ALBERT

Description: The tension between the meaning of causality in science and law or public policy is well-known; however, defendants in product liability cases or industries that might be affected by a government regulation may try to convince the factfinder to require evidence of a causal relationship that meets the standards of science. From the perspective of public health, however, people may be exposed unnecessarily to a health risk during the time period between the establishment of reasonably strong evidence of a causal relationship and the overwhelming evidence required for scientific causality. The Bayesian paradigm enables one to update information from epidemiologic studies as they accumulate, providing estimates of the probability that the relative risk of a particular harm from exposure exceeds a threshold value, e.g. 2.0 or 4.0 that is sufficient to meet the preponderance of the evidence standard or to support a health initiative. In order to diminish the role of the initial prior distribution, which may be quite subjective, the first case-control study or an analysis of adverse event and case reports is used to determine two prior distributions. One is the most favourable to the defendant, or industry that might be regulated, which is consistent with the previous data. The other is centred on or near the estimated relative risk from the first study. The method is applied to the studies that linked aspirin use to Reye syndrome and demonstrates that the evidence of a causal association was sufficiently strong in 1982, when the Food and Drug Administration first proposed that the public be warned of the risk, to support the regulation. Thus, lives would have been saved had the warning been given at the end of 1982 rather than in early 1985.

Description: In law, inferences of causation are sometimes made through a structured process in which multiple participants play various roles, and make decisions concerning various logical components of the overall inference (such as legal rules, policy objectives, presumptions, evidence, burdens of proof and findings of fact). This article illustrates such a process using empirical research into compensation decisions in the USA for injuries allegedly caused by vaccinations. Empirical research into actual legal processes is essential, in order to discover how various players approach their sub-tasks of decision-making. It also provides insights for areas outside of law, such as non-monotonic logic, cognitive science, sociology and artificial intelligence.

Description: Situations of causal factual uncertainty are relatively common in law. The problems and difficulties regarding ‘factual causation’ in law point to the need of ‘evidence’ and ‘proof’ models that are adequate and capable to accommodate the tests and methodologies used to explain and demonstrate it in a legal context. Given the configuration of the situations of causal factual uncertainty and the available ‘evidence’ and ‘proof’ models, I argue that it is justified to use an ‘argumentative-narrative’ model for ‘proving causation’ in law. However, considering that each model of ‘evidence’ and ‘proof’ reveals a different kind of ‘rationality’ that can still be viewed in different ways, I also argue that we must try to match the perspective we have on the ‘rationality’ behind the chosen model of ‘evidence’ and ‘proof’ with the ‘rationality’ underlying ‘causation’ in law.

Description: At least in some cases, the values confronted in legal decision-making appear to be incommensurable. Some legal theorists resist incommensurability because they fear that this presents an overwhelming obstacle to rational decision-making. By offering a close analysis of proportionality and, more particularly, measures of proportional value satisfaction, I show that this fear is unfounded. Comparative measures of proportional value satisfaction do not require the values to be commensurable. However, assuming incommensurability presents us with the problem of public significance in the proportional satisfaction of values. When two values are commensurable, this public significance is provided by the mediating effects of the overarching third value that provides the common measure of the values. However, when this common measure is removed, then the public significance of value satisfaction must be otherwise achieved. This is why I propose an equal proportional value satisfaction as the most appropriate proportionality maximand. Under equal proportional value satisfaction, the proportional satisfaction of any one value has significance for each and every other value. This kind of public significance is interpersonal rather than impersonal (or second-personal rather than third-personal). The article then shows that the legal process that is most appropriate to equal proportionality is a process that implements defeasible legal rules.

Description: In order to allocate the risk between parties in legal adjudication, we use evidentiary techniques with the main device among them being the standard of proof (SoP). The traditional view holds the grade of probability to be the parameter that shifts when moving to different standards. However, as soon as we dig slightly deeper, an incoherent picture is being revealed. In this article, I challenge the accepted view and try to show that it faces insurmountable problems concerning the rationality, the grammatical consistency and the impact of the SoP for the acceptability of verdicts. At the end of the article, I shortly discuss the theory of epistemological contextualism and propose a framework that allows rational distinctions to be drawn between different standards of proof. In the second part of this project (forthcoming), I will defend a contextualist view according to which shifting parameter is not the grade of (aleatory) probability, but instead the Set of Epistemic Defeaters in play.

Description: The aim of this paper is to study the relations between the Hausdorff dimensions of k -quasilines and the theory of extremal quasiconformal mappings. We show that there is an open and dense subset (Strebel points) of the universal Teichmüller space T (H) such that, for every [ f ] in the set, the Hausdorff dimension of the k -quasiline determined by [ f ] is strictly less than 1 + k 2 . We also show that there are some points [ f ] != [id] outside the open and dense set in the universal Teichmüller space such that the Hausdorff dimension of the quasiline determined by [ f ] is 1. Moreover, some results on the Hausdorff dimensions of the quasilines varying in the asymptotic Teichmüller space are also obtained.

