612 results
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2. Correction to a paper on wreath products
- Author
-
J. D. P. Meldrum
- Subjects
Pure mathematics ,Wreath product ,General Mathematics ,Mathematics - Abstract
Some time ago, Dr I. B. S. Passi pointed out to me the connexion between the structure of the terms of the α-central series of a wreath product and that of polynomial groups which he had defined and studied. In discussing this problem with him, we looked closely at a paper (1) which I had written on nilpotent wreath products. In the process of this, I found a mistake in Theorem 3·7, the main result of (1). The results in (2) depend onthis theorem. Fortunately, it is possible to obtain a correct version of Theorem 3·7 and to modify the proof in (2) which depends on this theorem in such a way that all the results of (2) remain valid.
- Published
- 1974
3. Note on the paper 'On the combination of subalgebras'
- Author
-
Garrett Birkhoff
- Subjects
Pure mathematics ,General Mathematics ,Dedekind cut ,Relation (history of concept) ,Mathematics - Abstract
Professor Oystein Ore of Yale University has kindly called my attention to the close relation between my paper and some earlier researches of Dedekind. I should like to correlate, as far as possible, my results with his.
- Published
- 1934
4. On generation of the root lattice by roots
- Author
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Simon M. Goodwin
- Subjects
Pure mathematics ,Lattice (module) ,General Mathematics ,Short paper ,Root (chord) ,Basis (universal algebra) ,Root system ,Mathematics - Abstract
Let Φ be a root system and let Γ ⊆ Φ. In this short paper we prove that Γ contains a ${\mathbb Z}$-basis of the lattice that it generates.
- Published
- 2007
5. On the line-comitants of a space cubic
- Author
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R. W. Weitzenböck
- Subjects
Pure mathematics ,General Mathematics ,Short paper ,Cubic form ,Projective invariants ,Invariant (mathematics) ,Mathematics - Abstract
In a short paper recently published in these Proceedings, E. J. F. Primrose determines amongst other things a projective invariant N of a space cubic K and a linear complex L, N being of the third degree in the coefficients cik of the complex L. The purpose of the following remarks is to establish the connexion between this invariant N and the well-known projective comitants of the cubic K.
- Published
- 1951
6. Higher horospherical limit sets for G-modules over CAT(0)-spaces
- Author
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Ross Geoghegan and Robert Bieri
- Subjects
Pure mathematics ,Discrete group ,Euclidean space ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Space (mathematics) ,01 natural sciences ,Action (physics) ,Zeroth law of thermodynamics ,010201 computation theory & mathematics ,Tropical geometry ,Limit (mathematics) ,0101 mathematics ,Group theory ,Mathematics - Abstract
The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.
- Published
- 2021
7. The factorisation property ofl∞(Xk)
- Author
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Paul F. X. Müller, Thomas Schlumprecht, Pavlos Motakis, and Richard Lechner
- Subjects
Pure mathematics ,Property (philosophy) ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Diagonal ,Banach space ,01 natural sciences ,Identity (music) ,Bounded operator ,Factorization ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we consider the following problem: letXk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by${(e_{(k,j)}^*)_j}$, for$k\in\N$, let$Z=\ell^\infty(X_k:k\kin\N)$be theirl∞-sum, and let$T:Z\to Z$be a bounded linear operator with a large diagonal,i.e.,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$Under which condition does the identity onZfactor throughT? The purpose of this paper is to formulate general conditions for which the answer is positive.
- Published
- 2020
8. Obstructions for semigroups of partial isometries to be self-adjoint
- Author
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Janez Bernik and Alexey I. Popov
- Subjects
Pure mathematics ,General Mathematics ,Self-adjoint operator ,Mathematics - Abstract
In this paper we study the following question: given a semigroup ${\mathcal S}$ of partial isometries acting on a complex separable Hilbert space, when does the selfadjoint semigroup ${\mathcal T}$ generated by ${\mathcal S}$ again consist of partial isometries? It has been shown by Bernik, Marcoux, Popov and Radjavi that the answer is positive if the von Neumann algebra generated by the initial and final projections corresponding to the members of ${\mathcal S}$ is abelian and has finite multiplicity. In this paper we study the remaining case of when this von Neumann algebra has infinite multiplicity and show that, in a sense, the answer in this case is generically negative.
- Published
- 2016
9. The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers
- Author
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Marcelo Laca and Siegfried Echterhoff
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,Mathematics::Operator Algebras ,Semigroup ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Algebraic number field ,Primary 46L05, 46L80, Secondary 20Mxx, 11R04 ,Space (mathematics) ,01 natural sciences ,Ring of integers ,Primitive ideal ,Prime (order theory) ,Crossed product ,0103 physical sciences ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Operator Algebras (math.OA) ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.
- Published
- 2012
10. On the formality and strong Lefschetz property of symplectic manifolds
- Author
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Jin Hong Kim and Taek Gyu Hwang
- Subjects
Pure mathematics ,Symplectic group ,General Mathematics ,Mathematical analysis ,Symplectic representation ,Mathematics::Geometric Topology ,Connected sum ,Symplectic vector space ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Mathematics ,Symplectic manifold ,Symplectic geometry - Abstract
The main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kähler and symplectic manifolds. To be precise, it is known that simply connected symplectic manifolds of dimension greater than 8, in general, do not satisfy the formality satisfied by all Kähler manifolds. In this paper we show that such non-formality of simply connected symplectic manifolds occurs even in dimension 8. We do this by some complicated but explicit construction of a simply connected non-formal symplectic manifold of dimension 8. In this construction we essentially use a variation of the construction of a simply connected symplectic manifold by Gompf. As a consequence, we can give infinitely many simply connected non-formal symplectic manifolds of any even dimension no less than 8.Secondly, we show that every compact symplectic manifold admitting a semi-free Hamiltonian circle action with only isolated fixed points must satisfy the strong Lefschetz property satisfied by all Kähler manifolds. This result shows that the strong Lefschetz property for the symplectic manifold admitting Hamiltonian circle actions is closely related to their fixed point set, as expected.
