The Quarterly Journal of Mathematics - Advance Access

SUMS OF OPERATOR LOGARITHMS

Clark, S. — 2008-07-17

Let A and B be a pair of resolvent commuting invertible sectorial operators. We shall show that, under Kalton–Weis-type conditions, the operator log A + log B is closed and equal to log (AB).

FLOWS OF G2-STRUCTURES, I

Karigiannis, S. — 2008-07-14

This is a foundational paper on flows of G₂-structures. We use local coordinates to describe the four torsion forms of a G₂ and derive the evolution equations for a general flow of a G₂-structure on a 7-manifold M. Specifically, we compute the evolution of the metric g, the dual 4-form and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernández–Gray.

As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G₂-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G₂-geometry.

CONTACT 5-MANIFOLDS WITH SU(2)-STRUCTURE

De Andres, L. C., Fernandez, M., Fino, A., Ugarte, L. — 2008-07-14

We consider 5-manifolds with a contact form arising from a hypo structure, which we call hypo-contact. We provide existence conditions for such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU(3)-structure. For half-flat manifolds with a Killing vector field X preserving the SU(3)-structure we study the geometry of the orbits space. Moreover, we describe the solvable Lie algebras admitting a hypo-contact structure. This allows us to exhibit examples of Sasakian -Einstein manifolds, as well as to prove that such structures give rise to new metrics with holonomy SU(3) and G₂.

TWISTOR SPACES, PLURIHARMONIC MAPS AND HARMONIC MORPHISMS

Simoes, B. A., Svensson, M. — 2008-07-10

The application of twistor methods to construct harmonic morphisms has proved to be a fruitful approach in the 4-dimensional case, where a variety of examples and, in some cases, even a complete classification of harmonic morphisms have been found. In this paper, we generalize this construction to obtain higher-dimensional analogues of these maps. We also prove several results on twistor lifts of pluriharmonic and (1, 1)-geodesic maps.

SOME Z/2-GRADED REPRESENTATION THEORY

Parshall, B. J., Scott, L. L. — 2008-06-18

In representation theory, the existence of a Z⁺-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category O associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p≥n, though this grading would suffice to establish Lusztig's character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Astérique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces Z/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full Z⁺-grading, while being possibly much easier to find. We define the notion of a Z/2-based Kazhdan–Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan–Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan–Lusztig theories, Tôhoku Math. J. 45 (1993), 511–534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Extⁿ-groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

ETA-INVARIANTS FROM MOLIEN SERIES

Degeratu, A. — 2008-06-14

We look at the orbifold Cⁿ/ with a finite subgroup of U(n) from two perspectives: from a differential point of view it is a non-compact orbifold with boundary at infinity S^2n–1/, while from an algebraic point of view it is a scheme with coordinate ring the -invariant polynomials in n variables. The main result is a relation between the -invariant of the boundary (an analytical object) and the Molien series of the singularity (an algebraic object).

A QUADRIC WITH ARITHMETIC PAUCITY

Blomer, V., Brudern, J. — 2008-06-10

The quadric given by the equations x₁²+x₂²+x₃² = y₁²+y₂²+y₃², x₁+x₂+x₃ = y₁+y₂+y₃ has almost all its solutions with prime coordinates on the diagonals. This is shown in quantitative form. A similar statement holds for integral solutions whose coordinates can be written as the sum of two squares.

FREIHEITSSATZE FOR ONE-RELATOR QUOTIENTS OF SURFACE GROUPS AND OF LIMIT GROUPS

Howie, J., Saeed, M. S. — 2008-06-10

Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups. Two of the proofs are new even in that restricted context.

LARGE INDECOMPOSABLE MINIMAL GROUPS

Shelah, S., Strungmann, L. — 2008-06-10

Assuming V=L we prove that there exist indecomposable almost-free minimal groups of size for every regular cardinal below the first weakly compact cardinal. This is to say that there are indecomposable almost-free torsion-free abelian groups of cardinality which are isomorphic to all of their finite index subgroups.

THE EULER OBSTRUCTION AND BRUCE-ROBERTS' MILNOR NUMBER

De Goes Grulha, N. — 2008-06-10

In this work we determine relations between the local Euler obstruction of an analytic function f and the Milnor number of f defined by Bruce and Roberts for functions on singular spaces.

