Karma Theory,Determinism, Fatalism and Freedom of Will

The so-called theory of karma is one of the distinguishing aspects of Hinduism and other non-Hindu south-Asian traditions. At the same time that the theory can be seen as closely connected with the freedom of will and action that we humans supposedly have, it has many times been said to be determinist and fatalist. The purpose of this paper is to analyze in some deepness the relations that are between the theory of karma on one side and determinism, fatalism and free-will on the other side. In order to do that, I shall use what has been described as the best formal approach we have to indeterminism: branching time theory. More specifically, I shall introduce a branching time semantic framework in which, among other things, statements such as “state of affairs e is a karmic effect of agent a”, “a wills it to be the case that e” and “e is inevitable” could be properly represented.     

Dyadic sets S: a dichotomy for indecomposable S-matrices

Dyadic sets S (see [3]) give rise to S-matrices, which are important in the investigation of modules over finite-dimensional algebras. If S admits only finitely many isoclasses of indecomposable S-matrices, “most” of the indecomposables can be described by means of a reduction to a well known special case. We determine the “missing” indecomposables.     

Numbers as functions

In this survey I discuss A. Buium’s theory of “differential equations in the padic direction” ([8]) and its interrelations with “geometry over field with one element”, on the background of various approaches to p-adic models in theoretical physics (cf. [1, 30]).     

On monochromatic solutions of some nonlinear equations in ℤ/pℤ

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are x, y ∈ ? such that x + y, xy, and x have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group ?/p?, where p is a prime, and obtain an even stronger density result.     

On a nonlinear nonlocal problem of elliptic type

The solvability of a nonlinear nonlocal problem of the elliptic type that is a generalized Bitsadze–Samarskii-type problem is analyzed. Theorems on sufficient solvability conditions are stated. In particular, a nonlocal boundary value problem with p-Laplacian is studied. The results are illustrated by examples considered earlier in the linear theory (for p = 2). The examples show that, in contrast to the linear case under the same “nice” nonlocal boundary conditions, for p > 2, the problem can have one or several solutions, depending on the right-hand side.     

Dempster's rule of combination

The theory of belief functions is a generalization of probability theory; a belief function is a set function more general than a probability measure but whose values can still be interpreted as degrees of belief. Dempster's rule of combination is a rule for combining two or more belief functions; when the belief functions combined are based on distinct or “independent” sources of evidence, the rule corresponds intuitively to the pooling of evidence. As a special case, the rule yields a rule of conditioning which generalizes the usual rule for conditioning probability measures. The rule of combination was studied extensively, but only in the case of finite sets of possibilities, in the author's monograph A Mathematical Theory of Evidence. The present paper describes the rule for general, possibly infinite, sets of possibilities. We show that the rule preserves the regularity conditions of continuity and condensability, and we investigate the two distinct generalizations of probabilistic independence which the rule suggests.     

Orbit structure and countable sections for actions of continuous groups

It is shown that if a second countable locally compact group G acts nonsingularly on an analytic measure space (S, μ), then there is a Borel subset E ? S such that EG is conull in S and each sG ∩ E is countable. It follows that the measure groupoid constructed from the equivalence relation s ～ sg on E may be simply described in terms of the measure groupoid made from the action of some countable group. Some simplifications are made in Mackey's theory of measure groupoids. A natural notion of “approximate finiteness” (AF) is introduced for nonsingular actions of G, and results are developed parallel to those for countable groups; several classes of examples arising naturally are shown to be AF. Results on “skew product” group actions are obtained, generalizing the countable case, and partially answering a question of Mackey. We also show that a group-measure space factor obtained from a continuous group action is isomorphic (as a von Neumann algebra) to one obtained from a discrete group action.     

On various averaging methods for a nonlinear oscillator with slow time-dependent potential and a nonconservative perturbation

The main aim of the paper is to compare various averaging methods for constructing asymptotic solutions of the Cauchy problem for the one-dimensional anharmonic oscillator with potential V (x, τ) depending on the slow time τ = ?t and with a small nonconservative term ?g( $ \dot x $ , x, τ), ? ? 1. This problem was discussed in numerous papers, and in some sense the present paper looks like a “methodological” one. Nevertheless, it seems that we present the definitive result in a form useful for many nonlinear problems as well. Namely, it is well known that the leading term of the asymptotic solution can be represented in the form $ X\left( {\frac{{S\left( \tau \right) + \varepsilon \varphi \left( \tau \right)}} {\varepsilon },I\left( \tau \right),\tau } \right) $ , where the phase S, the “slow” parameter I, and the so-called phase shift ? are found from the system of “averaged” equations. The pragmatic result is that one can take into account the phase shift ? by considering it as a part of S and by simultaneously changing the initial data for the equation for I in an appropriate way.     