- Published
- 2008
11. Weak amenability of certain classes of Banach algebras without bounded approximate identities
- Author
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F. Ghahramani and A. T. M. Lau
- Subjects
Pure mathematics ,Fourier algebra ,Computer Science::Information Retrieval ,General Mathematics ,Subalgebra ,Group algebra ,C*-algebra ,Filtered algebra ,Algebra ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Division algebra ,Cellular algebra ,Banach *-algebra ,Mathematics - Abstract
In a recent paper [3] Dales and Pandey have shown that the class Sp of Segal algebras is weakly amenable. In this paper, for various classes of Segal algebras, we characterize derivations and multipliers from a Segal algebra into itself and into its dual module. In particular, we prove that every Segal algebra on a locally compact abelian group is weakly amenable and an abstract Segal subalgebra of a commutative weakly amenable Banach algebra is weakly amenable. We also introduce the Lebesgue–Fourier algebra of a locally compact group G and study its Arens regularity when G is discrete or compact.
- Published
- 2002
12. The invariant algebraic surfaces of the Lorenz system
- Author
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Sir Peter Swinnerton-Dyer
- Subjects
Discrete mathematics ,Pure mathematics ,Integrable system ,General Mathematics ,Algebraic surface ,Invariant (physics) ,Special values ,Lorenz system ,Mathematics - Abstract
The object of this paper is to find all the irreducible algebraic surfaces which (for special values of the parameters b, r, s) are invariant under the Lorenz systemx˙ = X(x, y, z) = s(y−x), y˙ = Y(x, y, z) = rx−y−xz, ż = Z(x, y, z) =−bz+xy. (1)It is customary in considering the Lorenz system to require the parameters b, r, s to be all strictly positive; however for this particular problem we shall follow previous practice in only imposing the condition s ≠ 0. (If s = 0 the equations are trivially integrable and x is constant on any trajectory; thus x should be regarded as a parameter and the question discussed in this paper ceases to be a natural one.)
- Published
- 2002
13. On the deep structure of the blowing-up of curve singularities
- Author
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Juan Elias
- Subjects
Pure mathematics ,Monomial ,Hilbert series and Hilbert polynomial ,Semigroup ,General Mathematics ,Mathematical analysis ,Tangent cone ,Discrete valuation ring ,Blowing up ,symbols.namesake ,symbols ,Invariant (mathematics) ,Quotient ,Mathematics - Abstract
Let C be a germ of curve singularity embedded in (kn, 0). It is well known that the blowing-up of C centred on its closed ring, Bl(C), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl(C) → C. We perform this by studying the structure of [Oscr ]Bl(C)/[Oscr ]C as W-module, where W is a discrete valuation ring contained in [Oscr ]C. Since [Oscr ]Bl(C)/[Oscr ]C is a torsion W-module, its structure is determined by the invariant factors of [Oscr ]C in [Oscr ]Bl(C). The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2).In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ]C (see Proposition 1·4).The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [6]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = {g[r,i+1]}i [ges ] 0 of the tangent cone of [Oscr ]C (see Section 2). The main property of g is that the ideals g[r,i+1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations.In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.
- Published
- 2001
14. An obstruction to slicing knots using the eta invariant
- Author
-
Carl F. Letsche
- Subjects
Knot complement ,Eta invariant ,Pure mathematics ,Knot (unit) ,Closed manifold ,Metabelian group ,General Mathematics ,Mathematical analysis ,Cobordism ,Homology (mathematics) ,Invariant (mathematics) ,Mathematics::Geometric Topology ,Mathematics - Abstract
We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.
- Published
- 2000
15. A systematic approach to symmetric presentations II: Generators of order 3
- Author
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John N. Bray and Robert T. Curtis
- Subjects
Classical group ,Algebra ,Monomial ,Pure mathematics ,Character table ,business.industry ,General Mathematics ,Simple group ,Order (group theory) ,Modular design ,business ,Mathematics - Abstract
In this paper we conduct a systematic computerized search for groups generated by small, but highly symmetric, sets of elements of order 3. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. Firstly, we introduce monomial modular representations as these will prove useful later in the paper. Then the techniques of symmetric generation developed elsewhere are described afresh. The results we obtain are presented in a convenient tabular form, together with relevant character tables.