ON MORITA THEORY FOR SELF-DUAL MODULES

Willems, W., Zimmermann, A. — 2008-06-10

Let G be a finite group and k be a field of characteristic p. It is known that a kG-module V carries a non-degenerate G-invariant bilinear form b if and only if V is self-dual. We show that whenever a Morita bimodule M that induces an equivalence between two blocks such as B(kG) and B(kH) of group algebras kG and kH is self-dual, then the correspondence preserves self-duality. Even more, if the bilinear form on M is symmetric, then, for p odd, the correspondence preserves the geometric type of simple modules. In characteristic 2, this holds also true for projective modules.

DIVISIBILITY OF EXPONENTIAL SUMS AND SOLVABILITY OF CERTAIN EQUATIONS OVER FINITE FIELDS

Castro, F. N., Rubio, I., Vega, J. M. — 2008-06-05

Carlitz [Solvability of certain equations in a finite field, Quart. J. Math. (Oxford) 7 (1956), 3–4] determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz's results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields.

MIXED WEAK TYPE INEQUALITIES FOR ONE-SIDED OPERATORS

Martin-Reyes, F. J., Ombrosi, S. J. — 2008-06-05

We discuss mixed weak type inequalities in weighted spaces for one-sided operators. In particular, we prove that if T_cf(x) = (x – c)^–1_c^x f(y) dy, x > c, is the Hardy averaging operator, u A₁^– (one-sided Muckenhoupt A₁ class), and v A₁⁺ (another one-sided Muckenhoupt A₁ class), then there exists a constant C such that sup_cR _{{x:|T_cf(x)|>v(x)}}uv ≤ C _R|f|u.

HODGE POLYNOMIALS OF THE MODULI SPACES OF TRIPLES OF RANK (2, 2)

Munoz, V., Ortega, D., Vazquez-Gallo, M.-J. — 2008-06-05

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple (E₁, E₂, ) on X consists of two holomorphic vector bundles E₁ and E₂ over X and a holomorphic map : E₂ -> E₁. There is a concept of stability for triples which depends on a real parameter . In this paper, we determine the Hodge polynomials of the moduli spaces of -stable triples with rk(E₁) = rk(E₂) = 2, using the theory of mixed Hodge structures (in the cases that these moduli spaces are smooth and compact). This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we also give the Hodge polynomial of the moduli space of even degree rank 2 stable vector bundles.

THE GAP BETWEEN LOCAL MULTIPLIER ALGEBRAS OF C*-ALGEBRAS

Argerami, M., Farenick, D., Massey, P. — 2008-06-05

The local multiplier algebra M_loc(A) of a C*-algebra A has the property that M_loc (A) M_loc(M_loc(A)). In this paper we show that there is a separable liminal C*-algebra A such that the inclusion is proper.

THE ADAPTED COMPLEXIFICATION OF THE TWO-SPHERE WITH A LIOUVILLE METRIC

Aguilar, R. M. — 2008-04-19

We show that the two-sphere with a Riemannian metric that is Liouville with finite isometry group does not admit an unbounded adapted complexification in the sense of Lempert and Szoke and of Guillemin and Stenzel; that is, its Grauert tube cannot have infinite radius. We prove this by first extending a classical theorem valid for umbilical geodesics in a triaxial ellipsoid to general Liouville metrics. Furthermore, we derive an isometric rigidity result for the Monge–Ampère foliation of a two-dimensional Grauert tube with infinite radius.

MODULI SPACES OF PARABOLIC U(p, q)-HIGGS BUNDLES

Garcia-Prada, O., Logares, M., Munoz, V. — 2008-04-19

Using the L²-norm of the Higgs field as a Morse function, we study the moduli space of parabolic U(p, q)-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. When the parabolic degree is zero this space is homeomorphic to the moduli space of representations of the fundamental group of the punctured surface in U(p, q), with fixed compact holonomy classes around the marked points. By means of this homeomorphism we count the number of connected components of this moduli space of representations. Finally, we apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.