Description: A new uniqueness result for a general n th order differential equation is obtained. We show that some previous results follow immediately from our theorem.

Description: Consider a discrete uniformly elliptic divergence form equation on the d ≥3 dimensional lattice Z d with random coefficients. In Conlon and Spencer [ Trans. Amer. Math. Soc. , http://www.math.lsa.umich.edu/~conlon/paper/hom10.pdf ], rate of convergence results in homogenization and estimates on the difference between the averaged Green's function and the homogenized Green's function for random environments which satisfy a Poincaré inequality were obtained. Here, these results are extended to certain environments in which correlations can have arbitrarily small power law decay. These environments are simply related via a convolution to environments which do satisfy a Poincaré inequality.

Description: In this paper, we prove that, for any a , M N with ( a , M ) = 1, there are infinitely many Carmichael numbers m such that m a mod M .

Description: We give a topological analogue for openness of a criterion for flatness that originates with Auslander. Over a normal base of dimension n , failure of openness is detected by a vertical component in the n th fibred power of the morphism.

Description: Investigated are regular maps from real algebraic varieties into real Fermat varieties. It is proved that under some natural assumptions, all such maps are null homotopic.

Description: Let ( M , g ) be a complete non-compact Riemannian manifold. We consider operators of the form g + V , where g is the non-negative Laplacian associated with the metric g , and V a locally integrable function. Let be a Riemannian covering, with Laplacian g and potential . If the operator + V is non-negative on ( M , g ), then the operator is non-negative on . In this note, we show that the converse statement is true provided that is a co-amenable subgroup of 1 ( M ).

Description: Peak interpolation is concerned with a foundational kind of mathematical task: building functions in a fixed algebra A , which have prescribed values or behaviour on a fixed closed subset (or on several disjoint subsets). In this paper, we do the same but now A is an algebra of operators on a Hilbert space. We briefly survey this noncommutative peak interpolation , which we have studied with coauthors in a long series of papers, and whose basic theory now appears to be approaching its culmination. This programme developed from, and is based partly on, theorems of Hay and Read whose proofs were spectacular, but therefore inaccessible to an uncommitted reader. We give short proofs of these results, using recent progress in noncommutative peak interpolation, and conversely give examples of the use of these theorems in peak interpolation. For example, we prove a useful new noncommutative peak interpolation theorem.

Description: We show how a parameterized family of maps of the spine of a manifold can be used to construct a family of homeomorphisms of the ambient manifold which have the inverse limits of the spine maps as global attractors. We describe applications to unimodal families of interval maps, to rotation sets, and to the standard family of circle maps.

Description: For every prime p , we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil–Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q that is not divisible by 4 in the Weil–Châtelet group. This gives a negative answer to a question of Cassels.

Description: We show that at least of the zeros of the Riemann zeta-function are simple, assuming the Riemann hypothesis. This was previously established by Conrey, Ghosh and Gonek [ Proc. London Math. Soc. 76 (1998) 497–522] under the additional assumption of the generalized Lindelöf hypothesis. We are able to remove this hypothesis by careful use of the generalized Vaughan identity.

Description: We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative-definite 4-manifold with trivial first homology and bounded second Betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.

Description: We observe that E -resultant of a very ample rank 2 vector bundle E on a real projective curve (with no real points) is nonnegative when restricted to the space of real sections. Moreover, we show that if E has a section vanishing at exactly two points and the degree d of E satisfies d ( d – 6) ≥ 4( g – 1), then this polynomial cannot be written as a sum of squares.

Description: Let G be one of the Ricci-flat holonomy groups SU( n ), Sp( n ), Spin(7) or G 2 , and M a compact manifold of dimension 2 n , 4 n , 8 or 7, respectively. We prove that the natural map from the moduli space of torsion-free G -structures on M to the moduli space of Ricci-flat metrics is open, and that the image is a smooth manifold. For the exceptional cases G = Spin(7) and G 2 , we extend the result to asymptotically cylindrical manifolds.

Description: We improve the results of Booker and Krishnamurthy ( Compos. Math. 147 (2011) 669–715) by allowing restricted sets of poles among the unramified twists. This allows for a clean statement of the GL(2) converse theorem which includes all cases of Eisenstein series.

Description: Let G be a solvable subgroup of the automorphism group Aut( X ) of a compact Kähler manifold X of complex dimension n , and let N ( G ) be the normal subgroup of G consisting of elements with null entropy. Let us denote by G * the image of G under the natural map from Aut( X ) to GL( V , R ), where V is the Dolbeault cohomology group H 1, 1 ( X , R ). Assume that the Zariski closure of G * in GL( V C ) is connected. The main aim of this paper is to show that, when the rank r ( G ) of the quotient group G / N ( G ) is equal to n – 1 and the identity component of Aut( X ) is trivial, then the normal subgroup N ( G ) of G is finite. This affirmatively answers a question in Invent. Math. posed by D.-Q. Zhang.

