ALBERT

Description: We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of non-autonomous Hamiltonian vector fields.

Description: We consider the Schur–Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well known to be extremely difficult, and in fact, it remains open for matrices of size greater than $3$ . We show that the infinite-dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I $_\infty$ , II, and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.

Description: We study the rate of convergence to zero of the tail entropy of $C^\infty$ maps. We give an upper bound of this rate in terms of the growth in $k$ of the derivative of order $k$ and give examples showing the optimality of the established rate of convergence. We also consider the case of multimodal maps of the interval. Finally, we prove that homoclinic tangencies give rise to $C^r$ $(r\geqslant 2)$ robustly non- $h$ -expansive dynamical systems.

Description: Let ${{\mathscr {C}}}^0_{{{\mathfrak {g}}}}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'({{\mathfrak {g}}})$ and let $R^{A_\infty }{\mbox {-}\mathrm {gmod}}$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty }$ . In this paper, we investigate the relationship between the categories ${{\mathscr {C}}}^0_{A_{N-1}^{(1)}}$ and ${{\mathscr {C}}}^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur–Weyl duality functors ${\mathcal {F}}^{(t)}$ from $R^{A_\infty }{\mbox {-}\mathrm {gmod}}$ to ${{\mathscr {C}}}^0_{A_{N-1}^{(t)}}\ (t=1,2)$ .

Description: We present new constructions of complex and $p$ -adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.

Description: We introduce a new framework for the analysis of the stability of solitons for the one-dimensional Gross–Pitaevskii equation. In particular, we establish the asymptotic stability of the black soliton with zero speed.

Description: Let $k$ and $n$ be positive even integers. For a cuspidal Hecke eigenform $h$ in the Kohnen plus space of weight $k-n/2+1/2$ for $\varGamma _0(4),$ let $I_n(h)$ be the Duke–Imamo $\bar {{\text {g}}}$ lu–Ikeda lift of $h$ in the space of cusp forms of weight $k$ for ${\rm Sp}_n({{\bf{Z}}}),$ and $f$ be the primitive form of weight $2k-n$ for ${\rm SL}_2({{\bf{Z}}})$ corresponding to $h$ under the Shimura correspondence. We then express the ratio $\displaystyle {\langle I_n(h), I_n(h) \rangle / \langle h, h \rangle }$ of the period of $I_n(h)$ to that of $h$ in terms of special values of certain $L$ -functions of $f$ . This proves the conjecture proposed by Ikeda concerning the period of the Duke–Imamo $\bar {{\text {g}}}$ lu–Ikeda lift.

Description: We present a proof system for the provability logic GLP in the formalism of nested sequents and prove the cut elimination theorem for it. As an application, we obtain the reduction of GLP to its important fragment called J syntactically.

Description: Intuitionistic sentential logic is shown to be sound and complete with respect to a semantics centered around extensions of atomic bases (i.e. sets of inference rules for atomic sentences). The result is made possible through a non-standard interpretation of disjunction, whereby, roughly speaking, a disjunction is taken to hold just in case every atomic sentence that follows from each of the disjuncts separately holds; it is argued that this interpretation makes good sense provided that rules in atomic bases are conceived of as being accepted hypothetically rather than categorically.

Description: Motivated by both established and new applications, we study navigational query languages for graphs (binary relations). The simplest language has only the two operators union and composition, together with the identity relation. We make more powerful languages by adding any of the following operators: intersection; set difference; projection; coprojection; converse; transitive closure; and the diversity relation. All these operators map binary relations to binary relations. We compare the expressive power of all resulting languages, both for binary-relation queries as well as for boolean queries. In the absence of transitive closure, a complete Hasse diagram of relative expressiveness has already been established [ 8 ]. Moreover, it has already been shown that for boolean queries over a single edge label, transitive closure does not add any expressive power when only projection and diversity may be present [ 11 ]. In the present article, we now complete the Hasse diagram in the presence of transitive closure, both for the case of a single edge label, as well as for the case of at least two edge labels. The main technical results are the following: (1) In contrast to the above-stated result [ 11 ] transitive closure does add expressive power when coprojection is present. (2) Transitive closure also adds expressive power as soon as converse is present. (3) Conversely, converse adds expressive power in the presence of transitive closure. In particular, the converse elimination result from [ 8 ] no longer works in the presence of transitive closure. (4) As a corollary, we show that the converse elimination result from [ 8 ] necessitates an exponential blow-up in the degree of the expressions.

Description: Many authors have argued that, when performing simultaneous statistical test procedures, one should seek for solutions that lead to decisions that are consistent and, consequently, easier to communicate to practitioners of statistical methods. In this way, the set of hypotheses that are rejected and the set of hypotheses that are not rejected by a testing procedure should be consistent from a logic standpoint. For instance, if hypothesis A implies hypothesis B , a procedure that rejects B should also reject A , a property not always met by multiple test procedures. We contribute to this discussion by exploring how far one can go in constructing coherent procedures while still preserving statistical optimality. This is done by studying four types of logical consistency relations. We show that although the only procedures that satisfy more than (any) two of these properties are simple tests based on point estimation, it is possible to construct various interesting methods that fulfil one or two of them while preserving different statistical optimality criteria. This is illustrated with several Bayesian and frequentist examples. We also characterize some of these properties under a decision-theoretic framework.

Description: The Svenonius theorem describes the (first-order) definability in a structure in terms of permutations preserving the relations of elementary extensions of the structure. In the present article, we prove a version of this theorem using permutations of sequences over the original structure (these are permutations of sequences of tuples of the structure elements as well). We say that such a permutation almost preserves a relation if for every sequence of its arguments the value of the relation on an n -th element of the sequence and on its image under coincide for almost all numbers n . We prove that a relation is definable in a structure iff the relation is almost preserved by all permutations almost preserving the relations of the structure. This version limits consideration to the original structure only and does not refer to any logical notion, such as ‘elementary equivalence’.

Description: The celebrated theorem proved by Goldblatt and Thomason in 1974 gives necessary and sufficient conditions for an elementary class of Kripke frames to be modally definable. Here we obtain a local analogue of this result, which deals with modal definability of classes of pointed frames. Furthermore, we generalize it to the case of n -frames , which are frames with n distinguished worlds. For talking about n -frames, we generalize modal formulas to modal expressions . While a modal formula is evaluated at a single world of a model, a modal expression with n individual variables is evaluated at an n -tuple of worlds, just as a first-order formula with n free variables. We introduce operations on n -frames that preserve validity of modal expressions, and show that closure under these operations is a necessary and sufficient condition for an elementary class of n -frames to be modally definable. We also discuss the relationship between modal expressions and hybrid logic and leave open questions.

Description: We add strong negation N to classical logic and interpret the attack relation of ‘ x attacks y ’ in argumentation as (x-〉Ny) . We write a corresponding object level (using N only) classical theory for each argumentation network and show that the classical models of this theory correspond exactly to the complete extensions of the argumentation network. We show by example how this approach simplifies the study of abstract argumentation networks. We compare with other translations of abstract argumentation networks into logic, such as classical predicate logic or modal logics, or logic programming, and we also compare with Abstract Dialectical Frameworks.

Description: This article uses possible-world semantics to model the changes that may occur in an agent's knowledge as she loses information. This builds on previous work in which the agent may forget the truth-value of an atomic proposition, to a more general case where she may forget the truth-value of a propositional formula. The generalization poses some challenges, since in order to forget whether a complex proposition is the case, the agent must also lose information about the propositional atoms that appear in it, and there is no unambiguous way to go about this. We resolve this situation by considering expressions of the form [] , which quantify over all possible (but ‘minimal’) ways of forgetting whether . Propositional atoms are modified non-deterministically, although uniformly, in all possible worlds. We then represent this within action model logic in order to give a sound and complete axiomatization for a logic with knowledge and forgetting. Finally, some variants are discussed, such as when an agent forgets (rather than forgets whether ) and when the modification of atomic facts is done non-uniformly throughout the model.