HOMOLOGICAL SYSTEMS IN MODULE CATEGORIES OVER PRE-ORDERED SETS

Mendoza, O., Saenz, C., Xi, C. — 2008-04-10

We introduce the so-called homological systems in a module category over a pre-ordered set, which generalize the notion of a stratifying system over a linearly ordered set, and study both the corresponding modules filtrated by the systems and algebras stratified by the systems. In particular, we recover the tilting theory for pre-standardly stratified algebras, and get a general formula for computing the Cartan determinants of pre-standardly stratified algebras in terms of standard modules and simple modules. Also, the finitistic dimension of a given algebra, and the relative homological dimensions of full subcategories of the modules related to a homological system, are discussed. As an application, we get a new bound for the finitistic dimension of a pre-standardly stratified algebra.

A NOTE ON THE LEAST TOTIENT OF A RESIDUE CLASS

Garaev, M. Z. — 2008-04-09

Let q be a large prime number, a be any integer and be a fixed small positive quantity. Friedlander and Shparlinksi (Least totient in a residue class, Bull. London Math. Soc. 39 (2007), 425–432) have shown that there exists a positive integer n<<q^5/2+ such that (n) falls into the residue class a±od q. Here, (n) denotes Euler's function. In the present paper we improve this bound to n<<q²⁺.

THE IDENTITIES OF A LIE ALGEBRA VIEWED AS A LIE RING

Krasilnikov, A. — 2008-04-09

Let G be a Lie algebra over the field Q of rationals. Then G can be viewed as a Lie ring as well. If the identities of the Lie ring G have a finite basis then so have the identities of the Lie algebra G. We give an example which shows that the converse is not always true.

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

Chan, Y.-M. — 2008-02-19

This is the second of two papers studying Calabi–Yau 3-folds with conical singularities and their desingularizations. In our first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181] we constructed the desingularization of the conically singular manifold M₀ by gluing an asymptotically conical (AC) Calabi–Yau 3-fold Y into M₀ at the singular point, thus obtaining a 1-parameter family of compact, non-singular Calabi–Yau 3-folds M_t for small t > 0. During the gluing process one may encounter a kind of cohomological obstruction to defining a 3-form _t on M_t which interpolates between the 3-form ₀ on M₀ and the scaled 3-form t³ _Y on Y if the rate at which the AC Calabi–Yau 3-fold Y converges to the Calabi–Yau cone is equal to–3. The first paper [3] studied the simpler case < –3 where there is no obstruction. This paper extends the result in the first one by considering a more complicated situtation when = –3. Assuming the existence of singular Calabi–Yau metrics on compact complex 3-folds with ordinary double points, our result in this paper can be applied to repairing such kinds of singularities, which is an analytic version of Friedman's result giving necessary and sufficient conditions for smoothing ordinary double points.

CLASSIFICATION OF A CLASS OF CONTINUOUS MAPS ON THE UNIT INTERVAL

Sawyer, D. J., Truss, J. K. — 2008-02-15

We give a classification of certain continuous maps from the unit interval (or any non-trivial closed bounded interval of the real line) to itself up to conjugacy. The maps studied are those for which there is a partition of the interval into finitely many subintervals whose endpoints form a cycle of the map, and on each of which the map is monotonic.

POSITIVE LAWS IN DERIVED SUBGROUPS OF FIXED POINTS

Shumyatsky, P. — 2008-02-12

Let A be an elementary abelian group of order at least q⁴ acting on a finite q'-group G in such a manner that the derived subgroup of C_G(a) satisfies a positive law of degree n for any a A^#. It is proved that G' satisfies a positive law of degree bounded by a function of q and n only.

ON DAVENPORT-STOTHERS INEQUALITIES AND ELLIPTIC SURFACES IN POSITIVE CHARACTERISTIC

Schutt, M., Schweizer, A. — 2008-02-06

We show that the Davenport–Stothers inequality from characteristic 0 fails in any characteristic p>3. The proof uses elliptic surfaces over P¹ and inseparable base change. We then present adjusted inequalities. These follow from results of Pesenti and Szpiro. For characteristics 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over P¹. Our ideas are also related to supersingular surfaces (in Shioda's sense).

