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In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study of certain principles of self-referential truth. [ABSTRACT FROM AUTHOR]

Time spent in national pandemic 'lockdowns' and 'tiers' for most of 2020 has created an opportunity to revisit some ideas previously committed to paper, but unfinished. I now return to one of them, to continue and share the line of thought, if not to complete it. [ABSTRACT FROM AUTHOR]

In [17], we introduced an extensional variant of generic realizability [22], where realizers act extensionally on realizers, and showed that this form of realizability provides inner models of $\mathsf {CZF}$ (constructive Zermelo–Fraenkel set theory) and $\mathsf {IZF}$ (intuitionistic Zermelo–Fraenkel set theory), that further validate $\mathsf {AC}_{\mathsf {FT}}$ (the axiom of choice in all finite types). In this paper, we show that extensional generic realizability validates several choice principles for dependent types , all exceeding $\mathsf {AC}_{\mathsf {FT}}$. We then show that adding such choice principles does not change the arithmetic part of either $\mathsf {CZF}$ or $\mathsf {IZF}$. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified Kudla–Rapoport conjecture for the Krämer model of unitary Rapoport–Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla–Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies the arithmetic Siegel–Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice. [ABSTRACT FROM AUTHOR]

We investigate sections of the arithmetic fundamental group $\pi _1(X)$ where X is either a smooth affinoid p-adic curve , or a formal germ of a p-adic curve , and prove that they can be lifted (unconditionally) to sections of cuspidally abelian Galois groups. As a consequence, if X admits a compactification Y , and the exact sequence of $\pi _1(X)$ splits , then $\text {index} (Y)=1$. We also exhibit a necessary and sufficient condition for a section of $\pi _1(X)$ to arise from a rational point of Y. One of the key ingredients in our investigation is the fact, we prove in this paper in case X is affinoid, that the Picard group of X is finite. [ABSTRACT FROM AUTHOR]

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$ , the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli. [ABSTRACT FROM AUTHOR]

The aim of this article is to provoke discussion concerning arithmetic properties of the function $p_{d}(n)$ counting partitions of a positive integer n into dth powers, where $d\geq 2$. Apart from results concerning the asymptotic behaviour of $p_{d}(n)$ , little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into dth powers. The second part of the paper is experimental and contains questions and conjectures concerning the arithmetic behaviour of the sequence $(p_{d}(n))_{n\in \mathbb {N}}$ , based on computations of $p_{d}(n)$ for $n\leq 10^5$ for $d=2$ and $n\leq 10^{6}$ for $d=3, 4, 5$. [ABSTRACT FROM AUTHOR]

Algebraic effects and handlers are a convenient method for structuring monadic effects with primitive effectful operations and separating the syntax from the interpretation of these operations. However, the scope of conventional handlers is limited as not all side effects are monadic in nature. This paper generalizes the notion of algebraic effects and handlers from monads to generalized monoids, which notably covers applicative functors and arrows as well as monads. For this purpose, we switch the category theoretical basis from free algebras to free monoids. In addition, we show how lax monoidal functors enable the reuse of handlers and programs across different computation classes, for example, handling applicative computations with monadic handlers. We motivate and present these handler interfaces in the context of build systems. Tasks in a build system are represented by a free computation and their interpretation as a handler. This use case is based on the work of Mokhov et al. [(2018). PACMPL2(ICFP), 79:1–79:29.]. [ABSTRACT FROM AUTHOR]

In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for a much weaker arithmetic theory, a version of Robinson's R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic. [ABSTRACT FROM AUTHOR]

Gireesh and Mahadeva Naika ['On 3-regular partitions in 3-colors', Indian J. Pure Appl. Math.50 (2019), 137–148] proved an infinite family of congruences modulo powers of 3 for the function $p_{\{3,3\}}(n)$ , the number of 3-regular partitions in three colours. In this paper, using elementary generating function manipulations and classical techniques, we significantly extend the list of proven arithmetic properties satisfied by $p_{\{3,3\}}(n).$ [ABSTRACT FROM AUTHOR]

In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations $\Omega ^{n}(\bar {\xi })$ defined with the help of numeral exponents. Since $Num$ (x) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with "insuperable difficulties." This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that the N-operator $N(\bar {\xi })$ and recursively defined arithmetic operators $\Omega ^{n}(\bar {\xi })$ are not different in kind, we can address Ramsey's problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction. [ABSTRACT FROM AUTHOR]