Description: This paper deals with the boundary control problems for the system described by first-order hyperbolic partial integro-differential equations with nonlocal boundary conditions by means of the deformation formulas. It is shown that the associated operator with nonlocal terms generates a C 0 -semigroup and the system defines an abstract boundary control system in L 2 -space. On the basis of the construction of integral kernel function, the ordinal and evolutional deformation formulas are established for a general class of Volterra integro-differential equations with nonlocal terms. By applying a special form of the formulas, a design of feedback boundary control law is given, which makes the scalar boundary control system null controllable in finite time. Furthermore, the null controllability result is extended to coupled nonsymmetric first-order hyperbolic systems by using appropriate linear transformations. Applications to physically important coupled systems are also given.

Description: Arguments need to be judged against other arguments. The decision to accept or reject an argument is generally a global decision that involves examining the same question for other arguments that oppose or can defend the argument in question. This article presents the acceptability semantics for abstract argumentation that through a recursive definition gives a global assignment of the acceptable and non-acceptable subsets of arguments. This semantics stems from the aim to formalize directly the generally accepted intuition that: ‘An argument can be accepted if and only if all its challenging arguments can be rejected.’ The acceptability semantics tightly integrates the notion of defending against a challenging argument by counter-attacking it with the notion of self-defeating (or self-rejecting) arguments that (help to) bring about their own non-acceptability. The proposal is motivated by earlier studies of the semantics of Logic Programming (LP) in terms of argumentation, where the basic well founded and stable model semantics of LP can be uniformly captured using a recursively defined argumentation semantics for Negation as Failure and where these standard semantics of LP can be further extended through argumentation.

Description: This article is concerned with the design and analysis of polynomial time algorithms for determining whether a Planar Quantified Integer Program (PQIP) is feasible. A PQIP can be described briefly as an integer program involving two variables, in which each variable can be either universally or existentially quantified. There are four types of PQIPs, depending on how the variables are quantified (existentially or universally). In this article, we present two new, simple, and efficient algorithms for the case as well as a detailed account of the complexity of the other cases. Moreover, we discuss certification with respect to the provided algorithms.

Description: In the present article, the quantifiers over propositions are first introduced into the language for reasoning about probability, then the complexity issues for validity problems dealing with the corresponding hierarchy of probabilistic sentences are investigated. We prove, among other things, the $${\Pi }_{1}^{1}$$ -completeness for the general validity and also indicate the least level in the hierarchy for which the validity problem is undecidable.

Description: We use mosaics to provide a simple, sound, complete and terminating tableau reasoning procedure for the temporal logic of until and since over general linear time.

Description: The aim of what semantic science is to have scientific ontologies, data and hypotheses represented and published in machine understandable forms that enable predictions on new cases. There is much work on developing scientific ontologies and representing scientific data in terms of these ontologies. The next step is to publish hypotheses that can make (probabilistic) predictions on the published data and can be used for prediction on new cases. The published data can be used to evaluate hypotheses. To make a prediction in a particular case, hypotheses are combined to form models. This article considers feature-based semantic science where the data and new cases are described in terms of features. A prediction for a new case is made by building a model made up of hypotheses that fit together, are consistent with the ontologies used, and are adequate for the case. We give some desiderata for such models, and show how the construction of such models is a form of abduction. We provide a definition for models that satisfies these criteria and prove that it produces a coherent probability distribution over the values of interest.

Description: In this article we present methods of transition from one perspective on logic to others, and apply this in particular to obtain a coalgebraic presentation of logic. The central ingredient in this process is to view consequence relations as morphisms in a category.

Description: We consider the expressive power of the first-order structure 〈, C 〉 where is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation ‘Region x is in contact with region y .’ We prove two main theorems: Let $$\mathcal{P}$$ [Q] be the domain of bounded, non-empty, rational polyhedra in two- or three-dimensional Euclidean space. A relation over $$\mathcal{P}$$ [Q] is definable in the structure 〈 $$\mathcal{P}$$ [Q], C 〉 if and only if is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions. Let $$\mathcal{R}$$ be the space of bounded, non-empty, closed regular regions in n -dimensional Euclidean space. Any analytical relation over lower dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈 $$\mathcal{R}$$ , C 〉.

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: This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The subject is central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither self-adjoint nor elliptic. To help deal with such impediments, earlier studies have established the well-posedness of an elastodynamic ITP under notably restrictive assumptions on the contrast in elastic and mass density parameters between the scatterer and the background solid. Due to lack of self-adjointness of the problem, these analyses were further successful in substantiating the discreteness of the relevant eigenvalue spectrum but not its existence. The aim of this work is to provide a systematic treatment of the ITP for elastic bodies that transcends the limitations of earlier analyses. Considering a broad range of material-contrast configurations, this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established via a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems.

Description: Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Description: In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.