Description: There are various ways of achieving an enlarged understanding of a concept of interest. One way is by giving its proper definition. Another is by giving something else a proper definition and then using it to model or formally represent the original concept. Between the two we find varying shades of grey. We might open up a concept by a direct lexical definition of the predicate that expresses it, or by a theory whose theorems define it implicitly. At the other end of the spectrum, the modelling-this-as-that option also admits of like variation, ranging from models rooted in formal representability theorems to models conceived of as having only heuristic value. There exist on both sides of this divide further differences still. In one of them, both the definiendum and definiens of a definition are words or phrases of some common natural language. In others, the item of interest is a natural language expression and its representation is furnished by the artificial linguistic system that models it. The modern history of these approaches is both very large and growing. Much of this evolution has given too short a shrift to the history of the demotion of ‘intuitive’ concepts in favour of the artificially contrived ones intended to model them. A working assumption of this article is that in the absence of a good understanding of what motivated the modelling-turn in the foundations of mathematics and the intuitive theory of truth, the whole notion of formal representability will have been inadequately understood. In the interests of space, I will concentrate on seminal issues in set theory as dealt with by Russell and Frege, and in the theory of truth in natural languages as dealt with by Tarski. The nub of the present focus is the representational role of model theory in the logics of formalized languages.

Description: An important distinction is that between selective abductions , which select an optimal candidate from given multitude of possible explanations, and creative abductions , which introduce new theoretical concepts and models. The article focuses on creative abductions, which are essential for scientific progress, although they are rarely discussed in the literature. Scientifically, fruitful creative abductions are demarcated from purely speculative abductions by means of three virtues which are possessed by the former but not by the latter: (i) providing unification , (ii) detecting common causes and (iii) generating novel predictions by means of which they can be independently tested . Based on historical examples it is demonstrated that common cause abduction from correlated dispositions is the fundamental abductive operation by which new theoretical concepts and models are scientifically generated. Statistical factor analysis can be regarded as a statistical generalization of common cause abduction.

Description: This article investigates the appropriateness of LP-based reasoning to machine ethics, an interdisciplinary field of inquiry that emerges from the need of imbuing autonomous agents with the capacity for moral decision making. The first contribution of the article is that of identifying morality viewpoints, as studied in moral philosophy and psychology, which are amenable to computational modelling, and then mapping them to appropriate LP-based reasoning features. The identified viewpoints are covered by two morality themes: moral permissibility and the dual-process model. In the second contribution, various LP-based reasoning features are applied to model these identified morality viewpoints, via classic moral examples taken offthe-shelf from the literature. For this purpose, our Q UALM system, which features a combination of LP abduction, updating and counterfactuals, supported by LP tabling mechanisms, are mainly employed. The applications are also supported by other existing LP-based systems, featuring preference handling and probabilistic reasoning, which complement Q UALM in addressing the morality viewpoints in question.

Description: Theories of truthlikeness (or verisimilitude) are currently being classified according to two independent distinctions: that between ‘content’ and ‘likeness’ accounts, and that between ‘conjunctive’ and ‘disjunctive’ ones. In this article, I present and discuss a new definition of truthlikeness, which employs Carnap's notion of the content elements entailed by a theory or proposition, and is then labelled ‘Carnapian’. After studying in detail the properties and shortcomings of this definition, I argue that it occupies a unique position in the landscape of different approaches to truthlikeness. More precisely, I show that it provides the only explication of truthlikeness which is both ‘conjunctive’ and ‘content-based’ in a suitably defined sense.

Description: In this article, we show that, in agreement with Alan Turing's original view, but contrary to many of his interpreters, high-level cognitive activities like algorithm execution are better described as dynamical activities, which involve the coordinate work of both internal (mental) resources and external ones, like paper and pencil or similar external tools. We, then, propose to exploit the basic dynamical features of a Turing machine in order to develop a dynamical approach to the cognitive explanation of human computational activities.

Description: A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert–Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty $ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko–Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that \[ f(A+B)-f(A)-\left.\frac{d}{dt}(f(A+tB))\right\vert_{t=0}\notin \mathcal S^1. \]

Description: Let $\mu $ be a probability measure on $ \mathbb R^n$ with a bounded density $f$ . We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely invariant extremal inequalities for certain averages of $f$ on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann–Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on $\mu $ , any marginal $\pi _E(\mu )$ , or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

Description: Let $\pi :X\to \mathbb {P}^1_{\mathbb {Q}}$ be a non-singular conic bundle over $\mathbb {Q}$ having $n$ non-split fibres and denote by $N(\pi ,B)$ the cardinality of the fibres of Weil height at most $B$ that possess a rational point. Serre showed in 1990 that a direct application of the large sieve yields \[ N(\pi,B)\ll B^2(\log B)^{-n/2} \] and raised the problem of proving that this is the true order of magnitude of $N(\pi ,B)$ under the necessary assumption that there exists at least one smooth fibre with a rational point. We solve this problem for all non-singular conic bundles of rank at most 3. Our method comprises the use of Hooley neutralisers, estimating divisor sums over values of binary forms, and an application of the Rosser–Iwaniec sieve.

Description: We associate a dimer algebra $A$ to a Postnikov diagram $D$ (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian ${\rm Gr}(k,n)$ . We show that $A$ is isomorphic to the endomorphism algebra of a corresponding Cohen–Macaulay module $T$ over the algebra $B$ used to categorify the cluster structure of ${\rm Gr}(k,n)$ by Jensen–King–Su. It follows that $B$ can be realised as the boundary algebra of $A$ , that is, the subalgebra $eAe$ for an idempotent $e$ corresponding to the boundary of the disc. The construction and proof uses an interpretation of the diagram $D$ , with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.

Description: We describe a ring whose category of Cohen–Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$ -planes in $n$ -space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective–injective object is Geiss–Leclerc–Schröer's category Sub $Q_k$ , which categorifies the coordinate ring of the big cell in this Grassmannian.

Description: We develop a way of seeing a complete orientable hyperbolic 4-manifold $ {\mathcal {M}}$ as an orbifold cover of a Coxeter polytope $ {\mathcal {P}} \subset \mathbb {H}^4$ that has a facet colouring. We also develop a way of finding a totally geodesic sub-manifold $ {\mathcal {N}}$ in $ {\mathcal {M}}$ , and describing the result of mutations along $ {\mathcal {N}}$ . As an application of our method, we construct an example of a complete orientable hyperbolic 4-manifold $ {\mathcal {X}}$ with a single non-toric cusp and a complete orientable hyperbolic 4-manifold ${\mathcal {Y}}$ with a single toric cusp. Both $ {\mathcal {X}}$ and $ {\mathcal {Y}}$ have twice the minimal volume among all complete orientable hyperbolic 4-manifolds.

Description: The covariogram $g_{K}$ of a convex body $K$ in $ \mathbb {R}^n$ is the function that associates to each $x\in \mathbb {R}^n$ the volume of the intersection of $K$ with $K+x$ . Determining $K$ from the knowledge of $g_K$ is known as the Covariogram Problem. It is equivalent to determining the characteristic function $1_K$ of $K$ from the modulus of its Fourier transform $\widehat {{1_K}}$ in $ \mathbb {R}^n$ , a particular instance of the Phase Retrieval Problem. We connect the Covariogram Problem to two aspects of the Fourier transform $\widehat {{1_K}}$ seen as a function in $\mathbb {C}^n$ . The first connection is with the problem of determining $K$ from the knowledge of the zero set of $\widehat {{1_K}}$ in $\mathbb {C}^n$ . To attack this problem T. Kobayashi studied the asymptotic behavior at infinity of this zero set. We obtain this asymptotic behavior assuming less regularity on $K$ and we use this result as an essential ingredient for proving that when $K$ is sufficiently smooth and in any dimension $n$ , $K$ is determined by $g_K$ in the class of sufficiently smooth bodies. The second connection is with the irreducibility of the entire function $\widehat {{1_K}}$ . This connection also shows a link between the Covariogram Problem and the Pompeiu Problem in integral geometry.