Z.-W. Sun ['Refining Lagrange's four-square theorem', J. Number Theory175 (2017), 169–190] conjectured that every positive integer n can be written as $ x^2+y^2+z^2+w^2\ (x,y,z,w\in \mathbb {N}=\{0,1,\ldots \})$ with $x+3y$ a square and also as $n=x^2+y^2+z^2+w^2\ (x,y,z,w \in \mathbb {Z})$ with $x+3y\in \{4^k:k\in \mathbb {N}\}$. In this paper, we confirm these conjectures via the arithmetic theory of ternary quadratic forms. [ABSTRACT FROM AUTHOR]

It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation. [ABSTRACT FROM AUTHOR]

This paper explores how a semantics for Prior’s infamous connective tonk should be, a connective defined by inference rules that trivialize the logic of a deductive system if that logic is supposed to be transitive. To avoid triviality, one must reject transitivity and in a relatively recent paper, Roy Cook develops a semantics for tonk with non-transitive entailment. However, I show in this paper that a cut-free sequent calculus for tonk - the arguably most natural and simplest deductive system for a non-transitive logic - can neither be complete with respect to Cook’s semantics nor with respect to a semantics with non-transitive entailment based on a semantics for vagueness and transparent truth developed by Cobreros et al. It is argued that the failure to adequately represent tonk is connected with the fact that tonk is not uniquely defined in a cut-free sequent calculus system unless the logic is in addition non-reflexive. To remedy this, the paper develops a semantics with non-transitive and non-reflexive entailment based on the idea that complex formulae are true or false relative to them being assessed as premise or as conclusion. [ABSTRACT FROM AUTHOR]

A Gödel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence of arithmetic to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin’s problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed points for the formulae are obtained. This paper is the first of two papers. In the present paper we focus on provability. In part II, we will consider other properties like Rosser provability and partial truth predicates. [ABSTRACT FROM AUTHOR]

William Harvey's famous quantitative argument from De motu cordis (1628) about the circulation of blood explained how a small amount of blood could recirculate and nourish the entire body, upending the Galenic conception of the blood's motion. This paper argues that the quantitative argument drew on the calculative and rhetorical skills of merchants, including Harvey's own brothers. Modern translations of De motu cordis obscure the language of accountancy that Harvey himself used. Like a merchant accounting for credits and debits, intake and output, goods and moneys, Harvey treated venous and arterial blood as essentially commensurate, quantifiable and fungible. For Harvey, the circulation (and recirculation) of blood was an arithmetical necessity. The development of Harvey's circulatory model followed shifts in the epistemic value of mercantile forms of knowledge, including accounting and arithmetic, also drawing on an Aristotelian language of reciprocity and balance that Harvey shared with mercantile advisers to the royal court. This paper places Harvey's calculations in a previously underappreciated context of economic crisis, whose debates focused largely on questions of circulation. [ABSTRACT FROM AUTHOR]

The aim of this corrigendum is to correct an error in Corollary 10.7 to Theorem 10.6, one of the main results in the paper 'On the cuspidal cohomology of $S$ -arithmetic subgroups of reductive groups over number fields'. This makes necessary a thorough investigation of the conditions under which a Cartan-type automorphism exists on $G_1=\mathrm {Res}_{\mathbb {C}/\mathbb {R}}G_0$ , where $G_0$ is a connected semisimple algebraic group defined over $\mathbb {R}$. [ABSTRACT FROM AUTHOR]

The following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle a ∈ 2ω we can ask the same question relativized to a. A negative answer for every a implies Vaught's Conjecture for Lω1ω [ABSTRACT FROM AUTHOR]

We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a function with |f(n)| ≤ 1 for all n and ‖f‖U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this.By combining the main result of the present paper with several previous results of the first two authors one obtains the generalised Hardy–Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p1 < p2 < p3 < p4 < p5 ≤ N of primes. [ABSTRACT FROM AUTHOR]

This paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various "paradoxical notions" for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell's paradox, a variant of Mirimanoff's paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth. [ABSTRACT FROM AUTHOR]

