In Defense of Logical Universalism: Taking Issue with Jean van Heijenoort

Van Heijenoort??s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schr?der, and L?wenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then I describe his way of arguing for the second view (Sect. 2); and finally I come down in favor of the first view (Sect. 3). There, I specify the version of universalism for which I am prepared to argue (Sect. 3, introduction). Choosing ZFC to play the part of universal, logical (in a nowadays forgotten sense) system, I show, through an example, how the usual model theory can be naturally given its proper place, from the universalist point of view, in the logical framework of ZFC; I outline another, not rival but complementary, semantics for admissible extensions of ZFC in the very same logical framework; I propose a way to get universalism out of the predicaments in which universalists themselves believed it to be (Sect. 3.1). Thus, if universalists of the classical period did not, in fact, construct these semantics, it was not that their universalism forbade them, in principle, to do so. The historical defeat of universalism was not technical in character. Neither was it philosophical. Indeed, it was hardly more than the victory of technicism over the very possibility of a philosophical dispute (Sect. 3.2).     

Highway improvement project selection by the joint consideration of cost-benefit and risk criteria

Since highway improvement project selection requires screening thousands of road segments with respect to crashes for further analysis and final project selection, we provide a two-step project selection methodology and describe an application case to demonstrate its advantages. In the first step of the proposed methodology, we will use odds against observing a given crash count, injury count, run-off road count and so on as measures of risk and a multi-criteria pre-selection technique with the objective to decrease the number of prospective improvement locations. In the second step, the final project selection is accomplished based on a composite efficiency measure of estimated cost, benefit and hazard assessment (odds) under budget constraints. To demonstrate the two-step methodology, we will analyze 4 years of accident data at 23?000 locations where the final projects are selected out of several hundred of potential locations.     

On a generalization of Kaplansky's game

In this paper, games of the following general kind are studied: Two players move alternately by selecting unselected integer coordinate points in the plane. On each move, the first player selects exactly r points and the second player selects exactly one point. The first player wins if he can select p points on a line having none of his opponent's points before his opponent selects q points on a line having none of his own. If this latter eventuality occurs first, the second player wins. It is shown that if p ? c(r)q, then the second player can always win.     

Son of the linear search problem

I wish to find something which is located on a certain road. I start at a point on the road, but I do not know in which direction the object sought is to be found. Somehow, I must incorporate in my way of searching the possibility that it is either to the right or to the left. Thus, I must search first to the right, and then to the left, and then to the right again until it is found. What is a good way of conducting this search, and what is a bad way? This general problem can be phrased in many ways mathematically, some of which are answered in the papers in the bibliography. In this paper, we consider three well-known assumptions concerning thea priori guesses for the probability distribution on where the object is located. These concern uniform distribution on an interval, triangular distribution around the original point, and normal distribution about that point. The uniform distribution has a simple answer. For the triangular distribution, we obtain qualitative results and calculate approximate values for the turning points.     

Extreme point and halving edge search in abstract order types

Many properties of finite point sets only depend on the relative position of the points, e.g., on the order type of the set. However, many fundamental algorithms in computational geometry rely on coordinate representations. This includes the straightforward algorithms for finding a halving line for a given planar point set, as well as finding a point on the convex hull, both in linear time. In his monograph Axioms and Hulls, Knuth asks whether these problems can be solved in linear time in a more abstract setting, given only the orientation of each point triple, i.e., the set?s chirotope, as a source of information. We answer this question in the affirmative. More precisely, we can find a halving line through any given point, as well as the vertices of the convex hull edges that are intersected by the supporting line of any two given points of the set in linear time. We first give a proof for sets realizable in the Euclidean plane and then extend the result to non-realizable abstract order types.     

Conglomerable coherence

We contrast Williams? and Walley?s theories of coherent lower previsions in the light of conglomerability. These are two of the most credited approaches to a behavioural theory of imprecise probability. Conglomerability is the notion that distinguishes them most: Williams? theory does not consider it, while Walley aims at embedding it in his theory. This question is important, as conglomerability is a major point of disagreement at the foundations of probability, since it was first defined by de Finetti in 1930. We show that Walley?s notion of joint coherence (which is the single axiom of his theory) for conditional lower previsions does not take all the implications of conglomerability into account. Considering also some previous results in the literature, we deduce that Williams? theory should be the one to use when conglomerability is not required; for the opposite case, we define the new theory of conglomerably coherent lower previsions, which is arguably the one to use, and of which Walley?s theory can be understood as an approximation. We show that this approximation is exact in two important cases: when all conditioning events have positive lower probability, and when conditioning partitions are nested.     

