A criterion for membership in archimedean semirings

We prove an extension of the classical Real Representation Theorem (going back to Krivine, Stone, Kadison, Dubois and Becker and often called Kadison-Dubois Theorem). It is a criterion for membership in subsemirings (sometimes called preprimes) of a commutative ring. Whereas the classical criterion is only applicable for functions which are positive on the representation space, the new criterion can under certain arithmetic conditions be applied also to functions which are only nonnegative. Only in the case of preorders (i.e., semirings containing all squares), our result follows easily from recent work of Scheiderer, Kuhlmann, Marshall and Schwartz. Our proof does not use (and therefore shows) the classical criterion. We illustrate the usefulness of the new criterion by deriving a theorem of Handelman from it saying inter alia the following: If an odd power of a real polynomial in several variables has only nonnegative coefficients, then so do all sufficiently high powers.Comment: 23 pages. See also: http://www.mathe.uni-konstanz.de/homepages/schweigh

Saddlepoint approximation to functional equations in queueing theory and insurance mathematics

2010 Fall.Includes bibliographical references.We study the application of saddlepoint approximations to statistical inference when the moment generating function (MGF) of the distribution of interest is an explicit or an implicit function of the MGF of another random variable which is assumed to be observed. In other words, let W (s) be the MGF of the random variable W of interest. We study the case when W (s) = h{G (s) ; λ}, where G (s) is an MGF of G for which a random sample can be obtained, and h is a smooth function. If Ĝ (s) estimates G (s), then Ŵ (s) = h{Ĝ (s) ; λ̂} estimates W (s). Generally, it can be shown that Ŵ (s) converges to W (s) by the strong law of large numbers, which implies that F̂ (t), the cumulative distribution function (CDF) corresponding to Ŵ (s), converges to F (t), the CDF of W, almost surely. If we set Ŵ* (s) = h{Ĝ* (s) ; λ̂}, where Ĝ* (s) and λ̂* are the empirical MGF and the estimator of λ from bootstrapping, the corresponding CDF F̂* (t) can be used to construct the confidence band of F(t). In this dissertation, we show that the saddlepoint inversion of Ŵ (s) is not only fast, reliable, stable, and accurate enough for a general statistical inference, but also easy to use without deep knowledge of the probability theory regarding the stochastic process of interest. For the first part, we consider nonparametric estimation of the density and the CDF of the stationary waiting times W and Wq of an M/G/1 queue. These estimates are computed using saddlepoint inversion of Ŵ (s) determined from the Pollaczek-Khinchin formula. Our saddlepoint estimation is compared with estimators based on other approximations, including the Cramér-Lundberg approximation. For the second part, we consider the saddlepoint approximation for the busy period distribution FB (t) in a M/G/1 queue. The busy period B is the first passage time for the queueing system to pass from an initial arrival (1 in the system) to 0 in the system. If B (s) is the MGF of B, then B (s) is an implicitly defined function of G (s) and λ, the inter-arrival rate, through the well-known Kendall-Takács functional equation. As in the first part, we show that the saddlepoint approximation can be used to obtain F̂B (t), the CDF corresponding to B̂(s) and simulation results show that confidence bands of FB (t) based on bootstrapping perform well

Teachers' and fourth graders' questions during a problem-solving lesson

Peer reviewe

A Tool for Producing Verified, Explainable Proofs

Mathematicians are reluctant to use interactive theorem provers. In this thesis I argue that this is because proof assistants don't emphasise explanations of proofs; and that in order to produce good explanations, the system must create proofs in a manner that mimics how humans would create proofs. My research goals are to determine what constitutes a human-like proof and to represent human-like reasoning within an interactive theorem prover to create formalised, understandable proofs. Another goal is to produce a framework to visualise the goal states of this system. To demonstrate this, I present HumanProof: a piece of software built for the Lean 3 theorem prover. It is used for interactively creating proofs that resemble how human mathematicians reason. The system provides a visual, hierarchical representation of the goal and a system for suggesting available inference rules. The system produces output in the form of both natural language and formal proof terms which are checked by Lean's kernel. This is made possible with the use of a structured goal state system which interfaces with Lean's tactic system which is detailed in Chapter 3. In Chapter 4, I present the subtasks automation planning subsystem, which is used to produce equality proofs in a human-like fashion. The basic strategy of the subtasks system is break a given equality problem in to a hierarchy of tasks and then maintain a stack of these tasks in order to determine the order in which to apply equational rewriting moves. This process produces equality chains for simple problems without having to resort to brute force or specialised procedures such as normalisation. This makes proofs more human-like by breaking the problem into a hierarchical set of tasks in the same way that a human would. To produce the interface for this software, I also created the ProofWidgets system for Lean 3. This system is detailed in Chapter 5. The ProofWidgets system uses Lean's metaprogramming framework to allow users to write their own interactive, web-based user interfaces to display within the VSCode editor and in an online web-editor. The entire tactic state is available to the rendering engine, and hence expression structure and types of subexpressions can be explored interactively. The ProofWidgets system also allows the user interface to interactively edit the proof document, enabling a truly interactive modality for creating proofs; human-like or not. In Chapter 6, the system is evaluated by asking real mathematicians about the output of the system, and what it means for a proof to be understandable to them. The user group study asks participants to rank and comment on proofs created by HumanProof alongside natural language and pure Lean proofs. The study finds that participants generally prefer the HumanProof format over the Lean format. The verbal responses collected during the study indicate that providing intuition and signposting are the most important properties of a proof that aid understanding.EPSR

Glossary of statistical terms: English-Lithuanian-French-Russian-German

This edition is devoted to standardization of Lithuanian statistical terms adequate to those used, first of all, in English, also in French, Russian and German. It amounts of 8 dictionaries: multilingual table of statistical terms and 7 bilingual dictionaries with Lithuanian language. The dictionaries were based on the corrected multilingual vocabularies of the International Statistical Institute on the web (http:/isi.cbs.nl/glossary/ index.htm).Leidinys skirtas lietuviškų statistikos terminų norminimui pagal atitikmenis anglų, taip pat prancūzų, rusų ir vokiečių kalbomis. Jį sudaro 8 žodynai: daugiakalbė statistikos terminų lentelė ir 7 dvikalbiai lietuvių ir nurodytų svetimų kalbų žodynai. Šie žodynai remiasi pakoreguotu daugiakalbiu Tarptautinio statistikos instituto terminų sąvadu internete (http:/isi.cbs.nl/glossary/index.htm)

The micro-culture of a mathematics classroom : Artefacts and Activity in Meaning making and Problem solving

Doktorgradsavhandlin

The relevance of Pólya's random-walk problem for the single-species reaction-diffusion system

The diffusion-limited reactions $A+A\to A$ and $A+A\to 0$ in dimension d > 2 are reconsidered from the point of view of the random-walk theory. It is pointed out that Pólya's theorem on the returning probability of a random walker to the origin, which would imply a probability less than one for the meeting of two typical particles, would predict the possibility of a state in which the reaction seems to have spontaneously ceased, in contradiction with the very well known asymptotic $N(t) \sim t^{-1}$ for the particles population of these reactions. In fact, a given condition is presented, in which the relative particle number N(t)/N(0) decays to a non-vanishing constant. The condition is that the initial distribution of particles in the d-dimensional space has a dimension γ, such that $0 < \gamma <d-2$