Summary: Path planning on a two-dimensional grid is a well-studied problem in robotics. It usually involves searching for a shortest path between two vertices on a grid given that some of the grid cells are impassable (occupied by obstacles). Single-source path planning finds shortest paths from a given source vertex to all other vertices of the grid. Singles-source path planning enhances robot autonomy by calculating multiple possible paths for various navigation scenarios when the destination state is unknown. A high-performance algorithm for single-source any-angle path planning on a grid called CWave is proposed here. Any-angle attribute implies that the algorithm calculates paths which can include line segments at any angle, as opposed to standard A* that runs on an 8-connected graph, which permits turns with 45° increments only. The key idea of CWave is to abandon the graph model and operate directly on the grid geometry using discrete geometric primitives (instead of individual vertices) to represent the wave front. In its most basic form (CWaveInt), CWave requires only integer arithmetics. CWaveInt , however, can accumulate the distance error at turning points. A modified version of CWave (CWaveFpuSrc) with minimal usage of floating-point calculations is also developed to eliminate any accumulative errors, which is proven mathematically and experimentally on several maps. The performance of the algorithm on most of the tested maps is demonstrated to be significantly faster than that of Theta*, Lazy Theta*, Field A*, ANYA, Block A*, and A* adapted for single-source planning (on maps with lower number of isolated obstacles, CWaveFpuSrc is 2−3 times faster than its fastest tested alternative Block A*). An N -threaded implementation (CWaveN) of CWave is presented and tested to demonstrate an improved performance (multithreaded implementation is 1.5−3 times faster than single-threaded CWave). The paper discusses foundations and experimental validation of CWave, and presents future work to address the limitations of the current implementations and obtain further performance enhancements. [ABSTRACT FROM AUTHOR]

The Tabula Alimentaria of Veleia records the details of two second-century a.d. imperial alimentary schemes at the northern Italian town of Veleia, providing a rare insight into the workings of these schemes. Imperial loans are made to local landowners in exchange for pledges of specified property. Interest paid by landowners is used to fund cash subsidies for the upbringing of selected local children. In the early twentieth century, the French scholar Félix de Pachtere came close to demonstrating a consistent arithmetical relationship between a landowner's declared property value and the loan received. However, anomalies remained. This article proposes a revised formula which establishes a precise and consistent linkage between loan amounts and property declarations. Based on this arithmetical dataset, the paper proposes some hypotheses about how these fractional computations might have been performed in second-century Rome. [ABSTRACT FROM AUTHOR]

We develop an untyped framework for the multiverse of set theory. $\mathsf {ZF}$ is extended with semantically motivated axioms utilizing the new symbols $\mathsf {Uni}(\mathcal {U})$ and $\mathsf {Mod}(\mathcal {U, \sigma })$ , expressing that $\mathcal {U}$ is a universe and that $\sigma $ is true in the universe $\mathcal {U}$ , respectively. Here $\sigma $ ranges over the augmented language, leading to liar-style phenomena that are analyzed. The framework is both compatible with a broad range of multiverse conceptions and suggests its own philosophically and semantically motivated multiverse principles. In particular, the framework is closely linked with a deductive rule of Necessitation expressing that the multiverse theory can only prove statements that it also proves to hold in all universes. We argue that this may be philosophically thought of as a Copernican principle that the background theory does not hold a privileged position over the theories of its internal universes. Our main mathematical result is a lemma encapsulating a technique for locally interpreting a wide variety of extensions of our basic framework in more familiar theories. We apply this to show, for a range of such semantically motivated extensions, that their consistency strength is at most slightly above that of the base theory $\mathsf {ZF}$ , and thus not seriously limiting to the diversity of the set-theoretic multiverse. We end with case studies applying the framework to two multiverse conceptions of set theory: arithmetic absoluteness and Joel D. Hamkins' multiverse theory. [ABSTRACT FROM AUTHOR]

The importance of an abstract approach to a computation theory over general data types has been stressed by Tucker in many of his papers. Berger and Seisenberger recently elaborated the idea for extraction out of proofs involving (only) abstract reals. They considered a proof involving coinduction of the proposition that any two reals in [−1, 1] have their average in the same interval, and informally extract a Haskell program from this proof, which works with stream representations of reals. Here we formalize the proof, and machine extract its computational content using the Minlog proof assistant. This required an extension of this system to also take coinduction into account. [ABSTRACT FROM PUBLISHER]

