The Hausdorff dimension of the level sets of Takagi’s function

Recently, Maddock (2006) [12] has conjectured that the Hausdorff dimension of each level set of Takagi’s function is at most 1/2. We prove this conjecture using the self-affinity of the function of Takagi and the existing relationship between the Hausdorff and box-counting dimensions.     

On an isoperimetric inequality for capacity conjectured by Pólya and Szegö

We study a conjecture by Pólya and Szegö on the approximation of the electrostatic capacity of convex bodies in terms of their surface measure. We prove that a “local version” of this conjecture holds true and we give some results which bring further evidence to its global validity.     

On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a two-dimensional manifold. With an explicit computation we examine a 3-web whose linearizability was claimed in [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654] and was contested later in [V.V. Goldberg, V.V. Lychagin, On the Blaschke conjecture for 3-webs, J. Geom. Anal. 16 (1) (2006) 69–115] and [V.V. Goldberg, V.V. Lychagin, On linearization of planar three-webs and Blaschke’s conjecture, C. R. Acad. Sci. Paris, Ser. I. 341 (3) (2005)]. On the basis of the theories of [J. Grifone, Z. Muzsnay, J. Saab, On the linearizability of 3-webs, Nonlinear Anal. 47 (2001) 2643–2654], we give an effective method for computing the linearizability criterion, and we prove that this particular web is linearizable by finding explicitly the affine deformation tensor and the corresponding flat linear connection.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their N-adic expansion

In this paper we apply the techniques and results from the theory of multifractal divergence points developed in [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103-122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] to give a systematic and detailed account of the Hausdorff dimensions of sets of d-tuples numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their N-adic expansion. Using the method and results from [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591-1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103-122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] we investigate and compute the Hausdorff dimension of several new sets of d-tuples of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of d-tuples numbers for which the limiting frequencies of the digits in their N-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of higher-dimensional and non-linear generalizations of known results, by Cajar and Volkmann and others, on the Hausdorff dimension of sets of normal and non-normal numbers.     

Stillman's question for exterior algebras and Herzog's conjecture on Betti numbers of syzygy modules

Let K be a field of characteristic 0 and consider exterior algebras of finite dimensional K-vector spaces. In this short paper we exhibit principal quadric ideals in a family whose Castelnuovo–Mumford regularity is unbounded. This negatively answers the analogue of Stillman's Question for exterior algebras posed by I. Peeva. We show that, via the Bernstein–Gel'fand–Gel'fand correspondence, these examples also yields counterexamples to a conjecture of J. Herzog on the Betti numbers in the linear strand of syzygy modules over polynomial rings.     

Congruences between derivatives of abelian <Emphasis Type="Italic">L</Emphasis>-functions at <Emphasis Type="Italic">s</Emphasis>=0

Let K/k be a finite abelian extension of global fields. We prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s=0 of derivatives of the Dirichlet L-functions associated to K/k. We also show that these congruences provide a universal approach to the ‘refined abelian Stark conjectures’ formulated by, inter alia, Stark, Gross, Rubin, Popescu and Tate. We thereby obtain the first proofs of, amongst other things, the Rubin–Stark conjecture and the ‘refined class number formulas’ of both Gross and Tate for all extensions K/k in which K is either an abelian extension of ℚ or is a function field. Mathematics Subject Classification (1991)  Primary 11G40; Secondary 11R65; 19A31; 19B28     

<Emphasis Type="Italic">q</Emphasis>-deformation of Witt-Burnside rings

In this paper, we construct a q-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where q ranges over the set of integers. When q = 1, it coincides with the Witt-Burnside ring introduced by Dress and Siebeneicher (Adv. Math. 70, 87–132 (1988)). To achieve our goal we first show that there exists a q-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the case q = 1, q-deformed Witt-Burnside rings and necklace rings always come equipped with inductions and restrictions. We also study their properties. As a byproduct, we prove a conjecture due to Lenart (J. Algebra. 199, 703-732 (1998)). Finally, we classify up to strict natural isomorphism in case where G is an abelian profinite group. The author gratefully acknowledges support from the following grants: KOSEF Grant # R01-2003-000-10012-0; KRF Grant # 2006-331-C00011.     

