Notion Of Orthogonality Research Articles

Orthogonality connected with integral means and characterizations of inner product spaces

Kikianty and Dragomir (Math Inequal Appl 13:1–32, 2010) introduced the p−HH norms on the Cartesian square of a normed space, which are equivalent, but are geometrically different, to the well-known p-norms. In this paper, notions of orthogonality in terms of the 2−HH norm are introduced; and their properties are studied. Some characterizations of inner product spaces are established, as well as a characterization of strictly convex spaces.

Towards a typed Geometry of Interaction

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut elimination. We introduce a typed version of GoI, called Multiobject GoI for both multiplicative linear logic (MLL) and multiplicative exponential linear logic (MELL) with units. We present a categorical setting that includes our previous (untyped) GoI models, as well as more general models based on monoidal *-categories. Our development of multiobject GoI depends on a new theory of partial traces and trace classes, which we believe is of independent interest, as well as an abstract notion of orthogonality (which is related to work of Hyland and Schalk). We develop Girard's original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of cut elimination (under some restrictions). We prove soundness theorems for the MGoI interpretation (for Multiplicative and Multiplicative Exponential Linear Logic) in partially traced *-categories with an orthogonality. Finally, we briefly discuss the relationship between our GoI interpretation and other categorical interpretations of GoI.

Orthogonality in complex martingale spaces and connections with the Beurling–Ahlfors transform

We introduce and analyze a notion of orthogonality and dimension for spaces of $\mathbb{C}^n$-martingales. In particular, the space of martingale transforms of heat-extensions of $L^2(\mathbb{R}^{2m})$ functions is shown to be the orthogonal sum of $2$ conformal subspaces. We show that a theorem and proof of D. L. Burkholder for the computation of $L^p$-norm of martingale transforms applies specially for $n$-conformal and for pairwise conformal $n$-martingales. This leads to estimates of the $L^p$-norms of singular integral operators associated with the second-order Riesz transforms, in particular of the Beurling–Ahlfors operator.

On the confluence of lambda-calculus with conditional rewriting

The confluence of untyped λ -calculus with unconditional rewriting is now well un- derstood. In this paper, we investigate the confluence of λ -calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Müller and Dougherty for unconditional rewriting. Two cases are considered, whether β -reduction is allowed or not in the evaluation of conditions. Moreover, Dougherty’s result is improved from the assumption of strongly normalizing β -reduction to weakly normalizing β -reduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve. Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules.

A Characterization of Orthogonality

Let X be a real inner product space of (finite or infinite) dimension greater than one. We proved (see Theorem 7, Chapter 1 of our book [1]) that if T is a separable translation group of X, and d an appropriate distance function of X which is supposed to be invariant under T and the orthogonal group O of X, then there are, up to isomorphism, exactly two solutions of geometries (X,G(T,O)), G the group generated by T ∪ O, namely euclidean and hyperbolic geometry over X. With the same geometrical definition for both geometries of arbitrary (finite or infinite) dimension > 1 we will characterize in this note the notion of orthogonality.

Proofs as Polynomials

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut-elimination. In previous work we introduced a typed version of GoI, called Multiobject GoI (MGoI) for multiplicative linear logic [Haghverdi, E. and Scott, P.J. (2005), Towards a Typed Geometry of Interaction, CSL2005 (Computer Science Logic), Luke Ong, Ed. SLNCS 3634, 216–231]. This was later extended to cover the exponentials by the first author [Haghverdi, E. (2006), Typed GoI for Exponentials. in: M. Bugliesi et al. (Eds.): Proc. of ICALP 2006, Part II, LNCS 4052, 384–395. Springer Verlag]. Our development of MGoI depends on a new theory of partial traces, as well as an abstract notion of orthogonality (related to work of Hyland and Schalk.) In this paper we recall the MGoI semantics for MLL, and discuss how it relates to denotational semantics of MLL in certain *-autonomous categories. Finally, we prove characterization theorems for the MGoI interpretation of MLL in partially traced categories with an orthogonality, and for the original untyped GoI interpretation of MLL in a traced unique decomposition category.

