The Theory of Classification, Part 2: The Scratch-Built Typechecker.

We develop rules for coalgebras in type theory, and give meaning explanations for them. We show that elements of coalgebras are determined by their elimination rules, whereas the introduction rules can be considered as derived. This is in contrast with algebraic data types, for which the opposite is true: elements are determined by their introduction rules, and the elimination rules can be considered as derived. In this sense, the function type from the logical framework is more like a coalgebraic data type, the elements of which are determined by the elimination rule. We illustrate why the simplest form of guarded recursion is nothing but the introduction rule originating from the formulation of coalgebras in category theory. We discuss restrictions needed in order to preserve decidability of equality.

This paper formulates a bilateral account of harmony that is an alternative to one proposed by Francez. It builds on an account of harmony for unilateral logic proposed by K\"urbis and the observation that reading the rules for the connectives of bilateral logic bottom up gives the grounds and consequences of formulas with the opposite speech act. I formulate a process I call 'inversion' which allows the determination of assertive elimination rules from assertive introduction rules, and rejective elimination rules from rejective introduction rules, and conversely. It corresponds to Francez's notion of vertical harmony. I also formulate a process I call 'conversion', which allows the determination of rejective introduction rules from assertive elimination rules and conversely, and the determination of assertive introduction rules from rejective elimination rules and conversely. It corresponds to Francez's notion of horizontal harmony. The account has a number of features that distinguishes it from Francez's.

The idea of an ‘inversion principle’, and the name itself, originated in the work of Paul Lorenzen in the 1950s, as a method to generate new admissible rules within a certain syntactic context. Some fifteen years later, the idea was taken up by Dag Prawitz to devise a strategy of normalization for natural deduction calculi (this being an analogue of Gentzen's cut-elimination theorem for sequent calculi). Later, Prawitz used the inversion principle again, attributing it with a semantic role. Still working in natural deduction calculi, he formulated a general type of schematic introduction rules to be matched – thanks to the idea supporting the inversion principle – by a corresponding general schematic Elimination rule. This was an attempt to provide a solution to the problem suggested by the often quoted note of Gentzen. According to Gentzen ‘it should be possible to display the elimination rules as unique functions of the corresponding introduction rules on the basis of certain requirements’. Many people have since worked on this topic, which can be appropriately seen as the birthplace of what are now referred to as “general elimination rules”, recently studied thoroughly by Sara Negri and Jan von Plato. In this study, we retrace the main threads of this chapter of proof-theoretical investigation, using Lorenzen's original framework as a general guide.

The basic idea behind that account is that we need to distinguish straightforward inconsistency, or self-contradiction, of a set of assumptions, from paradoxicality. Both involve proofs of absurdity (1). But the proofs of absurdity in connection with straightforward contradictions are normalizable, whereas those in connection with paradoxes are not. Proofs are normalizable when they can be brought into normal form by a finite sequence of applications of reduction procedures. These reduction procedures are designed to get rid of unnecessary prolixity. Such prolixity can arise, most importantly, by applying an introduction rule for a logical operator and then immediately applying the corresponding elimination rule. The result is a sentence occurrence within the proof standing as the conclusion of an application of the introduction rule and as the major premiss of an application of the corresponding elimination rule. Reductions get rid of such 'maximal' sentence occurrences, which stand as unwanted 'knuckles' in the proof. The reduction procedures for the logical operators are designed to eliminate such unnecessary detours within proofs.1

We explicate the different ways that a first-order sentence can be true (resp., false) in a model M, as formal objects, called (M-relative) truth-makers (resp., falsity-makers). M-relative truth-makers and falsity-makers are co-inductively definable, by appeal to the “atomic facts” in M, and to certain rules of verification and of falsification, collectively called rules of evaluation. Each logical operator has a rule of verification, much like an introduction rule; and a rule of falsification, much like an elimination rule. Applications of the rules (∀) and (∃) involve infinite furcation when the domain of M is infinite. But even in the infinite case, truth-makers and falsity-makers are tree-like objects whose branches are at most finitely long. A sentence φ is true (resp., false) in a model M (in the sense of Tarski) if and only if there existsπ such that π is an M-relative truth-maker (resp., falsity-maker) for φ. With “ways of being true” explicated as these logical truthmakers, one can re-conceive logical consequence between given premises and a conclusion. It obtains just in case there is a suitable method for transforming M-relative truthmakers for the premises into an M-relative truthmaker for the conclusion, whatever the model M may be.