Description: Let $\varphi :X\to S$ be a morphism between smooth complex analytic spaces and let $f=0$ define a free divisor on $S$ . We prove that if the deformation space $T^1_{X/S}$ of $\varphi $ is a Cohen–Macaulay $\mathcal {O}_X$ -module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi $ , then $f\circ \varphi =0$ defines a free divisor on $X$ ; this is generalized in several directions. Among applications we recover a result of Mond–van Straten, generalize a construction of Buchweitz–Conca, and show that a map $\varphi :\mathbb {C}^{n+1}\to \mathbb {C}^n$ with critical set of codimension 2 has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi $ is the algebraic quotient $X\to S=X\!{/\!/} G$ with $X\!{/\!/} G$ smooth, then we describe sufficient conditions for $T^1_{X/S}$ to be Cohen–Macaulay of codimension 2. In one such case, a free divisor on $\mathbb {C}^{n+1}$ lifts under the operation of ‘castling’ to a free divisor on $\mathbb {C}^{n(n+1)}$ , partially generalizing work of Granger–Mond–Schulze on linear free divisors. We give several other examples of such representations.

Description: Let $M^n$ be a compact manifold of dimension $n$ with free $T^k$ -action. We consider collapsings of $M$ on $N=M/T^k$ such that the sectional curvature and diameter of $M$ satisfy $|K(M)|\leq a$ and $ {\rm diam}(M) 〈 d$ , and give examples of collapsings for all $k$ such that the first non-zero eigenvalue of Laplacian acting on 1-forms and 2-forms of $M$ are bounded above by $c(M)\cdot \hbox {inj}(M)^{2k}$ . Moreover, we prove that the first non-zero eigenvalue of Laplacian acting on 1-forms of all principal $T^k$ -bundle $M$ over $N$ is bounded below by $c(n,a,d,N)\cdot {\rm Vol}(M)^2$ and $c\cdot \hbox {inj}(M)^{2k}$ when $M$ collapses on $N$ .

Description: Among the possible solutions to the paradoxes of collective preferences, single-peakedness is significant because it has been associated to a suggestive conceptual interpretation: a single-peaked preference profile entails that, although individuals may disagree on which option is the best, they conceptualize the choice along a shared unique dimension, i.e. they agree on the rationale of the collective decision. In this article, we discuss the relationship between the structural property of single-peakedness and its suggested interpretation as uni-dimensionality of a social choice. In particular, we offer a formalization of the relationship between single-peakedness and its conceptual counterpart, we discuss their logical relations, and we question whether single-peakedness provides a rationale for collective choices.

Description: In logic and epistemology, the concept of autoimmunity refers to the partial incapability of the human agent to distinguish between her knowledge and her ignorance, due to an involuntary mechanism which underlies the fixation and revision of beliefs. The idea originated within the project initiated by Dov Gabbay and John Woods of a Naturalization of Logic , which aims at informing elaborated notions of logic and epistemology with well-established results of cognitive science. The term autoimmunity follows from the consideration that the cognitive states of belief, doubt, knowledge and ignorance affect the epistemic status of the agent who experiences them in ways she cannot anticipate nor control. Thus, we contend that the concept of autoimmunity could be usefully employed beyond the epistemological and logical fieldwork, in order to describe the cognitive mechanism supporting what the philosophical literature calls ‘epistemic feelings’, explaining some problematic occurrences of them related to the incorrect analysis of the agent’s own cognition (tip-of-the-tongue experience, misplaced feeling of knowing, etc).

Description: Research in the field of human reasoning shows that the content of propositions involved in (deductive or inductive) logical tasks significantly influences task performance. Theoretical studies clearly show that reasoning is context sensitive and dependent on people goals and intentions. A large amount of experimental tasks in logic and reasoning have been produced in the last 50 years, in particular deductive tasks with syllogistic reasoning. Recently, research has moved to cognitive sciences and neurosciences, pointing at the different brain regions involved in reasoning and logical thinking; interesting findings concern the involvement in cognition of affective regions (e.g. amygdale), as well as the involvement in emotions of brain regions traditionally viewed as cognitive (e.g. frontal cortex). Cognition and emotion appear therefore integrated in the brain activity. We present here a review of the research developed in this field and conclude with the results of a study produced at the University of Chieti.

Description: I will analyse some properties of abduction that are essential from a logical standpoint. When dealing with the so-called ‘inferential problem’, I will opt for the more general concepts of input and output instead of those of premisses and conclusions, and show that in this framework two consequences can be derived that help clarify basic logical aspects of abductive reasoning: (i) it is more natural to accept the ‘multimodal’ and ‘context-dependent’ character of the inferences involved, (ii) inferences are not merely conceived of in the terms of the process leading to the ‘generation of an output’ or to the proof of it, as in the traditional and standard view of deductive proofs, but rather, from this perspective abductive inferences can be seen as related to logical processes in which input and output fail to hold each other in an expected relation, with the solution involving the modification of inputs, not that of outputs. I will also describe that if we wish to naturalize the logic of the abductive processes and its special consequence relation, we should refer to the following main aspects: ‘optimization of situatedness’, ‘maximization of changeability’ of both input and output, and high ‘information-sensitiveness’.

Description: In this article, we first study abduction in human dialogues. Given an utterance, objective abduction produces a hearer’s belief that could explain the utterance, while subjective abduction produces a hearer’s belief that could explain the belief state of a speaker. Different types of abduction are formulated using propositional epistemic logic. We next consider conversational implicature used as pragmatic inference in speech acts. Two conflicting implicatures, Q-implicature and I-implicature , are formulated and contrasted with abduction. We also argue how speakers could use abduction or conversational implicature for the purpose of misleading hearers.

Description: Coherentism is the view that in order for beliefs to be justified they have to belong to a system of beliefs the elements of which hang together or dovetail with each other. In recent years, approaches to coherence within Bayesian epistemology have revived the interest in coherentism. In this context, a considerable number of arguments have been put forward relating coherence to diverse epistemic goals that on closer inspection turn out to be sensitive to choice of measure of coherence. In this article, it is shown that for all these arguments there is at least one coherence measure discussed in recent literature for which the argument is not formally sound. Furthermore, it is shown that for purely mathematical reasons there can be no measure that turns every argument valid. In the light of these findings, we argue that a moderate pluralism as regards probabilistic measures of coherence is the appropriate conclusion.

Description: First, I will introduce the distinction between analytic and synthetic as a partition of propositions, proofs, programs and definitions, as an extension of the distinction between analytic categorical propositions and synthetic categorical propositions in ancient Logic. Secondly, I will discuss some logical questions concerning the distinction between analytic and synthetic, and I will present main results obtained in mathematical logic as answers to the logical questions concerning the relationships between analytic and synthetic.