Confirmation,Increase in Probability,and the Likelihood Ratio Measure: a Reply to Glass and McCartney

Bayesian confirmation theory is rife with confirmation measures. Zalabardo (2009) focuses on the probability difference measure, the probability ratio measure, the likelihood difference measure, and the likelihood ratio measure. He argues that the likelihood ratio measure is adequate, but each of the other three measures is not. He argues for this by setting out three adequacy conditions on confirmation measures and arguing in effect that all of them are met by the likelihood ratio measure but not by any of the other three measures. Glass and McCartney (2015), hereafter “G&M,” accept the conclusion of Zalabardo’s argument along with each of the premises in it. They nonetheless try to improve on Zalabardo’s argument by replacing his third adequacy condition with a weaker condition. They do this because of a worry to the effect that Zalabardo’s third adequacy condition runs counter to the idea behind his first adequacy condition. G&M have in mind confirmation in the sense of increase in probability: the degree to which E confirms H is a matter of the degree to which E increases H’s probability. I call this sense of confirmation “IP.” I set out four ways of precisifying IP. I call them “IP1,” “IP2,” “IP3,” and “IP4.” Each of them is based on the assumption that the degree to which E increases H’s probability is a matter of the distance between p(H | E) and a certain other probability involving H. I then evaluate G&M’s argument (with a minor fix) in light of them.     

Precise asymptotics – A general approach

The legendary 1947-paper by Hsu and Robbins, in which the authors introduced the concept of “complete convergence”, generated a series of papers culminating in the like-wise famous Baum–Katz 1965-theorem, which provided necessary and sufficient conditions for the convergence of the series $\sum_{n=1}^{\infty}n^{r/p-2}P (|S_{n}| \geqq \varepsilon n^{1/p})$ for suitable values of r and p, in which S _n denotes the n-th partial sum of an i.i.d. sequence. Heyde followed up the topic in his 1975-paper where he investigated the rate at which such sums tend to infinity as ε↘0 (for the case r=2 and p=1). The remaining cases have been taken care later under the heading “precise asymptotics”. An abundance of papers have since then appeared with various extensions and modifications of the i.i.d.-setting. The aim of the present paper is to show that the basis for the proof is essentially the same throughout, and to collect a number of examples. We close by mentioning that Klesov, in 1994, initiated work on rates in the sense that he determined the rate, as ε↘0, at which the discrepancy between such sums and their “Baum–Katz limit” converges to a nontrivial quantity for Heyde’s theorem. His result has recently been extended to the complete set of r- and p-values by the present authors.     

A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems

We make a contribution to the theory of embeddings of anisotropic Sobolev spaces into L ^p -spaces (Sobolev case) and spaces of Hölder continuous functions (Morrey case). In the case of bounded domains the generalized embedding theorems published so far pose quite restrictive conditions on the domain’s geometry (in fact, the domain must be “almost rectangular”). Motivated by the study of some evolutionary PDEs, we introduce the so-called “semirectangular setting”, where the geometry of the domain is compatible with the vector of integrability exponents of the various partial derivatives, and show that the validity of the embedding theorems can be extended to this case. Second, we discuss the a priori integrability requirement of the Sobolev anisotropic embedding theorem and show that under a purely algebraic condition on the vector of exponents, this requirement can be weakened. Lastly, we present a counterexample showing that for domains with general shapes the embeddings indeed do not hold.     

Can a Single Property Be Both Dispositional and Categorical? The “Partial Consideration Strategy”, Partially Considered

One controversial position in the debate over dispositional and categorical properties maintains that our concepts of these properties are the result of partially considering unitary properties that are both dispositional and categorical. As one of its defenders (Heil 2005, p. 351) admits, this position is typically met with “incredulous stares”. In this paper, I examine whether such a reaction is warranted. This thesis about properties is an instance of what I call “the Partial Consideration Strategy”—i.e., the strategy of claiming that what were formerly thought of as distinct entities are actually a unified entity, partially considered. By evaluating its use in other debates, I uncover a multi-layered prima facie case against the use of the Partial Consideration Strategy in the dispositional/categorical properties debate. In closing, I describe how the Partial Consideration Strategy can be reworked in a way that would allow it to sidestep this prima facie case.     

Projective modules of finite type over the supersphere S2,2

In the spirit of noncommutative geometry we construct all inequivalent superline bundles over the (2,2)-dimensional supersphere S^2,2 by means of global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding “rank 1” supervector bundle over S^2,2. The canonical connection ∇=p∘d is used to compute the Chern numbers by means of the Berezin integral on S^2,2. The associated connection 1-forms are graded extensions of monopoles with not trivial topological charge. Supertransposed projectors gives opposite values for the charges. We also comment on the K-theory of S^2,2.     

The failure of the Hardy inequality and interpolation of intersections

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=?1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L _p(w₀), N∩L_p(w₁)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L _p(w₀),N∩L _p (w₁)), which turns out to be essentially different from theK-functional for (L _p(w₀), L_p(w₁)), even for the case whenN∩L _p(w_i) is dense inL _p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.     