NON-COMMUTATIVE VITALI-HAHN-SAKS THEOREM HOLDS PRECISELY FOR FINITE W*-ALGEBRAS

Chetcuti, E., Hamhalter, J. — 2008-02-05

It is shown that the bona fide generalization of the Vitali–Hahn–Saks theorem to von Neumann algebras is possible if, and only if, the algebra is finite. This settles the problem on the non-commutative Vitali–Hahn–Saks theorem completely and provides new means of characterizing finite von Neumann algebras.

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

Choi, Y. S., Garcia, D., Maestre, M., Martin, M. — 2008-01-31

We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K, X) and L(µ, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every -finite measure µ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k ≥ slant 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X** is the least possible, namely k^k/(1–k). We also give new examples of Banach spaces with the polynomial Daugavet property, namely L(µ, X) when µ is atomless, and C_w(K, X), C_w*(K, X*) when K is perfect.

WHEN JORDAN SUBMODULES ARE BIMODULES

Bresar, M., Kissin, E., Shulman, V. S. — 2008-01-31

Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submodule of X if Ay + yA Y for all A A and y Y (if X = A, then this coincides with the classical concept of a Jordan ideal). When is a Jordan A-submodule a submodule? We give a thorough analysis of this question in both algebraic and analytic context. In the first part of the paper, we consider general algebras and general Banach algebras. In the second part, we treat some more specific topics, such as symmetrically normed Jordan A-submodules. Some of our results are of interest also in the classical situation; in particular, we show that there exist C*-algebras having Jordan ideals that are not ideals.

MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Buchweitz, R.-O., Green, E. L., Snashall, N., Solberg, O. — 2008-01-18

Let = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultiplicative’ structure in [E. L. Green, G. Hartman, E. N. Marcos and Ø. Solberg, Resolutions overKoszul algebras, Arch. Math. 85 (2005), 118–127.], and this is used to find a minimal projective resolution P of over the enveloping algebra ^e. Using these results, we showthat the multiplication in the Hochschild cohomology ring of relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of is shown to be surjective onto the graded centre of the Koszul dual.

ON THE SUM OF THE FIRST n PRIMES

Cilleruelo, J., Luca, F. — 2008-01-17

In this note, we show that the set of n such that the arithmetic mean of the first n primes is an integer is of asymptotic density zero. We use the same method to show that the set of n such that the sum of the first n primes is a square is also of asymptotic density zero. We also prove that both the arithmetic mean of the first n primes as well as the square root of the sum of the first n primes are well distributed modulo 1.

ON THE ERROR TERMS AND EXCEPTIONAL SETS IN CONJECTURAL SECOND MAIN THEOREMS

Levin, A., McKinnon, D., Winkelmann, J. — 2008-01-11

We study the error terms and exceptional sets in conjectural Second Main Theorems in Nevanlinna theory and Diophantine approximation (Vojta's conjecture). In particular, we give a geometric description of the exceptional set in the case of surfaces and the trivial divisor. Examples are given which show that, in general, the exceptional sets in conjectural Second Main Theorems must depend on the parameter in these conjectures. As a consequence, we obtain counterexamples to a conjecture of S. Lang on the forms of the error terms in conjectural Second Main Theorems.

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

Zhou, H.-S., Zhu, H. — 2007-12-12

In this paper, we consider the following semilinear elliptic problem:

where f(x, t) tends to p(x) and q(x) L(R^N), respectively, as t -> 0 and t -> +. We prove that there exist two numbers l and L with L < l such that problem (P) has at least one positive solution for (–l, –L) and has no positive solution for all [–l,–L]. The existence and non-existence of positive solutions for problem (P) at = –l and = –L are also discussed.

SIMULTANEOUS DIOPHANTINE APPROXIMATION BY SQUARE-FREE NUMBERS

Dietmann, R. — 2007-12-04

Let ₁, ...,_r R be ‘not very well approximable’, for example, Q-linearly independent real algebraic numbers. Then there are infinitely many positive square-free integers n such that $$\Vert n{\alpha }_{i}\Vert \ll {n}^{-(2/3r)+\epsilon}(1\le i\le r)$$, where ||·|| denotes distance to the nearest integer.