- Published
- 2000
16. Brill–Noether theory for vector bundles on projective curves
- Author
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E. Ballico
- Subjects
Section (fiber bundle) ,Pure mathematics ,Mathematics::Algebraic Geometry ,Chern class ,Line bundle ,General Mathematics ,Associated bundle ,Mathematical analysis ,Vector bundle ,Stiefel–Whitney class ,Brill–Noether theory ,Principal bundle ,Mathematics - Abstract
In this paper we will study the Brill–Noether theory of vector bundles on a smooth projective curve X. As usual in papers on this topic we are mainly interested in stable or at least semistable bundles. Let Wkr, d(X) be the scheme of all stable vector bundles E on X with rank (E)=r, deg (E)=d and h0(X, E)[ges ]k+1. For a survey of the main known results, see the introduction of [6]. The referee has pointed out that the results in [6] were improved by V. Mercat in [14]; he proved that Wkr, d(X) is non-empty for d
- Published
- 1998
17. Fourier duality in the Brascamp–Lieb inequality
- Author
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Jonathan Bennett and Eunhee Jeong
- Subjects
Pure mathematics ,Brascamp–Lieb inequality ,Property (philosophy) ,General Mathematics ,Duality (optimization) ,symbols.namesake ,Fourier transform ,Euclidean geometry ,symbols ,Mathematics::Metric Geometry ,Dual polyhedron ,Locally compact space ,Abelian group ,Mathematics - Abstract
It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.
- Published
- 2021
18. On Ramanujan's cubic transformation formula for 2 F 1(1/3, 2/3; 1; z)
- Author
-
Heng Huat Chan
- Subjects
Algebra ,symbols.namesake ,Pure mathematics ,Transformation (function) ,General Mathematics ,symbols ,Elliptic function ,Mathematical proof ,Signature (topology) ,Ramanujan's sum ,Mathematics - Abstract
The main aim of this paper is to provide two new proofs of Ramanujan's cubic transformation formula for 2F1(1/3, 2/3; 1; z) (see (1·8) below). For our first proof, we have to develop Ramanujan's elliptic functions in the theory of signature 3 using a different approach from that given in a recent paper by Berndt, Bhargava and Garvan. For our second proof, we use two of Goursat's formulas.
- Published
- 1998
19. Lagrangian isometric immersions of a real-space-form Mn(c) into a complex-space-form M˜n(4c)
- Author
-
Franki Dillen, Luc Vrancken, Leopold Verstraelen, and Bang-Yen Chen
- Subjects
Twistor theory ,Pure mathematics ,Complex space ,Geodesic ,General Mathematics ,Mathematical analysis ,Immersion (mathematics) ,Holomorphic function ,Space form ,Isometric exercise ,Sectional curvature ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
It is well known that totally geodesic Lagrangian submanifolds of a complex-space-form M˜n(4c) of constant holomorphic sectional curvature 4c are real-space-forms of constant sectional curvature c. In this paper we investigate and determine non-totally geodesic Lagrangian isometric immersions of real-space-forms of constant sectional curvature c into a complex-space-form M˜n(4c). In order to do so, associated with each twisted product decomposition of a real-space-form of the form f1I1×… ×fkIk×1Nn−k(c), we introduce a canonical 1-form, called the twistor form of the twisted product decomposition. Roughly speaking, our main result says that if the twistor form of such a twisted product decomposition of a simply-connected real-space-form of constant sectional curvature c is twisted closed, then it admits a ‘unique’ adapted Lagrangian isometric immersion into a complex-space-form M˜n(4c). Conversely, if L: Mn(c)→ M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the Lagrangian immersion L is given by the corresponding adapted Lagrangian isometric immersion of the twisted product. In this paper we also provide explicit constructions of adapted Lagrangian isometric immersions of some natural twisted product decompositions of real-space-forms.
- Published
- 1998
20. Minimal normal systems of compact right topological groups
- Author
-
Alan Moran
- Subjects
Discrete mathematics ,Normal subgroup ,Pure mathematics ,Compact group ,Group (mathematics) ,General Mathematics ,Hausdorff space ,Topological ring ,Topological group ,Locally compact group ,Mathematics ,Haar measure - Abstract
We introduce in this paper the notion of a normal system for a compact, Hausdorff, right topological group, G. This generalizes the notion of a strong normal system as introduced in [7], in which it was shown that closed subgroups of compact right topological groups with dense topological centres possess strong normal systems and have Haar measure. Their method involved the construction of a strong normal system, the Furstenberg–Namioka system. In this paper we study a variation of the Furstenberg–Namioka system which we call the N-system. The N-system for a group G is a normal system and we show that if G has dense topological centre, it possesses an N-system, though the two systems do not in general coincide. In Section 2 we give examples of groups with arbitrarily long N-systems and show that a dense topological centre is not a necessary requirement for the existence of normal systems. In Section 3 we introduce the notion of minimality of normal systems and we show that if the N-system for a group, G, is minimal then the N-system and the Furstenberg–Namioka system, if the latter exists, coincide (and therefore G has Haar measure!). Finally in our main theorem, we show that if G has a minimal N-system then the closed normal subgroups of G must lie between successive terms of the N-system.
- Published
- 1998
21. Exceptions to the multifractal formalism for discontinuous measures
- Author
-
Benoit B. Mandelbrot and Rudolf H. Riedi
- Subjects
Pure mathematics ,General Mathematics ,Multifractal formalism ,Degenerate energy levels ,Mathematical analysis ,Inverse ,Multifractal spectra ,Mathematics - Abstract
In an earlier paper [MR] the authors introduced the inverse measure μ[dagger](dt) of a given measure μ(dt) on [0, 1] and presented the ‘inversion formula’ f[dagger](α)=αf(1/α) which was argued to link the respective multifractal spectra of μ and μ[dagger]. A second paper [RM2] established the formula under the assumption that μ and μ[dagger] are continuous measures.Here, we investigate the general case which reveals telling details of interest to the full understanding of multifractals. Subjecting self-similar measures to the operation μ[map ]μ[dagger] creates a new class of discontinuous multifractals. Calculating explicitly we find that the inversion formula holds only for the ‘fine multifractal spectra’ and not for the ‘coarse’ ones. As a consequence, the multifractal formalism fails for this class of measures. A natural explanation is found when drawing parallels to equilibrium measures. In the context of our work it becomes natural to consider the degenerate Holder exponents 0 and ∞.