The Combined Power of Conditions and Information on Failures to Solve Asynchronous Set Agreement

To cope with the impossibility of solving agreement problems in asynchronous systems made up of $n$ processes and prone to $t$ process crashes, system designers tailor their algorithms to run fast in “normal” circumstances. Two orthogonal notions of “normality” have been studied in the past through failure detectors that give processes information about process crashes, and through conditions that restrict the inputs to an agreement problem. This paper investigates how the two approaches can benefit from each other to solve the $k$-set agreement problem, where processes must agree on at most $k$ of their input values (when $k=1$ we have the famous consensus problem). It proposes novel failure detectors for solving $k$-set agreement and a protocol that combines them with conditions, establishing a new bridge among asynchronous, synchronous, and partially synchronous systems with respect to agreement problems. The paper also proves a lower bound when solving the $k$-set agreement problem with a condition.

Orthogonal Design for Calibration of Multiples of Kilogram

Orthogonality is considered to be one of the most important properties for designing an experiment. Introducing the notion of orthogonality into mass calibration techniques has beneficial results, ...

Strengthening Conditional Presuppositions

In this paper it will be shown how conditional presuppositions can be strengthened to unconditional ones if we assume that the antecedent and consequent of a conditional presupposition are independent of one another. Our notion of independence is very weak, and based on Lewis' (1988) notion of orthogonality of questions. It will be argued that our way of strengthening these presuppositions does not give rise to some wrong predictions Geurts (1996) argued other proposed strengthening accounts do.

The support of a uniform semigroup valued measure

In this paper we introduce the notion of orthogonality for arbitrary families of elements of a partial abelian monoid, we study uniform semigroup valued measures on a partial abelian monoid, and we establish interesting results about the existence of the support of such measures.

Orthogonality in 2-normed spaces revisited

In this paper we discuss some existing notions of orthogonality in 2-normed spaces and their drawback. We also formulate new definitions of orthogonality that improve the existing ones. In the standard 2-normed space, our notions of orthogonality coincide with the usual one.

From Geometry of Interaction to Denotational Semantics

We analyze the categorical foundations of Girard's Geometry of Interaction Program for Linear Logic. The motivation for the work comes from the importance of viewing GoI as a new kind of semantics and thus trying to relate it to extant semantics. In an earlier paper we showed that a special case of Abramsky's GoI situations–ones based on Unique Decomposition Categories (UDC's)–exactly captures Girard's functional analytic models in his first GoI paper, including Girard's original Execution formula in Hilbert spaces, his notions of orthogonality, types, datum, algorithm, etc. Here we associate to a UDC-based GoI Situation a denotational model (a *- autonomous category (without units) with additional exponential structure). We then relate this model to some of the standard GoI models via a fully-faithful embedding into a double-gluing category, thus connecting up GoI with earlier Full Completeness Theorems.

A characterisation of Hilbert spaces via orthogonality and proximinality

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

Natural computation and non-Turing models of computation

We propose certain non-Turing models of computation, but our intent is not to advocate models that surpass the power of Turing machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of application and are ill-suited to use outside that domain. Hence we review the presuppositions and context of the TM model and show that it is unsuited to natural computation (computation occurring in or inspired by nature). Therefore we must consider an expanded definition of computation that includes alternative (especially analog) models as well as the TM. Finally we present an alternative model, of continuous computation, more suited to natural computation. We conclude with remarks on the expressivity of formal mathematics.