Description: In this paper, we extend certain key results from the classical theory of isotropic elasticity to the generalized theory of elasticity for decagonal quasicrystaline composites. These results include: (i) the dependence of the solution on the number of elastic constants, (ii) Green's functions for bimaterials consisting of two bonded half-planes, (iii) Green's functions for a circular elastic inclusion, (iv) the oscillatory singular stress field in the vicinity of an interface crack tip and (v) the inverse problem corresponding to the design of harmonic shapes.

Description: The note develops an approximate approach to mixed boundary value problems in linear elasticity starting from an explicit asymptotic model for the Rayleigh surface wave. It is demonstrated that the original vector mixed problem may be reduced to a scalar problem for the Laplace equation. As an illustration, the steady-state motion of a rigid stamp is analysed. Comparison of asymptotic and exact results is presented.

Description: We show the existence and uniqueness of a solution for the nonlocal vector-valued Allen–Cahn variational inequality in a formulation involving Lagrange multipliers for local and nonlocal constraints. Furthermore, we propose and analyse a primal–dual active set (PDAS) method for local and nonlocal vector-valued Allen–Cahn variational inequalities. The local convergence behaviour of the PDAS algorithm is studied by interpreting the approach as a semismooth Newton method and numerical simulations are presented demonstrating its efficiency.

Description: In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to nonuniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the main application of our new theory, we will consider the wave equation in exterior domains which is formulated as a retarded boundary integral equation.

Description: We address the error control of Galerkin discretization (in space) of linear second-order hyperbolic problems. More specifically, we derive a posteriori error bounds in the L ( L 2 ) norm for finite element methods for the linear wave equation, under minimal regularity assumptions. The theory is developed for both the space-discrete case and for an implicit fully discrete scheme. The derivation of these bounds relies crucially on carefully constructed space and time reconstructions of the discrete numerical solutions, in conjunction with a technique introduced by Baker (1976, Error estimates for finite element methods for second-order hyperbolic equations. SIAM J. Numer. Anal. , 13 , 564–576) in the context of a priori error analysis of Galerkin discretization of the wave problem in weaker-than-energy spatial norms.

Description: The surface finite element method can be used to approximate curvatures on embedded hypersurfaces and to discretize geometric partial differential equations. In this paper, we present a definition of discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension based on the discretization of a weak formulation with isoparametric finite elements. We prove that for a piecewise quadratic approximation of a two- or three-dimensional hypersurface R n +1 , this definition approximates the Ricci curvature of with a linear order of convergence in the L 2 ( ) norm. By using a smoothing scheme in the case of a piecewise linear approximation of , we still get a convergence of order 2/3 in the L 2 ( ) norm and of order 1/3 in the W 1, 2 ( ) norm.

Description: We give general conditions which guarantee that the sequence generated by a descent algorithm converges to an equilibrium point. The convergence result is based on the Lojasiewicz gradient inequality; optimal convergence rates are also derived, as well as a stability result. We show how our results apply to a large variety of standard time discretizations of gradient-like flows. Schemes with variable time step are considered and optimal conditions on the maximal step size are derived. Applications to time and space discretizations of the Allen–Cahn equation, the sine–Gordon equation and a damped wave equation are given.

Description: This paper presents quadratic finite-volume methods for elliptic and parabolic problems on quadrilateral meshes that use Barlow points (optimal stress points) for dual partitions. Introducing Barlow points into the finite-volume formulations results in better approximation properties at the cost of loss of symmetry. The novel ‘symmetrization’ technique adopted in this paper allows us to derive optimal-order error estimates in the H 1 - and L 2 -norms for elliptic problems and in the L ( H 1 )- and L ( L 2 )-norms for parabolic problems. Superconvergence of the difference between the gradients of the finite-volume solution and the interpolant can also be derived. Numerical results confirm the proved error estimates.

Description: A linear parabolic differential equation on a moving surface is discretized in space by evolving-surface finite elements and in time by backward difference formulas (BDFs). Using results from Dahlquist's G-stability theory and Nevanlinna & Odeh's multiplier technique together with properties of the spatial semidiscretization, stability of the full discretization is proved for BDF methods up to order 5 and optimal-order convergence is shown. Numerical experiments illustrate the behaviour of the fully discrete method.

Description: In this article, we develop the a priori and a posteriori error analysis of hp -version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain R d , d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm, which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp -adaptive refinement algorithm.

Description: In recent years, it has become increasingly clear that the critical issue in gradient methods is the choice of the step length, whereas using gradient as the search direction may lead to very effective algorithms, whose surprising behaviour has only been partially explained, mostly in terms of the spectrum of the Hessian matrix. On the other hand, the convergence of the classical Cauchy steepest descent (SD) method has been analysed extensively and related to the spectral properties of the Hessian matrix, but the connection with the spectrum of the Hessian has not been exploited much to modify the method in order to improve its behaviour. In this work, we show how, for convex quadratic problems, moving from some theoretical properties of the SD method, second-order information provided by the step length can be exploited to dramatically improve the usually poor practical behaviour of this method. This allows us to achieve computational results comparable with those of the Barzilai and Borwein algorithm, with the further advantage of monotonic behaviour.