- Published
- 1998
22. Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity
- Author
-
D. Rees and D. Kirby
- Subjects
Discrete mathematics ,Pure mathematics ,Exact sequence ,Noetherian ring ,General Mathematics ,Graded ring ,Free module ,Multiplicity (mathematics) ,Finitely-generated abelian group ,Mathematics - Abstract
While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].
- Published
- 1996
23. Centers and Azumaya loci for finite W-algebras in positive characteristic
- Author
-
Bin Shu and Yang Zeng
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .
- Published
- 2021
24. Generalized interpolation in finite maximal subdiagonal algebras
- Author
-
Kichi-Suke Saito
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Mathematics::Operator Algebras ,General Mathematics ,Mathematics ,Interpolation - Abstract
Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.
- Published
- 1995
25. Examples in non-commutative projective geometry
- Author
-
J. J. Zhang and J. T. Stafford
- Subjects
Discrete mathematics ,Pure mathematics ,Collineation ,General Mathematics ,Complex projective space ,Projective space ,Projective differential geometry ,Pencil (mathematics) ,Projective variety ,Mathematics ,Twisted cubic ,Projective geometry - Abstract
Let A = k ⊕ ⊕n ≥ 1An connected graded, Noetherian algebra over a fixed, central field k (formal definitions will be given in Section 1 but, for the most part, are standard). If A were commutative, then the natural way to study A and its representations would be to pass to the associated projective variety and use the power of projective algebraic geometry. It has become clear over the last few years that the same basic idea is powerful for non-commutative algebras; see, for example, [ATV1, 2], [AV], [Sm], [SS] or [TV] for some of the more significant applications. This suggests that it would be profitable to develop a general theory of ‘non-commutative projective geometry’ and the foundations for such a theory have been laid down in the companion paper [AZ]. The results proved there raise a number of questions and the aim of this paper is to provide negative answers to several of these.
- Published
- 1994
26. Free quotients of subgroups of the Bianchi groups whose kernels contain many elementary matrices
- Author
-
A. W. Mason and R. W. K. Odoni
- Subjects
Pure mathematics ,Elementary matrix ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematical economics ,Quotient ,Mathematics - Abstract
Let d be a square-free positive integer and let be the ring of integers of the imaginary quadratic number field ℚ(√ − d) The Bianchi groups are the groups SL2() (or PSL2(). Let m be the order of index m in . In this paper we prove that for each d there exist infinitely many m for which SL2(m)/NE2(m) has a free, non-cyclic quotient, where NE2(m) is the normal subgroup of SL2(m) generated by the elementary matrices. When d is not a prime congruent to 3 (mod 4) this result is true for all but finitely many m. The proofs are based on the fundamental paper of Zimmert and its generalization due to Grunewald and Schwermer.The results are used to extend earlier work of Lubotzky on non-congruence subgroups of SL2(), which involves the concept of the ‘non-congruence crack’. In addition the results highlight a number of low-dimensional anomalies. For example, it is known that [SLn(m), SLnm)] = En(m), when n ≥ 3, where [SLn(m), SLn(m)] is the commutator subgroup of SL(m) and En(m) is the subgroup of SLn(m) generated by the elementary matrices. Our results show that this is not always true when n = 2.
- Published
- 1994
27. Double point surfaces of smooth immersions Mn → ℝ2n−2
- Author
-
András Szücs
- Subjects
Multiple point ,Pure mathematics ,Double point ,General Mathematics ,Mathematical analysis ,Euclidean geometry ,Mathematics - Abstract
In recent years many papers have dealt with the zero-dimensional multiple point sets of smooth immersions of closed manifolds in Euclidean spaces (see, for example, [2, 6, 7, 8, 9, 10, 13, 14, 15, 21]). In the present paper we deal with the case when the double points form a two-dimensional surface and consider the following question: Which surfaces can occur as double point surfaces of self-transverse immersions of closed n-manifolds in ℝ2n−2? We prove the following
- Published
- 1993
28. Elliptic functions, theta function and hypersurfaces satisfying a basic equality
- Author
-
Jie Yang and Bang-Yen Chen
- Subjects
Riemann curvature tensor ,Mean curvature flow ,Pure mathematics ,General Mathematics ,Prescribed scalar curvature problem ,Mathematical analysis ,symbols.namesake ,symbols ,Curvature form ,Mathematics::Differential Geometry ,Sectional curvature ,Exponential map (Riemannian geometry) ,Ricci curvature ,Scalar curvature ,Mathematics - Abstract
In previous papers [ 4 , 6 ], B.-Y. Chen introduced a Riemannian invariant δ M for a Riemannian n -manifold M n , namely take the scalar curvature and subtract at each point the smallest sectional curvature. He proved that every submanifold M n in a Riemannian space form R m (e) satisfies: δ M [les ][ n 2 ( n −2)]/ 2( n −1) H 2 +[half]( n +1)( n −2)e. In this paper, first we classify constant mean curvature hypersurfaces in a Riemannian space form which satisfy the equality case of the inequality. Next, by utilizing Jacobi's elliptic functions and theta function we obtain the complete classification of conformally flat hypersurfaces in Riemannian space forms which satisfy the equality.