In the Trinity Term of 1977 I had the privilege of hearing Michael Dummett lecture in the Old Library at All Souls on the justification and criticism of logical laws, and subsequently I also assisted Mark Helme in the production of some unauthorised, not altogether successful, notes on these lectures. I can vividly recall my surprise when Prof. Dummett stated, in connection with the idea with which he had up till then been mainly concerned, namely the idea that meaning is primarily determined in terms of the Gentzen introduction rules, that my old tutor, the Swedish logician Per Martin-Lof, in correspondence, held that “the meanings of the logical constants CAN IN NO WAY be given by the introduction rules, rather they are given by the elimination rules”.1 This was, as far as I knew from tutorials and conversations with Martin-Lof, not a position he would defend at the time and certainly not today. In fact, he has steadfastly held by the introduction-rule orthodoxy in all relevant publications from 1979 onwards.2 Taken jointly those publications constitute, as far as I know, the only sustained effort towards a realisation of a constructivist theory of meaning for a sizeable interpreted language serving the needs of pure mathematics on a scale comparable to that of Frege’s Grundgesetze. From this circumstance alone it seems clear that much of Dummettian interest can be found in the works of Martin-Lof.

A general formulation of inductive and recursive definitions in Martin-Löf's type theory is presented. It extends Backhouse's ‘Do-It-Yourself Type Theory’ to include inductive definitions of families of sets and definitions of functions by recursion on the way elements of such sets are generated. The formulation is in natural deduction and is intended to be a natural generalisation to type theory of Martin-Löf's theory of iterated inductive definitions in predicate logic.Formal criteria are given for correct formation and introduction rules of a new set former capturing definition by strictly positive, iterated, generalised induction. Moreover, there is an inversion principle for deriving elimination and equality rules from the formation and introduction rules. Finally, there is an alternative schematic presentation of definition by recursion.The resulting theory is a flexible and powerful language for programming and constructive mathematics. We hint at the wealth of possible applications by showing several basic examples: predicate logic, generalised induction, and a formalisation of the untyped lambda calculus.

The battle against cancer has intensified in the last decade. New experimental techniques and theoretical models have been been proposed to understand the behavior, growth, and evolution of different types of brain tumors. Unfortunately, for glioblastoma multiforme (GBM), except for methylation of the O6-methylguanine-DNA methyltransferase (MGMT) promoter that has some benefit in the local control of tumors using alkylating agents such as temozolomide, to date personalized treatments do not exist. In this article, we present a comprehensive review of different aspects intertwined in the mathematical growth modeling applied to high-grade gliomas. We briefly cover the following fundamental aspects related to the conventional imaging in GBM: defining the tumor regions in GBM, segmentation of the tumor regions using magnetic resonance imaging (MRI) of the brain, response assessment using the neuro-oncology response criteria versus the Macdonald criteria, availability of software for the segmentation of MRI of the brain, mathematical modeling applied to tumor growth, principles of mathematical modeling, factors involved in tumor growth models, mathematical modeling based on imaging data, most common equations used in high-grade glioma growth modeling, integration of mathematical growth models in computer simulators, tumor growth modeling as a part of brain's complex system, and challenges in mathematical growth modeling. We conclude by saying that it is the combination of biomedical imaging and mathematical modeling that allows the assembling of clinically relevant models of tumor growth and treatment response; the most appropriate model will depend on the premise and findings of each experiment.

A global review for the hydrodynamic response investigation method of submerged floating tunnels

Copper (II) oxide nanoparticles as additives in RBD palm olein: Experimental analysis and mathematical modelling

Phagocytosis is a fascinating process whereby a cell surrounds and engulfs particles such as bacteria and dead cells. This is crucial both for single-cell organisms (as a way of acquiring nutrients) and as part of the immune system (to destroy foreign invaders). This whole process is hugely complex and involves multiple coordinated events such as membrane remodelling, receptor motion, cytoskeleton reorganisation and intracellular signalling. Because of this, phagocytosis is an excellent system for theoretical study, benefiting from biophysical approaches combined with mathematical modelling. Here, we review these theoretical approaches and discuss the recent mathematical and computational models, including models based on receptors, models focusing on the forces involved, and models employing energetic considerations. Along the way, we highlight a beautiful connection to the physics of phase transitions, consider the role of stochasticity, and examine links between phagocytosis and other types of endocytosis. We cover the recently discovered multistage nature of phagocytosis, showing that the size of the phagocytic cup grows in distinct stages, with an initial slow stage followed by a much quicker second stage starting around half engulfment. We also address the issue of target shape dependence, which is relevant to both pathogen infection and drug delivery, covering both one-dimensional and two-dimensional results. Throughout, we pay particular attention to recent experimental techniques that continue to inform the theoretical studies and provide a means to test model predictions. Finally, we discuss population models, connections to other biological processes, and how physics and modelling will continue to play a key role in future work in this area.

