Quasi-parabolic analytic transformations of <Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>. Parabolic manifolds

In [Rong, F., Quasi-parabolic analytic transformations of C ⁿ , J. Math. Anal. Appl. 343 (2008), 99–109], we showed the existence of “parabolic curves” for certain quasi-parabolic analytic transformations of C ⁿ . Under some extra assumptions, we show the existence of “parabolic manifolds” for such transformations.     

LP（Rn）-Boundedness of certain commutators

In this paper, the Lp(Rn)-boundedness of the commutators generalized by BMO(Rn) function and the singular integral operator T with rough kernel Ω∈ Llog+ L(Sn-1) is proved by using the Bony's formula for the paraproduct of two functions.     

<Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript>(R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)-boundedness for maximal functions

In this paper, we get W ^1,p (R ⁿ )-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results. Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005 and the Zhejiang Provincial Natural Science Foundation of China.     

Wavelet Filter Functions,the Matrix Completion Problem,and Projective Modules Over <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Bold">T</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We discuss how one can use certain filters from signal processing to describe isomorphisms between certain projective C(T ⁿ )-modules. Conversely, we show how cancellation properties for finitely generated projective modules over C(T ⁿ ) can often be used to prove the existence of continuous high pass filters, of the kind needed for multivariate wavelets, corresponding to a given continuous low-pass filter. However, we also give an example of a continuous low-pass filter for which it is impossible to find corresponding continuous high-pass filters. In this way we give another approach to the solution of the matrix completion problem for filters of the kind arising in wavelet theory.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

<Emphasis Type="Italic">n</Emphasis>-Flat and <Emphasis Type="Italic">n</Emphasis>-FP-injective modules

In this paper, we study the existence of the n-flat preenvelope and the n-FP-injective cover. We also characterize n-coherent rings in terms of the n-FP-injective and n-flat modules.     

Hyperbolic <Emphasis Type="Italic">n</Emphasis>-dimensional manifolds with automorphism group of dimension <Emphasis Type="Italic">n</Emphasis><Superscript>2</Superscript>

We obtain a complete classification of complex Kobayashihyperbolic manifolds of dimension n ≥ 2, for which the dimension of the group of holomorphic automorphisms is equal to n². Received: May 2005 Accepted: November 2005     

Free <Emphasis Type="Bold">C</Emphasis><Subscript>+</Subscript>-actions on <Emphasis Type="Bold">C</Emphasis><Superscript>3</Superscript> are translations

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C₊-action on it such that S=X//C₊ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C₊-action on C³ is a translation in a suitable coordinate system.     

Weierstrass Type Representation of Willmore Surfaces in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we reformulate the Euler-Lagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebra-valued 1-forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n, R^n, H^n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n, R^n, H^n.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Cyclic Functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Bold">R</Emphasis>), 1 \le <Emphasis Type="Italic">p</Emphasis> < \infty

Functions whose translates span L ^p (R) are called L ^p-cyclic functions. For a fixed p \memb [1, \infty], we construct Schwartz-class functions which are L ^r -cyclic for r > p and not L ^r - cyclic for r \le p. We then construct Schwartz-class functions which are L ^r -cyclic for r \ge p and not L ^r -cyclic for r < p. The constructions differ for p \memb (1, 2) and p > 2.     

Integral Modular Categories of Frobenius-Perron Dimension <Emphasis Type="Italic">pq</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Integral modular categories of Frobenius-Perron dimension pq ⁿ , where p and q are primes, are considered. It is already known that such categories are group-theoretical in the cases of 0 ≤ n ≤ 4. In the general case we determine that these categories are either group-theoretical or contain a Tannakian subcategory of dimension q ⁱ for i > 1. We then show that all integral modular categories \(\mathcal {C}\) with \(\text {FPdim}(\mathcal {C})=pq^{5}\) are group-theoretical, and, if in addition p < q, all with \(\text {FPdim}(\mathcal {C})=pq^{6}\) or pq ⁷ are group-theoretical. In the process we generalize an existing criterion for an integral modular category to be group-theoretical.     

Some cyclic codes of length 2<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Explicit expressions for 4n + 2 primitive idempotents in the semi-simple group ring $R_{2p^{n}}\equiv \frac{GF(q)[x]}{p and q are distinct odd primes; n ≥ 1 is an integer and q has order \fracf(2pⁿ)2{\frac{\phi(2p^{n})}{2}} modulo 2p ⁿ . The generator polynomials, the dimension, the minimum distance of the minimal cyclic codes of length 2p ⁿ generated by these 4n + 2 primitive idempotents are discussed. For n = 1, the properties of some (2p, p) cyclic codes, containing the above minimal cyclic codes are analyzed in particular. The minimum weight of some subset of each of these (2p, p) codes are observed to satisfy a square root bound.     

Bordism of semi-free <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-actions

We calculate geometric and homotopical bordism rings associated to semi-free S¹ actions on complex manifolds, giving explicit generators for the geometric theory. The classification of semi-free actions with isolated fixed points up to cobordism complements similar results from symplectic geometry.     

Compact embedded rotation hypersurfaces of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

In this paper, we prove that and round geodesic spheres are the only n-dimensional compact embedded rotation hypersurfaces with H_m = 0 (1 ≤ m ≤ n − 1) in a unit sphere Sⁿ⁺¹(1). When m = 1, our result reduces to the result of T. Otsuki [O1], [O2], Brito and Leite [BL]. The project is supported by the grant No. 10531090 of NSFC.     

Surfaces with Vanishing Moebius Form in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

An important Moebius invariant in the theory of Moebius surfaces in S^n is the so-called Moebius form. In this paper,we give a complete classification of surfaces in S^n with vanishing Moebius form under the Moebius transformation group.     

The existence of eigenvalues for operators acting in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(<Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We present conditions that allow us to prove the existence of eigenvalues and characteristic values for operator F(D) ? C(λ): L ²(R ^m ) → L ²(R ^m ), where F(D) is a pseudo-differential operator with a symbol F(iξ) and C(λ): L ²(R ^m ) → L ²(R ^m ) is a linear continuous operator.     

Further Results for the Queueing System <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">x</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis>

For the multi-channel bulk-arrival queue, M ^x /M/c, Abol'nikov and Kabak independently obtained steady state results. In this paper the results of these authors are extended, corrected and simplified. A number of measures of efficiency are calculated for three cases where the arrival group size has: (i) a constant value, (ii) a geometric distribution, or (iii) a positive Poisson distribution. The paper also shows how to calculate fractiles for both the queue length and the waiting time distribution. Examples of extensive numerical results for certain measures of efficiency are presented in tabular and chart form.     

Thomae formulae for general fully ramified <Emphasis Type="Italic">Z</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> curves

We generalize the elementary methods presented in several examples in the book [FZ] to obtain the Thomae formulae for general fully ramified Z _n curves.     

Remarks on the <Emphasis Type="Italic">k</Emphasis>-error linear complexity of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-periodic sequences

Recently the first author presented exact formulas for the number of 2 ⁿ -periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k ≥ 2, of a random 2 ⁿ -periodic binary sequence. A crucial role for the analysis played the Chan–Games algorithm. We use a more sophisticated generalization of the Chan–Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for p ⁿ -periodic sequences over prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of p ⁿ -periodic sequences over .