Die Limes-Uniformisierbarkeit der Limesräume

Convergence structures (Limitierungen) defined by H. R. Fischer [1] in 1959 anduniform convergence structures (uniforme Limitierungen) introduced by C. H. Cook and H. R. Fischer [2] in 1965 are generalizations of the concepts of topology and uniformity respectively. A convergence structure, induced by some uniform convergence structure, will be called (L)-uniformizable (Limes-uniformisierbar). In this paper a necessary and sufficient condition for (L)-uniformizability of a convergence structure will be given. As a consequence any separated (Hausdorff) convergence structure on a set turns out to be (L)-uniformizable. Also any compatible convergence structure (separated or not) on a group is (L)-uniformizable.     

Uniform Distributions on the Natural Numbers

We compare the following three notions of uniformity for a finitely additive probability measure on the set of natural numbers: that it extend limiting relative frequency, that it be shift-invariant, and that it map every residue class mod m to 1/m. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set L of extensions of limiting relative frequency is a proper subset of the set S of shift-invariant measures and that S is a proper subset of the set R of measures which map residue classes uniformly. Moreover, we show that there are subsets G of ℕ for which the range of possible values μ(G) for μ∈L is properly contained in the set of values obtained when μ ranges over S, and that there are subsets G which distinguish S and R analogously.     

Entropy and Convergence on Compact Groups

We investigate the behaviour of the entropy of convolutions of independent random variables on compact groups. We provide an explicit exponential bound on the rate of convergence of entropy to its maximum. Equivalently, this proves convergence of the density to uniformity, in the sense of Kullback–Leibler. We prove that this convergence lies strictly between uniform convergence of densities (as investigated by Shlosman and Major), and weak convergence (the sense of the classical Ito–Kawada theorem). In fact it lies between convergence in L ¹⁺ and convergence in L ¹.     

Linear Forms and Higher-Degree Uniformity for Functions On {\mathbb{F}^{n}_{p}}

In [GW1] we began an investigation of the following general question. Let L ₁, . . . , L _m be a system of linear forms in d variables on Fⁿ_p{F^n_p}, and let A be a subset of Fⁿ_p{F^n_p} of positive density. Under what circumstances can one prove that A contains roughly the same number of m-tuples L ₁(x ₁, . . . , x _d ), . . . , L _m (x ₁, . . . , x _d ) with x₁,?, x_d ? \mathbb Fⁿ_p{x_1,\ldots, x_d \in {\mathbb F}^n_p} as a typical random set of the same density? Experience with arithmetic progressions suggests that an appropriate assumption is that ||A - d1||_U^k{||A - \delta 1||_{U{^k}}} should be small, where we have written A for the characteristic function of the set A, δ is the density of A, k is some parameter that depends on the linear forms L ₁, . . . , L _m , and || ·||_U^k{|| \cdot ||_U{^k}} is the kth uniformity norm. The question we investigated was how k depends on L ₁, . . . , L _m . Our main result was that there were systems of forms where k could be taken to be 2 even though there was no simple proof of this fact using the Cauchy–Schwarz inequality. Based on this result and its proof, we conjectured that uniformity of degree k − 1 is a sufficient condition if and only if the kth powers of the linear forms are linearly independent. In this paper we prove this conjecture, provided only that p is sufficiently large. (It is easy to see that some such restriction is needed.) This result represents one of the first applications of the recent inverse theorem for the U ^k norm over Fⁿ_p{F^n_p} by Bergelson, Tao and Ziegler [TZ2], [BTZ]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.     

Uniformity in double designs

In constructing two-level fractional factorial designs, the so-called doubling method has been employed. In this paper, we study the problem of uniformity in double designs. The centered L2-discrepancy is employed as a measure of uniformity. We derive results connecting the centered L2-discrepancy value of D（X） and generalized wordlength pattern of X, which show the uniformity relationship between D（X） and X. In addition, we also obtain lower bounds of centered L2-discrepancy value of D（X）, which can be used to assess uniformity of D（X）.     

Lattice uniformities on pseudo-D-lattices

Let L be a pseudo-D-lattice. We prove that the lattice uniformities on L which make uniformly continuous the operations of L are uniquely determined by their system of neighbourhoods of 0 and form a distributive lattice. Moreover we prove that every such uniformity is generated by a family of weakly subadditive [0,+∞]-valued functions on L.     

A note on lower bound of centered L2-discrepancy on combined designs

This note provides a theoretical justification of optimal foldover plans in terms of uniformity. A new lower bound of the centered L ₂-discrepancy values of combined designs is obtained, which can be used as a benchmark for searching optimal foldover plans. Our numerical results show that this lower bound is sharper than existing results when more factors reverse the signs in the initial design.     

Modal Multilattice Logic

A modal extension of multilattice logic, called modal multilattice logic, is introduced as a Gentzen-type sequent calculus \(\hbox {MML}_n\). Theorems for embedding \(\hbox {MML}_n\) into a Gentzen-type sequent calculus S4C (an extended S4-modal logic) and vice versa are proved. The cut-elimination theorem for \(\hbox {MML}_n\) is shown. A Kripke semantics for \(\hbox {MML}_n\) is introduced, and the completeness theorem with respect to this semantics is proved. Moreover, the duality principle is proved as a characteristic property of \(\hbox {MML}_n\).     

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions on higher rank complex simple groups, we establish estimates for which are of the form in the spectral parameter and have uniform exponential decay in regular directions in the group variable a _t . Here is an explicit constant depending on G, and may be singular, for instance.?The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L ^p for singular sphere averages on complex semisimple Lie groups for all p in . We use these to establish singular differentiation theorems and pointwise ergodic theorems in L ^p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions. Submitted: May 2000, Revised version: October 2000.     