- Published
- 2001
29. The ideal structure of some Banach algebras
- Author
-
M. Filali
- Subjects
Discrete mathematics ,Annihilator ,Pure mathematics ,General Mathematics ,Subalgebra ,Locally compact group ,Abelian group ,Invariant (mathematics) ,Character group ,Commutative property ,Linear subspace ,Mathematics - Abstract
Let G be a locally compact group and let G be its character group. Among other results, the minimal left ideals of L m (G)* and LUC(G)* are all found when G is abelian. The main tool used for this study is the set of (topological) ^-invariant functionals (^eG), defined in this paper. Let Gbe a locally compact group. The conjugate space of L°°((r) and some of its subspaces like LUC (6?) can be made into a Banach algebra by the use of an Arens product. The principal concern of this paper is to study the minimal ideals of these algebras. The paper is organized as follows. In the first section, some basic definitions and notation are given. In Section 2, we introduce the ^-invariant functionals (where ^ is a character of G). This is an extension of the notion of the invariant functionals. It provides us with more minimal left ideals and more elements of the radicals of L X (G)* and LUC(G)* when G is amenable. In Section 3, G is assumed to be abelian (so amenable). We first see how the maximal modular ideals of a Banach algebra si are related to those of a closed commutative subalgebra of si'. We then establish a 'duality' between the minimal and the maximal modular ideals of an arbitrary algebra. The combination of these two results leads to the complete determination of the minimal left ideals of L X (G)*, LUC((?)* and other algebras. We also obtain, by this technique, some information about the minimal right ideals. However, it is still unknown whether such ideals exist in either L CO (G ! )* or LUC (G)* when G is non-compact. A study of minimal ideals in some special cases like annihilator algebras can be found in chapter 5 of [19] or chapter 2 of [22].
- Published
- 1992
30. m-full ideals II
- Author
-
Junzo Watanabe
- Subjects
Discrete mathematics ,Pure mathematics ,Hilbert series and Hilbert polynomial ,Mathematics::Commutative Algebra ,Betti number ,General Mathematics ,Polynomial ring ,Local ring ,Binomial number ,symbols.namesake ,Artin algebra ,symbols ,Ideal (ring theory) ,Mathematics ,Hilbert–Poincaré series - Abstract
Introduction In his paper [10] the author investigated the structure of m-full ideals by analysing their syzygies and, as one special case, showed how the Betti numbers of Borel stable ideals over polynomial rings can be computed. The same result, among other things, was also obtained by Eliahou and Kervaire[l] by a different method. Let a be a Borel stable ideal in a polynomial ring R and let V{ be the ideal generated by i generic linear forms and let tn_i_l be the type of the ideal a+ VJVt over R/Vt. Then the Betti numbers of a are linear combinations of to,...,tn_l with certain binomial numbers as coefficients, and conversely from the numbers t0, ...,tn_1 the Betti numbers can be recovered (see [10], corollary 9 and proposition 4). The present paper grew out of the question of deciding what sequences t0, ...,tn_1 of integers can arise from a Borel stable ideal as above. Our goal of this paper is to prove Theorems 4l and 4-2 of Section 4, in which we show that if we make the restriction on the ideal that it be generated by monomials of a fixed degree then the sequence t0,..., tn_^ is precisely the same as what has been known as the Hilbert series of a graded Artin algebra. In an earlier paper [11] the author showed some properties of m-primary m-full ideals. In this present paper we need to generalize these results to ideals which are not necessarily m-primary. Most results of [11] can be generalized to general m-full ideals. The Oth local cohomology module Ua:mYa of R/a plays an important role. These are the contents of Section 1 and have independent interest. In Section 2 we summarize some combinatorial formulae derived from Macaulay's theorem which characterizes the Hilbert function of homogeneous algebras (Theorem 2-2). In Section 3 we consider m-full ideals o in regular local rings which satisfy the condition m (lo = ma. The result (Theorem 3-1) will be applied, in Section 4, to Borel stable ideals generated by monomials all of degree d to obtain the theorems mentioned above.
- Published
- 1992
31. Maximal prime homomorphic images of mod-p Iwasawa algebras
- Author
-
William Woods
- Subjects
Normal subgroup ,Ring (mathematics) ,Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Minimal prime ideal ,Mathematics - Rings and Algebras ,01 natural sciences ,Matrix ring ,Prime (order theory) ,Finite field ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Subquotient ,Mathematics - Representation Theory ,Group ring ,Mathematics - Abstract
Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.
- Published
- 2021
32. Twisted Donaldson invariants
- Author
-
Hang Wang, Hirofumi Sasahira, and Tsuyoshi Kato
- Subjects
Mathematics - Differential Geometry ,Fundamental group ,Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,Mathematics - Operator Algebras ,Picard group ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology ,01 natural sciences ,Manifold ,Mathematics - Geometric Topology ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,57R55, 57R57, 58B34, 46L87 ,Mathematics::Differential Geometry ,Gauge theory ,Abelian group ,Invariant (mathematics) ,Operator Algebras (math.OA) ,Mathematics::Symplectic Geometry ,Commutative property ,Smooth structure ,Mathematics - Abstract
Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang-Mills gauge theory from twisting by the Picard group of a four manifold in the case when the fundamental group is free abelian. We then generalize it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic four manifolds., Comment: typos fixed, Rewrite Section 4 to reduce technicality
- Published
- 2021
33. Paracompact locales and metric spaces
- Author
-
Jan Paseka
- Subjects
Pure mathematics ,General Mathematics ,Injective metric space ,010102 general mathematics ,0102 computer and information sciences ,Equivalence of metrics ,Pseudometric space ,Topology ,01 natural sciences ,Convex metric space ,Metric space ,010201 computation theory & mathematics ,Metric (mathematics) ,Isometry ,Paracompact space ,0101 mathematics ,Mathematics - Abstract
This paper deals with the category of paracompact locales (‘pointless topologies’), defined in the classic paper [6] of Isbell. A full discussion concerning paracompact locales can be found in Dowker and Strauss[2] and in Pultr[10, 11]. We shall provide a description of paracompact Tychonoff locales by means of a system of suitably chosen metric spaces. This answers Pultr's question whether each paracompact Tychonoff locale is a closed sublocale of a (localic) product of metric spaces.