On the Loebl–Komlós–Sós conjecture

The Loebl–Komlós–Sós conjecture says that any graph G on n vertices with at least half of vertices of degree at least k contains each tree of size k. We prove that the conjecture is true for paths as well as for large values of k(k ≥ n − 3). © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 269–276, 2000     

On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval

In this paper, we propose a new iteration, called the SP-iteration, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the SP-iteration of continuous functions on an arbitrary interval is given. We also compare the convergence speed of Mann, Ishikawa, Noor and SP-iterations. It is proved that the SP-iteration is equivalent to and converges faster than the others. Our results extend and improve the corresponding results of Borwein and Borwein [D. Borwein, J. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991) 112-126], Qing and Qihou [Y. Qing, L. Qihou, The necessary and sufficient condition for the convergence of Ishikawa iteration on an arbitrary interval, J. Math. Anal. Appl. 323 (2006) 1383-1386], Rhoades [B.E. Rhoades, Comments on two fixed point iteration methods, J. Math. Anal. Appl. 56 (1976) 741-750], and many others. Moreover, we also present numerical examples for the SP-iteration to compare with the Mann, Ishikawa and Noor iterations.     

Uniform boundedness of <Emphasis Type="Italic">p</Emphasis>-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E _s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.     

A new lower bound on the strong connectivity of an oriented graph. Application to diameters with a particular case related to Caccetta Häggkvist conjecture

In this paper, by using minimum out-degree and minimum in-degree, we give a new lower bound on the vertex-strong connectivity of an oriented graph. In the case of a tournament, our lower bound improves that of Thomassen obtained in 1980 and which use the notion of irregularity (see [C. Thomassen, Hamiltonian-connected tournaments, J. Combin. Theory Ser. B 28 (1980) 142–163]). As application, we determine a pertinent upper bound on the diameter of some oriented graphs, and in a particular case, related to Caccetta Häggkvist conjecture, we improve a result of Broersma and Li obtained in 2002 (see [H.J. Broersma, X. Li, Some approaches to a conjecture on short cycles in digraphs, Discrete Appl. Math. 120 (2002) 45–53]).     

Proof of a conjecture by Đoković on the Poincaré series of the invariants of a binary form

?okovi? (2006)  [3] gave an algorithm for the computation of the Poincaré series of the algebra of invariants of a binary form, where the correctness proof for the algorithm depended on an unproven conjecture. Here we prove this conjecture.     

On global asymptotic stability of nonlinear higher-order difference equations

In this paper, we generalize the main theorem of Liz and Ferreiro [E. Liz, J.B. Ferreiro, A note on the global stability of generalized difference equations, Appl. Math. Lett. 15 (2002) 655–659] and some other global stability results for nonautonomous higher-order difference equations to the case when contraction-type steps are incorporated together with the steps when the difference sequence can increase.     

On the Block Independence in G-Inverse and Reflexive Inner Inverse of A Partitioned Matrix

By applying the multiple quotient singular value decomposition QQQQQ-SVD, we study the block independence in g-inverse and reflexive inner inverse of 2× 2 partitioned matrices, and prove a conjecture in [Yiju Wang, SIAM J. Matrix Anal. Appl., 19（2）, 407-415（1998）].     

Cyclicity of several quadratic reversible systems with center of genus one

By using the Chebyshev criterion to study the number of zeros of Abelian integrals, developed by&nbsp;M. Grau, F. Ma\(\~n\)osas and J. Villadelprat in [2], we prove&nbsp;that the cyclicity of period annulus of the quadratic reversible systems with center of genus one, classified as (r8), (r13) and&nbsp;(r16) by S. Gautier, L. Gavrilov and I. D. Iliev in [1],&nbsp;under quadratic perturbations is two. These results partially give a&nbsp;positive answer to the conjecture 1 in [1].     

Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov, and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile’s monotonicity conditions are not satisfied.     

The isogeny conjecture for <Emphasis Type="Italic">A</Emphasis>-motives

We prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree ≤1. This conjecture says that for any semisimple A-motive M over K, there exist only finitely many isomorphism classes of A-motives M′ over K for which there exists a separable isogeny M′→M. The result is in precise analogy to known results for abelian varieties and for Drinfeld modules and will have strong consequences for the \mathfrak p{\mathfrak {p}}-adic and adelic Galois representations associated to M. The method makes essential use of the Harder–Narasimhan filtration for locally free coherent sheaves on an algebraic curve.     

On the 7 Total Colorability of Planar Graphs with Maximum Degree 6 and without 4-cycles

Vizing and Behzad independently conjectured that every graph is (Δ + 2)-totally-colorable, where Δ denotes the maximum degree of G. This conjecture has not been settled yet even for planar graphs. The only open case is Δ = 6. It is known that planar graphs with Δ ≥ 9 are (Δ + 1)-totally-colorable. We conjecture that planar graphs with 4 ≤ Δ ≤ 8 are also (Δ + 1)-totally-colorable. In addition to some known results supporting this conjecture, we prove that planar graphs with Δ = 6 and without 4-cycles are 7-totally-colorable. Supported by the Natural Science Foundation of Department of Education of Zhejiang Province, China, Grant No. 20070441.