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1.
We prove analogues for reductive algebraic groups of some resultsfor finite groups due to Knörr and Robinson from Someremarks on a conjecture of Alperin, J. London Math. Soc(2) 39 (1989), 48–60, which play a central rôlein their reformulation of Alperin's conjecture for finite groups. 相似文献
2.
Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for hypercentralor nilpotent-by-Chernikov. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [13, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]). 相似文献
3.
Let S(X, B) be a symmetric (palindromic) wordin two letters X and B. A theorem due to Hillar and Johnsonstates that for each pair of positive definite matrices B andP, there is a positive definite solution X to the word equationS(X, B)=P. They also conjectured that these solutions are finiteand unique. In this paper, we resolve a modified version ofthis conjecture by showing that the Brouwer degree of such anequation is equal to 1 (in the case of real matrices). It followsthat, generically, the number of solutions is odd (and thusfinite) in the real case. Our approach allows us to addressthe more subtle question of uniqueness by exhibiting equationswith multiple real solutions, as well as providing a secondproof of the result of Hillar and Johnson in the real case. 相似文献
4.
A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, NG(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a perfectisometry and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of NG(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an isotypy. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a splendid tilting complexfor p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same p-local structure, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and lifting. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and cutting by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods. 相似文献
5.
We introduce the concept of geometrical spinefor 3-manifolds with natural metrics, in particular, for lensmanifolds. We show that any spine of Lp,q that is close enoughto its geometrical spine contains at least E(p,q) – 3vertices, which is exactly the conjectured value for the complexityc(Lp,q). As a byproduct, we find the minimal rotation distance(in the Sleator–Tarjan–Thurston sense) between atriangulation of a regular p-gon and its image under rotation. 相似文献
6.
Motivated in part by the first author's work [23] on the Weyl-Berryconjecture for the vibrations of fractal drums(that is, drums with fractal boundary), M. L.Lapidus and C. Pomerance [31] have studied a direct spectralproblem for the vibrations of fractal strings(that is, one-dimensional fractal drums) and establishedin the process some unexpected connections with the Riemannzeta-function = (s) in the critical interval0 < s < 1. In this paper we show, in particular, thatthe converse of their theorem (suitably interpreted as a naturalinverse spectral problem for fractal strings, with boundaryof Minkowski fractal dimension D (0,1)) is not true in themidfractal case when D = , but that it is true for all other D in the criticalinterval (0,1) if and only if the Riemann hypothesis is true.We thus obtain a new characterization of the Riemann hypothesisby means of an inverse spectral problem. (Actually, we provethe following stronger result: for a given D (0,1), the aboveinverse spectral problem is equivalent to the partialRiemann hypothesis for D, according to which = (s)does not have any zero on the vertical line Re s = D.) Therefore,in some very precise sense, our work shows that the question(à la Marc Kac) "Can one hear the shape of a fractalstring?" – now interpreted as a suitable converse (namely,the above inverse problem) – is intimately connected withthe existence of zeros of = (s) in the critical strip 0 <Res < 1, and hence to the Riemann hypothesis. 相似文献
7.
A theorem of K. W. Roggenkamp and L. L. Scott shows that fora finite group G with a normal p-subgroup containing its owncentralizer, any two group bases of the integral group ringZG are conjugate in the units of ZpG. Though the theorem presentsitself in the work of others and appears to be needed, thereis no published account. There seems to be a flaw in the proof,because a theorem appearing in the survey [K.W. Roggenkamp, The isomorphism problem for integral grouprings of finite groups, Progress in Mathematics 95 (Birkhäuser,Basel, 1991), pp. 193--220], where the main ingredients of aproof are given, is false. In this paper, it is shown how toclose this gap, at least if one is only interested in the conclusionmentioned above. Therefore, some consequences of the resultsof A. Weiss on permutation modules are stated. The basic stepsof which any proof should consist are discussed in some detail.In doing so, a complete, yet short, proof of the theorem isgiven in the case that G has a normal Sylow p-subgroup. 2000Mathematical Subject Classification: primary 16U60; secondary20C05. 相似文献
8.
de Bobadilla J. Fernandez; Luengo-Velasco I.; Melle-Hernandez A.; Nemethi A. 《Proceedings London Mathematical Society》2006,92(1):99-138
In 2002, L. Nicolaescu and the fourth author formulated a verygeneral conjecture which relates the geometric genus of a Gorensteinsurface singularity with rational homology sphere link withthe Seiberg--Witten invariant (or one of its candidates) ofthe link. Recently, the last three authors found some counterexamplesusing superisolated singularities. The theory of superisolatedhypersurface singularities with rational homology sphere linkis equivalent with the theory of rational cuspidal projectiveplane curves. In the case when the corresponding curve has onlyone singular point one knows no counterexample. In fact, inthis case the above Seiberg--Witten conjecture led us to a veryinteresting and deep set of compatibility propertiesof these curves (generalising the Seiberg--Witten invariantconjecture, but sitting deeply in algebraic geometry) whichseems to generalise some other famous conjectures and propertiesas well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yauinequalities). Namely, we provide a set of compatibilityconditions which conjecturally is satisfied by a localembedded topological type of a germ of plane curve singularityand an integer d if and only if the germ can be realized asthe unique singular point of a rational unicuspidal projectiveplane curve of degree d. The conjectured compatibility propertieshave a weaker version too, valid for any rational cuspidal curvewith more than one singular point. The goal of the present articleis to formulate these conjectured properties, and to verifythem in all the situations when the logarithmic Kodaira dimensionof the complement of the corresponding plane curves is strictlyless than 2. 2000 Mathematics Subject Classification 14B05,14J17, 32S25, 57M27, 57R57 (primary), 14E15, 32S45, 57M25 (secondary). 相似文献
9.
