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351. CMB 2007 (vol 50 pp. 588)

Labute, John; Lemire, Nicole; Mináč, Ján; Swallow, John
 Cohomological Dimension and Schreier's Formula in Galois Cohomology Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\in \N$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is at most $n$ if and only if the corestriction maps $H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open subgroups $H$ of index $p$. Using this result, we generalize Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp} H^n(H,\Fp)$. Keywords:cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groupsCategories:12G05, 12G10

352. CMB 2007 (vol 50 pp. 567)

Joshi, Kirti
 Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications. Keywords:threefolds, Frobenius splitting, Hodge--Witt, crystalline cohomology, slope spectral sequence, exotic torsionCategories:14F30, 14J30

353. CMB 2007 (vol 50 pp. 434)

Õzarslan, M. Ali; Duman, Oktay
 MKZ Type Operators Providing a Better Estimation on $[1/2,1)$ In the present paper, we introduce a modification of the Meyer-K\"{o}nig and Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and $f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the $r-$th order generalization of our operators and study their approximation properties. Keywords:Meyer-KÃ¶nig and Zeller operators, Korovkin type approximation theorem, modulus of continuity, Lipschitz class functionalsCategories:41A25, 41A36

354. CMB 2007 (vol 50 pp. 474)

Zhou, Jiazu
 On Willmore's Inequality for Submanifolds Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total square mean curvature $\int_M H^2 d\sigma$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$. Keywords:submanifold, mean curvature, kinematic formul, scalar curvatureCategories:52A22, 53C65, 51C16

355. CMB 2007 (vol 50 pp. 447)

Śniatycki, Jędrzej
 Generalizations of Frobenius' Theorem on Manifolds and Subcartesian Spaces Let $\mathcal{F}$ be a family of vector fields on a manifold or a subcartesian space spanning a distribution $D$. We prove that an orbit $O$ of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$ and it has constant rank on $O$. This result implies Frobenius' theorem, and its various generalizations, on manifolds as well as on subcartesian spaces. Keywords:differential spaces, generalized distributions, orbits, Frobenius' theorem, Sussmann's theoremCategories:58A30, 58A40

356. CMB 2007 (vol 50 pp. 234)

Kuo, Wentang
 A Remark on a Modular Analogue of the Sato--Tate Conjecture The original Sato--Tate Conjecture concerns the angle distribution of the eigenvalues arising from non-CM elliptic curves. In this paper, we formulate a modular analogue of the Sato--Tate Conjecture and prove that the angles arising from non-CM holomorphic Hecke eigenforms with non-trivial central characters are not distributed with respect to the Sate--Tate measure for non-CM elliptic curves. Furthermore, under a reasonable conjecture, we prove that the expected distribution is uniform. Keywords:$L$-functions, Elliptic curves, Sato--TateCategories:11F03, 11F25

357. CMB 2007 (vol 50 pp. 206)

Golasiński, Marek; Gonçalves, Daciberg Lima
 Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$ Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well. Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space formCategories:55M35, 55P15, 20E22, 20F28, 57S17

358. CMB 2007 (vol 50 pp. 284)

McIntosh, Richard J.
 Second Order Mock Theta Functions In his last letter to Hardy, Ramanujan defined 17 functions $F(q)$, where $|q|<1$. He called them mock theta functions, because as $q$ radially approaches any point $e^{2\pi ir}$ ($r$ rational), there is a theta function $F_r(q)$ with $F(q)-F_r(q)=O(1)$. In this paper we establish the relationship between two families of mock theta functions. Keywords:$q$-series, mock theta function, Mordell integralCategories:11B65, 33D15

359. CMB 2007 (vol 50 pp. 11)

Borwein, David; Borwein, Jonathan
 van der Pol Expansions of L-Series We provide concise series representations for various L-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying L-series. Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-seriesCategories:11M35, 11M41, 30B50

360. CMB 2007 (vol 50 pp. 149)

Śliwa, Wiesław
 On Quotients of Non-Archimedean KÃ¶the Spaces We show that there exists a non-archimedean Fr\'echet-Montel space $W$ with a basis and with a continuous norm such that any non-archimedean Fr\'echet space of countable type is isomorphic to a quotient of $W$. We also prove that any non-archimedean nuclear Fr\'echet space is isomorphic to a quotient of some non-archimedean nuclear Fr\'echet space with a basis and with a continuous norm. Keywords:Non-archimedean KÃ¶the spaces, nuclear FrÃ©chet spaces, pseudo-basesCategories:46S10, 46A45

361. CMB 2007 (vol 50 pp. 113)

Li, ZhenYang; Zhang, Xi
 Hermitian Harmonic Maps into Convex Balls In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary. Keywords:Hermitian harmonic map, Hermitian manifold, convex ballCategories:58E15, 53C07

362. CMB 2007 (vol 50 pp. 85)

Han, Deguang
 Classification of Finite Group-Frames and Super-Frames Given a finite group $G$, we examine the classification of all frame representations of $G$ and the classification of all $G$-frames, \emph{i.e.,} frames induced by group representations of $G$. We show that the exact number of equivalence classes of $G$-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number $L$ such that there exists an $L$-tuple of strongly disjoint $G$-frames. Keywords:frames, group-frames, frame representations, disjoint framesCategories:42C15, 46C05, 47B10