Cuplength estimates on lagrangian intersections

We prove the following estimate on Lagrangian intersections: If L is a Lagrangian submanifold of P with π₂(P, L) = 0 and L' is obtained from L by an exact diffeomorphism of P, then the number of elements of L ∩ L' is greater than or equal to the cuplength of P.     

Translation-Invariant Bilinear Operators with Positive Kernels

We study L ^r (or L ^{r, ∞}) boundedness for bilinear translation-invariant operators with nonnegative kernels acting on functions on \mathbb Rⁿ{\mathbb {R}^n}. We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on \mathbb R²ⁿ{\mathbb R^{2n}}, while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some sufficient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L ¹ to L ¹ and from L ¹ to L ^{1, ∞} are equivalent properties, boundedness from L ¹ × L ¹ to L ^1/2 and from L ¹ × L ¹ to L ^{1/2, ∞} may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels.     

Remarks on Normal Generation of Special Line Bundles on Algebraic Curves

Let L be a very ample line bundle with h ¹(L) ≥2 on a curve of genus g. We prove that L is normally generated if deg(L) ≥2g ? 1 ? 4h ¹(L) for large enough genus g.     

Desarguesian finite generalized quadrangles are classical or dual classical

Let S = (P, B, I) be a finite generalized quadrangle of order (s, t), s > 1, t > 1. Given a flag (p, L) of S, a (p, L)-collineation is a collineation of S which fixes each point on L and each line through p. For any line N incident with p, N L, and any point u incident with L, u p, the group G(p, L) of all (p, L)-collineations acts semiregularly on the lines M concurrent with N, p not incident with M, and on the points w collinear with u, w not incident with L. If the group G(p, L) is transitive on the lines M, or equivalently, on the points w, then we say that S is (p, L)-transitive. We prove that the finite generalized quadrangle S is (p, L)-transitive for all flags (p, L) if and only if S is classical or dual classical. Further, for any flag (p, L), we introduce the notion of (p, L)-desarguesian generalized quadrangle, a purely geometrical concept, and we prove that the finite generalized quadrangle S is (p, L)-desarguesian if and only if it is (p, L)-transitive.Research Associate of the National Fund for Scientific Research (Belgium).     

Varieties and Vector Measures

A locally convex space L has the property ? if equicontinuous subsets of L* are weak-star sequentially compact. (L*, σ(L*, L)) is a MAZUR space if given F ∈ L** with F weak-star sequentially continuous then F∈ L. If L is complete with the property ∈, then (L*, σ (L*, L)) is a MAZUR space. The class of locally convex spaces with the property ? forms a variety ??_? and this variety is generated by the BANACH spaces it contains. Weakly compactly generated locally convex spaces and SCHWARTZ spaces belong to ??_?. MAZUR spaces are used to give a characterization of GROTHENDIECK BANACH spaces. The last section contains a characterization of the variety generated by the reflexive BANACH spaces.     

On derivations arising in differential equations

Let L be a linear transformation on the set of all n×n matrices over an algebraically closed field of characteristic 0. It is shown that if AB=BA implies L(A)L(B)=L(B)L(A) and if either L is nonsingular or the implication in the hypothesis can also be reversed, then L is a sum of a scalar multiple of a similarity transformation and a linear functional times the identity transformation.     

Geometry of kinematicK-loops

AK-loop is called kinematic, if a further condition (K7) is valid. Such a loop (L, ⊕) can be provided in a natural way with a left and right structureL andG such that (L,L) and (L,G) become incidence (linear) spaces. For (L,L) andt∈L, each left translationt ⁺:L→L;x→b⊕x is a collineation and (L,G) can be turned in an incidence space with parallelism (L,G, ‖). Examples of kinematicK-loops are given for which the corresponding automorphisms δ_a,b are either the identity or fixed point free.     

An axiomatic characterization of multilattices

The author presents an axiomatic characterization of M. Benado's definition of a multilattice.     

On uniqueness of invariant measures for finite‐ and infinite‐dimensional diffusions

We prove uniqueness of “invariant measures,” i.e., solutions to the equation L*μ = 0 where L = Δ + B · ∇ on ℝⁿ with B satisfying some mild integrability conditions and μ being a probability measure on ℝⁿ. This solves an open problem posed by S. R. S. Varadhan in 1980. The same conditions are shown to imply that the closure of L on L¹(μ) generates a strongly continuous semigroup having μ as its unique invariant measure. The question whether an extension of L generates a strongly continuous semigroup on L¹(μ) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on B. The special case when B is a gradient of a function (i.e., the “symmetric case”) in particular is studied and conditions are identified ensuring that L*μ = 0 implies that L is symmetric on L²(μ) or L*μ = 0 has a unique solution. We also prove infinite‐dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in infinite dimensions. © 1999 John Wiley & Sons, Inc.     

On sernisimple alternative loop algebras

Let QL be the loop algebra of an R.A. loop L over the rational field Q. Assume that the non-trivial commutator in L is not a square of a central element. In [1]: it is shown that QL determines L. In this paper we characterize all fields K, with char K ≠ 2, such that KL determines L for all R..A. 2-loops L with the above property.     

The Vector-Valued Big <Emphasis Type="Italic">q</Emphasis>-Jacobi Transform

Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L ²-space of functions on ℝ with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform ℱ with two different big q-Jacobi functions as a kernel, and we construct the inverse of ℱ.