The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

On Defining Ideals or Subrings of Hall Algebras

Let A be a finitary algebra over a finite field k, and A- \textmod\text{mod} the category of finite dimensional left A-modules. Let H(A)\mathcal{H}(A) be the corresponding Hall algebra, and for a positive integer r let D _r (A) be the subspace of H(A)\mathcal{H}(A) which has a basis consisting of isomorphism classes of modules in A- \textmod\text{mod} with at least r + 1 indecomposable direct summands. If A is the path algebra of the quiver of type A _n with linear orientation, then D _r (A) is known to be the kernel of the map from the twisted Hall algebra to the quantized Schur algebra indexed by n + 1 and r. For any A, we determine necessary and sufficient conditions for D _r (A) to be an ideal and some conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A). For A the path algebra of a quiver, we also determine necessary and sufficient conditions for D _r (A) to be a subring of H(A)\mathcal{H}(A).     

Piecewise Hereditary Nakayama Algebras

Let k be an algebraically closed field. Let Λ be the path algebra over k of the linearly oriented quiver \mathbb A_n\mathbb A_n for n ≥ 3. For r ≥ 2 and n > r we consider the finite dimensional k −algebra Λ(n,r) which is defined as the quotient algebra of Λ by the two sided ideal generated by all paths of length r. We will determine for which pairs (n,r) the algebra Λ(n,r) is piecewise hereditary, so the bounded derived category D ^b (Λ(n,r)) is equivalent to the bounded derived category of a hereditary abelian category H\mathcal H as triangulated category.     

Eigenvalue problem for an irregular λ-matrix

The solution of the eigenvalue problem is examined for the polyomial matrixD()=A_o^s+A₁^s–1+...+A_s when the matricesA ₀ andA ₂ (or one of them) are singular. A normalized process is used for solving the problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigenvalue of matrixD() and to the zero eigenvalue of matrixA ₀. The computation of the other eigenvalues ofD() is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 80–92, 1976.     

Formule d'homotopie entre les complexes de Hochschild et de De Rham

Let k be the field or let M be the space k ⁿ and let A be the algebra of polynomials over M. We know from Hochschild and co-workers that the Hochschild homology H _·(A,A) is isomorphic to the de Rham differential forms over M: this means that the complexes (C _·(A,A),b) and (^·(M), 0) are quasi-isomorphic. In this work, I produce a general explicit homotopy formula between those two complexes. This formula can be generalized when M is an open set in a complex manifold and A is the space of holomorphic functions over M. Then, by taking the dual maps, I find a new homotopy formula for the Hochschild cohomology of the algebra of smooth fonctions over M (when M is either a complex or a real manifold) different from the one given by De Wilde and Lecompte. I will finally show how this formula can be used to construct an homotopy for the cyclic homology.     

Some remarks on ample line bundles on abelian varieties

Let L be an ample line bundle on an abelian variety A. We show that L² is very ample if (A,L) is not isomorphic to (A₁×A₂,o(D₁×A₂+A₁×D₂)) where A_i is an abelian variety (i=1,2), D_i is an ample divisor on A_i (i=1,2) and (A₁,o(D₁))=1, and if (A,L)2. As an application we show that L² is base point free if L is an ample line bundle on bielliptic surface.In conclusion, the author would like to thank the referee for very helpful advice.     

Classes caracteristiques de Γ(G,H)-structures et finitude de leur evaluation

LetG be a Lie group andH a closed subgroup ofG. We denote by (G,H) the groupoïd of germs of left translations ofG over the homogeneous spaceG/H.LetV be a compact manifold andx the universal characteristic class of dimensionk which belongs to the vector spaceH _cont ^k ((G, H)).The evaluation ofx over all the (G, H)-structures overV determines a subsetA _{(G, H)} (x, V) of the vector spaceH ^k (V;).We show that in some cases this set is finite.     

On cyclic caps in 4-dimensional projective spaces

For any divisor k of q ⁴−1, the elements of a group of k th-roots of unity can be viewed as a cyclic point set C _k in PG(4,q). An interesting problem, connected to the theory of BCH codes, is to determine the spectrum A(q) of maximal divisors k of q ⁴−1 for which C _k is a cap. Recently, Bierbrauer and Edel [Edel and Bierbrauer (2004) Finite Fields Appl 10:168–182] have proved that 3(q ² + 1)∈A(q) provided that q is an even non-square. In this paper, the odd order case is investigated. It is proved that the only integer m for which m(q ² + 1)∈A(q) is m = 2 for q ≡ 3 (mod 4), m = 1 for q ≡ 1 (mod 4). It is also shown that when q ≡ 3 (mod 4), the cap is complete.      