Description: Research into the cognition of conditionals has predominantly focused on conditional reasoning, producing a range of theories which explain associated phenomena with considerable success. However, such theories have been less successful in accommodating experimental data concerning how agents assess the probability of indicative conditionals. Since an acceptable account of conditional reasoning should be compatible with evidence regarding how we evaluate conditionals' likelihoods, this constitutes a failing of such theories. Section 1 introduces the most dominant established approach to conditional reasoning: mental models theory. Surveying a range of experimental results, I show that mental models theory (along with competing theories) is incapable of fully accounting for findings regarding judgements about conditionals' probabilities. Section 2 introduces an alternative account of deductive reasoning, the erotetic theory, recently proposed by Koralus and Mascarenhas (2013). Section 3 argues that, given a natural extension, this theory is able to explain the otherwise unaccounted for data.

Description: We give explicit atomic bases of arbitrary coefficient-free cluster algebras of types A and à . This entails showing that the minimal elements of the positive semiring of these cluster algebras form a linear basis over the integers for the cluster algebra.

Description: We prove that strongly F -regular and F -pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F -injective singularities fail to satisfy even the most basic Bertini-type results.

Description: This is the second of a pair of papers on the Delta-group structure on the braid and mapping class groups of a surface. We obtain a description of the homotopy groups of these Delta-groups and generalize to an arbitrary surface the Berrick–Cohen–Wong–Wu exact sequence relating the Brunnian braid groups of the 2-sphere to its homotopy groups. We prove a similar result for Brunnian mapping class groups.

Description: We construct a geometric realization of the Khovanov–Lauda–Rouquier algebra R associated with a symmetric Borcherds–Cartan matrix A = ( a ij ) i , j I via quiver varieties. As an application, if a ii != 0 for any i I , we prove that there exists a one-to-one correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of U A – (g) (respectively, V A ( )) and the set of isomorphism classes of indecomposable projective graded modules over R (respectively, R ).

Description: The purpose of this paper is to study the nature of quasi-invariant measures for finitely generated non-discrete subgroups of Diff ( S 1 ). For this, we apply ideas involving the closure of these groups to find out that the regularity of the measure depends on a ‘measurable version’ of well-known problems concerning stable self-intersection of Cantor sets. As applications, we prove that every d -quasiconformal probability measure for a non-solvable and non-discrete group must be absolutely continuous. Concerning singular quasi-invariant measures, it is also proved that their associated Hausdorff measures must either be zero or of infinite mass, a result contrasting with the case of dynamically defined Cantor sets and also applicable to the examples of singular stationary measures constructed by Kaimanovich and Le Prince. As a further application of our methods, a theorem of rigidity for measurable conjugations between groups as above is obtained.

Description: We study the space of period polynomials associated with modular forms of integral weight for finite-index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite-index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for 1 ( N ): an extension of the Eichler–Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficient theorem of Manin.

Description: We present a logical system for reasoning about types of actions (processes) and about agent capabilities to execute types of actions. The syntax of the system is based on that of Propositional Dynamic Logic ( PDL ), though the semantics we define is different (interpreting process terms as types, i.e. sets of binary relations). The standard PDL syntax is extended with capabilities statements, as in the KARO framework, atomic process types specified as precondition-effect pairs, written as , as well as backwards possibility operators. The resulting system is shown (by filtration) to have a decidable satisfiability problem and a sound and complete Gentzen-style proof system is presented.

Description: It is known that an epsilon-invariant sentence has a first-order reformulation, although it is not in an explicit form, since, the proof uses the non-constructive interpolation theorem. We make an attempt to describe the explicit meaning of sentences containing epsilon-terms, adopting the strong assumption of their first-order reformulability. We will prove that, if a monadic predicate is syntactically independent from an epsilon-term and if the sentence obtained by substituting the variable of the predicate with the epsilon-term is epsilon-invariant, then the sentence has an explicit first-order reformulation. Finally, we point out that the formula gives a contextual-quantificational meaning for the indefinite descriptions, provided that one accepts Kneebone’s read of epsilon-terms.

Description: Filtration constructions are among the oldest and best known methods for obtaining finite model properties for modal logics, and appear in the literature in both model-theoretic and algebraic versions. In this article we investigate definitions of algebraic filtrations by means of different types of binary relations on modal algebras, and the relationships between these. We generalize the notion of a model-theoretic filtration somewhat while simultaneously lifting it to the level of frames. We proceed to link algebraic filtrations with their model- or frame-theoretic counterparts by showing how our filtration notions interface neatly with the well-known duality theory of modal algebras and Kripke frames. We illustrate, by means of some examples, how this theory enables one to easily translate between algebraic and model-theoretic versions of some well-known filtrations. We obtain some order theoretic insights regarding the (usually model-theoretically specified) smallest and largest filtrations by considering their algebraic versions, thus demonstrating the utility of having ready access to both versions.

Description: It is generally assumed that Conservativity is a necessary ingredient for interpreting collective and cumulative predicates or, more in general, for interpreting—what I termed as—Independent Set (IS) readings (a.k.a. Scopeless readings). Under this assumption, several approaches aiming at formally providing logical representations of collective/cumulative readings implement Conservativity. This is done also in the frameworks defined in Winter (2001, Flexibility Principles in Boolean Semantics: Coordination, Plurality, and Scope in Natural Language , MIT Press) and in Robaldo (2011, J. Logic Lang. Infor. , 20, 233–271). This article investigates further the role played by Conservativity in the interpretation of IS readings, and argues that its need is not specifically due to the use of collective/cumulative predicates. Rather, it serves to maximize witness sets for quantifiers thereby guaranteeing correct truth conditions in several cases.

Description: We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt (5) and SL 2 (5) is a Beauville group. In particular, we settle a conjecture of Bauer, Catanese and Grunewald which asserts that all non-abelian finite quasisimple groups except for the alternating group Alt (5) are Beauville groups.

Description: Let U R d be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f : U -〉 R can be approximated by real analytic convex functions, uniformly on all of U . We also show that C 0 -fine approximation of convex functions by smooth (or real analytic) convex functions on R d is possible in general if and only if d = 1. Nevertheless, for d ≥ 2, we give a characterization of the class of convex functions on R d which can be approximated by real analytic (or just smoother) convex functions in the C 0 -fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every C 1 convex and proper function on U can be approximated by C convex functions in the C 1 -fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

Description: We investigate the extent of second-order characterizable structures by extending Shelah's Main Gap dichotomy to second-order logic. For this end we consider a countable complete first-order theory T . We show that all sufficiently large models of T have a characterization up to isomorphism in the extension of second-order logic obtained by adding a little bit of infinitary logic if and only if T is shallow superstable with NDOP and NOTOP. Our result relies on cardinal arithmetic assumptions. Under weaker assumptions we get consistency results or alternatively results about second-order logic with Henkin semantics. Mathematics Subject Classification: 03C85, 03C75.

Description: In this article, I present a procedure that generates proofs for finally derivable adaptive logic consequences. A proof procedure for the inconsistency adaptive logic CLuN r is already presented in [ 7 ]. In this article a procedure for CLuN m is presented and the results for both logics are generalized to all adaptive logics, on the presupposition that there exists a proof procedure for the lower limit logic. The generated proofs are so called goal-directed proofs, i.e. proofs that (i) start with the formula (the goal) of which one wants to know whether it is a consequence of a certain premise set and (ii) only consist of lines that may potentially be useful for proving or disproving the goal. The goal-directed proofs form good explications of actual problem-solving reasoning processes.

Description: We show how various modal systems considered by Malinowski as extensions of classical propositional calculus may be obtained as fibrings of classical propositional calculus and corresponding implicative modal logics, using the fibring framework for combining logics of Fern'andez and Coniglio. Taking advantage of this construction and known results of Malinowski, we draw some useful conclusions concerning some limitations of the fibring process. Finally, Malinowski's constructions are extended to obtain some modal extensions of arbitrary equivalential logics in the context of abstract algebraic logic. These are studied with respect to their algebraic character.

Description: It is shown, directly and in two ways, that Brouwer’s fan theorem for detachable bars can be proved within Bishop-style constructive mathematics supplemented by (countably many applications of) the omniscience principle LLPO .