- Published
- 1991
34. Compactifications of discrete versions of semitopological semigroups by filters of zero sets
- Author
-
Talin Budak
- Subjects
Discrete mathematics ,Pure mathematics ,Compact group ,Semigroup ,General Mathematics ,Metrization theorem ,Hausdorff space ,Topological semigroup ,Special classes of semigroups ,Compactification (mathematics) ,Topological space ,Mathematics - Abstract
The maximal proper prime filters together with the ultrafilters of zero sets of any metrizable compact topological space are shown to have a compact Hausdorff topology in which the ultrafilters form a discrete, dense subspace. This gives a general theory of compactifications of discrete versions of compact metrizable topological spaces and some of the already known constructions of compact right topological semigroups are special cases of the general theory. In this way, simpler and more elegant proofs for these constructions are obtained.In [8], Pym constructed compactifications for discrete semigroups which can be densely embedded in a compact group. His techniques made extensive use of function algebras. In [4] Helmer and Isik obtained the same compactifications by using the existence of Stone ech compactifications. The aim of this paper is to present a general theory of compactifications of semitopological semigroups so that Helmer and Isik's results in [4] are a simple consequence. Our proofs are different and are based on filters which provide a natural way of getting compactifications. Moreover we present new insights by emphasizing maximal proper primes which are not ultrafilters.We start by defining filters of zero sets (called z-filters) on a given topological space X, and their convergence. In the case of compact metrizable topological spaces, we establish the connections between proper maximal prime z-filters on X and zultrafilters in β(X\{x})\(X\{x}) where β(X\{x}) is the Stone-ech compactification of X\{x}. We then define a topology on the set of all prime z-filters on X such that the subspace of all proper maximal primes is compact Hausdorff. We denote by the set of all proper maximal prime z-filters on X together with the z-ultrafilters and show that when X is a compact metrizable cancellative semitopological semigroup, is a compact right topological semigroup with dense topological centre. Also, when is considered for a compact Hausdorff metrizable group, the semigroup obtained is exactly the same (algebraically and topologically) as the semigroup obtained in [4]. Hence the result in [4] is just a consequence of the general theory presented in this paper.
- Published
- 1991
35. Kähler groups and rigidity phenomena
- Author
-
F. E. A. Johnson and E. G. Rees
- Subjects
Pure mathematics ,Class (set theory) ,Group (mathematics) ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Rigidity (psychology) ,Projective test ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
The class of fundamental groups of non-singular complex projective varieties is an interesting, but as yet imperfectly understood, class of finitely presented groups. Membership of is known to be extremely restricted (see [22, 23]). In this paper, we employ geometrical rigidity properties to realize some group extensions as elements of as in our previous papers, we find it convenient to work simultaneously with the class ℋ of fundamental groups of compact Kähler manifolds.
- Published
- 1991
36. The flow near non-trivial minimal sets on 2-manifolds
- Author
-
Konstantin Athanassopoulos
- Subjects
Set (abstract data type) ,Continuation ,Work (thermodynamics) ,Pure mathematics ,Flow (mathematics) ,Continuous flow ,General Mathematics ,Mathematics - Abstract
In this paper we give a description of the qualitative behaviour of the orbits near a non-trivial compact minimal set of a continuous flow on a 2-manifold. The first results in this direction were obtained in [1] and the present paper can be regarded as a continuation of that work. The main result can be stated as follows:Theorem 1·1. Let (ℝ, M, f) be a continuous flow on a 2-manifold M and A ⊂ M a non-trivial compact minimal set.
- Published
- 1990
37. Fibrewise separation axioms for locales
- Author
-
P. T. Johnstone
- Subjects
Pure mathematics ,General Mathematics ,Separation axiom ,Mathematics - Abstract
In this paper we study the weak versions of the fibrewise separation axioms for locales over a base locale, whose introduction has been made possible by the development, in a previous paper by the author, of the fibrewise notion of closure for sublocales of locales over a base. We establish the implications which hold between these axioms and the traditional separation axioms for locales, and give counter-examples to show that some of these implications are irreversible.