Description: I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.

Description: In Plato's Phaedrus , Socrates offers two speeches, the first portraying madness as mere disease, the second celebrating madness as divine inspiration. Each speech is correct, says Socrates, though neither is complete. The two kinds of madness are like the left and right sides of a living body: no account that focuses on just one half can be adequate. In a recent paper, Hugh Benson gives a left-handed speech about a psychic condition endemic among mathematicians: dianoia . Benson acknowledges that his account is one-sided, but only hints at the virtues of right-handed dianoia . This note sketches a somewhat fuller picture.

Description: In a recent paper, Aidan Lyon and Mark Colyvan have proposed an explanation of the structure of the bee's honeycomb based on the mathematical Honeycomb Conjecture. This explanation has instantly become one of the standard examples in the philosophical debate on mathematical explanations of physical phenomena. In this critical note, I argue that the explanation is not scientifically adequate. The reason for this is that the explanation fails to do justice to the essentially three-dimensional structure of the bee's honeycomb.

Description: Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can explain the consistency of T and suggest that an analogous explanation can be provided for the consistency of Peano arithmetic.

Description: In this paper I examine some difficulties with the argument presented as a topological sorites in Z. Weber and M. Colyvan, ‘A topological sorites’, Journal of Philosophy 107, 311–325. In particular, I suggest that the argument may be used to support the claim that sorites-type paradoxes cannot arise in a cohesive environment.

Description: Mistaken reasons for thinking diagrammatic proofs aren't rigorous are explored. The main result is that a confusion between the contents of a proof procedure (what's expressed by the referential elements in a proof procedure) and the unarticulated mathematical aspects of a proof procedure (how that proof procedure is enabled) gives the impression that diagrammatic proofs are less rigorous than language proofs. An additional (and independent) factor is treating the impossibility of naturally generalizing a diagrammatic proof procedure as an indication of lack of rigor.

Description: This paper examines the ontological commitments of the second-order language of arithmetic and argues that they do not extend beyond the first-order language. Then, building on an argument by George Boolos, we develop a Tarski-style definition of a truth predicate for the second-order language of arithmetic that does not involve the assignment of sets to second-order variables but rather uses the same class of assignments standardly used in a definition for the first-order language.

Description: This note has three goals. First, we discuss a presentation of Bertrand's paradox in a recent issue of Philosophia Mathematica , which we believe to be a subtle but important misinterpretation of the problem. We compare claims made about Bertrand with his 1889 Calcul des Probabilités . Second, we use this source to understand Bertrand's true intention in describing what we now call his paradox, comparing it both to another problem he describes in the same section and to a modern treatment. Finally, we briefly consider the importance of knowing when a specific example represents a general case.

Description: This article focuses on the question of how decision makers with no relevant scientific background can (if at all) legitimately evaluate conflicting scientific expert testimonies and determine their relative reliability. Sceptics argue that non-experts can never reach justifiable conclusions regarding the merits of conflicting expert testimonies because they lack the fundamental epistemic capacity to make such judgement calls. In this article, I draw on works on epistemology, philosophy of practical reasoning, philosophy of science, science and technology studies, and legal theory in order to scrutinize recent proposals to solve the problem of conflicting scientific expert testimonies. Addressing this question is of ultimate importance due to the idea that immanent in the idea of rule-of-law there is an intellectual due process norm, which articulates that epistemically arbitrary legal decisions are also not legally justified. This article is divided into two Sections. In Section 2 , I describe the basic philosophical inquiries underlying the debate about expert testimony. In particular, I first elaborate on the philosophy of testimony and its epistemic justifications, then move to the idea of epistemic deference, and finish with philosophical accounts of expertise. Section 3 presents the problem of conflicting scientific expert testimonies and analyses recent attempts to solve it as formulated by Ward Jones, Alvin Goldman and Scott Brewer. I argue that there is no single criterion (or set of criteria) upon which the non-expert could rely in order to make a rationally justified decision in each and every case in which he faces conflicting scientific expert testimonies. The alternative view here defended is to stop looking for an epistemic panacea and accept the idea that testimonial reliability operates differently within different kinds of testimony—and differently within the same kind of testimony at different times.

Description: The so-called inductive McKay condition on finite simple groups, due to Isaacs–Malle–Navarro, has recently been reformulated by Späth. We show that this reformulation applies to the reduction theorem for Alperin's weight conjecture, due to Navarro–Tiep. This also simplifies the checking of the inductive condition for Alperin's weight conjecture in the case of simple groups of Lie type with regard to the defining prime.