Description: Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$ -space. In this paper, we analyze the homotopy type of the homotopy fixed point set $X^{hG}$ , and the natural injection $k\colon X^G\hookrightarrow X^{hG}$ . We show that if $X$ is elliptic, that is, it has finite-dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, that for $G$ a torus, $\pi _* (k)$ is injective in rational homotopy but, often, far from being a rational homotopy equivalence.

Description: We employ the ergodic-theoretic machinery of scenery flows to address classical geometric measure-theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.

Description: A notion of tangential thickness of a manifold is introduced. An extensive calculation within the class of lens and fake lens spaces leads to a classification of such manifolds with thickness 1, 3 or 2 $k$ , for $k\geq 1$ . On the other hand, calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants.

Description: Extending a classical result of Widom from 1969, polynomials with small supremum norms are constructed for a large family of compact sets $\Gamma$ : their norm is at most a constant times the theoretical lower limit ${{\rm cap}}(\Gamma )^n$ , where ${{\rm cap}}(\Gamma )$ denotes logarithmic capacity. The construction is based on a discretization of the equilibrium measure, and the polynomials have the additional property that outside the given set $\Gamma$ they increase as fast as possible, namely as ${{\rm cap}}(\Gamma )^n\exp (ng_{ \overline {{{}C}}\setminus \Gamma }(z))$ , with the Green's function with pole at infinity in the exponent. This latter fact allows us to use these polynomials as building blocks in constructing Dirac delta-type polynomials around corners: if a compact set $K$ has a corner at some point $z_0$ , then Dirac delta-type polynomials (fast decreasing polynomials) peaking at $z_0$ are polynomials $P_n(z)$ with $P_n(z_0)=1$ that decrease as $|P_n(z)|\prec \exp (-n^ \beta |z-z_0|^ \gamma )$ on the set $K$ as $z$ moves away from $z_0$ . The possible $(\beta , \gamma )$ pairs are completely described in turn of the angle $\alpha \pi$ at $z_0$ ( $\beta \lt 1$ and $\gamma \ge \beta /(2- \alpha )$ or $\beta =1$ and $\gamma 〉 \beta /(2- \alpha )$ ). As application of these fast decreasing polynomials sharp Nikolskii- and Markov-type inequalities are proved for Jordan domains with corners. The paper uses distortion properties of conformal maps, potential theoretic techniques as well as the theory of weighted logarithmic potentials.

Description: We consider for every $n\in \mathbb {N}$ an algebra $\mathcal {A}_{n}$ of germs at $0\in \mathbb {R}^{n}$ of continuous real-valued functions, such that we can associate to every germ $f\in \mathcal {A}_{n}$ a (divergent) series $\mathcal {T}(f)$ with non-negative real exponents, which can be thought of as an asymptotic expansion of $f$ . We require that the $\mathbb {R}$ -algebra homomorphism $f\mapsto \mathcal {T}(f)$ be injective (quasianalyticity property). In this setting, we prove analogue results to Denef and van den Dries’ quantifier elimination theorem and Hironaka's rectilinearization theorem for subanalytic sets.

Description: Suppose that $F(x)\in \mathbb {Z}[\![x]\!]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$ for an integer $b\geq 2$ . In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If, in addition, $F(x)$ is regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$ -number or a $T$ -number in Mahler's classification of real numbers.

Description: We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$ -adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.

Description: We give a bordism-theoretic characterization of those closed almost contact $(2q{+ }1)$ -manifolds (with $q\geq 2$ ) that admit a Stein fillable contact structure. Our method is to apply Eliashberg's $h$ -principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion-free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.

Description: Casson-type invariants emerging from Donaldson theory over certain negative-definite four-manifolds were recently suggested by Teleman. These are defined by an algebraic count of points in a zero-dimensional moduli space of flat instantons. Motivated by the cobordism programme of proving Witten's conjecture, we use a moduli space of ${\rm PU}(2)$ Seiberg–Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.

Description: The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems and includes a positive voltage parameter $\lambda$ . It is shown that there is a threshold value $\lambda _* 〉 0$ of the voltage parameter such that no radially symmetric stationary solution exists for $\lambda 〉 \lambda _* $ , while at least two such solutions exist for $\lambda \in (0,\lambda _* )$ . Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $\lambda 〉 \lambda _* $ .

Description: Assuming the generalized Riemann hypothesis, we prove a quantitative estimate for the number of simple zeros on the critical line for $L$ -functions attached to classical holomorphic newforms.

Description: In this paper, we consider a $\mathbb {Q}$ -Fano $3$ -fold weighted complete intersection of codimension $2$ in the $85$ families listed in Iano-Fletcher's list and determine which cycle is a maximal center or not. For each maximal center, we construct either a birational involution which untwists the maximal singularity or a Sarkisov link centered at the cycle to another explicitly described Mori fiber space. As a consequence, nineteen families are proved to be birationally rigid and the remaining $66$ families are proved to be birationally non-rigid.

Description: In this article, we consider propositional calculi, which are finitely axiomatizable extensions of intuitionistic implicational propositional calculus together with the rules of modus ponens and substitution. We give a proof of undecidability of the following problem for these calculi: whether a given finite set of propositional formulas constitutes an adequate axiom system for a fixed propositional calculus. Moreover, we prove the same for the following restriction of this problem: whether a given finite set of theorems of a fixed propositional calculus derives all theorems of this calculus. The proof of these results is based on a reduction of the undecidable halting problem for the tag systems introduced by Post.

Description: This article describes the use of continuous vector space models for reasoning with a formal knowledge base. The practical significance of these models is that they support fast, approximate but robust inference and hypothesis generation, which is complementary to the slow, exact, but sometimes brittle behaviour of more traditional deduction engines such as theorem provers. The article explains the way logical connectives can be used in semantic vector models, and summarizes the development of Predication-based Semantic Indexing, which involves the use of Vector Symbolic Architectures to represent the concepts and relationships from a knowledge base of subject-predicate-object triples. Experiments show that the use of continuous models for formal reasoning is not only possible, but already demonstrably effective for some recognized informatics tasks, and showing promise in other traditional problem areas. Examples described in this article include: predicting new uses for existing drugs in biomedical informatics; removing unwanted meanings from search results in information retrieval and concept navigation; type inference from attributes; comparing words based on their orthography; and representing tabular data, including modelling numerical values. The algorithms and techniques described in this article are all publicly released and freely available in the Semantic Vectors open-source software package. 1

Description: This article is concerned with the problem of transforming a formula of the modal μ-calculus into an equivalent guarded formula. Many procedures in the literature explicitly require their input to be guarded. There are procedures that effectively transform a formula into guarded form. Kupferman, Vardi and Wolper (2000, J. ACM, 47, 312–360) as well as Mateescu (2002, LNCS, 2280, 281–295) present such a transformation which they independently claim to produce formulas of linear, respectively, quadratic size when measured as the size of the Fischer–Ladner closure of the formula, respectively, the size of its syntax-DAG (directed acyclic graph). We first show that these claims are wrong: they can produce formulas of quadratic closure size and exponential DAG size. This also implies that both size measures can differ by up to one exponential. We then study different forms of syntax for the modal μ-calculus—vectorial form and hierarchical modal equation systems—and related size measures. We show that guarded transformation for formulas in vectorial form is as hard as solving parity games, thus unlikely to be polynomial in the DAG size. We also investigate the guarded transformation procedure by Seidl and Neumann (1999, LNCS, 1683, 484–498) which is polynomial for formulas from the modal μ-calculus, but produces genuine equation systems. Transforming these back into plain formulas is again as hard as solving parity games. Thus, it is likely to need an exponential blow-up. Based on this we argue that closure size may not be an appropriate measure for the representation of a formula. The question of whether or not polynomial guarded transformation for formulas of the modal μ-calculus is possible must be considered as still open. There is some evidence that this problem is reasonably hard.