An Approximate Truthful Mechanism for Combinatorial Auctions with Single Parameter Agents

Mechanism design seeks algorithms whose inputs are provided by selfish agents who would lie if it were to their advantage. Incentive-compatible mechanisms compel the agents to tell the truth by making it in their self-interest to do so. Often, as in combinatorial auctions, such mechanisms involve the solution of NP-hard problems. Unfortunately, approximation algorithms typically destroy incentive compatibility. Randomized rounding is a commonly used technique for designing approximation algorithms. We devise a version of randomized rounding that is incentivecompatible, giving a truthful mechanism for combinatorial auctions with single parameter agents (e.g., "single minded bidders") that approximately maximizes the social value of the auction. We discuss two orthogonal notions of truthfulness for a randomized mechanism–truthfulness with high probability and in expectation–and give a mechanism that achieves both simultaneously. We consider combinatorial auctions where multiple copies of many different items are on sale, and each bidder _i_ desires a subset _S__<em>i</em>. Given a set of bids, the problem of finding the allocation of items that maximizes total valuation is the well-known SetPacking problem. This problem is NP-hard, but for the case of items with many identical copies, the optimum can be approximated very well. To turn this approximation algorithm into a truthful auction mechanism, we overcome two problems: We show how to make the allocation algorithm _monotone_, and give a method to compute the appropriate payments efficiently.

Natural computation and non-Turing models of computation

We propose certain non-Turing models of computation, but our intent is not to advocate models that surpass the power of Turing machines (TMs), but to defend the need for models with orthogonal notions of power. We review the nature of models and argue that they are relative to a domain of application and are ill-suited to use outside that domain. Hence we review the presuppositions and context of the TM model and show that it is unsuited to natural computation (computation occurring in or inspired by nature). Therefore we must consider an expanded definition of computation that includes alternative (especially analog) models as well as the TM. Finally we present an alternative model, of continuous computation, more suited to natural computation. We conclude with remarks on the expressivity of formal mathematics.

Explicit characterizations of orthogonality in lSp( C)

In a previous paper, the author used a notion of orthogonality introduced in another article to establish characterizations for orthogonality in the spaces l S p ( C), 1⩽ p<∞, thus obtaining generalizations of the usual characterization of orthogonality in the Hilbert spaces l S 2( C), via inner products. In this paper we make explicit these characterizations for some of the spaces l S p ( C). We finish by presenting some remarks and open problems.

An Extension of the Notion of Orthogonality to Banach Spaces

We extend the usual notion of orthogonality to Banach spaces. We show that the extension is quite rich in structure by establishing some of its main properties and consequences. Geometric characterizations and comparison results with other extensions are established. Also, we establish a characterization of compact operators on Banach spaces that admit orthonormal Schauder bases. Finally, we characterize orthogonality in the spaces l2p(C).

Characterisation of orthogonality in certain Banach spaces

In this paper we adopt the notion of orthogonality introduced by the author in a previous article. We establish a characterisation for orthogonality in the spaces , where S is a set of positive integers and C is the field of complex numbers. Generalisations of the usual characterisation of orthogonality in the Hilbert spaces , via inner products, are obtained.

A Numerical Scheme for Impact Problems II: The Multidimensional Case

We consider a mechanical system with impact and n degrees of freedom, written in generalized coordinates. The system is not necessarily Lagrangian. The representative point is subject to a constraint: it must stay inside a closed set K with boundary of class C3 . We assume that, at impact, the tangential component of the impulsion is conserved, while its normal coordinate is reflected and multiplied by a given coefficient of restitution $e\in[0,1]$: the mechanically relevant notion of orthogonality is defined in terms of the local metric for the impulsions (local cotangent metric). We define a numerical scheme which enables us to approximate the solutions of the Cauchy problem: this is a generalization of the scheme presented in the companion paper [L. Paoli and M. Schatzman, SIAM J. Numer. Anal., 40 (2002), pp. 702--733]. We prove the convergence of this numerical scheme to a solution, which also yields an existence result. Without any a priori estimates, the convergence and the existence are local; with some a priori estimates, the convergence and the existence are proved on intervals depending exclusively on these estimates. The technique of proof uses a localization of the scheme close to the boundary of K; this idea is classical for a differential system studied in the framework of flows of a vector field. It is much more difficult to implement here because finite differences schemes are only approximately local: straightening the boundary creates quadratic terms which cause all the difficulties of the proof.