Description: We construct a family of minimal smooth surfaces of general type with K 2 = 3 and p g = 0, which are finite (Z/2Z) 2 -covers of the 4-nodal cubic surface. This turns out to be a five-dimensional subfamily of the six-dimensional family constructed by Mendes Lopes and Pardini, which realizes the Keum–Naie surfaces with K 2 = 3 as degenerations. We show that the base of the Kuranishi family of a general surface in our subfamily is smooth. We prove that the closure of the corresponding subset of the Keum–Naie–Mendes Lopes–Pardini surfaces is an irreducible component of the Gieseker moduli space. As an important byproduct, it is shown that, for the surfaces in this irreducible component, the degree of the bicanonical map can only be 2 or 4.

Description: We show that the isometry group of the bounded Urysohn space is a simple group.

Description: In this paper we study Koszul cohomology and the Green and Prym–Green conjectures for canonical and Prym-canonical binary curves. We prove that, if property N p holds for a canonical or a Prym-canonical binary curve of genus g , then it holds for a generic canonical or Prym-canonical binary curve of genus g + 1. We also verify the Green and Prym–Green conjectures for generic canonical and Prym-canonical binary curves of low genus (6 ≤ g ≤ 15, g != 8 for Prym-canonical and 3 ≤ g ≤ 12 for canonical).

Description: We construct an infinite-dimensional Banach space of continuous functions C ( K ) such that every one-to-one operator T : C ( K )-〉 C ( K ) is onto.

Description: Adaptive sliding-mode strategies of first and second sliding orders are developed for single-input single-output systems of the first and second relative degrees, respectively. Since the only concrete known uncertainty bounds are assumed to be the upper bounds of logarithmic derivatives, no standard sliding-mode technique can solve the problem.

Description: This paper investigates the robust guaranteed cost control problem of uncertain stochastic hybrid complex delayed networks with Markovian jump parameters and external stochastic disturbances. The main goal of this paper is to design state-feedback controllers such that the complex networks achieve mean-square stability, and meanwhile the optimal upper bound of cost function is guaranteed. Based on Lyapunov–Krasovskii stability theory and Dynkin's formula, a sufficient condition for the existence of the optimal guaranteed cost control law is given in terms of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed method.

Description: This paper investigates the forced second-order consensus problem in directed networks with time delays. First, a general forced consensus protocol is studied and some sufficient and necessary conditions on control parameters are obtained to guarantee the forced consensus; based on these conditions, several convenient methods for choosing parameters are provided. Then, by analysing the poles of the closed-loop system, the delay sensitivity of the protocol is investigated. The maximal upper bound of the allowable delays is provided. Simulation examples are presented to illustrate the effectiveness of the theoretical results.

Description: Let A + be the ring of Laplace transforms of complex Borel measures on R with support in [0,+) which do not have a singular nonatomic part. We compare the -metric d A + for stabilizable plants over A + given in Ball & Sasane (2012, Complex Analysis and Operator Theory , 6 , 65–89). with yet another metric d H | A + , namely the one induced by the metric d H for the set of stabilizable plants over H given in Sasane (2012, Mathematical Control and Related Fields , 2 , 29–44). Both d A + and d H coincide with the classical Vinnicombe metric defined for rational transfer functions, but we show here by means of an example that these two possible extensions of the classical -metric for plants over A + do not coincide on the set of stabilizable plants over A + . We also prove that they nevertheless give rise to the same topology on stabilizable plants over A + .

Description: The problems of delay-dependent robust stability and stabilization for discrete singular time-delay systems with norm-bounded uncertainties are investigated in this paper. By introducing some non-positive items to compensate the redundant energy caused by the positive items, an improved Lyapunov functional, namely partial non-positive (PNP) Lyapunov functional, is presented for achieving less conservative results. On the basis of the forward difference of this PNP Lyapunov functional, modified strict linear matrix inequality (LMI)-based delay-dependent conditions are established for the singular time-delay system to be regular, causal and stable. By solving these LMIs, an explicit expression for the desired state-feedback control law is obtained. Finally, numerical examples are given to illustrate that the proposed results are effective and less conservative than the existing ones.

Description: argumentation frameworks nowadays provide the most popular formalization of argumentation on a conceptual level. Numerous semantics for this paradigm have been proposed, whereby the cf2 semantics has shown to solve particular problems concerned with odd-length cycles in such frameworks. Due to the complicated definition of this semantics it has somehow been neglected in the literature. In this article, we introduce an alternative characterization of the cf2 semantics which, roughly speaking, avoids the recursive computation of subframeworks. This facilitates further investigation steps, like a complete complexity analysis. Furthermore, we show how the notion of strong equivalence can be characterized in terms of the cf2 semantics. In contrast to other semantics, it turns out that for the cf2 semantics strong equivalence coincides with syntactical equivalence. We make this particular behaviour more explicit by defining a new property for argumentation semantics, called the succinctness property. If a semantics satisfies the succinctness property, then for every framework F , all its attacks contribute to the evaluation of at least one framework F ' containing F . We finally characterize strong equivalence also for the stage and the naive semantics. Together with known results these characterizations imply that none of the prominent semantics for abstract argumentation, except the cf2 semantics, satisfies the succinctness property.

