Force,deflection, and time: Proposition VI of Newton's Principia

In this extended study of Proposition VI, and its first corollary, in Book I of Newton's Principia, we clarify both the statements and the demonstrations of these fundamental results. We begin by tracing the evolution of this proposition and its corollary, to see how their texts may have changed from their initial versions. To prepare ourselves for some of the difficulties our study confronts, we then examine certain confusions which arise in two recent commentaries on Proposition VI. We go on to note other confusions, not in any particular commentary, but in Newton's demonstration and, especially, in his statement of the proposition. What, exactly, does Newton mean by a “body [that] revolves … about an immobile center”? By a “just-nascent arc”? By the “sagitta of the arc”? By the “centripetal force”? By “will be as”? We search for the mathematical meanings that Newton has in mind for these fragments of the Proposition VI statement, a search that takes us to earlier sections of the Principia and to discussions of the “method of first and last ratios,” centripetal force, and the second law of motion. The intended meaning of Proposition VI then emerges from the combined meanings of these fragments. Next we turn to the demonstration of Proposition VI, noting first that Newton's own argument could be more persuasive, before we construct a modern, more rigorous proof. This proof, however, is not as simple as one might expect, and the blame for this lies with the “sagitta of the arc,” Newton's measure of deflection in Proposition VI. Replacing the sagitta with a more natural measure of deflection, we obtain what we call Platonic Proposition VI, whose demonstration has a Platonic simplicity. Before ending our study, we examine the fundamental first corollary of Proposition VI. In his statement of this Corollary 1, Newton replaces the sagitta of Proposition VI by a not quite equal deflection from the tangent and the area swept out (which represents the time by Proposition I) by a not quite equal area of a triangle. These two approximations create small errors, but are these errors small enough? Do the errors introduced by these approximations tend to zero fast enough to justify these replacements? Newton must believe so, but he leaves this question unasked and unanswered, as have subsequent commentators on this crucial corollary. We end our study by asking and answering this basic question, which then allows us to give Corollary 1 a convincing demonstration.     

Kepler's area law in the Principia: filling in some details in Newton's proof of Proposition 1

During the past 30 years there has been controversy regarding the adequacy of Newton's proof of Prop. 1 in Book 1 of the Principia. This proposition is of central importance because its proof of Kepler's area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the Principia. It is shown here that the critics of Prop. 1 have misunderstood Newton's continuum limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well-grounded.     

Passage to the limit in Proposition I, Book I of Newton's Principia

The purpose of this paper is to analyze the way in which Newton uses his polygon model and passes to the limit in Proposition I, Book I of his Principia. It will be evident from his method that the limit of the polygon is indeed the orbital arc of the body and that his approximation of the actual continuous force situation by a series of impulses passes correctly in the limit into the continuous centripetal force situation. The analysis of the polygon model is done in two ways: (1) using the modern concepts of force, linear momentum, linear impulse, and velocity, and (2) using Newton's concepts of motive force and quantity of motion. It should be clearly understood that the term “force” without the adjective “motive,” is used in the modern sense, which is that force is a vector which is the time rate of change of the linear momentum. Newton did not use the word “force” in this modern sense. The symbol F denotes modern force. For Newton “force” was “motive force,” which is measured by the change in the quantity of motion of a body. Newton's “quantity of motion” is proportional to the magnitude of the modern vector momentum. Motive force is a scalar and the symbol F_m is used for motive force.     

Discrete time Galerkin approximations to the nonlinear filtering solution

This paper concerns discrete time Galerkin approximations to the solution of the filtering problem for diffusions. Two families of schemes approximating the unnormalized conditional density, respectively, in an “average” and in a “pathwise” sense, are presented. L² error estimates are derived and it is shown that the rate of convergence is linear in the time increment or linear in the modulus of continuity of the sample path.     

The realization of networks of convex polyhedra with equiangular vertices. 4

A fundamental theorem on closed polyhedra with equiangular vertices is presented. The proof of the theorem was begun in parts 1–3 of this paper. Here, in part 4, we find the polyhedra containing faces of type (4, 4,n) and (4, 5,n)-a total of 44 polyhedra and one infinite series of polyhedra dual to prisms. One table. Seven figures.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 32–47.     