Cartan calculus for quantum differentials on bicrossproducts

We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups k(M) k G associated to finite group factorizations X = GM and a field k. The irreducible calculi are associated to certain conjugacy classes in X and representations of isotropy groups. We find the full exterior algebras and show that they are inner by a bi-invariant 1-form which is a generator in the noncommutative de Rham cohomology H ¹. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codouble D ^*(S ₃)k(S ₃) k₆ and the quantum double D(S ₃) \triangleleft $$ " align="middle" border="0"> k S ₃, finding respectively a natural calculus and a unique calculus with H ⁰ = k.1.     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

On the domain monotonicity of the first eigenvalue of a boundary value problem

Summary This paper is concerned with the monotonicity of the first eigenvalue ₁ (D) of (1) as a functional of the domainD.For 0<k< we prove that ₁(D) is increasing withD in several cases, but a counterexample is given showing that this is not true in general. In the same cases we prove that, for –<k<0, ₁(D) decreases whenD increases.

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Zusammenfassung Diese Arbeit behandelt die Monotonie des ersten Eigenwertes ₁(D) von (1) als Funktional des GebietesD. Für 0<k< beweisen wir, dass ₁(D) in mehreren Fällen mitD wächst; aber ein Gegenbeispiel zeigt, dass dies nicht allgemein gilt. In denselben Fällen beweisen wir, dass ₁(D) für –<k<0 bei wachsendemD abnimmt.

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Technical Report TR-75-13.     

Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

Discrepancy in generalized arithmetic progressions

Estimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problems in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D(N)=min_χmax_A|∑_xAχ(x)|=Θ(N^1/4), where the minimum is taken over all colorings χ:[N]→{−1,1} and the maximum over all arithmetic progressions in [N]={0,…,N−1}.Sumsets of k arithmetic progressions, A₁++A_k, are called k-arithmetic progressions and they are important objects in additive combinatorics. We define D_k(N) as the discrepancy of the set {P∩[N]:P is a k-arithmetic progression}. The second author proved that D_k(N)=Ω(N^k/(2k+2)) and Přívětivý improved it to Ω(N^1/2) for all k≥3. Since the probabilistic argument gives D_k(N)=O((NlogN)^1/2) for all fixed k, the case k=2 remained the only case with a large gap between the known upper and lower bounds. We bridge this gap (up to a logarithmic factor) by proving that D_k(N)=Ω(N^1/2) for all k≥2.Indeed we prove the multicolor version of this result.     

Lacunary Interpolation by Antiperiodic Trigonometric Polynomials

The problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial T exist which satisfies T(x _k) = a _k, D ^m T(x _k) = b _k, 0 k n – 1, where x _k = k/n is a nodal set, a _k and b _k are prescribed complex numbers, and m N. Results obtained by several authors for the periodic case are extended to the antiperiodic case. In particular solvability is established when n as well as m are even. In this case a periodic solution does not exist.     

Proof of Oppenheim's area inequalities for triangles and quadrilaterals

Leta ₁,b ₁,c ₁,A ₁ anda ₂,b ₂,c ₂,A ₂ be the sides and areas of two triangles. Ifa=(a ₁ ^p +a ₂ ^p )^1/p ,b=(b ₁ ^p +b ₂ ^p )^1/p ,c=(c ₁ ^p +c ₂ ^p )^1/p , and 1p4, thena, b, c are the sides of a triangle and its area satisfiesA ^p/2A ₁ ^p/2 +A ₂ ^p/2 . If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaA _f, which satisfies (4A _f/3)^1/2f((4A/3)^1/2).     

Upper recollements and triangle expansions

By using the stable t-structure induced by an adjoint pair, we extend several results concerning recollements to upper (resp. lower) recollements. These include the fundamental results of Parshall and Scott on comparisons of recollements, Wiedemann’s result on the global dimension and Happel’s result on the finitistic dimension, occurring in a recollement (D ^b (A′),D ^b (A),D ^b (A″)) of bounded derived categories of Artin algebras. We introduce and describe a triangle expansion of a triangulated category and illustrate it by examples.     

Detailed Error Analysis for a Fractional Adams Method

We investigate a method for the numerical solution of the nonlinear fractional differential equation D _* y(t)=f(t,y(t)), equipped with initial conditions y ^(k)(0)=y ₀ ^(k), k=0,1,...,–1. Here may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.