Description: We study the following problem: determine which almost structurally complete quasivarieties are structurally complete. We propose a general solution to this problem and then a solution in the semisimple case. As a consequence, we obtain a characterization of structurally complete discriminator varieties. An interesting corollary in logic follows: Let L be a propositional logic/deductive system in the language with formulas for verum , which is a theorem, and falsum , which is not a theorem. Assume also that L has an adequate semantics given by a discriminator variety. Then L is structurally complete if and only if it is maximal. All such logics/deductive systems are almost structurally complete.

Description: Explicit substitution calculi propose solutions to the main drawback of the -calculus: substitution defined as a meta-operation in the system. By making explicit the process of substitution, the theoretical system gets closer to an eventual implementation. Furthermore, for implementation purposes, many explicit substitution systems are written with de Bruijn indices. The -calculus with de Bruijn indices, called dB , assembles each α-class of -terms in a unique term, which is more ‘machine-friendly’ than the classical version with variables. Intersection types (IT) provide finitary type polymorphism satisfying important properties like principal typing (PT), which allows the type system to include features such as data abstraction (modularity) and separate compilation. Although some explicit substitution calculi with simple type systems are well investigated, providing nice applications such as specialized implementations of higher order unification, more elaborated type systems such as IT have not been proposed/studied for these calculi. In an earlier work, we introduced IT systems for two explicit substitution calculi, and s e , conjecturing them to satisfy the basic property of subject reduction (SR), which guarantees the preservation of types during computations. In this article, we take a deeper look at these systems, providing an insight into their development which helps us construct for the first time the proofs of SR omitted before. This new result also (i) enables us to prove another new result: SR for an IT system for dB ; and (ii) allows us to introduce for the first time an IT system for the -calculus.

Description: This article provides formal accounts of dishonest attitudes of agents. We introduce a propositional multi-modal logic that can represent an agent's belief and intention as well as communication between agents. Using the language, we formulate different categories of dishonesty. We first provide two different definitions of lies and provide their logical properties. We then consider an incentive behind the act of lying and introduce lying with objectives. We subsequently define bullshit , withholding information and half-truths , and analyse their formal properties. We compare different categories of dishonesty in a systematic manner, and examine their connection to deception . We also propose maxims for dishonest communication that agents should ideally try to satisfy.

Description: Suppose that a sequence of numbers $x_n$ (a ‘signal’) is transmitted through a noisy channel. The receiver observes a noisy version of the signal with additive random fluctuations, $x_n + \xi _n$ , where $\xi _n$ is a sequence of independent standard Gaussian random variables. Suppose further that the signal is known to come from some fixed space ${\mathscr {X}}$ of possible signals. Is it possible to fully recover the transmitted signal from its noisy version? Is it possible to at least detect that a non-zero signal was transmitted? In this paper, we consider the case in which signals are infinite sequences and the recovery or detection are required to hold with probability 1. We provide conditions on the space ${\mathscr {X}}$ for checking whether detection or recovery are possible. We also analyze in detail several examples including spaces of Fourier transforms of measures, spaces with fixed amplitudes and the space of almost periodic functions. Many of our examples exhibit critical phenomena, in which a sharp transition is made from a regime in which recovery is possible to a regime in which even detection is impossible.

Description: In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in $\mathbb {Q}^n$ to any acyclic quiver $Q$ , and this simplicial fan allows one to completely determine the canonical presentation of any element in $\mathbb {Z}^n$ . This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-1 subsets of $\mathbb {Q}^n$ , each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.

Description: The $j$ -multiplicity plays an important role in the intersection theory of Stückrad–Vogel cycles, while recent developments confirm the connections between the $\epsilon$ -multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between the $j$ -multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$ -multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$ - and $\epsilon$ -multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

Description: Let $\pi : X \to Y$ be a morphism of projective varieties and suppose that $\alpha $ is a pseudo-effective numerical cycle class satisfying $\pi _{*}\alpha =0$ . A conjecture of Debarre, Jiang, and Voisin predicts that $\alpha $ is a limit of classes of effective cycles contracted by $\pi $ . We establish new cases of the conjecture for higher codimension cycles. In particular, we prove a strong version when $X$ is a fourfold and $\pi $ has relative dimension 1.

Description: Let $R$ be a group of prime order $r$ that acts on the $r'$ -group $G$ , let $RG$ be the semidirect product of $G$ with $R$ , let ${\mathbb {F}}$ be a field and $V$ be a faithful completely reducible $\mathbb {F}[{RG}]$ -module. Trivially, $C_{G}({R})$ acts on $C_{V}({R})$ . Let $K$ be the kernel of this action. What can be said about $K$ ? This question is considered when $G$ is soluble. It turns out that $K$ is subnormal in $G$ or $r$ is a Fermat or half-Fermat prime. In the latter cases, the subnormal closure of $K$ in $G$ is described. Several applications to the theory of automorphisms of soluble groups are given.

Description: The mono-epi (ME) exact structure on the morphisms of an exact category $(\mathcal {A}; \mathcal {E})$ is introduced and used to prove ideal versions of Salce's Lemma, Christensen's (Ghost) Lemma, and Wakamatsu's Lemma for an exact category. Salce's Lemma establishes a bijective correspondence $\mathcal {I} \mapsto \mathcal {I}^{\perp }$ between the class of special precovering ideals of $(\mathcal {A}; \mathcal {E})$ and that of its special preenveloping ideals. ME-extensions of morphisms are used to define an extension $\mathcal {I} \diamond \mathcal {J}$ of ideals. Christensen's Lemma asserts that the class of special precovering (respectively, special preenveloping) ideals is closed under products and extensions and that the bijective correspondence of Salce's Lemma satisfies $(\mathcal {I} \mathcal {J})^{\perp } = \mathcal {J}^{\perp } \diamond \mathcal {I}^{\perp }$ and $(\mathcal {I} \diamond \mathcal {J})^{\perp } = \mathcal {J}^{\perp } \mathcal {I}^{\perp }.$ Wakamatsu's Lemma asserts that if a covering ideal $\mathcal {I}$ is closed under ME-extensions, then it is a special precovering ideal. As an application, it is proved that if $G$ is a finite group and $\Phi $ is the ideal of phantom morphisms in the category $k[G]$ - $\rm Mod,$ then $\Phi ^{n-1}$ is the object ideal generated by projective modules, where $n$ is the nilpotency index of the Jacobson radical $J.$ If $R$ is a semiprimary ring, with $J^n =0,$ then $\Phi ^n$ is generated by projective modules. For a right coherent ring $R$ over which every cotorsion left $R$ -module has a coresolution of length $n$ by pure injective modules, $\Phi ^{n+1}$ is generated by flat modules.

Description: We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics.

Description: Let $P_{n}(x)= \sum _{i=0}^n \xi _i x^i$ be a Kac random polynomial where the coefficients $\xi _i$ are i.i.d. copies of a given random variable $\xi $ . Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a real double root. As an application, we consider the problem of estimating the number of real roots of $P_n$ , which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables $\xi $ , that the expected number of real roots of $P_n(x)$ is exactly $({2}/{\pi }) \log n +C +o(1)$ , where $C$ is an absolute constant depending on the atom variable $\xi $ . Prior to this paper, such a result was known only for the case when $\xi $ is Gaussian.