Sub-Laplacians of Holomorphic L-type on Rank One AN-Groups and Related Solvable Groups

Consider a right-invariant sub-Laplacian L on an exponential solvable Lie group G, endowed with a left-invariant Haar measure. Depending on the structure of G and possibly also that of L, L may admit differentiable L^p-functional calculi, or may be of holomorphic L^p-type for a given p≠2, as recent studies of specific classes of groups G and sub-Laplacians L have revealed. By “holomorphic L^p-type” we mean that every L^p-spectral multiplier for L is necessarily holomorphic in a complex neighborhood of some point in the L²-spectrum of L. This can only arise if the group algebra L¹(G) is non-symmetric. In this article we prove that, for large classes of exponential groups, including all rank one AN-groups, a certain Lie algebraic condition, which characterizes the non-symmetry of L¹(G) [37], also suffices for L to be of holomorphic L¹-type. Moreover, if this condition, which was first introduced by J. Boidol [6] in a different context, holds for generic points in the dual * of the Lie algebra of G, then L is of holomorphic L^p-type for every p≠2. Besides the non-symmetry of L¹(G), also the closedness of coadjoint orbits plays a crucial role. We also discuss an example of a higher rank AN-group. This example and our results in the rank one case suggest that sub-Laplacians on exponential Lie groups may be of holomorphic L¹-type if and only if there exists a closed coadjoint orbit Ω * such that the points of Ω satisfy Boidol's condition. In the course of the proof of our main results, whose principal strategy is similar as in [8], we develop various tools which may be of independent interest and largely apply to more general Lie groups. Some of them are certainly known as “folklore” results. For instance, we study subelliptic estimates on representation spaces, the relation between spectral multipliers and unitary representations, and develop some “holomorphic” and “continuous” perturbation theory for images of sub-Laplacians under “smoothly varying” families of irreducible unitary representations.     

Notes on subtlety and ineffability in P κ λ

A type of subtlety for P_κλ called “strongly subtle” is introduced to show almost ineffability is consistencywise stronger than Shelah property. The following are also shown: is strongly subtle” has rather strong consequences. (ii) The ideal is not strongly subtle} is not λ-saturated , and completely ineffable ideal is not precipitous. (iii) In case that λ^<κ=2^λ, almost λ-ineffability coincides with λ-ineffability. (iv) It is not provable that κ is λ^<κ-ineffable whenever κ is λ-ineffable.Research partially supported by “Grant-in-Aid for Scientific research (C), The Ministry of Education, Science, Sports and Culture of Japan 09640299”, and “Japan Society for the Promotion of Science 14540142”.The author is very grateful to the referee for his correcting many errors and helpful suggestions.     

Reverse mathematics and well-ordering principles: A pilot study

The larger project broached here is to look at the generally sentence “if X is well-ordered then f(X) is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory T_f whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, we prove in this paper that the statement “if X is well-ordered then ε_X is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search (deduction chains) [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a (small) fragment of .     

Self-intersection local time for Gaussian ′( )-processes: Existence, path continuity and examples

In this paper we develop a criterion for existence or non-existence of self-intersection local time (SILT) for a wide class of Gaussian ′( ^d)-valued processes, we show that quite generally the SILT process has continuous paths, and we give several examples which illustrate existence of SILT for different ranges of dimensions (e.g., d ≤ 3, d ≤ 7 and 5 ≤ d ≤ 11 in the Brownian case). Some of the examples involve branching and exhibit “dimension gaps”. Our results generalize the work of Adler and coauthors, who studied the special case of “density processes” and proved that SILT paths are cadlag in the Brownian case making use of a “particle picture” approximation (this technique is not available for our general formulation).     