This chapter introduces a general overview of four major concepts and models in quantum theory: quantum mechanics, quantum field theory, Feynman’s path integral, and quantum anharmonic oscillators. These four concepts with models in quantum theory form critical mass to establish the basic theoretical and mathematical models in quantum finance theory in next chapter, namely, Feynman’s path integral model of quantum finance (also known as first generation of quantum finance) and quantum anharmonic model of quantum finance (also known as second generation of quantum finance). To avoid complex mathematical derivations, this chapter focuses on the basic concepts of quantum theory with its related models that are crucial to understand the mathematical and quantum model in quantum finance.

A mathematical model was developed to compute the motions of semisubmersible drilling vessels in waves for a wide variety of semisubmersible configurations. The model was derived from a linear representation of motions, ocean waves, and forces. The semisubmersible is represented as a rigid space frame composed of a number of cylindrical members with arbitrary diameters, lengths and orientations. Forces on the semisubmersible are derived from anchorline properties, and hydrostatic hydrodynamic principles. A solution is obtained for motions in six degrees of freedom for a sinusoidal wave train of arbitrary height, period, direction and water depth. Results from the analysis of three semisubmersibles are compared with results from available model test data to verily the mathematical model. Introduction An accurate and complete representation of the response of a drilling vessel to waves is a valuable engineering tool for predicting vessel performance and designing drilling equipment. The performance and designing drilling equipment. The wave response for a floating vessel may be obtained to various degrees of accuracy from model tests or analytical means, as described by Barkley and Korvin-Kroukovsky and as applied by Bain. A review of the works cited shows that the evaluation of the wave response for a particular vessel requires considerable time and effort, either in model construction and testing or in computer programming and calculations. In order to reduce programming and calculations. In order to reduce the amount of time and effort required to evaluate a particular vessel, means were investigated to generalize and automate, on a digital computer, methods for evaluating wave response for vessels of arbitrary configuration. The mathematical model described in this paper is the result of such an investigation for semisubmersible-type drilling vessels. The paper presents a general description of the mathematical model and the basic principles and assumptions from which it was derived. The validity of the model is evaluated by comparing results of the analysis of three semisubmersibles with available model test data. MATHEMATICAL MODEL The mathematical model for calculating the motions of a semisubmersible in waves is derived from basic principles and empirical relationships in classical mechanics. All equations are derived for "small amplitude" waves and motions. The nonlinear equations that appear in the problem are replaced by "equivalent" linear equations in order to conform to the linear analysis method used in obtaining a solution. The model is implemented in a computer program that computes vessel response in all six degrees of freedom for a broad range of semisubmersible configurations and wave parameters. The basic elements in the theoretical model are outlined, with a more detailed discussion of the principles and derivations used to obtain the model principles and derivations used to obtain the model presented in the Appendix. presented in the Appendix. SEMISUBMERSIBLE DESCRIPTION AND EQUATIONS OF MOTION The semisubmersible is characterized as a space-frame of cylindrical members and is described geometrically by specifying end-coordinates and diameters for all of the members. Specification of the mass, moments of inertia, center of gravity and floating position are required to complete the description. The six equations of motion for the semisubmersible derive from Newton's second law for a rigid body. These differential equations, when written in matrix form, equate the product of the six-component acceleration vector, {x}, and the inertia matrix, I, to a six-component, force-moment vector, {FT}. SPEJ P. 311

Radicalization and terrorism are global events affecting different countries and present in diverse historical contexts. A historical account indicates the development of terrorism in human history to shed some light on radicalization. There are elements of radicalization in the dark history of humanity that still today bear consequences on their victims and their progenies. However, at first analysis, terrorist attacks occur unexpectedly. We used a theoretical approach to mathematical and statistical instruments to make a forecast from available data. Delphi groups of psychiatrists, experts in the study of radicalization, met regularly to provide joint predictions about topics of concern presented as case scenarios. The data from peergroup discussions were analyzed for drawing consensus and forecasting radicalization and terrorism. Mathematical prediction models included probability estimates, Kappa statistics, Boolean logics, fractal mathematics, and geometrical progressions. The study supports the hypothesis that with sufficient data analysis and mathematical models, radicalization and terrorism could be prevented using forecasting methods and prompt prevention.

Resident Perspectives 56-5