By work of Looijenga and others, one understands the relationship between Geometric Invariant Theory (GIT) and Baily–Borel compactifications for the moduli spaces of degree-2 K3 surfaces, cubic fourfolds, and a few other related examples. The similar-looking cases of degree-4 K3 surfaces and double Eisenbud–Popescu–Walter (EPW) sextics turn out to be much more complicated for arithmetic reasons. In this paper, we refine work of Looijenga in order to handle these cases. Specifically, in analogy with the so-called Hassett–Keel program for the moduli space of curves, we study the variation of log canonical models for locally symmetric varieties of Type IV associated to D-lattices. In particular, for the 19-dimensional case, we conjecturally obtain a continuous one-parameter interpolation between the GIT and Baily–Borel compactifications for the moduli of degree-4 K3 surfaces. The analogous 18-dimensional case, which corresponds to hyperelliptic degree-4 K3 surfaces, can be verified by means of Variation of Geometric Invariant Theory (VGIT) quotients. [ABSTRACT FROM AUTHOR]

Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x , y , z and w ; let X be the generic element of the family of surfaces in ℙ given by X : w2 = x6 + y6 + z6 + tx2y2z2. The surface X is a K3 surface over the function field ℚ(t). In this paper, we explicitly compute the geometric Picard lattice of X, together with its Galois module structure, as well as derive more results on the arithmetic of X and other elements of the family X. [ABSTRACT FROM AUTHOR]

Let $h(n)$ denote the largest product of distinct primes whose sum does not exceed $n$. The main result of this paper is that the property for all $n\geq 1$ , we have $\log h(n) (where $\text{li}^{-1}$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis. [ABSTRACT FROM AUTHOR]

This paper is a follow-up to [4], in which a mistake in [6] (which spread also to [9]) was corrected. We give a strenghtening of the main result on the semantical nonconservativity of the theory of PT− with internal induction for total formulae (PT− + INT(tot), denoted by PT− in [9]). We show that if to PT− the axiom of internal induction for all arithmetical formulae is added (giving PT− + INT), then this theory is semantically stronger than PT−+ INT(tot). In particular the latter is not relatively truth definable (in the sense of [11]) in the former. Last but not least, we provide an axiomatic theory of truth which meets the requirements put forward by Fischer and Horsten in [9]. The truth theory we define is based on Weak Kleene Logic instead of the Strong one. [ABSTRACT FROM AUTHOR]

In 2012, Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math. 88 (2012), 223–235] introduced a special class of totally symmetric plane partitions, called $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$-shell totally symmetric plane partitions. Let $f(n)$ denote the number of $1$-shell totally symmetric plane partitions of weight $n$. More recently, Hirschhorn and Sellers [‘Arithmetic properties of 1-shell totally symmetric plane partitions’, Bull. Aust. Math. Soc. to appear. Published online 27 September 2013] discovered a number of arithmetic properties satisfied by $f(n)$. In this paper, employing some results due to Cui and Gu [‘Arithmetic properties of $l$-regular partitions’, Adv. Appl. Math. 51 (2013), 507–523], and Hirschhorn and Sellers, we prove several new infinite families of congruences modulo 4 and 8 for $1$-shell totally symmetric plane partitions. For example, we find that, for $n\geq 0$ and $\alpha \geq 1$, $$\begin{equation*} f(8 \times 5^{2\alpha } n+39\times 5^{2\alpha -1})\equiv 0 \pmod 8. \end{equation*}$$ [ABSTRACT FROM AUTHOR]

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel's first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn't give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics. [ABSTRACT FROM AUTHOR]

Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i\leq ~k\} - \mathrm{ht} (I)$, for all $n\geq 1$. In addition we show that if $I$ is MHC and $w$ is an indeterminate which is not in the monomials generating $I$, then $\mathrm{reg} (S/ \mathop{(I+ {w}^{d} S)}\nolimits ^{n} )\leq \mathrm{reg} (S/ I)+ nd- 1$ for all $n\geq 1$ and $d$ large enough. Finally, we implement an algorithm for the computation of $m(I)$. [ABSTRACT FROM AUTHOR]