Description: In this paper, we investigate the advantages of the 2D Navier–Stokes–Voight (NSV) turbulence model for use in algorithms and explore its limits in the context of image inpainting. We begin by giving a brief review of the work of Bertalmio et al. in 2001 when an elegant analogy between the image intensity function for the image inpainting problem and the stream function in 2D incompressible fluid was established. An approximate solution to the inpainting problem was then obtained by numerically approximating the steady-state solution of the 2D Navier–Stokes vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics. Recently, the 3D NSV model of viscoelastic fluid was suggested by Cao et al. as an inviscid regularization to the 3D Navier–Stokes equations (NSEs). We give some background on the NSV model, describe why it is a good candidate sub-grid scale turbulence model and then we propose this model as an alternative partial differential equation for image inpainting. We describe an implementation of the inpainting procedure using the NSV model and then present numerical results comparing the images obtained when using the NSE versus the NSV model. Our results show that the NSV model allows for a larger time step to converge to the steady-state solution, yielding a more efficient numerical process when automating the inpainting process.We compare the quality of the resulting images using a subjective measure (human evaluation) and an objected measure (by calculating the peak signal-to-noise ratio). We also present some new theoretical results based on energy methods comparing the sufficient conditions for numerical stability for the two model equations. These theoretical and numerical studies shed some light on what can be expected from this category of approach when automating the inpainting problem.

Description: On the basis of the general non-linear theory of a hyperelastic material with initial stress, initially without consideration of the origin of the initial stress, we determine explicit expressions for the stress-dependent tensor of incremental elastic moduli. In considering three special cases of initial stress within the general framework, namely hydrostatic stress, uniaxial stress and planar shear stress, we then elucidate in general form the dependence of various elastic moduli on the initial stress. In each case, the effect of initial stress on the wave speed of homogeneous plane waves is studied and it is shown how various special theories from the earlier literature fit within the general framework. We then consider the situation in which the initial stress is a pre-stress associated with a finite deformation and, in particular, we discuss the specialization to the second-order theory of elasticity and highlight connections between several classical approaches to the topic, again with special reference to the influence of higher-order terms on the speed of homogeneous plane waves. Some discrepancies arising in the earlier literature are noted.

Description: In this paper, we introduce a macroscopic model for road traffic accidents along highway sections. We discuss the motivation and the derivation of such a model, and we present its mathematical properties. The results are presented by means of examples where a section of a crowded one-way highway contains in the middle a cluster of drivers whose dynamics are prone to road traffic accidents. We discuss the coupling conditions and present some existence results of weak solutions to the associated Riemann Problems. Furthermore, we illustrate some features of the proposed model through some numerical simulations.

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: In recent years, there has been an enormous interest in developing methods for the approximation of manifold-valued functions. In this paper, we focus on the manifold of symmetric positive-definite (SPD) matrices. We investigate the use of SPD-matrix means to adapt linear positive approximation methods to SPD-matrix-valued functions. Specifically, we adapt corner-cutting subdivision schemes and Bernstein operators. We present the concept of admissible matrix means and study the adapted approximation schemes based on them. Two important cases of admissible matrix means are treated in detail: the exp–log and the geometric matrix means. We derive special properties of the approximation schemes based on these means. The geometric mean is found to be superior in the sense of preserving more properties of the data, such as monotonicity and convexity. Furthermore, we give error bounds for the approximation of univariate SPD-matrix-valued functions by the adapted operators.

Description: We present a mass-preserving scheme for the stochastic nonlinear Schrödinger equation with multiplicative noise of Stratonovich type. It is a splitting scheme and we present an explicit formula for solving the sub-step related to the nonlinear part. The scheme is unconditionally stable in the L 2 norm. For the linear stochastic Schrödinger equation, we prove that the scheme has a strong convergence rate in time equal to 1, which is not common for stochastic partial differential equations with noise depending on space and time.

Description: A numerical scheme for the approximation of the elastic flow of inextensible curves is devised and convergence of approximations to exact solutions of the nonlinear time-dependent partial differential equation is proved. The nonlinear, pointwise constraint of local length preservation is linearized about a previous solution in each time step which leads to a sequence of linear saddle-point problems. The spatial discretization is based on piecewise Bézier curves and the resulting semiimplicit scheme is unconditionally stable and convergent.

