of rings that have FIP., Comment: The paper entitled FIP and FCP products of ring morphisms (arXiv: 1312.1250 [math.AC]) is now split into three papers. The present paper contains the last section of the original paper and many other results on etale FIP extensions
- Published
- 2016
34. Uniqueness of meromorphic functions whose nonlinear differential polynomials share a polynomial
- Author
-
Pulak Sahoo and Himadri Karmakar
- Subjects
Discrete mathematics ,Pure mathematics ,Polynomial ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Polynomial matrix ,Nonlinear system ,Uniqueness ,0101 mathematics ,Differential (mathematics) ,Mathematics ,Meromorphic function - Abstract
In this paper, we study some uniqueness problems of meromorphic functions when certain nonlinear differential polynomials generated by them share a nonconstant polynomial. The results of the paper improve the concerning results due to Xu et al. (Mat Vesnik 64:1–16, 2012).
- Published
- 2016
35. Metrically and topologically projective ideals of Banach algebras
- Author
-
N. T. Nemesh
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Identity (mathematics) ,Banach algebra ,Bounded function ,0103 physical sciences ,Metric (mathematics) ,Ideal (order theory) ,010307 mathematical physics ,0101 mathematics ,Projective test ,Commutative property ,Approximate identity ,Mathematics - Abstract
In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary C*-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.
- Published
- 2016
36. Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems
- Author
-
Raffaella Servadei, Giovanni Molica Bisci, Alessio Fiscella, Fiscella, A, Molica Bisci, G, and Servadei, R
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,variational techniques ,010102 general mathematics ,Multiplicity (mathematics) ,integrodifferential operators ,01 natural sciences ,Dirichlet distribution ,Fractional Laplacian ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,critical nonlinearities ,Operator (computer programming) ,Fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,Bounded function ,best fractional critical Sobolev constant ,fractional Laplacian, critical nonlinearities, best fractional critical Sobolev constant, variational techniques, integrodifferential operators ,symbols ,Exponent ,0101 mathematics ,Bifurcation ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.
- Published
- 2016
37. $\bm{p}$-frames, Hilbert-Schauder frames and $\bm{\sigma}$-frame operators
- Author
-
Lin Liqiong, Zhu Yucan, and Zhang Yunnan
- Subjects
Discrete mathematics ,symbols.namesake ,Pure mathematics ,General Mathematics ,Hilbert space ,symbols ,Banach space ,Sigma ,Mathematics - Abstract
In view of the fact that there are not appropriate frame operators of frames in Banach spaces, this paper considers a class of sequences satisfying certain conditions in Banach spaces which is called as the $\sigma$-frame, and the corresponding concept of frame operators is given. The $\sigma$-frames and $\sigma$-frame operators are natural generalizations of frames and frame operators in Hilbert spaces. This paper illustrates that $\sigma$-frame operators are positive, self-adjoint and they can be decomposed through $l_2$. The perturbation result under operators of $\sigma$-frame is obtained. This paper also shows that the kind of $\sigma$-frames contains two other kinds of frames in Banach space---$p$-frames ($1<p\leq 2$) and $\sigma {\rm HS}$ frames which are a kind of frames according to the definition of the Hilbert-Schauder frames. The perturbation results under operators of $p$-frames ($1<p\leq 2$) and $\sigma {\rm HS}$ frames are obtained.
- Published
- 2016
38. Degeneration of the Hilbert pairing in formal groups over local fields
- Author
-
O. Yu. Podkopaeva, Sergei V. Vostokov, and Regina P. Vostokova
- Subjects
Discrete mathematics ,Ring (mathematics) ,Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Formal group ,Field (mathematics) ,Subring ,01 natural sciences ,010305 fluids & plasmas ,Formal derivative ,Pairing ,0103 physical sciences ,0101 mathematics ,Symbol (formal) ,Mathematics - Abstract
For an arbitrary local field K (a finite extension of the field Qp) and an arbitrary formal group law F over K, we consider an analog cF of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing cF has a certain fundamental symbol property for all Lubin–Tate formal groups, then cF = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring OK of integers of K and have a fixed ring O0 of endomorphisms, where O0 is a subring of OK. We prove that if the symbol cF has the above-mentioned symbol property, then cF = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.
- Published
- 2016
39. Results on uniqueness of entire functions whose certain difference polynomials share a small function
- Author
-
Pulak Sahoo and Himadri Karmakar
- Subjects
Discrete mathematics ,Pure mathematics ,Difference polynomials ,General Mathematics ,Entire function ,Function (mathematics) ,Uniqueness ,Type (model theory) ,Mathematics - Abstract
In the paper, using the concept of weakly weighted sharing and relaxed weighted sharing, we investigate the uniqueness problems of certain type of difference polynomials that share a small function. The results of the paper improve and extend some recent results due to C. Meng [Math. Bohem., 139(2014), 89–97] and the present first author [Commun. Math. Stat., 3(2015), 227–238].