On the Analytic Order-Preserving Discrete-Time Dynamical Systems in Rn with every Fixed Point Stable
This paper studies the asymptotic behaviour of an analytic order-preservingdiscrete-time dynamical system in Rn, which is usually generatedby a periodic cooperative system. The author proves that forsuch a dynamical system, if every fixed point is Liapunov stableand every positive semi-orbit has compact closure, then everypositive semi-orbit converges. This result does not requirethe assumption strongly and gives an affirmativeanswer to the conjecture proposed by the author in [17] forthe analytic case. 相似文献
10.
The paper contains a final identification theorem for the genericK*-groups of finite Morley rank. 相似文献
11.
The HallPaige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa trivial product. These two conditions are known to be equivalent,but the first direct, elementary proof that (i) implies (ii)is given here. It is also shown that the HallPaige conjecture impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a double transversal,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60. 相似文献
12.
Let E be an elliptic curve defined over the rational numbersand r a fixed integer. Using a probabilistic model consistentwith the Chebotarev density theorem for the division fieldsof E and the Sato–Tate distribution, Lang and Trotterconjectured an asymptotic formula for the number of primes upto x which have Frobenius trace equal to r, where r is a fixedinteger. However, as shown in this note, this asymptotic estimatecannot hold for all r in the interval with a uniform bound for the error term, becausean estimate of this kind would contradict the Chebotarev densitytheorem as well as the Sato–Tate conjecture. The purposeof this note is to refine the Lang–Trotter conjecture,by taking into account the "semicircular law," to an asymptoticformula that conjecturally holds for arbitrary integers r inthe interval , with auniform error term. We demonstrate consistency of our refinementwith the Chebotarev density theorem for a fixed division field,and with the Sato–Tate conjecture. We also present numericalevidence for the refined conjecture. 相似文献
13.
Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a symplecticblowup of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X. 相似文献
14.
Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order 总被引:4,自引:0,他引:4
Bourgain J.; Glibichuk A. A.; Konyagin S. V. 《Journal London Mathematical Society》2006,73(2):380-398
Our first result is a sum-product theorem forsubsets A of the finite field Fp, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums 相似文献
15.
Let b 2 be an integer. According to a conjecture of ÉmileBorel, the b-adic expansion of any irrational algebraic numberbehaves in some respects like a random sequence.We give a contribution to the following related problem: let and ' be irrational algebraic numbers, then prove that theirb-adic expansions either have the same tail, or behave in somerespects like independent random sequences. 相似文献
16.
On sait associer à certaines structures de Poisson surRn, de 1-jet nul en 0, des actions de R2 sur Rn, donnéespar le rotationnel de leur partie quadratiqueet un autre champ de vecteurs. Lorsque ces actions sont nonrésonantes et hyperboliques, onmontre que ces structures sont quadratisables,en ce sens qu'il existe des coordonnées dans lesquelles,elles sont quadratiques. Dans le cas de la dimension 3, nosrésultats mènent à la non-dégénérescencegénérique des structures de Poisson quadratiquesà rotationnels inversibles. We can associate with some Poisson structures defined on Rnwith a zero 1-jet at zero, actions from R2 on Rn, given by thecurl of their quadratic part and another vectorfield. Assuming that those actions are hyperbolicsand without resonances, we give a normal formfor those structures. On R3, we prove that every quadratic Poissonstructure with invertible curl, is generically non degenerate. 相似文献
17.
Inspired by the work of Bloch and Kato in [2], David Burns constructedseveral equivariant Tamagawa invariants associatedto motives of number fields. These invariants lie in relativeK-groups of group-rings of Galois groups, and in [3] Burns gaveseveral conjectures (see Conjecture 3.1) about their values.In this paper I shall verify Burns' conjecture concerning theinvariant Tloc(N/Q,1) for some families of quaternion extensionsN/Q. Using the results of [9] I intend in a subsequent paperto verify Burns' conjecture for those families of quaternionfields which are not covered here. 相似文献
18.
Nochetto Ricardo H.; Veeser Andreas; Verani Marco 《IMA Journal of Numerical Analysis》2009,29(1):126-140
19.
The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is mobile (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from relative or total-space irreducibilityto absolute or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05. 相似文献
20.
Sola Conde Luis Eduardo; Wisniewski Jaroslaw A. 《Proceedings London Mathematical Society》2004,89(2):273-290
A line bundle over a complex projective variety is called bigand 1-ample if a large multiple of it is generated by globalsections and a morphism induced by the evaluation of the spanningsections is generically finite and has at most 1-dimensionalfibers. A vector bundle is called big and 1-ample if the relativehyperplane line bundle over its projectivisation is big and1-ample. The main theorem of the present paper asserts that any complexprojective manifold of dimension 4 or more, whose tangent bundleis big and 1-ample, is equal either to a projective space orto a smooth quadric. Since big and 1-ample bundles are almostample, the present result is yet another extension of the celebratedMori paper Projective manifolds with ample tangent bundles(Ann. of Math. 110 (1979) 593606). The proof of the theorem applies results about contractionsof complex symplectic manifolds and of manifolds whose tangentbundles are numerically effective. In the appendix we re-provethese results. 2000 Mathematics Subject Classification 14E30,14J40, 14J45, 14J50. 相似文献