Computability has its origins in Logic within the framework formed along the original path laid down by the founding fathers of the modern foundational analysis for Mathematics (Frege and Hilbert). This theoretical itinerary, which was largely focused on Logic and Arithmetic, departed in principle from the renewed relations between Geometry and Physics occurring at the time. In particular, the key issue of physical measurement, as our only access to ‘reality’, played no part in its theoretical framework. This is in stark contrast to the position in Physics, where the role of measurement has been a core theoretical and epistemological issue since Poincaré, Planck and Einstein. Furthermore, measurement is intimately related to unpredictability, (in-)determinism and the relationship with physical space–time. Computability, despite having exact access to its own discrete data type, provides a unique tool for the investigation of ‘unpredictability’ in both Physics and Biology through its fine-grained analysis of undecidability – note that unpredictability coincides with physical randomness in both classical and quantum frames. Moreover, it now turns out that an understanding of randomness in Physics and Biology is a key component of the intelligibility of Nature. In this paper, we will discuss a few results following along this line of thought. [ABSTRACT FROM AUTHOR]

In the past years, analyzers have been introduced to detect classes of non-terminating queries for definite logic programs. Although these non-termination analyzers have shown to be rather precise, their applicability on real-life Prolog programs is limited because most Prolog programs use non-logical features. As a first step towards the analysis of Prolog programs, this paper presents a non-termination condition for Logic Programs containing integer arithmetics. The analyzer is based on our non-termination analyzer presented at International Logic Programming Conference (ICLP) 2009. The analysis starts from a class of queries and infers a subclass of non-terminating ones. In first phase, we ignore the outcome (success or failure) of the arithmetic operations, assuming success of all arithmetic calls. In second phase, we characterize successful arithmetic calls as a constraint problem, the solution of which determines the non-terminating queries. [ABSTRACT FROM AUTHOR]

This paper is devoted to the rational approximation of automatic real numbers, that is, real numbers whose expansion in an integer base can be generated by a finite automaton. We derive upper bounds for the irrationality exponent of famous automatic real numbers associated with the Thue-Morse, Rudin-Shapiro, paper folding and Baum-Sweet sequences. These upper bounds arise from the construction of some explicit Padé or Padé type approximants for the generating functions of these sequences. In particular, we prove that the Thue-Morse-Mahler numbers have an irrationality exponent at most equal to 4. We also obtain an explicit description of infinitely many convergents to these numbers. [ABSTRACT FROM AUTHOR]

Let [ a 1 ( x ), a 2 ( x ), . . .] be the continued fraction expansion of x ϵ [0,1). Write T - n ( x )=max{ a - k ( x ):1 ⩽ k ⩽ n }. Philipp [ 6 ] proved that - - $$T(x)=:\liminf\limits•{n\to \infty}{ T•n(x)\log\log n\over {n}}=\frac{1}{\log 2}\ \ \ a.e..$$ Okano [ 5 ] showed that for any k ⩾ 2, there exists x ϵ [0, 1) such that T ( x )=1/log k . In this paper we show that, for any α ⩾ 0, the set - - $$E(\alpha)=\left\{x\in [0,1), \lim\limits•{n\to \infty}{T•n(x)\log\log n\over {n}}=\alpha\right\}$$ is of Hausdorff dimension 1. [ABSTRACT FROM AUTHOR]

AbstractSuppose that Aand A? are subsets of /N. We write A+A? for the set {a+a?:aAand a? A?} and call it the sumsetof Aand A?. In this paper we address the following question. Suppose that A1,.?.?.,Amare subsets of /N. Does A1+?????+Amcontain a long arithmetic progression?The situation for m=2 is rather different from that for m? 3. In the former case we provide a new proof of a result due to Green. He proved that A1+A2contains an arithmetic progression of length roughly $\exp (c\sqrt{\alpha_1\alpha_2 \log N})$where ?1and ?2are the respective densities of A1and A2. In the latter case we improve the existing estimates. For example we show that if A/Nhas density $\alpha \gg \sqrt{\log \log N/\log N}$then A+A+Acontains an arithmetic progression of length Nc?. This compares with the previous best of Nc?2+.Two main ingredients have gone into the paper. The first is the observation that one can apply the iterative method to these problems using some machinery of Bourgain. The second is that we can localize a result due to Chang regarding the large spectrum of L2-functions. This localization seems to be of interest in its own right and has already found one application elsewhere. [ABSTRACT FROM AUTHOR]