Description: The goal of this article was to study the Iwasawa theory of an abelian variety $A$ that has complex multiplication by a complex multiplication (CM) field $F$ that contains the reflex field of $A$ , which has supersingular reduction at every prime above $p$ . To do so, we make use of the signed Coleman maps constructed in our companion article [Kâzım Büyükboduk and Antonio Lei, ‘Integral Iwasawa theory of motives for non-ordinary primes’, 2014, in preparation, draft available upon request] to introduce signed Selmer groups as well as a signed $p$ -adic $L$ -function via a reciprocity conjecture that we formulate for the (conjectural) Rubin–Stark elements (which is a natural extension of the reciprocity conjecture for elliptic units). We then prove a signed main conjecture relating these two objects. To achieve this, we develop along the way a theory of Coleman-adapted rank- $g$ Euler–Kolyvagin systems to be applied with Rubin–Stark elements and deduce the main conjecture for the maximal $\mathbb {Z}_p$ -power extension of $F$ for the primes failing the ordinary hypothesis of Katz.

Description: We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way, we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold-valued rough paths. Finally, this framework is used to show that there is a one-to-one correspondence between rough paths on a $d$ -dimensional manifold and rough paths on $d$ -dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eells and Elworthy and Malliavin.

Description: We show that several important normal subgroups $\Gamma $ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure-preserving action $\Gamma \curvearrowright X$ is stably $OE$ -superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from Chifan–Ioana–Kida [‘ $W^*$ -superrigidity for arbitrary actions of central quotients of braid groups’, Math. Ann. 361 (2015) 925–959], we deduce that any free, ergodic, probability measure-preserving action of almost any surface braid group is stably $W^*$ -superrigid, that is, it can be completely reconstructed from its von Neumann algebra.

Description: Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting contractions on a finite-dimensional Hilbert space and studying their joint generalized eigenspaces. Non-tangential regularity of rational functions on the polydisk is also studied. One result states that rational inner functions on the polydisk have non-tangential limits at every point of the $n$ -torus. An algebraic characterization of higher non-tangential regularity is given. We also make some connections with the earlier material and prove that rational functions on the bidisk which are square integrable on the two-torus are non-tangentially bounded at every point. Several examples are provided.

Description: We propose a construction of a tensor exact category $\mathcal {F}_X^m$ of Artin–Tate motivic sheaves with finite coefficients $\mathbb {Z}/m$ over an algebraic variety $X$ (over a field $K$ of characteristic prime to $m$ ) in terms of étale sheaves of $\mathbb {Z}/m$ -modules over $X$ . Among the objects of $\mathcal {F}_X^m$ , in addition to the Tate motives $\mathbb {Z}/m(j)$ , there are the cohomological relative motives with compact support $\mathcal {M}_{cc}^m(Y/X)$ of varieties $Y$ quasi-finite over $X$ . Exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi-finite morphisms of varieties $Y\longrightarrow X$ act on the exact categories $\mathcal {F}_X^m$ . Assuming the existence of triangulated categories of motivic sheaves $\mathcal {D}\mathcal {M}(X,\mathbb {Z}/m)$ over algebraic varieties $X$ over $K$ and a weak version of the ‘six operations’ in these categories, we identify $\mathcal {F}_X^m$ with the exact subcategory in $\mathcal {D}\mathcal {M}(X,\mathbb {Z}/m)$ consisting of all the iterated extensions of the Tate twists $\mathcal {M}_{cc}^m(Y/X)(j)$ of the motives $\mathcal {M}_{cc}^m(Y/X)$ . An isomorphism of the $\mathbb {Z}/m$ -modules ${\rm Ext}$ between the Tate motives $\mathbb {Z}/m(j)$ in the exact category $\mathcal {F}_X^m$ with the motivic cohomology modules predicted by the Beilinson–Lichtenbaum étale descent conjecture (recently proved by Voevodsky, Rost et al. ) holds for smooth varieties $X$ over $K$ if and only if the similar isomorphism holds for Artin–Tate motives over fields containing $K$ . When $K$ contains a primitive $m$ -root of unity, the latter condition is equivalent to a certain Koszulity hypothesis, as shown in our previous paper [Positselski, ‘Mixed Artin–Tate motives with finite coefficients’, Mosc. Math. J. 11 (2011) 317–402].

Description: We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let $X$ be an arithmetic scheme (scheme of finite type over $\textbf {Z}$ ), and for a prime $p$ let $\zeta _{X_p}(s)$ be the local factor of its zeta function. We present an algorithm that computes $\zeta _{X_p}(s)$ for a single prime $p$ in time $p^{1/2+o(1)}$ , and another algorithm that computes $\zeta _{X_p}(s)$ for all primes $p 〈 N$ in time $N \log ^{3+o(1)} N$ . These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.

Description: Discovering Cellular Automata (CAs) rules able to generate a desired global behaviour is a highly challenging problem due to the local nature of rules combined with the expected global effect. This article investigates the evolution and dynamics of small-world networks for the density classification task in CAs. Both unweighted and node-weighted networks are evolved to serve as the CA topology in density classification using the majority rule over the neighbouring nodes. Allowing a weight for each node of the network offers the means to induce flexibility in the application of the majority rule. Computational experiments indicate a better performance of node-weighted networks compared with their weight-free versions for the density classification task. The best evolved networks are analysed in terms of their tolerance to dynamic network changes understood as node additions and removals. Results indicate a good performance and robustness of the obtained small-world networks for CA density problem.

Description: In this article, we consider the capacitated fixed-charge transportation problem in a two-stage supply chain network, in which the manufacturer can ship to any of the existing $$m$$ distribution centres and each of the distribution centres can ship to any of the $$n$$ customers aiming at minimizing the cost of the physical distribution flow. Due to the complexity of the problem, we propose an improved hybrid algorithm combining the Nearest Neighbour search heuristic with a powerful local search procedure. The extensive computational experiments on benchmark instances from the literature show that our hybrid algorithm is competitive in comparison with the known heuristics published to date.

Description: A popular and easy to implement classifier is the $$k$$ -Nearest Neighbour (k-NN). However, sequentially searching for nearest neighbours in large datasets leads to inefficient classification because of the high computational cost involved. This article presents an adaptive hybrid and cluster-based method for speeding up the $$k$$ -NN classifier. The proposed method reduces the computational cost as much as possible while maintaining classification accuracy at high levels. The method is based on the well-known $$k$$ -means clustering algorithm and consists of two main parts: (i) a pre-processing algorithm that builds a two-level, cluster-based data structure, and (ii) a hybrid classifier that classifies new items by accessing either the first or the second level of the data structure. The proposed approach was tested on seven real life datasets and the experiential measurements were statistically validated by the Wilcoxon signed ranks test. The results show that the proposed classification method can be used either to achieve high accuracy with slightly higher cost or to reduce the cost at a minimum level with slightly lower accuracy.

Description: Qualitative description of movements can be very important for representation and reasoning about dynamic systems which are complex in structure or whenever numerical data are incomplete or inaccessible. For this reason, we present a hybrid approach based on the combination of qualitative reasoning, quantitative data and logical methods. In this article, we introduce a new propositional dynamic logic QM for representation and reasoning with relative movements of objects. In this way, we can infer additional information about movements by using axioms and the logic apparatus. We present a sound and complete deduction system in dual tableaux style for the logic QM . The system can be used for verification of validity of formulas of the logic in question.

Description: In this article a novel encryption scheme for RGB digital images is introduced. The proposed protocol consists of two iterative phases: the confusion phase ruled by the 2D chaotic Cat map, and the diffusion phase governed by a reversible memory cellular automata over $${\mathbb{F}}_{2}^{8}$$ . This algorithm is shown to be secure against the more important cryptanalytic attacks on image encryption methods (statistical attacks, sensitivity dependence, differential attack, etc). More specifically, it is shown that: (i) the correlation coefficient of two adjacent pixels (horizontal, vertical and diagonal adjacency) of the encrypted image is approximately $$k\cdot {10}^{-3}$$ , whereas the same parameter for the original image is near 0.99; (ii) approximately the 99.5% of pixels of the encrypted image are changed when only one bit of the secret key is modified; (iii) the NPCR coefficient of the encrypted image if near 99.6%, and the UACI coefficient is about 33.3%. Moreover, it supposes an improvement of the CA-based methods to encrypt digital images using chaotic maps.