- Published
- 2015
40. The Fundamental Theorem of Algebra: A Visual Approach
- Author
-
Daniel J. Velleman
- Subjects
Discrete mathematics ,Pure mathematics ,Fundamental theorem of algebra ,Color scheme ,History and Philosophy of Science ,Fundamental theorem ,General Mathematics ,Fundamental theorem of linear algebra ,Color wheel ,Mathematical proof ,Complex plane ,Complex number ,Mathematics - Abstract
Some version of the statement of the Fundamental Theorem of Algebra first appeared early in the 17th century in the writings of several mathematicians, including Peter Roth, Albert Girard, and Rene Descartes. The first proof of the Fundamental Theorem was published by Jean Le Rond d’Alembert in 1746 [2], but his proof was not very rigorous. Carl Friedrich Gauss is often credited with producing the first correct proof in his doctoral dissertation of 1799 [15], although this proof also had gaps. (For a comparison of these two proofs, see [26, pp. 195–200].) Today there are many known proofs of the Fundamental Theorem of Algebra, including proofs using methods of algebra, analysis, and topology. (The references include many papers and books containing proofs of the Fundamental Theorem; [14] alone contains 11 proofs.) Our focus in this paper will be on the use of pictures to see why the theorem is true. Of course, if we want to use pictures to display the behavior of polynomials defined on the complex numbers, we are immediately faced with a difficulty: the complex numbers are two-dimensional, so it appears that a graph of a complex-valued function on the complex numbers will require four dimensions. Our solution to this problem will be to use color to represent some dimensions. We begin by assigning a color to every number in the complex plane. Figure 1 is a picture of the complex plane in which every point has been assigned a different color. The origin is colored black. Traveling counterclockwise around a circle centered at the origin, we go through the colors of a standard color wheel: red, yellow, green, cyan, blue, magenta, and back to red. Points near the origin have dark colors, with the color assigned to a complex number z approaching black as z approaches 0. Points far from the origin are light, with the color of z approaching white as |z| approaches infinity. Every complex number has a different color in this picture, so a complex number can be uniquely specified by giving its color. We can now use this color scheme to draw a picture of a function f : C → C as follows: we simply color each point z in the complex plane with the color corresponding to the value of f(z). From such a picture, we can read off the value of f(z), for any complex number z, by determining the color of the point z in the picture, and then consulting Figure 1 to see what complex number is represented by that color.
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- 2015
41. RESULTS ON MEROMORPHIC FUNCTIONS SHARING THREE VALUES WITH THEIR DIFFERENCE OPERATORS
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Cong-Yun Kang, Xiao-Min Li, and Hong-Xun Yi
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Discrete mathematics ,Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,Order (group theory) ,Value (computer science) ,Uniqueness ,Shift operator ,Positive real numbers ,Complex plane ,Measure (mathematics) ,Mathematics ,Meromorphic function - Abstract
Under the restriction of finite order, we prove two uniqueness theorems of nonconstant meromorphic functions sharing three values with their difference operators, which are counterparts of Theorem 2.1 in (6) for a finite-order meromorphic function and its shift operator. 1. Introduction and main results In this paper, by meromorphic functions we will always mean meromorphic functions in the complex plane. We adopt the standard notations of the Nevan- linna theory of meromorphic functions as explained in (5), (10) and (16). It will be convenient to let E denote any set of positive real numbers of finite lin- ear measure, not necessarily the same at each occurrence. For a nonconstant meromorphic function h, we denote by T(r,h) the Nevanlinna characteristic of h and by S(r,h) any quantity satisfying S(r,h) = o(T(r,h)), as r → ∞,r 6∈E. Let f and g be two nonconstant meromorphic functions, and let a be a value in the extended plane. We say that f and g share the value a CM, provided that f and g have the same a-points with the same multiplicities. We say that f and g share the value a IM, provided that f and g have the same a-points ignoring multiplicities (cf. (16)). Throughout this paper, we denote by �(f) the order of f (cf. (5), (10) and (16)). We also need the following two definitions: Definition 1.1 ((15)). Let f be a nonconstant meromorphic function. We define difference operators of f as
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- 2015
42. Simultaneous uniformization for uniformly quasisymmetric circle dynamical systems
- Author
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Yunping Jiang and Frederick P. Gardiner
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Discrete mathematics ,Pure mathematics ,Conjecture ,General Mathematics ,Riemann surface ,Riemann sphere ,Quasicircle ,Computer Science Applications ,symbols.namesake ,symbols ,Branched covering ,Uniqueness ,Invariant (mathematics) ,Probability measure ,Mathematics - Abstract
In the 1960s Bers showed how to uniformize simultaneously two Riemann surfaces of the same finite analytic type by using a single quasi-Fuchsian group of the first kind. In this paper, we show how to uniformize simultaneously two uniformly quasisymmetric circle endomorphisms of the same degree by a unique normalized branched covering of the Riemann sphere of the same degree such that this branched covering has a unique normalized quasicircle as an invariant limit set. We use this simultaneous uniformization to define a transformation between their spaces of probability invariant measures and formulate several equivalent conjectures to the uniqueness conjecture for symmetric invariant probability measures. In a subsequent paper, we study these conjectures.