On the problem of the finiteness of the number of the combinatorial types of infinite convex polyhedra with equiangular faces

An infinite (complete) convex polyhedron with equiangular faces, that is, such that all the angles of each of its faces are equal to one another, is called irreducible if the number of monogonal faces belonging to it cannot be decreased (by identifying their sides) while preserving the equiangularity of all of the other faces and the convexity of the polyhedron itself (a lack of conditional edges). Any infinite convex polyhedron with equiangular faces can be obtained from the corresponding irreducible one by pasting in the missing number of monogons. It is proved that the number of combinatorially different irreducible polyhedra is finite, not counting three infinite series of frusta of cones with finite or infinite bases and right prisms with infinite bases. It is also established that, without exception, all infinite convex polyhedra with equiangular faces and total curvature 2are the derivatives of closed convex polyhedra with equiangular faces. The proof is carried out with the help of A. D. Milka's method from Ross. Zh. Mat., 1988, 3A830. Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 75–83, 1992.     

Uniformly generated submodules of permutation modules: Over fields of characteristic 0

This paper is motivated by a link between algebraic proof complexity and the representation theory of the finite symmetric groups. Our perspective leads to a new avenue of investigation in the representation theory of S_n. Most of our technical results concern the structure of “uniformly” generated submodules of permutation modules. For example, we consider sequences of submodules of the permutation modules M^(n−k,1k) and prove that if the sequence W_n is given in a uniform (in n) way – which we make precise – the dimension p(n) of W_n (as a vector space) is a single polynomial with rational coefficients, for all but finitely many “singular” values of n. Furthermore, we show that dim(W_n)<p(n) for each singular value of n≥4k. The results have a non-traditional flavor arising from the study of the irreducible structure of the submodules W_n beyond isomorphism types. We sketch the link between our structure theorems and proof complexity questions, which are motivated by the famous NP vs. co-NP problem in complexity theory. In particular, we focus on the complexity of showing membership in polynomial ideals, in various proof systems, for example, based on Hilbert's Nullstellensatz.     

On the construction of equiangular frames from graphs

We give details of the 1-1 correspondence between equiangular frames of n vectors for and graphs with n vertices. This has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory. The nontight examples given here (which correspond to graphs with more than 2 eigenvalues) have the potential for similar applications, e.g., the frame corresponding to the 5-cycle graph is the unique Grassmannian frame of 5 vectors in openR³. Further, the associated canonical tight frames have a small number of angles in many cases.     

Newton'sPrincipiaand Inverse-Square Orbits in a Resisting Medium: A Spiral of Twisted Logic

Proposition XV/Theorem XII in Book Two of Newton'sPrincipiadeals with the spiral path of a body attracted by an inverse-square force toward a fixed center and retarded by the medium in which it travels. This article examines the argument offered by Newton as proof of the proposition/theorem and finds it fallacious. Also presented here are accounts of how Newton's purported proof is dealt with in each of three late-20th-century publications—none of which reports detection of the fallacy.Copyright 1998 Academic Press.La Proposition XV (théorème XI) du livre II desPrincipiade Newton traite de la trajectoire en forme de spirale d'un corps attiré vers un centre immobile par une force d'intensité proportionnelle à l'inverse du carré de la distance, et ralenti par le milieu qu'il traverse. Cet article examine le raisonnement offert par Newton pour preuve de la proposition (du théorème) et montre qu'il est incorrect. L'article présente aussi la façon dont la prétendue preuve de Newton a été traitée dans trois publications de la fin du vingtième siècle—aucune d'entres elles ne découvre l'erreur du raisonnement.Copyright 1998 Academic Press.Proposition XV/Theorem XII derPrincipiaBuch Zwei von Newton befaßt sich mit dem spiralförmigen Weg eines Körpers der durch einer Kraft gemäß dem inversen Quadratgesetz zu einen bestimmten Mittlepunkt, und durch das Medium verzögert wird, worin er sich bewegt. Dieser Aufsatz untersucht das von Newton vorgebrachte Argument für den Beweis den Satzes/Theorems, und findet es durch einen Trugschluß beeinträchtigt. Auch wird hier dargelegt, wie der von Newton gegebene Beweiß in jeden der drie Veröffentlichungen des späten zwanzigsten Jahrhunderts behandelt wird—keiner von ihnen berichtet von der Entdeckung des Trugschlußes.Copyright 1998 Academic Press.MSC 1991 subject classifications: 01A45, 70M20     

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.     

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of “mixed” products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type . A realization of the basic representation that is of “”-type plays the central role.     