Geometric and analytic problems for a real submanifold in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with CR singularities

In this survey article, we present some known results and also propose some open questions related to the analytic and geometric aspects of Bishop submanifolds in a complex space. We mainly focus on those problems that the author and his coauthors have recently worked on. The article also contains an example of a Bishop submanifold in ?³ of real codimension two, which cannot be quadratically flattened at a CR singular point but is CR non-minimal at any CR point. This provides a counter-example to a question asked in a private communication by Zaistev (2013).     

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

The classical problem of Apollonius is to construct circles that are tangent to three given circles in the plane. This problem was posed by Apollonius of Perga in his work “Tangencies.” The Sylvester problem, which was introduced by the English mathematician J.J. Sylvester, asks for the smallest circle that encloses a finite collection of points in the plane. In this paper, we study the following generalized version of the Sylvester problem and its connection to the problem of Apollonius: given two finite collections of Euclidean balls, find the smallest Euclidean ball that encloses all the balls in the first collection and intersects all the balls in the second collection. We also study a generalized version of the Fermat–Torricelli problem stated as follows: given two finite collections of Euclidean balls, find a point that minimizes the sum of the farthest distances to the balls in the first collection and shortest distances to the balls in the second collection.     

On the bivariate Shepard-Lidstone operators

We propose a new combination of the bivariate Shepard operators (Coman and Trîmbi?a?, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given (Kraaijpoel and van Leeuwen, 2010 [13]). Numerical comparison with other well known combinations is presented.     

Holomorphic curvature of Finsler metrics and complex geodesics

In his famous 1981 paper, Lempert proved that given a point in a strongly convex domain the complex geodesics (i.e., the extremal disks) for the Kobayashi metric passing through that point provide a very useful fibration of the domain. In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifoldM, it is possible to find purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesies ofM. We first discuss at some length the notion of holomorphic sectional curvature for a complex Finsler metric; then, using the differential equation of complex geodesies we obtained in [AP], we show that for every pair (p;v) ∈T M, withv ≠ 0, there is a (only a segment if the metric is not complete) complex geodesic passing throughp tangent tov iff the Finsler metric is Kähler, has constant holomorphic sectional curvature ?4, and its curvature tensor satisfies a specific simmetry condition—which are the differential geometric conditions we were after. Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature ?4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.     

Über ein Paradoxon aus der Verkehrsplanung

Zusammenfassung Für die Straßenverkehrsplanung möchte man den Verkehrsfluß auf den einzelnen Straßen des Netzes abschätzen, wenn die Zahl der Fahrzeuge bekannt ist, die zwischen den einzelnen Punkten des Straßennetzes verkehren. Welche Wege am günstigsten sind, hängt nun nicht nur von der Beschaffenheit der Straße ab, sondern auch von der Verkehrsdichte. Es ergeben sich nicht immer optimale Fahrzeiten, wenn jeder Fahrer nur für sich den günstigsten Weg heraussucht. In einigen Fällen kann sich durch Erweiterung des Netzes der Verkehrsfluß sogar so umlagern, daß größere Fahrzeiten erforderlich werden.

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Summary For each point of a road network let be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of the traffic flow. Whether a street is preferable to another one depends not only upon the quality of the road but also upon the density of the flow. If every driver takes that path which looks most favorable to him, the resultant running times need not be minimal. Furthermore it is indicated by an example that an extension of the road network may cause a redistribution of the traffic which results in longer individual running times.

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Vorgel. v.:W. Knödel.     