POWERS OF OPERATORS DOMINATED BY STRICTLY SINGULAR OPERATORS

Flores, J., Hernandez, F. L., Tradacete, P. — 2007-12-04

It is proved that every positive operator R on a Banach lattice E dominated by a strictly singular operator T:E -> E satisfies that the R⁴ is strictly singular. Moreover, if E is order continuous then the R² is already strictly singular.

DEFORMATIONS OF HYPERCOMPLEX STRUCTURES ASSOCIATED TO HEISENBERG GROUPS

Grantcharov, G., Pedersen, H., Poon, Y. S. — 2007-12-04

Let X be a compact quotient of the product of the real Heisenberg group H_4m+1 of dimension 4m + 1 and the three-dimensional real Euclidean space R³. A left-invariant hypercomplex structure on H_4m+1 x R³ descends onto the compact quotient X. The space X is a hyperholomorphic fibration of 4-tori over a 4m-torus. We calculate the parameter space and obstructions to deformations of this hypercomplex structure on X. Using our calculations, we show that all small deformations generate invariant hypercomplex structures on X but not all of them arise from deformations of the lattice. This is in contrast to the deformations on the 4m-torus.

PLANE CURVE DIAGRAMS AND GEOMETRICAL APPLICATIONS

Dias, F. S., Nuno-Ballesteros, J. — 2007-12-04

We look at plane curve diagrams (f,), which are given by a plane curve multigerm : (R, S) -> R² and a function on it f:(R, S) -> R. We obtain a classification of all such diagrams, where has A_e-codimension ≤ 2 and f has finite order. Then we define an equivalence between plane curves which we call A_h-equivalence and which is determined by the class of the diagram (h, ). Here, h denotes the height function of with respect to its normal vector. This is an equivalence which not only takes into account the topology of the singularity of , but also its flat geometry. Finally, we apply our results in order to obtain a classification of all the plane projections of a generic space curve embedded in R³

SCALED ASYMPTOTICS FOR SOME q-SERIES

Zhang, R. — 2007-10-30

This work, investigates the asymptotics for Euler’s q-exponential E_q(z), Ramanujan’s function A_q(z), Jackson’s q-Bessel function $${J}_{\nu }^{(2)}(z;q)$$, the Stieltjes–Wigert orthogonal polynomials S_n(x; q) and q-Laguerre polynomials $${L}_{n}^{(\alpha )}(x;q)$$ as q approaches 1.

THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Blomer, V. — 2007-10-30

Let (n) denote the Fourier coefficients of cusp forms or the number of divisors of n. Estimates of the type $${\displaystyle \sum _{b(q)}^{\ast }\hbox{ \hspace{0.17em} }}|{\displaystyle \sum _{\begin{array}{c}n\le X\\ n\equiv b(q)\end{array}}\alpha }(n)-\hbox{ main term }{|}^{2}{\ll }_{\epsilon }{X}^{1+\epsilon }$$ are shown, uniformly in q ≤ X. The methods can be extended to other arithmetic functions, for example, the number of representations of n as a sum of two squares or k-free numbers. As an application, sums of the type _{n ≤ X}(n) (n) for any q-periodic function can be estimated non-trivially.

ALPERIN'S CONJECTURE FOR ALGEBRAIC GROUPS

Rohrle, G., Rouquier, R. — 2007-10-30

We prove analogues for reductive algebraic groups of some results for finite groups due to Knörr and Robinson from ‘Some remarks on a conjecture of Alperin’, J. London Math. Soc (2) 39 (1989), 48–60, which play a central rôle in their reformulation of Alperin's conjecture for finite groups.

ORTHOGONALLY ADDITIVE POLYNOMIALS ON C*-ALGEBRAS

Palazuelos, C., Peralta, A. M., Villanueva, I. — 2007-10-30

We show that for every orthogonally additive scalar n-homogeneous polynomial P on a C*-algebra A there exists in A* satisfying P(x)= (xⁿ), for each element x in A. The vector-valued analogue follows as a corollary.

BIALGEBRAS AND CATERPILLARS

Turner, W. — 2007-10-23

We describe a double construction, which associates a symmetric associative algebra to a bialgebra. We show how a block of a finite group with cyclic defect can be realised via this double construction, after a felicitous choice of bialgebra.