This paper is an introduction to the RealLib package for exact real number computations. The library provides certified accuracy, but tries to achieve this at performance close to the performance of hardware floating point for problems that do not require higher precision. The paper gives the motivation and features of the design of the library and compares it with other packages for exact real arithmetic. [ABSTRACT FROM AUTHOR]

Applying circle inversion on a square grid filled with circles, we obtain a configuration that we call a fabric of kissing circles. We focus on the curvature inside the individual components of the fabric, which are two orthogonal frames and two orthogonal families of chains. We show that the curvatures of the frame circles form a doubly infinite arithmetic sequence (bi-sequence), whereas the curvatures in each chain are arranged in a quadratic bi-sequence. We also prove a sufficient condition for the fabric to be integral. [ABSTRACT FROM AUTHOR]

In this paper, we treat the specification problem in Krivine classical realizability (Krivine 2009Panoramas et synthèses27), in the case of arithmetical formulæ. In the continuity of previous works from Miquel and the first author (Guillermo 2008Jeux de réalisabilité en arithmétique classique, Ph.D. thesis, Université Paris 7; Guillermo and Miquel 2014Mathematical Structures in Computer Science, Epub ahead of print), we characterize the universal realizers of a formula as being the winning strategies for a game (defined according to the formula). In the first sections, we recall the definition of classical realizability, as well as a few technical results. In Section 5, we introduce in more details the specification problem and the intuition of the game-theoretic point of view we adopt later. We first present a game 1, that we prove to be adequate and complete if the language contains no instructions ‘quote’ (Krivine 2003Theoretical Computer Science308 259–276), using interaction constants to do substitution over execution threads. We then show that as soon as the language contain ‘Quote,’ the game is no more complete, and present a second game 2 that is both adequate and complete in the general case. In the last Section, we draw attention to a model-theoretic point of view and use our specification result to show that arithmetical formulæ are absolute for realizability models. [ABSTRACT FROM PUBLISHER]

It’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of second-order Peano Arithmetic in the theory $H{P^2}$, whose sole axiom is Hume’s principle. What’s less well known is that, in Die Grundlagen Der Arithmetic §82–83 Boolos (2011), George Boolos provided a converse interpretation of $H{P^2}$ in $P{A^2}$. Boolos’ interpretation can be used to show that the Frege’s construction allows for any model of $P{A^2}$ to be recovered from some model of $H{P^2}$. So the space of possible arithmetical universes is precisely characterized by Hume’s principle.In this paper, I show that a large class of second-order theories admit characterization by an abstraction principle in this sense. The proof makes use of structural abstraction principles, a class of abstraction principles of considerable intrinsic interest, and categories of interpretations in the sense of Visser (2003). [ABSTRACT FROM PUBLISHER]

Gödel introduced the original provability predicate in the proofs of Gödel’s incompleteness theorems, and Rosser defined a new one. They are equivalent in the standard model ${\mathbb N}$ of arithmetic or any nonstandard model of ${\rm PA} + {\rm Con_{PA}} $, but the behavior of Rosser’s provability predicate is different from the original one in nonstandard models of ${\rm PA} + \neg {\rm Con_{PA}} $. In this paper, we investigate several properties of the derivability conditions for Rosser provability predicates, and prove the existence of a Rosser provability predicate with which we can define any consistent complete extension of ${\rm PA}$ in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. We call it a universal Rosser predicate. It follows from the theorem that the true arithmetic ${\rm TA}$ can be defined as the set of theorems of ${\rm PA}$ in terms of a universal Rosser predicate in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. By using this theorem, we also give a new proof of a theorem that there is a nonstandard model M of ${\rm PA} + \neg {\rm Con_{PA}} $ such that if N is an initial segment of M which is a model of ${\rm PA} + {\rm Con_{PA}} $ then every theorem of ${\rm PA}$ in N is a theorem of $\rm PA$ in ${\mathbb N}$. In addition, we prove that there is a Rosser provability predicate such that the set of theorems of $\rm PA$ in terms of the Rosser provability predicate is inconsistent in any nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $. [ABSTRACT FROM PUBLISHER]