- Published
- 1990
38. Directions in orbits of geometrically finite hyperbolic subgroups
- Author
-
Christopher Lutsko
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Distribution (number theory) ,Discrete group ,General Mathematics ,Hyperbolic space ,010102 general mathematics ,Boundary (topology) ,Dynamical Systems (math.DS) ,Lattice (discrete subgroup) ,01 natural sciences ,Correlation function (statistical mechanics) ,Fractal ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
We prove a theorem describing the limiting fine-scale statistics of orbits of a point in hyperbolic space under the action of a discrete subgroup. Similar results have been proved only in the lattice case, with two recent infinite-volume exceptions by Zhang for Apollonian circle packings and certain Schottky groups. Our results hold for general Zariski dense, non-elementary, geometrically finite subgroups in any dimension. Unlike in the lattice case, orbits of geometrically finite subgroups do not necessarily equidistribute on the whole boundary of hyperbolic space. But rather, they may equidistribute on a fractal subset. Understanding the behaviour of these orbits near the boundary is central to Patterson-Sullivan theory and much further work. Our theorem characterizes the higher order spatial statistics and thus addresses a very natural question. As a motivating example our work applies to sphere packings (in any dimension) which are invariant under the action of such discrete subgroups. At the end of the paper we show how this statistical characterization can be used to prove convergence of moments and to write down the limiting formula for the two-point correlation function and nearest neighbor distribution. Moreover we establish an formula for the 2 dimensional limiting gap distribution (and cumulative gap distribution) which was not known previously even in the lattice case., 33 pages, 1 figure; Accepted for Publication in Mathematical Proceedings of the Cambridge Philosophical Society
- Published
- 2020
39. On commensurability of right-angled Artin groups II: RAAGs defined by paths
- Author
-
Alexander Zakharov, Montserrat Casals-Ruiz, and Ilya Kazachkov
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Commensurability (mathematics) ,Mathematics - Abstract
In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to a RAAG defined by a tree of diameter 4 if and only if n ≡ 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.
- Published
- 2019
40. M-embedded symmetric operator spaces and the derivation problem
- Author
-
Jinghao Huang, Fedor Sukochev, and Galina Levitina
- Subjects
Pure mathematics ,Trace (linear algebra) ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Noncommutative geometry ,Operator space ,Symmetric function ,symbols.namesake ,Von Neumann algebra ,Norm (mathematics) ,Bounded function ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.
- Published
- 2019
41. Hilbert’s 16th problem on a period annulus and Nash space of arcs
- Author
-
Lubomir Gavrilov, Dongmei Xiao, and Jean-Pierre Françoise
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Perturbation (astronomy) ,Exceptional divisor ,01 natural sciences ,010101 applied mathematics ,Arc (geometry) ,Planar ,Quadratic equation ,Vector field ,0101 mathematics ,Bifurcation ,Mathematics - Abstract
This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.
- Published
- 2019
42. Critical groups for Hopf algebra modules
- Author
-
Darij Grinberg, Jia Huang, and Victor Reiner
- Subjects
Pure mathematics ,Finite group ,Quantitative Biology::Neurons and Cognition ,General Mathematics ,010102 general mathematics ,Regular representation ,05E10, 16T05, 16T30, 15B48, 20C20 ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,Hopf algebra ,01 natural sciences ,Cardinality ,Critical group ,Rings and Algebras (math.RA) ,FOS: Mathematics ,Greatest common divisor ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Indecomposable module ,Mathematics - Representation Theory ,Mathematics - Abstract
This paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalizes the critical groups of complex finite group representations studied by Benkart, Klivans, Reiner and Gaetz. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations., 24 pages. Comments are welcome! The ancillary file is a longer version with some more details and alternative proofs (compiled from the same source). The version on http://www.cip.ifi.lmu.de/~grinberg/algebra/ will probably get quicker updates. v3: Minor corrections
- Published
- 2018
43. On the SL(2, C)-representation rings of free abelian groups
- Author
-
Takao Satoh
- Subjects
Pure mathematics ,Ring (mathematics) ,General Mathematics ,Filtration (mathematics) ,Structure (category theory) ,Component (group theory) ,Homomorphism ,Abelian group ,Automorphism ,Mathematics ,Free abelian group - Abstract
In this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.
- Published
- 2018
44. Expected value of high powers of trace of frobenius of biquadratic curves over a finite field
- Author
-
Patrick Meisner
- Subjects
Class (set theory) ,Pure mathematics ,Trace (linear algebra) ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,010103 numerical & computational mathematics ,Expected value ,Infinity ,01 natural sciences ,Prime (order theory) ,Finite field ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Mathematics ,media_common - Abstract
Denote ΘCas the Frobenius class of a curveCover the finite field 𝔽q. In this paper we determine the expected value of Tr(ΘCn) whereCruns over all biquadratic curves whenqis fixed andgtends to infinity. This extends work done by Rudnick [15] and Chinis [5] who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda [1] who looked at ℓ-cyclic curves, for ℓ a prime, as well as cubic non-Galois curves.
- Published
- 2018
45. Whitney equisingularity of families of surfaces in ℂ3
- Author
-
O.N. Silva and M.A.S. Ruas
- Subjects
010101 applied mathematics ,Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,Germ ,0101 mathematics ,01 natural sciences ,Triviality ,Mathematics ,Counterexample - Abstract
In this paper, we study families of singular surfaces in ℂ3 parametrised by $\mathcal {A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding F of a finitely determined map germ f : (ℂ2, 0) → (ℂ3, 0). We investigate the following question: topological triviality implies Whitney equisingularity of the unfolding F? We provide a complete answer to this question, by giving counterexamples showing how the conjecture can be false.
- Published
- 2018
46. Fano manifolds of index n − 2 and the cone conjecture
- Author
-
Artie Prendergast-Smith and Izzet Coskun
- Subjects
Automorphism group ,Pure mathematics ,Mathematics::Algebraic Geometry ,Conjecture ,Cone (topology) ,Group (mathematics) ,Computer Science::Information Retrieval ,General Mathematics ,Fano plane ,Mathematics::Symplectic Geometry ,Action (physics) ,Mathematics - Abstract
The Morrison–Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi–Yau pair have rational, polyhedral fundamental domains. In [CPS], we proved the conjecture for certain blowups of Fano manifolds of index n - 1. In this paper, we consider the Morrison–Kawamata conjecture for blowups of Fano manifolds of index n - 2.
