1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem "P ? NP?".- 8 Newton's Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bezout's Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

Complexity and Real Computation

Checkpoint inhibitor-based immunotherapies that target cytotoxic T lymphocyte antigen 4 (CTLA4) or the programmed cell death 1 (PD1) pathway have achieved impressive success in the treatment of different cancer types. Yet, only a subset of patients derive clinical benefit. It is thus critical to understand the determinants driving response, resistance and adverse effects. In this Review, we discuss recent work demonstrating that immune checkpoint inhibitor efficacy is affected by a combination of factors involving tumour genomics, host germline genetics, PD1 ligand 1 (PDL1) levels and other features of the tumour microenvironment, as well as the gut microbiome. We focus on recently identified molecular and cellular determinants of response. A better understanding of how these variables cooperate to affect tumour-host interactions is needed to optimize the implementation of precision immunotherapy.

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The evolving landscape of biomarkers for checkpoint inhibitor immunotherapy

V. I. Arnold, on behalf of the International Mathematical Union has written to a number of mathematicians with a suggestion that they describe some great problems for the next century. This report is my response. Arnold's invitation is inspired in part by Hilbert's list of 1900 (see e.g. [Browder, 1976]) and I have used that list to help design this essay. I have listed 18 problems, chosen with these criteria:

Mathematical problems for the next century

Advances in mathematics

The emergence of electronic computers in the last thirty years has given rise to many interesting questions. Many of these questions are technical, relating to a machine’s ability to perform complex operations in a variety of circumstances. While some of these questions are not without philosophical interest, the one question which above all others has stimulated philosophical interest is explicitly non-technical and it can be expressed crudely as follows: Can a machine be said to think and, if so, in what sense? The issue has received much attention in the scholarly journals with articles and arguments appearing in great profusion, some resolutely answering this question in the affirmative, some, equally resolutely, answering this question in the negative, and others manifesting modified rapture. While the ramifications of the question are enormous I believe that the issue at the heart of the matter has gradually emerged from the forest of complications. It is easy to answer the question “Can machines/computers think?” Easy, that is, once we know with some degree of precision what thinking is, what machine/computers are, and, last but not least, how to understand the word “can.” The possibility of answering the question, “Can a computer think?” is rendered either question-beggingly trivial or mind-bogglingly impossible unless one is able to give some independent signification to its key terms. To define a computer as a sort of machine, and then to define machine in such a narrow way that it becomes meaningless to ask if it thinks viciously begs the question unless the definitions can be given independent justification. Of course, a similar consideration applies to arguments in favour of artificial intelligence. I assume that the majority of those who would put this question intend by “think” something like

https://researchrepository.ucd.ie/bitstream/10197/5469/1/Minds_and_Machines.pdf

Minds and machines

We describe Richardson's functor from the Diophantine equations and Diophantine problems into elementary real-valued functions and problems. We then derive a general undecidability and incompleteness result for elementary functions within ZFC set theory, and apply it to some problems in Hamiltonian mechanics and dynamical systems theory. Our examples deal with the algorithmic impossibility of deciding whether a given Hamiltonian can be integrated by quadratures and related questions; they lead to a version of Godel's incompleteness theorem within Hamiltonian mechanics. A similar application to the unsolvability of the decision problem for chaotic dynamical systems is also obtained.

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Undecidability and incompleteness in classical mechanics

We expose and discussPenrose's thesis: “Nature produces harnessable noncomputable processes, but none at the classical level.” We then suggest a partial counterexample to it, based on aGedanken experiment about an undecidable family of integrable Hamiltonian systems that could lead to a sort of idealized solution to the Halting problem for Turing machines.

Classical physics and Penrose's thesis

We expose the main ideas, concepts and results about Jaśkowski's discussive logic, and apply that logic to the concept of pragmatic truth and to the Dalla Chiara-di Francia view of the foundations of physics.

On Jaśkowski's Discussive Logics

We first state a few previously obtained results that lead to general undecidability and incompleteness theorems in axiomatized theories that range from the theory of finite sets to classical elementary analysis. Out of those results we prove several incompleteness theorems for axiomatic versions of the theory of noncooperative games with Nash equilibria; in particular, we show the existence of finite games whose equilibria cannot be proven to be computable.

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Francisco Antonio Doria

Papers

Undecidability and incompleteness in classical mechanics

Classical physics and Penrose's thesis

On Jaśkowski's Discussive Logics

The Incompleteness of Theories of Games

A Weyl-like equation for the gravitational field