Two-dimensionalism and the “Knowing Which” Requirement

Two-dimensional semantics aims to eliminate the puzzle of necessary a posteriori and contingent a priori truths. Recently many argue that even assuming two-dimensional semantics we are left with the puzzle of necessary and a posteriori propositions. Stephen Yablo (Pacific Philosophical Quarterly, 81, 98–122, 2000) and Penelope Mackie (Analysis, 62(3), 225–236, 2002) argue that a plausible sense of “knowing which” lets us know the object of such a proposition, and yet its necessity is “hidden” and thus a posteriori. This paper answers this objection; I argue that given two-dimensional semantics you cannot know a necessary proposition without knowing that it is true.

Duality, Vector Spaces and Absolutely Convex Modules

It is well-known (see Semadeni, Queen Pap. Pure Appl. Math., 33:1–98, 1973 and Pumplün and Röhrl, Commun. Algebra, 12(8):953–1019, 1984, 1985) that the embedding of vector spaces into the category of absolutely convex modules is reflective. As we will show, under a separatedness condition on these modules it is at the same time coreflective. This is a peculiar situation, see Kannan, Math. Ann., 195:168–174, (1972) and Hu $\textrm {\u{s}}It is well-known (see Semadeni, Queen Pap. Pure Appl. Math., 33:1–98, 1973 and Pumplün and R?hrl, Commun. Algebra, 12(8):953–1019, 1984, 1985) that the embedding of vector spaces into the category of absolutely convex modules is reflective. As we will show, under a separatedness condition on these modules it is at the same time coreflective. This is a peculiar situation, see Kannan, Math. Ann., 195:168–174, (1972) and Huek, Reflexive and coreflexive subcategories of unif and top, Seminar Uniform Spaces, Prague, 113–126, (1973), but we do find it also in the embedding (Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad. Oxford Mathematical Monographs, Oxford University Press, London, UK, 1997) and, by extension, in the embedding (see Lowen and Verwulgen, Houst. J. Math, 30(4):1127–1142, 2004, and Sioen and Verwulgen, Appl. Gen. Topol., 4(2):263–279, 2003. We demonstrate that, in this setting, by duality arguments, absolutely convex modules are indeed the numerical counterpart of vector spaces. All these, at first sight unrelated facts, are comprised in the commutative scheme below with natural dualisation functors and their left adjoints.      

Organizational learning: effects of (network) structure and (individual) strategy

Earlier theoretical accounts of collective learning relied on rules and operating procedures as the organizational memory (March in Organ. Sci. 2(1):71–87, 1991; Rodan in Scand. J. Manag. 21:407–428, 2005). This paper builds on this tradition drawing on ideas from social network theory. Learning is modeled as a social-psychological process (Darr and Kurtzberg in Organ. Behav. Hum. Decis. Process. 82(1):28–44, 2000; Rulke et al. in Organ. Behav. Hum. Decis. Process. 82(1):134–149, 2000), in which organizations learn by exchanging information internally between their members (Argote et al. in Organ. Behav. Hum. Decis. Process. 82(1):1–8, 2000; Carley in Am. Soc. Rev. 56(3):331–354, 1991; Carley in Soc. Perspect. 48(4):547–571, 1995). Learning is also characterized as stochastic and creative (Gruenfeld et al. in Organ. Behav. Hum. Decis. Process. 82(1):45–59, 2000). This model is used to explore predictions about the effect social networks have on idea generation and learning and alternative strategies for choosing from whom to seek information.

On Pairs of Dual Consequence Operations

In the paper, the authors discuss two kinds of consequence operations characterized axiomatically. The first one are consequence operations of the type Cn ⁺ that, in the intuitive sense, are infallible operations, always leading from accepted (true) sentences of a deductive system to accepted (true) sentences of the deductive system (see Tarski in Monatshefte für Mathematik und Physik 37:361–404, 1930, Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23:22–29, 1930; Pogorzelski and Słupecki in Stud Logic 9:163–176, 1960, Stud Logic 10:77–95, 1960). The second kind are dual consequence operations of the type Cn ⁻ that can be regarded as anti-infallible operations leading from non-accepted (rejected, false) sentences of a deductive system to non-accepted (rejected, false) sentences of the system (see Słupecki in Funkcja Łukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5–131, Zeszyty Naukowe Wyższej Szkoły Inżynierskiej w Opolu, Seria Matematyka 4(81):35–61, 1983, Ann Pure Appl Logic 127:243–266, 2004, in On the notion and function of rejected propositions, 179–202, 2005). The operations of the types Cn ⁺ and Cn ⁻ can be ordinary finitistic consequence operations or unit consequence operations. A deductive system can be characterized in two ways by the following triple:

Approximate Isomorphisms in <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

This paper is a survey on the Hyers–Ulam–Rassias stability of the following Cauchy–Jensen functional equation in C ^*-algebras:

On the Expected Probability of Constraint Violation in Sampled Convex Programs

In this note, we derive an exact expression for the expected probability V of constraint violation in a sampled convex program (see Calafiore and Campi in Math. Program. 102(1):25–46, 2005; IEEE Trans. Autom. Control 51(5):742–753, 2006 for definitions and an introduction to this topic):

Anchor Maps and Stable Modules in Depth Two

An algebra extension A ∣ B is right depth two if its tensor-square is in the Dress category . We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids and over the centralizer R = C _A (B) are the modules _S R and R _T studied in Kadison (J. Alg. & Appl., 2005, preprint), Kadison (Contemp. Math., 391: 149–156, 2005), and Kadison and Külshammer (Commun. Algebra, 34: 3103–3122, 2006), which provide information about the bialgebroids and the extension (Kadison, Bull. Belg. Math. Soc. Simon Stevin, 12: 275–293, 2005). The anchor maps for the Hopf algebroids in Khalkhali and Rangipour (Lett. Math. Phys., 70: 259–272, 2004) and Kadison (2005, preprint) reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in Van Oystaeyen and Panaite (Appl. Categ. Struct., 2006, in press). We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in Kadison (Proc. Am. Math. Soc., 131: 2993–3002, 2003 Prop. 1.1) to any A-module. We observe that Schneider’s coGalois theory in Schneider (Isr. J. Math., 72: 167–195, 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.      

Supertropical matrix algebra

The objective of this paper is to develop a general algebraic theory of supertropical matrix algebra, extending [11]. Our main results are as follows:

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> Hermite interpolation by Pythagorean hodograph quintics in Minkowski space

Curves in the Minkowski space are very well suited to describe the medial axis transform (MAT) of planar domains. Among them, Minkowski Pythagorean hodograph (MPH) curves correspond to domains where both the boundaries and their offsets admit rational parameterizations (Choi et al., Comput Aided Design 31:59–72, 1999; Moon, Comput Aided Geom Design 16:739–753; 1999). We construct MPH quintics which interpolate two points with associated first derivative vectors and analyze the properties of the system of solutions, including the approximation order of the ‘best’ interpolant.      

Weak Type Inequality for a Family of Singular Integral Operators Related with the Gaussian Measure

In this paper we study a family of singular integral operators that generalizes the higher order Gaussian Riesz Transforms and find the right weight w to make them continuous from L ¹(wdγ) into L ^1, ∞(dγ), being Some boundedness properties of these operators had already been derived by Urbina (Ann Scuola Norm Sup Pisa Cl Sci 17(4):531–567, 1990) and Pérez (J Geom Anal 11(3):491–507, 2001).      

The Algebraic Integrability of the Quantum Toda Lattice and the Radon Transform

We study the maximal commutative ring of partial differential operators which includes the quantum completely integrable system defined by the quantum Toda lattice. Kostant shows that the image of the generalized Harish-Chandra homomorphism of the center of the enveloping algebra is commutative (Kostant in Invent. Math. 48:101–184, 1978). We demonstrate the commutativity of the ring of partial differential operators whose principal symbols are -invariant. Our commutative ring includes the commutative system of Kostant (Invent. Math. 48:101–184, 1978). The main tools in this paper are Fourier integral operators and Radon transforms.      

Branch rings, thinned rings, tree enveloping rings

We develop the theory of “branch algebras”, which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field % MathType!End!2!1! we contruct a % MathType!End!2!1! which

The behavior of Riesz means of the coefficients of a symmetric square L-function

Let f(z) be a holomorphic Hecke eigencuspform of even weight k with respect to SL(2, Z) and let L(s, sym ² f) = ∑ _n=1 ^∞ c_nn^−s, Re s > 1, be the symmetric square L-function associated with f. Represent the Riesz mean (ρ ≥ 0)

On a class of secant-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided.     