- Published
- 2017
47. Free actions ofp-groups on affine varieties in characteristicp
- Author
-
Peter Fleischmann and Chris F. Woodcock
- Subjects
Discrete mathematics ,Pure mathematics ,Finite group ,Ring (mathematics) ,Group (mathematics) ,General Mathematics ,Polynomial ring ,010102 general mathematics ,Field (mathematics) ,01 natural sciences ,Invariant theory ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,Abelian category ,0101 mathematics ,Algebraically closed field ,QA ,Mathematics - Abstract
LetKbe an algebraically closed field and$\mathbb{A}$n≅Knaffinen-space. It is known that a finite group$\frak{G}$can only act freely on$\mathbb{A}$nifKhas characteristicp> 0 and$\frak{G}$is ap-group. In that case the group action is “non-linear” and the ring of regular functionsK[$\mathbb{A}$n] must be atrace-surjectiveK−$\frak{G}$-algebra.Now letkbe an arbitrary field of characteristicp> 0 and letGbe a finitep-group. In this paper we study the category$\mathfrak{Ts}$of all finitely generated trace-surjectivek−Galgebras. It has been shown in [13] that the objects in$\mathfrak{Ts}$are precisely those finitely generatedk−GalgebrasAsuch thatAG≤Ais a Galois-extension in the sense of [7], with faithful action ofGonA. Although$\mathfrak{Ts}$is not an abelian category it has “s-projective objects”, which are analogues of projective modules, and it has (s-projective) categorical generators, which we will describe explicitly. We will show thats-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category$\mathfrak{Ts}$also has “weakly initial objects”, which are closely related to the essential dimension ofGoverk. Our results yield a geometric structure theorem for free actions of finitep-groups on affinek-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory ofp-groups and, potentially, to a new constructive approach to homogeneous invariant theory.
- Published
- 2017
48. Analytic spread and non-vanishing of asymptotic depth
- Author
-
Cleto B. Miranda Neto
- Subjects
Pure mathematics ,Monomial ,General Mathematics ,Prime ideal ,Polynomial ring ,010102 general mathematics ,Field (mathematics) ,Monomial ideal ,01 natural sciences ,Prime (order theory) ,0103 physical sciences ,Prime factor ,010307 mathematical physics ,Ideal (ring theory) ,0101 mathematics ,Mathematics - Abstract
LetSbe a polynomial ring over a fieldKof characteristic zero and letM⊂Sbe an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case whereMis a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime idealP⊂ScontainingMbe such that the localisationMPhasnon-maximal analytic spread. Our technique describes, in fact, a concrete obstruction forPto be an asymptotic prime divisor ofMwith respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes.
- Published
- 2017
49. Generalised Bohr compactification and model-theoretic connected components
- Author
-
Anand Pillay and Krzysztof Krupiński
- Subjects
Normal subgroup ,Pure mathematics ,medicine.medical_specialty ,Group (mathematics) ,Discrete group ,General Mathematics ,010102 general mathematics ,Hausdorff space ,Bohr compactification ,Topological dynamics ,0102 computer and information sciences ,01 natural sciences ,Kernel (algebra) ,010201 computation theory & mathematics ,medicine ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
For a group G first order definable in a structure M, we continue the study of the “definable topological dynamics” of G (from [9] for example). The special case when all subsets of G are definable in the given structure M is simply the usual topological dynamics of the discrete group G; in particular, in this case, the words “externally definable” and “definable” can be removed in the results described below.Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant G*/(G*)000M of G, which appears to be “new” in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalised Bohr compactification of G; [externally definable] strong amenability. Among other things, we essentially prove: (i) the “new” invariant G*/(G*)000M lies in between the externally definable generalised Bohr compactification and the definable Bohr compactification, and these all coincide when G is definably strongly amenable and all types in SG(M) are definable; (ii) the kernel of the surjective homomorphism from G*/(G*)000M to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup; (iii) when Th(M) is NIP, then G is [externally] definably amenable iff it is externally definably strongly amenable.In the situation when all types in SG(M) are definable, one can just work with the definable (instead of externally definable) objects in the above results.
- Published
- 2016
50. An Inhomogeneous Jarník type theorem for planar curves
- Author
-
Dzmitry Badziahin, Stephen Harrap, and Mumtaz Hussain
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Type (model theory) ,Diophantine approximation ,01 natural sciences ,Dual (category theory) ,Planar ,Homogeneous ,0103 physical sciences ,Convergence (routing) ,Metric (mathematics) ,11J83, 11J13, 11K60 ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In metric Diophantine approximation there are two main types of approximations: simultaneous and dual for both homogeneous and inhomogeneous settings. The well known measure-theoretic theorems of Khintchine and Jarn\'ik are fundamental in these settings. Recently, there has been substantial progress towards establishing a metric theory of Diophantine approximations on manifolds. In particular, both the Khintchine and Jarn\'ik type results have been established for planar curves except for only one case. In this paper, we prove an inhomogeneous Jarn\'ik type theorem for convergence on planar curves and in so doing complete the metric theory for both the homogeneous and inhomogeneous settings for approximation on planar curves., Comment: 24 pages. Updated to include bibliography, which was incorrectly compiled initially
- Published
- 2016
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