A semantics for quantified modal logic is presented that is based on Kleene’s notion of realizability. This semantics generalizes Flagg’s 1985 construction of a model of a modal version of Church’s Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church’s Thesis and a variant of a modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra’s generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott’s graph model (for the untyped lambda calculus) which witnesses the failure of the stability of nonidentity. [ABSTRACT FROM PUBLISHER]

This paper is a contribution to the growing investigation of strong reducibilities between ${\rm{\Pi }}_2^1$ statements of second-order arithmetic, viewed as an extension of the traditional analysis of reverse mathematics. We answer several questions of Hirschfeldt and Jockusch [13] about Weihrauch (uniform) and strong computable reductions between various combinatorial principles related to Ramsey’s theorem for pairs. Among other results, we establish that the principle $SRT_2^2$ is not Weihrauch or strongly computably reducible to $D_{ < \infty }^2$, and that COH is not Weihrauch reducible to $SRT_{ < \infty }^2$, or strongly computably reducible to $SRT_2^2$. The last result also extends a prior result of Dzhafarov [9]. We introduce a number of new techniques for controlling the combinatorial and computability-theoretic properties of the problems and solutions we construct in our arguments. [ABSTRACT FROM PUBLISHER]

No natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) and Ramsey’s theorem for pairs ($RT_2^2$) in reverse mathematics. The tree theorem for pairs ($TT_2^2$) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle $TT_2^2$ is known to lie between ACA0 and $RT_2^2$ over RCA0, but its exact strength remains open. In this paper, we prove that $RT_2^2$ together with weak König’s lemma (WKL0) does not imply $TT_2^2$, thereby answering a question of Montálban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics. [ABSTRACT FROM PUBLISHER]

We define and study quasidialectical systems, which are an extension of Magari’s dialectical systems, designed to make Magari’s formalization of trial and error mathematics more adherent to the real mathematical practice of revision: our proposed extension follows, and in several regards makes more precise, varieties of empiricist positions à la Lakatos. We prove several properties of quasidialectical systems and of the sets that they represent, called quasidialectical sets. In particular, we prove that the quasidialectical sets are ${\rm{\Delta }}_2^0$ sets in the arithmetical hierarchy. We distinguish between “loopless” quasidialectal systems, and quasidialectical systems “with loops”. The latter ones represent exactly those coinfinite c.e. sets, that are not simple. In a subsequent paper we will show that whereas the dialectical sets are ω-c.e., the quasidialectical sets spread out throughout all classes of the Ershov hierarchy of the ${\rm{\Delta }}_2^0$ sets. [ABSTRACT FROM AUTHOR]

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture. [ABSTRACT FROM AUTHOR]

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series. [ABSTRACT FROM AUTHOR]

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ , where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi)$ , where $f$ is a (varying) newform on $D^\times$ of level $p^n$ , and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$ -adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property. [ABSTRACT FROM AUTHOR]

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory (Parsons, 1990; Parsons, 2008, sec. 49; McGee, 1997; Lavine, 1999; Väänänen & Wang, 2014). Another great enterprise in contemporary philosophy of mathematics has been Wright’s and Hale’s project of founding mathematics on abstraction principles (Hale & Wright, 2001; Cook, 2007). In Walsh (2012), it was noted that one traditional abstraction principle, namely Hume’s Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to (i) stability-like acceptability criteria on abstraction principles (cf. Cook, 2012), (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli (2010b) and Fine (2002), and (iii) supervaluational ideas coming out of the work of Hodes (1984, 1990, 1991). [ABSTRACT FROM AUTHOR]

Let Z2, Z3, and Z4 denote 2nd, 3rd, and 4th order arithmetic, respectively. We let Harrington’s Principle, HP, denote the statement that there is a real x such that every x-admissible ordinal is a cardinal in L. The known proofs of Harrington’s theorem “$Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies 0♯ exists” are done in two steps: first show that $Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies HP, and then show that HP implies 0♯ exists. The first step is provable in Z2. In this paper we show that Z2 + HP is equiconsistent with ZFC and that Z3 + HP is equiconsistent with ZFC + there exists a remarkable cardinal. As a corollary, Z3 + HP does not imply 0♯ exists, whereas Z4 + HP does. We also study strengthenings of Harrington’s Principle over 2nd and 3rd order arithmetic. [ABSTRACT FROM PUBLISHER]