Description: Class imbalanced data constitute difficulties for most classifiers. Standard ensembles fail to sufficiently recognize the minority class. The role of classification strategies in rule set based component classifiers inside bagging is studied. We argue that introducing abstaining in bagging, i.e. allowing component classifiers to refrain from predicting class labels in ambiguous situations, improves classification of imbalanced data. Comparative experiments with 5 different strategies and 2 rule induction algorithms confirm this hypothesis.

Description: In the early design phase of steel structures, the maximum capacity of the connections is one of the critical parameters that ought to be calculated first. Determining this force by means of analytical methods usually leads to overestimations, given its highly non-linear behaviour. The finite element (FE) method, a hard computing approach, represents one of the best alternatives to obtain more realistic connection responses, except for its high costs in terms of computation time. The present study proposes the use of a hybrid artificial intelligence system to accurately predict the response of a particular type of steel connection: the bolted lap joint. The proposed hybrid system is composed of hard and soft computing components. First, a set of 800 FE simulations of different joint configurations was conducted to generate the training and testing data sets for the development of the system. Secondly, a multilayer perceptron network ensemble model was trained and tested. Additionally, a procedure based on genetic algorithms was included to optimize simultaneously both the settings of the model and the number of input variables involved in the process. The optimized ensemble model is compared to other soft computing alternatives and shows higher generalization capacity when it deals with testing data. Finally, the results support the use of this hybrid system to create prediction models with similar performance to the FE method in terms of accuracy but with a greatly reduced computational effort.

Description: This article introduces a methodology for intelligent character recognition based on a new feature extraction technique that relies on the number of vertical strokes and the relative positions of the horizontal strokes connected to them. We also propose a new grammar in order to feed these features into a Deterministic Finite Automaton (DFA) that will generate string representations of every character. For a single writer, a set of training characters let us generate a XML-formatted Knowledge Base that, in combination with an Inference Engine, will provide a system to identify unknown characters' string representations. Our approach allows the construction of Knowledge Bases and their use with other writers without any previous training. The experiments with several writers and, hence, several Knowledge Bases, show promising performance rates, especially for the recognition of characters written by the owner of the training set.

Description: Learning white box models is a very challenging task that, if successful, it extracts knowledge from a process; this learning process allows developing embedded applications with the outcome of hybrid artificial intelligent models. In previous studies, both with crisp or low-quality data, it has been shown that learning white box models using Genetic Programming (GP) and Genetic Algorithm Programming (GAP) is still penalized with several well-known problems: bloat, over fitting and population diversity. This research describes two very simple and intuitive techniques to deal with the two former problems making use of heuristics. The underlying idea is to avoid or to limit the surplus computation needed to reduce the problems' effects with simple human like rules. In the case of bloat, a heuristic to deal with incoherent node sequences is proposed; in the case of over fitting, the models are allowed a small error and those performing with higher bias are then penalized. These simple techniques are evaluated in a carefully designed test bed, which enables the analysis of their behaviour. Results show the proposed bloat preventing heuristic, which enhances the results both in the genotype and in the phenotype landscapes, while the over fitting technique slightly improves the evolutionary process.

Description: Reliability is one of the key performance factors in data centres. The out-of-scale energy costs of these facilities lead data centre operators to increase the ambient temperature of the data room to decrease cooling costs. However, increasing ambient temperature reduces the safety margins and can result in a higher number of anomalous events. Anomalies in the data centre need to be detected as soon as possible to optimize cooling efficiency and mitigate the harmful effects over servers. This article proposes the usage of clustering-based outlier detection techniques coupled with a trust and reputation system engine to detect anomalies in data centres. We show how self-organizing maps or growing neural gas can be applied to detect cooling and workload anomalies, respectively, in a real data centre scenario with very good detection and isolation rates, in a way that is robust to the malfunction of the sensors that gather server and environmental information.

Description: The heart is truly successor to the brain in being the most significant vital organ in the human body. The heart, being a magnificent pump, has its performance orchestrated via a group of valves and highly sophisticated neural control. While the kinetics of the heart are accompanied by sound production, sound waves produced by the heart are reliable diagnostic tools to check heart activity. Chronologically, several data sets have been put forward to observe heart performance and lead to medical intervention whenever necessary. The heart sounds data set utilized in this article provides researchers with an abundance of sound signals classified using different classification algorithms; neural network, rotation forest and random forest are a few that can be mentioned. This article proposes an approach based on rough sets and a local transfer function classifier for heart valve disease detection. In order to achieve this objective, and to increase the efficiency of the predication model, a Boolean reasoning discretization algorithm is introduced to discrete the heart signal data set, then the rough set reduction technique is applied to find all reducts of the data which contain the minimal subset of attributes that are associated with a class label for classification. Then, the rough sets dependency rules are generated directly from all generated reducts. A rough confusion matrix is used to evaluate the performance of the predicted reducts and classes. Finally, a local transfer function classifier was employed to evaluate the ability of the selected descriptors to discriminate whether they represent healthy or unhealthy. Alternative classifiers were applied to the same data for comparison including Support Vector Machine (SVM), Hidden Naive Bayesian Network (HNB), Bayesian Network (BN), Naive Bayesian Tree (NBT), Decision Tree (DT), Sequential Minimal Optimization (SMO), Decision Table (DT), Rotation Forest (RoF), and Random Forest (RF); however, their performance for the same diagnostic problems was lower than the proposed rough local transfer function.

Description: Logistics distribution is adaptive, dynamic and open self-organizing system, which is maintained by flows of information, materials, goods, funds and energy. This article presents biological intelligence for modelling and optimization on vehicle routing problem (VRP) of logistics distribution. The aim of this research is to create a novel hybrid model including genetic and firefly algorithms in routing heterogeneous fleet of vehicles in logistics distribution system. The proposed hybrid model in routing heterogeneous fleet VRP is tested on realistic data set in Serbian company ‘Carnex Ltd’. The experimental results of this method are compared with the empirical results and show that the experimental results obtained by new hybrid genetic–firefly model are better then empirical results in the company.

Description: We undertake a systematic study of asymptotically hereditarily aspherical (AHA) groups, the class of groups introduced by Tadeusz Januszkiewicz and the second author as a tool for exhibiting exotic properties of systolic groups. We provide many new examples of AHA groups, also in high dimensions. We relate the AHA property with the topology at infinity of a group, and deduce in this way some new properties of (weakly) systolic groups. We also exhibit an interesting property of boundaries at infinity for a few classes of AHA groups.

Description: We show that certain geometrically defined higher codimension cycles are extremal in the effective cone of the moduli space ${\overline {\mathcal M}}_{g,n}$ of stable genus $g$ curves with $n$ ordered marked points. In particular, we prove that codimension 2 boundary strata are extremal and exhibit extremal boundary strata of higher codimension. We also show that the locus of hyperelliptic curves with a marked Weierstrass point in ${\overline {\mathcal M}}_{3,1}$ and the locus of hyperelliptic curves in ${\overline {\mathcal M}}_4$ are extremal cycles. In addition, we exhibit infinitely many extremal codimension 2 cycles in ${\overline {\mathcal M}}_{1,n}$ for $n\geq 5$ and in ${\overline {\mathcal M}}_{2,n}$ for $n\geq 2$ .