Description: We study the coercivity properties and the norm dependence on the wave-number k of certain regularized combined field boundary integral operators that we recently introduced for the solution of two- and three-dimensional acoustic scattering problems with Neumann boundary conditions. We show that in the case of circular and spherical boundaries, our regularized combined field boundary integral operators are L 2 coercive for large enough values of the coupling parameter, and that the norms of these operators are bounded by constant multiples of the coupling parameter. We establish that the norms of the regularized combined field boundary integral operators grow modestly with the wave-number k for smooth boundaries and we provide numerical evidence that these operators are L 2 coercive for two-dimensional starlike boundaries. We present and analyse a fully discrete collocation (Nyström) method for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions based on regularized combined field integral equations. In particular, for analytic boundaries and boundary data, we establish pointwise superalgebraic convergence rates of the discrete solutions.

Description: The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited to highly oscillatory integrals because they attain optimal asymptotic order. We show that, for the Fourier oscillator, this approach yields the numerical method of steepest descent, a method with optimal asymptotic order that has previously been proposed for this class of integrals. However, the approach readily extends to more general kernels, such as Bessel functions that appear as the kernel of the Hankel transform.

Description: We consider anisotropic Allen–Cahn equations with interfacial energy induced by an anisotropic surface energy density . Assuming that is positive, positively homogeneous of degree 1, strictly convex in tangential directions to the unit sphere and sufficiently smooth, we show the stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semiimplicit time discretizations of the double-well potential. In the semiimplicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer invariant under rotation of the initial configuration for a fixed orientation of the anisotropy.

Description: The criterion of bioequivalence of two drugs in infringement cases may differ from the requirements used for drug approval by the FDA. In Adams v . Perrigo , 1 the Federal Circuit examined three different sets of criteria for judging bioequivalence. The statistical properties of those criteria are explored and evaluated. Our results support the appellate court’s decision to impose less stringent requirements for bioequivalence in infringement cases.

Description: The North Carolina Racial Justice Act allows defendants to submit statistical studies of prosecutorial actions pertaining to their seeking the death penalty or in making peremptory challenges. These studies may consider data from four geographical regions: the state, county, judicial division or prosecutorial district. A study of the effect of race on peremptory challenges in death penalty cases demonstrating statistically significant disparities disadvantaging Black defendants has been submitted in several cases. This comment shows that a more appropriate statistical analysis yields much stronger statistical evidence that race entered into the peremptory challenge process in Randolph County than the affidavit submitted by the authors of the study. A subsequent sensitivity analysis indicates that in order for a characteristic to explain the highly statistically significant disparity, it would need to increase the odds of an individual being challenged by a factor of three and more than twice as many Black venire members would need to possess that characteristic as non-Blacks. Since the data examined excluded potential jurors who had been removed for cause, it may be difficult for the state to find a legitimate reason justifying the racial disparity.

Description: This article critically evaluates experiments used to justify inferences of specific source attribution (‘individualization’) to ‘100% certainty’ and ‘near-zero’ rates of error claimed by firearm toolmark examiners in court testimonies, and suggests approaches for establishing statistical foundations for firearm toolmarks practice that two recent National Academy of Science reports confirm do not currently exist. Issues that should be considered in the earliest stages of statistical foundational development for firearm toolmarks are discussed.

Description: Much debate exists between Frequentist and Bayesian methods in statistics. In the evaluation of evidence, the likelihood ratio is credited with quantifying the value of evidence in favour of one or other proposition by considering the probability of the evidence conditional on each proposition and this then converts the Bayesian prior odds into the posterior odds. Motivated by this approach, this paper considers an alternative p -value-based likelihood ratio by explicitly taking into account the behaviour of the Frequentist p -value under both hypotheses, rather than restricting focus solely on the null hypothesis. It is shown that by accommodating the alternative hypothesis, analysis leads to inferential conclusions which are consistent with Bayesian methods.

Description: This essay explores the implications of complexity for understanding both the law of evidence and the nature of the legal system. Among the propositions critically analysed is that one significant way to understand the general problem of the meaning of rationality is as a multivariate search for tools to understand and regulate a hostile environment. The law of evidence is conceptualized as a subset of this effort, at least in part, as involving a search for tools to regulate the almost infinitely complex domain of potentially relevant evidence and at the same time to accommodate policy demands. The proposition is then considered that the legal system of which the evidentiary system is a part has emergent properties that may not be deducible from its component parts, which suggests in turn that it may be, or at least has properties highly analogous to, a complex adaptive system. One implication of this analysis is that the tools of standard academic research that rely heavily on the isolation and reduction of analytical problems to manageable units to permit them to be subjected to standard deductive methodologies may need to be supplemented with analytical tools that facilitate the regulation of complex natural phenomena such as fluid dynamics. This has direct implications for such things as the conception of law as rules, and thus for the Hart–Dworkin debate that has dominated jurisprudence for 50 years. That debate may have mischaracterized the object of its inquiry, and thus the Dworkinian solution to the difficulties of positivism is inapplicable. It can certainly be shown that the Dworkinian solution is not achievable and cannot rationally be approximated. Solutions to legal problems within the legal system as a whole (as compared to any particular node within the legal system) are arrived at through a process of inference to the best explanation that occurs within a highly interconnected set of nodes similar to a neural or social network.