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- 2015
43. Probabilistically nilpotent Hopf algebras
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Sara Westreich and Miriam Cohen
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Discrete mathematics ,Pure mathematics ,Ring (mathematics) ,Quantum group ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,MathematicsofComputing_GENERAL ,Commutator (electric) ,Quasitriangular Hopf algebra ,Hopf algebra ,law.invention ,16T05 ,Nilpotent ,Invertible matrix ,law ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Nilpotent group ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A A which depends only on the Grothendieck ring of H . H. When H H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are.
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- 2015
44. Baire classes of complex L 1-preduals
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Jiří Spurný and Pavel Ludvík
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Discrete mathematics ,Unit sphere ,Pure mathematics ,Baire function ,General Mathematics ,Bounded function ,Baire category theorem ,Ball (mathematics) ,Property of Baire ,Baire space ,Baire measure ,Mathematics - Abstract
Let X be a complex L1-predual, non-separable in general. We investigate extendability of complex-valued bounded homogeneous Baire-α functions on the set ext BX* of the extreme points of the dual unit ball BX* to the whole unit ball BX*. As a corollary we show that, given α ∈ [1, ω1), the intrinsic α-th Baire class of X can be identified with the space of bounded homogeneous Baire-α functions on the set ext BX* when ext BX* satisfies certain topological assumptions. The paper is intended to be a complex counterpart to the same authors’ paper: Baire classes of non-separable L1-preduals (2015). As such it generalizes former work of Lindenstrauss and Wulbert (1969), Jellett (1985), and ourselves (2014), (2015).
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- 2015
45. Parametric properties of irreducibility sets of linear differential systems
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N. A. Izobov and S. A. Mazanik
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Lyapunov function ,Discrete mathematics ,Pure mathematics ,Partial differential equation ,General Mathematics ,Lyapunov exponent ,symbols.namesake ,Ordinary differential equation ,Bounded function ,Piecewise ,symbols ,Irreducibility ,Coefficient matrix ,Analysis ,Mathematics - Abstract
In the paper [Differ. Uravn., 2007, vol. 43, no. 2, pp. 191–202], we defined the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) bounded on the half-line [0,+∞) with norms ||A(t)|| ≤ a < +∞ for each of which there exists a linear differential system that cannot be reduced to it by Lyapunov transformations and whose coefficient matrix B(t) satisfies the condition ||B(t) - A(t)|| ≤ const × e −σt , t ≥ 0, or the more general condition that the Lyapunov exponent of the difference B(t) - A(t) does not exceed -σ, respectively. In the present paper, we study the properties of irreducibility sets treated as functions of the parameters σ and a.