Spectral bundle methods for non-convex maximum eigenvalue functions: second-order methods

We study constrained and unconstrained optimization programs for nonconvex maximum eigenvalue functions. We show how second order techniques may be introduced as soon as it is possible to reliably guess the multiplicity of the maximum eigenvalue at a limit point. We examine in which way standard and projected Newton steps may be combined with a nonsmooth first-order method to obtain a globally convergent algorithm with a fair chance to local superlinear or quadratic convergence. Dedicated to R. T. Rockafellar on the occasion of his 70th anniversary     

Order and characteristic of parabolically differentiable points

Summary Conditions are given for the union of two arcs of parabolic order four in the affine plane to be an arc of parabolic order four, and for an arc of parabolic order four to be extensible to a larger arc of parabolic order four. A characteristic of a parabolically differentiable interior point p of an arc is defined, and it is shown that the parabolic order of p is at least as great as the sum of the digits of the characteristic of p. If, in addition, p is the common end-point of two arcs of minimal parabolic order, then the parabolic order of p is equal to the sum of the digits of its characteristic. In memory of Guido Castelnuovo in the recurrence of the first centenary of his birth. The second author gratefully acknowledges the support of the National Research Council of Canada. Part of this research was done while the first author was a Fellow of the Summer Research Institute of the Canadian Mathematical Congress.     

Examples of non-convergence of conjugate descent algorithms with exact line-searches

Examples are given in which quasi-Newton and other conjugate descent algorithms fail to converge to a minimum of the object function. In the first there is convergence to a point where the gradient is infinite; in the second, a region characterized by a fine terraced structure causes the iterates to spiral indefinitely. A modification of the second construction gives convergence to a point where the gradient is non-zero, but the gradient is not continuous at this point. The implication of these examples is discussed.     

On Locations and Distances

Regional science is about integrating different spatial disciplines, in particular spatial economics and economic geography. In this spirit a first study bears on the envelopment of production units inside an urban technological complex, comparing natural and logarithmic envelopments from the point of view of their respective representative efficiency. A second and related study treats the reverse problem: given Manhattan distances between production units, which are their respective locations?     

Some variational aspects of an automatic control problem

Techniques of the calculus of variations are used in order to examine a two-point boundary value problem whose boundary condition depends on the value of the solution at an interior point.The author wishes to express his deep appreciation to the referee for his suggestion of a much briefer and more elegant proof of Theorem 2.1 than that originally given.     

Packing tripods: Narrowing the density gap

In 1984, Stein and his co-authors posed a problem concerning simple three-dimensional shapes, known as semicrosses, or tripods. By definition, a tripod of order n is formed by a corner and the three adjacent edges of an integer n×n×n cube. How densely can one fill the space with non-overlapping tripods of a given order? In particular, is it possible to fill a constant fraction of the space as tripod order tends to infinity? In this paper, we settle the second question in the negative: the fraction of the space that can be filled with tripods must be infinitely small as the order grows. We also make a step towards the solution of the first question, by improving the currently known asymptotic lower bound on tripod packing density, and by presenting some computational results on low-order packings.     

Modelling and assessing local area differences in road casualties: a case study in England

The context of this research is the governmental targeting of road casualty reductions in England. Local authorities have a major role to play in achieving such targets and this research explores the relationships between road casualties and local authority characteristics such as population, road networks, traffic, and geography. Regression models are identified for types of road casualty to provide local area expectations against which observed numbers can be compared. A number of issues are identified that raise concerns about the current methods adopted for assessing road safety performances and about the implementation of this approach at local levels. These issues include the large impact on assessments of judgements of casualty severity and of estimates of starting point casualties when estimating reductions over time. This study leads to recommendations for ways in which current approaches to road casualty appraisal might be improved.     

Geometry:Circlesand Circumference

<正>When am I ever going to use this?The London EyeFerris wheel measures 450 feet across.1.Which point appears to be the center of the Ferris wheel?2.Is the distance fromG to F greater than,less than,or equal to the distance from G to J?Acircle is the set of all points in a plane that arc the same distance from a given point,called the center.     

Utility function assessment on a micro-computer: An interactive procedure

The procedure UFAP is presented which allows a decision maker to interactively assess his von Neumann/Morgenstern single attribute utility function. UFAP puts special emphasis on potential biases in the assessment process. In the first part of the procedure three different assessment methods are used to derive ranges for the utility function. Using different methods enables us to point out a possible bias in the elicitation process. In the second part a consistent class of utility functions is derived based on the ranges assessed in the first part. In case inconsistencies between methods arise the decision maker has to reconsider selected preference statements previously given.