We give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σ α -measurable functions (for every fixed 1 ď α ă ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous f...

The Darbouxian theory of integrability allows to determine when a polynomial differential system in C2 has a first integral of the kind f λ1 1 · · ·f λp p exp(g/h) where fi , g and h are polynomials in C[x, y], and λi ∈ C for i = 1, . . . , p. The functions of this form are called Darbouxian functions. Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in C2 h...

We propose a simple algorithm for lattice point counting in rational polygons. A rational polygon is one whose vertices have rational coordinates. The algorithm decomposes a given polygon into right trapezoids and counts the number of lattice points in the right trapezoids. Each right trapezoid can be dissected into a rectangle and a right-angled triangle in the obvious way. The number of latti...

In goal-directed proofs, such as those used in HOL, tactics often must operate on one or more speciic assumptions. But goals often have many assumptions. Currently there is no good way to select or denote assumptions in HOL88. Most mechanisms are sensitive to inconsequential changes or are diicult to use. Denoting assumptions by lters (matching) makes it easier to maintain large proofs and reus...

A certificate of non-membership for a Boolean function f with respect to a class C, f 6∈ C, is a set S of input strings such that the values of f on strings from S are inconsistent with any function h ∈ C. We study certificates of non-membership with respect to several classes of read-once functions, generated by their bases. For the basis {&,∨,¬}, we determine the optimal certificate size for ...

This paper presents a thorough analysis of the capabilities of the pseudorelevance feedback (PRF) technique applied to Distributed Information Retrieval (DIR). This work emphasizes the effectiveness of PRF applied to the collection fusion problem. Usually, DIR Systems apply PRF in the same way of traditional Information Retrieval systems. For each collection, local results are improved through ...

A function F on the space of n × n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument. Spectral functions are just symmetric functions of the eigenvalues. We show that a spectral function is twice (continuously) differentiable at a matrix if and only if the corresponding symmetric function is twice (continuously) differentiable at the vector of eige...

The method of partitionable sets for constructing large sets of t-designs have now been used for nearly a decade. The method has resulted in some powerful recursive constructions and also existence results especially for large sets of prime sizes. Perhaps the main feature of the approach is its simplicity. In this paper, we describe the approach and show how it is employed to obtain some of the...

We give a characterization, with respect to a large class of models of untyped λcalculus, of those models that are fully abstract for head-normalization, i.e., whose equational theory is H. An extensional K-model D is fully abstract if and only if it is hyperimmune, i.e., non-well founded chains of elements of D cannot be captured by any recursive function. This article, together with its compa...

In 1976, Chisholm et al. 1 published a paper concerning the location of poles and zeros of Padé approximants of ln 1 − z developed at the complex point ζ : ln 1 − z ln 1 − ζ − ∑∞ n 1 1/n z − ζ/1 − ζ . They claimed that all poles and zeros of diagonal Padé approximants n/n interlace on the cut z ζ t 1 − ζ , t ∈ 1,∞ . Unfortunately, this result is only partially true, for poles. Klarsfeld remarke...