On Spherical Averages of Radial Basis Functions

A radial basis function (RBF) has the general form

The Upper Envelope of Positive Self-Similar Markov Processes

We establish integral tests in connection with laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation and on the study of the upper envelope of the future infimum due to the author (see Pardo in Stoch. Stoch. Rep. 78:123–155, [2006]). These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdős (Proceedings of the Second Berkeley Symposium, [1951]) and stable Lévy processes with no positive jumps conditioned to stay positive due to Bertoin (Stoch. Process. Appl. 55:91–100, [1995]). Research supported by a grant from CONACYT (Mexico).     

Learning Electricity with NIELS: Thinking with Electrons and Thinking in Levels

Electricity is regarded as one of the most challenging topics for students of all ages. Several researchers have suggested that na?ve misconceptions about electricity stem from a deep incommensurability (Slotta and Chi 2006; Chi 2005) or incompatibility (Chi et al. 1994) between na?ve and expert knowledge structures. In this paper we argue that adopting an emergent levels-based perspective as proposed by Wilensky and Resnick (1999), allows us to reconceive commonly noted misconceptions in electricity as behavioral evidences of “slippage between levels,” i.e., these misconceptions appear when otherwise productive knowledge elements are sometimes activated inappropriately due to certain macro-level phenomenological cues only. We then introduce NIELS (NetLogo Investigations In Electromagnetism), a curriculum of emergent multi-agent-based computational models. NIELS models represent phenomena such as electric current and resistance as emergent from simple, body-syntonic interactions between electrons and other charges in a circuit. We discuss results from a pilot implementation of NIELS in an undergraduate physics course, that highlight the ability of an emergent levels-based approach to provide students with a deep, expert-like understanding of the relevant phenomena by bootstrapping, rather than discarding their existing repertoire of intuitive knowledge.

Rational points on the unit sphere

It is known that the unit sphere, centered at the origin in ℝ ⁿ , has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝ ⁿ , and every ν > 0; there is a point r = (r ₁; r ₂;…;r _n) such that:

Do we self-regulate actions or perceptions? A test of two computational models

Self-regulation theories in applied psychology disagree about whether action or perceptions are the focus of regulation. Computational models based on the two conceptualizations were constructed and simulated. In one scenario, they performed identically and in conjunction with participants in a study of the goal-level effect (Vancouver et al., Organ Res Methods 8:100–127, 2005). In another scenario they created differentiating predictions and only the computational model based on the self-regulation of perceptions matched the data of participants. Implications for research and practice are discussed.

Duals Invert

Monoidal objects (or pseudomonoids) in monoidal bicategories share many of the properties of the paradigmatic example: monoidal categories. The existence of (say, left) duals in a monoidal category leads to a dualization operation which was abstracted to the context of monoidal objects by Day et al. (Appl Categ Struct 11:229–260, 2003). We define a relative version of this called exact pairing for two arrows in a monoidal bicategory; when one of the arrows is an identity, the other is a dualization. In this context we supplement results of Day et al. (Appl Categ Struct 11:229–260, 2003) (and even correct one of them) and only assume the existence of biduals in the bicategory where necessary. We also abstract recent work of Day and Pastro (New York J Math 14:733–742, 2008) on Frobenius monoidal functors to the monoidal bicategory context. Our work began by focusing on the invertibility of components at dual objects of monoidal natural transformations between Frobenius monoidal functors. As an application of the abstraction, we recover a theorem of Walters and Wood (Theory Appl Categ 3:25–47, 2008) asserting that, for objects A and X in a cartesian bicategory , if A is Frobenius then the category Map(X,A) of left adjoint arrows is a groupoid. Also, the characterization in Walters and Wood (Theory Appl Categ 3:25–47, 2008) of left adjoint arrows between Frobenius objects of a cartesian bicategory is put into our current setting. In the same spirit, we show that when a monoidal object admits a dualization, its lax centre coincides with the centre defined in Street (Theory Appl Categ 13:184–190, 2004). Finally we look at the relationship between lax duals for objects and adjoints for arrows in a monoidal bicategory.