363. CMB 2006 (vol 49 pp. 624)

Teragaito, Masakazu
 On Non-Integral Dehn Surgeries Creating Non-Orientable Surfaces For a non-trivial knot in the $3$-sphere, only integral Dehn surgery can create a closed $3$-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two. Keywords:knot, Dehn surgery, non-orientable surfaceCategory:57M25

364. CMB 2006 (vol 49 pp. 560)

Luijk, Ronald van
 A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire NÃ©ron-Severi group of this surface and find all low degree curves on it. Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theoryCategories:14G05, 14J28, 11D41

365. CMB 2006 (vol 49 pp. 536)

Dostál, Petr; Lukeš, Jaroslav; Spurný, Jiří
 Measure Convex and Measure Extremal Sets We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened. Keywords:measure convex set, measure extremal set, faceCategories:46A55, 52A07

366. CMB 2006 (vol 49 pp. 481)

Browkin, J.; Brzeziński, J.
 On Sequences of Squares with Constant Second Differences The aim of this paper is to study sequences of integers for which the second differences between their squares are constant. We show that there are infinitely many nontrivial monotone sextuples having this property and discuss some related problems. Keywords:sequence of squares, second difference, elliptic curveCategories:11B83, 11Y85, 11D09

367. CMB 2006 (vol 49 pp. 472)

Silvester, Alan K.; Spearman, Blair K.; Williams, Kenneth S.
 Cyclic Cubic Fields of Given Conductor and Given Index The number of cyclic cubic fields with a given conductor and a given index is determined. Keywords:Discriminant, conductor, index, cyclic cubic fieldCategories:11R16, 11R29

368. CMB 2006 (vol 49 pp. 371)

Floricel, Remus
 Inner $E_0$-Semigroups on Infinite Factors This paper is concerned with the structure of inner $E_0$-semigroups. We show that any inner $E_0$-semigroup acting on an infinite factor $M$ is completely determined by a continuous tensor product system of Hilbert spaces in $M$ and that the product system associated with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant. Keywords:von Neumann algebras, semigroups of endomorphisms, product systems, cocycle conjugacyCategories:46L40, 46L55

369. CMB 2006 (vol 49 pp. 358)

Khalil, Abdelouahed El; Manouni, Said El; Ouanan, Mohammed
 On the Principal Eigencurve of the $p$-Laplacian: Stability Phenomena We show that each point of the principal eigencurve of the nonlinear problem $$-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega,$$ is stable (continuous) with respect to the exponent $p$ varying in $(1,\infty)$; we also prove some convergence results of the principal eigenfunctions corresponding. Keywords:$p$-Laplacian with indefinite weight, principal eigencurve, principal eigenvalue, principal eigenfunction, stabilityCategories:35P30, 35P60, 35J70

370. CMB 2006 (vol 49 pp. 185)

 On the Inequality for Volume and Minkowskian Thickness Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon. Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratioCategories:52A40, 46B20

371. CMB 2006 (vol 49 pp. 203)

Çömez, Doğan
 The Ergodic Hilbert Transform for Admissible Processes It is shown that the ergodic Hilbert transform exists for a class of bounded symmetric admissible processes relative to invertible measure preserving transformations. This generalizes the well-known result on the existence of the ergodic Hilbert transform. Keywords:Hilbert transform, admissible processesCategories:28D05, 37A99

372. CMB 2006 (vol 49 pp. 281)

Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.
 Correction to a Theorem on Total Positivity A well-known theorem states that if $f(z)$ generates a PF$_r$ sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f(z)$ generates a PF sequence then $1/f(-z)$ generates a PF sequence. Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur functionCategories:15A48, 15A45, 15A57, 05E05

373. CMB 2006 (vol 49 pp. 270)

Occhetta, Gianluca
 A Characterization of Products of Projective Spaces We give a characterization of products of projective spaces using unsplit covering families of rational curves. Keywords:Rational curves, Fano varietiesCategories:14J40, 14J45

374. CMB 2006 (vol 49 pp. 265)

Nicholson, W. K.; Zhou, Y.
 Endomorphisms That Are the Sum of a Unit and a Root of a Fixed Polynomial If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in C[x]$, Camillo and Sim\'{o}n called a ring$g(x)$-clean if every element is the sum of a unit and a root of$g(x)$. If$V$is a vector space of countable dimension over a division ring$D,$they showed that$\end {}_{D}V$is$g(x)$-clean provided that$g(x)$has two roots in$C(D)$. If$g(x)=x-x^{2}$this shows that$\end {}_{D}V$is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that$\Mend {}_{R}M$is$g(x)$-clean for any semisimple module$M$over an arbitrary ring$R$provided that$g(x)\in (x-a)(x-b)C[x]$where$a,b\in C$and both$b$and$b-a$are units in$R$. Keywords:Clean rings, linear transformations, endomorphism ringsCategories:16S50, 16E50 375. CMB 2006 (vol 49 pp. 256) Neelon, Tejinder  A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for$C^{\infty }$functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak--Siciak theorem: a$C^{\infty }$\ function on$\mathbb{R}^{n}$that is real analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a double power series$F(x,y)$\ converges on a set of lines of positive capacity then$F(x,y)\$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero. Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power seriesCategories:32A05, 26E05
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