Weak compactness of wave maps and harmonic maps

We show that a weak limit of a sequence of wave maps in (1 + 2) dimensions with uniformly bounded energy is again a wave map. Essential ingredients in the proof are Hodge structures related to harmonic maps, ¹ estimates for Jacobians, ¹-BMO duality, a “monotonicity” formula in the hyperbolic context and the concentration compactness method. Application of similar ideas in the elliptic context yields a drastically shortened proof of recent results by Bethuel on Palais-Smale sequences for the harmonic map functional on two dimensional domains and on limits of almost H-surfaces.     

On the rate of approximation in the central limit theorem for dependent random variables and random vectors

Large “O” and small “o” approximations of the expected value of a class of smooth functions (f C^r(R)) of the normalized partial sums of dependent random variable by the expectation of the corresponding functions of normal random variables have been established. The same types of approximations are also obtained for dependent random vectors. The technique used is the Lindberg-Levy method generalized by Dvoretzky to dependent random variables.     

A general framework of compactly supported splines and wavelets

Let {V_k} be a nested sequence of closed subspaces that constitute a multiresolution analysis of L²( ). We characterize the family Φ = {φ} where each φ generates this multiresolution analysis such that the two-scale relation of φ is governed by a finite sequence. In particular, we identify the ε Φ that has minimum support. We also characterize the collection Ψ of functions η such that each η generates the orthogonal complementary subspaces W_k of V_k, . In particular, the minimally supported ψ ε Ψ is determined. Hence, the “B-spline” and “B-wavelet” pair (, ψ) provides the most economical and computational efficient “spline” representations and “wavelet” decompositions of L² functions from the “spline” spaces V_k and “wavelet” spaces W_k, k . A very general duality principle, which yields the dual bases of both {(·−j):j and {η(·−j):j } for any η ε Ψ by essentially interchanging the pair of two-scale sequences with the pair of decomposition sequences, is also established. For many filtering applications, it is very important to select a multiresolution for which both and ψ have linear phases. Hence, “non-symmetric” and ψ, such as the compactly supported orthogonal ones introduced by Daubechies, are sometimes undesirable for these applications. Conditions on linear-phase φ and ψ are established in this paper. In particular, even-order polynomial B-splines and B-wavelets φ_m and ψ_m have linear phases, but the odd-order B-wavelet only has generalized linear phases.     

Expansions for the multivariate chi-square distribution

Three classes of expansions for the distribution function of the χ_k²(d, R)-distribution are given, where k denotes the dimension, d the degree of freedom, and R the “accompanying correlation matrix.” The first class generalizes the orthogonal series with generalized Laguerre polynomials, originally given by Krishnamoorthy and Parthasarathy [12]. The second class contains always absolutely convergent representations of the distribution function by univariate chi-square distributions and the third class provides also the probabilities for any unbounded rectangular regions. In particular, simple formulas are given for the three-variate case including singular correlation matrices R, which simplify the computation of third order Bonferroni inequalities, e.g., for the tail probabilities of max{χ_i²|1 ≤ i ≤ k} (k > 3).     

Kernel estimates and -spectral independence of generators of -semigroups

After the appearance of W. Arendt's result that “Gaussian estimate of a semigroup implies the L^p-spectral independence of the generator,” various generalizations have been obtained. This paper shows that a certain kernel estimate of a semigroup implies the L^p-spectral independence of the generator, generalizing the case of upper Gaussian estimate and “Gaussian estimate of order α(0,1] [S. Miyajima, H. Shindoh, Gaussian estimates of order α and L^p-spectral independence of generators of C₀-semigroups, Positivity 11 (1) (2007) 15–39], Definition 3.1.” The proof uses S. Karrmann's result about the L^p-spectral independence and B.A. Barnes' theorem about the spectrum of integral operators. As an application, the L^p-spectral independence of −[(−Δ)^α+V] (α(0,1]) for a suitable V is proved with the help of a recent result by V. Liskevich, H. Vogt and J. Voigt [V. Liskevich, H. Vogt, J. Voigt, Gaussian bounds for propagators perturbed by potentials, J. Funct. Anal. 238 (2006) 245–277].