Boundary values of holomorphic semigroups of unbounded operators and similarity of certain perturbations

We obtain sufficient conditions for a “holomorphic” semigroup of unbounded operators to possess a boundary group of bounded operators. The theorem is applied to generalize to unbounded operators results of Kantorovitz about the similarity of certain perturbations. Our theory includes a result of Fisher on the Riemann-Liouville semigroup in L^p(0, ∞) 1 < p < ∞. In this particular case we give also an alternative approach, where the boundary group is obtained as the limit of groups in the weak operator topology.     

Computability-theoretic and proof-theoretic aspects of partial and linear orderings

Szpilrajn’s Theorem states that any partial orderP=〈S,<p〉 has a linear extensionP=〈S,<L〉. This is a central result in the theory of partial orderings, allowing one to define, for instance, the dimension of a partial ordering. It is now natural to ask questions like “Does a well-partial ordering always have a well-ordered linear extension?” Variations of Szpilrajn’s Theorem state, for various (but not for all) linear order typesτ, that ifP does not contain a subchain of order typeτ, then we can chooseL so thatL also does not contain a subchain of order typeτ. In particular, a well-partial ordering always has a well-ordered extension.We show that several effective versions of variations of Szpilrajn’s Theorem fail, and use this to narrow down their proof-theoretic strength in the spirit of reverse mathematics.     

A High Token Indicativity Account of Knowledge

In this paper, I provide a probabilistic account of factual knowledge, based on the notion of chance, which is a function of an event (or a fact — I will use ‘fact’ to cover both) given a prior history. This account has some affinity with my chance account of token causation, but it neither relies on it nor presupposes it. Here, I concentrate on the core cases of perceptual knowledge and of knowledge by memory (based on perception). (The account can be extended to the other modes of knowledge, but not in this paper.) The analysis of knowledge presented below is externalist. The underlying intuition guiding the treatment of knowledge in this paper is that knowledge boils down to high token discriminative indicativeness. Type indicativeness or type discriminability are neither necessary nor sufficient for knowledge: the token aspect comes out in the strong dependence on the specific circumstances and chances of the case. The main condition of the first section, the indicativity condition (KI), properly refined, yields pertinent (token) indicativity as a main constituent. Very roughly, it involves the chance of the content clause p being higher given the subject's believing that p than otherwise. The discriminability condition in question (section 3) captures the sense of discriminability appropriate for knowledge and yield the indicativity condition: it is an extension of the indicativity condition KI. Roughly, the subject’s ability to discriminate the object in front of her being red from its being green is captured by holding fixed, in the indicativity condition, the condition “the object in front of her is red or green.” A major element in the analysis is the so-called Contrast Class, which governs the scope of discriminability. This is the class of features that have to be taken into account in the discriminability condition, and it is characterized by two constraints. Very roughly, according to the first constraint, for a feature to be in the contrast class, it must not represent a sub-type of (the feature specified by) the predicate in the content clause. According to the second constraint, which is a central condition with many implications, the chance that the object specified in the content clause has a feature represented in the contrast class must not under the circumstances be too low. This constraint, within the framework of the discriminability condition, brings out a major constitutive aspect of knowledge: knowledge amounts to a limited vulnerability to mistakes of the belief in question under the circumstances at hand. The contrast class plays a major role in my treatment of skepticism. The second constraint on the Contrast Class together with the VHP condition below bring out precisely the way in which perceptual knowledge is fallible.     

Embeddings of covering projections of graphs

It is shown that any given k-fold covering projection of graphs p: G₁ → G₂ can be embedded in a k-fold covering projection of closed orientable surfaces π: S₁ → S₂ in the sense that there are embeddings of G₁ in S₁ and G₂ in S₂ such that p is the restriction of π. In the case of a regular covering projection p, which is the quotient map with respect to some group action on G₁, it is shown that there is a regular covering projection π of surfaces in which p can be embedded.     

Connecting theories in mathematics education: challenges and possibilities

This paper is a commentary on the problem of networking theories. My commentary draws on the papers contained in this ZDM issue and is divided into three parts. In the first part, following semiotician Yuri Lotman, I suggest that a network of theories can be conceived of as a semiosphere, i.e., a space of encounter of various languages and intellectual traditions. I argue that such a networking space revolves around two different and complementary “themes”—integration and differentiation. In the second part, I advocate conceptualizing theories in mathematics education as triplets formed by a system of theoretical principles, a methodology, and templates of research questions, and attempt to show that this tripartite view of theories provides us with a morphology of theories for investigating differences and potential connections. In the third part of the article, I discuss some examples of networking theories. The investigation of limits of connectivity leads me to talk about the boundary of a theory, which I suggest defining as the “limit” of what a theory can legitimately predicate about its objects of discourse; beyond such an edge, the theory conflicts with its own principles. I conclude with some implications of networking theories for the advancement of mathematics education.