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- 2015
46. THE LIMITING CASE OF SEMICONTINUITY OF AUTOMORPHISM GROUPS
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Steven G. Krantz
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Discrete mathematics ,Pure mathematics ,Smoothness (probability theory) ,Complex space ,Biholomorphism ,General Mathematics ,Boundary (topology) ,Limiting case (mathematics) ,Automorphism ,Domain (mathematical analysis) ,Mathematics - Abstract
In this paper we study the semicontinuity of the automor-phism groups of domains in multi-dimensional complex space. We giveexamples to show that known results are sharp (in terms of the requiredboundary smoothness). 1. IntroductionThepaper [4] wasthe firstworktostudy the semicontinuityofautomorphismgroups of domains in complex space. The main result there is as follows:Theorem 1.1. LetΩ ∗ ⊆ C n be a strongly pseudoconvex domain with smoothboundary. ThenthereisaneighborhoodU ofΩ ∗ intheC ∞ topologyondomains(thatistosay,U isacollectionofdomains) sothat,ifΩ ∈ U,thenAut(Ω) isasubgroupofAut(Ω ∗ ). Moreover, thereisaC ∞ mappingΨ fromΩ toΩ 0 sothatAut(Ω) ∋ ϕ −→ Ψ◦ϕ ◦Ψ −1 isaninjectivegrouphomomorphism fromAut(Ω) toAut(Ω ∗ ).Over the years, the hypothesis of smooth or C ∞ boundary in this theoremhas been weakened. In the paper [5], the hypothesis was weakened (using anentirely different argument) to C 2 boundary smoothness. In the paper [3],yet another approach to the C 2 boundary smoothness situation was described.The paper [2] treats the case of C
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- 2015
47. THE STRONG MORI PROPERTY IN RINGS WITH ZERO DIVISORS
- Author
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Fanggui Wang and Dechuan Zhou
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Domain (ring theory) ,Commutative ring ,Ideal (ring theory) ,Ascending chain condition ,Regular ideal ,Quotient ring ,Zero divisor ,Mathematics ,Integral domain - Abstract
An SM domain is an integral domain which satisfies the as-cending chain condition on w-ideals. Then an SM domain also satisfiesthe descending chain condition on those chains of v-ideals whose intersec-tion is not zero. In this paper, a study is begun to extend these propertiesto commutative rings with zero divisors. A Q 0 -SM ring is defined to be aring which satisfies the ascending chain condition on semiregular w-idealsand satisfies the descending chain condition on those chains of semiregularv-ideals whose intersection is semiregular. In this paper, some propertiesof Q 0 -SM rings are discussed and examples are provided to show the dif-ference between Q 0 -SM rings and SM rings and the difference betweenQ 0 -SM rings and Q 0 -Mori rings. 1. IntroductionIn this paper, we assume that Ris a commutative ring with an identity andSis a multiplicatively closed set of R. Let Z(R) denote the set of zero divisorsof Rand let T(R) be the total quotient ring of R.Let I be an ideal of R. Then I is called a regular ideal if I contains aregular element and I is called a semiregular ideal of Rif I contains a finitelygenerated ideal I
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- 2015
48. On the Convolution of a Finite Number of Analytic Functions Involving a Generalized Srivastava–Attiya Operator
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Janusz Sokół, Ravinder Krishna Raina, and Poonam Sharma
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Discrete mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Convolution power ,01 natural sciences ,Convexity ,Circular convolution ,Riemann zeta function ,Convolution ,symbols.namesake ,Operator (computer programming) ,symbols ,0101 mathematics ,Finite set ,Mathematics ,Analytic function - Abstract
The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.
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- 2015
49. The Structure of Translation-Invariant Spaces on Locally Compact Abelian Groups
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Marcin Bownik and Kenneth A. Ross
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Discrete mathematics ,Pointwise ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Dimension function ,Second-countable space ,Linear subspace ,Euclidean geometry ,Locally compact space ,Abelian group ,Invariant (mathematics) ,Analysis ,Mathematics - Abstract
Let \(\Gamma \) be a closed co-compact subgroup of a second countable locally compact abelian (LCA) group \(G\). In this paper we study translation-invariant (TI) subspaces of \(L^2(G)\) by elements of \(\Gamma \). We characterize such spaces in terms of range functions extending the results from the Euclidean and LCA setting. The main innovation of this paper, which contrasts with earlier works, is that we do not require that \(\Gamma \) be discrete. As a consequence, our characterization of TI-spaces is new even in the classical setting of \(G=\mathbb {R}^n\). We also extend the notion of the spectral function in \(\mathbb {R}^n\) to the LCA setting. It is shown that spectral functions, initially defined in terms of \(\Gamma \), do not depend on \(\Gamma \). Several properties equivalent to the definition of spectral functions are given. In particular, we show that the spectral function scales nicely under the action of epimorphisms of \(G\) with compact kernel. Finally, we show that for a large class of LCA groups, the spectral function is given as a pointwise limit.
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- 2015
50. Graded Annihilators and Uniformly F-Compatible Ideals
- Author
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Rodney Y. Sharp
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,13A35, 16S36 ,Semiprime ring ,Excellent ring ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Associated prime ,Cohen–Macaulay ring ,Module ,FOS: Mathematics ,Krull dimension ,Ideal (ring theory) ,Tight closure ,Mathematics - Abstract
Let $R$ be a commutative (Noetherian) local ring of prime characteristic $p$ that is $F$-pure. This paper is concerned with comparison of three finite sets of radical ideals of $R$, one of which is only defined in the case when $R$ is $F$-finite (that is, is finitely generated when viewed as a module over itself via the Frobenius homomorphism). Two of the afore-mentioned three sets have links to tight closure, via test ideals. Among the aims of the paper are a proof that two of the sets are equal, and a proposal for a generalization of I. M. Aberbach's and F. Enescu's splitting prime., 17 pages. This paper has been accepted for publication in Acta Mathematica Vietnamica
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- 2015
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