arXiv subjects

Daniel Lehmann

Publications and source records attributed to Daniel Lehmann.

At least 37 records · Page 2Linked to original sources

arXiv2015

Courbure des tissus planaires définis implicitement par une équation différentielle polynomiale en y'. Programmation

The aim of this paper is mainly, after some theoretical explanations, to provide a program on Maple for computing, whatever be d, the curvature of the planar d-web implicitely defined by a differential equation F(x,y,y')=0, F being polynomial of degree d with respect to y'. Moreover, we prove in the appendix a "concentration theorem" for any calibrated ordinary $d$-web of codimension one in n-dimensional manifold (in particular for any planar web). Its curvature matrix, relatively to an "adapted" trivialization, is concentrated on the (n-2+k0)!/(n-2)!k0! last lines (the last line if n=2), k0 denoting the integer such that d=(n-1+k0)!/(n-1)!(k0)!$.

math.DG↗

arXiv2014

Des séries de tissus ordinaires de rang maximum en toute dimension

In this paper, we define, from a finite set E of functions, a family of holomorphic webs ${\cal W}(n;E)$ of codimension one in any dimension $ n $. We prove that it is sufficient to check a finite number of conditions for these webs to be all ordinary and to have all maximal rank. Moreover, these conditions may be practically checked by computer with the program on Maple published in [DL](arXiv1408.3909v1, DG, 18 August 2014), the time of computation being acceptable, as far as some integer $k_0$ given in the data be "not too big".

math.DG↗

arXiv2014

Calcul explicite de la courbure des tissus calibrés ordinaires

We present a MAPLE program for the explicit computation of the curvature of calibrated ordinary webs in codimension one in any dimension (recall that all planar webs are calibrated and ordinary). The vanishing of this curvature means the maximality of the rank. We give some first examples.

math.DG↗

arXiv2014

Generalized Qualitative Probability: Savage Revisited

Preferences among acts are analyzed in the style of L. Savage, but as partially ordered. The rationality postulates considered are weaker than Savage's on three counts. The Sure Thing Principle is derived in this setting. The postulates are shown to lead to a characterization of generalized qualitative probability that includes and blends both traditional qualitative probability and the ranked structures used in logical approaches.

cs.AI↗

arXiv2012

Some properties of n-party entanglement under LOCC operations

Nielsen characterized in full those 2-party quantum protocols of local operations and classical communication that transform, with probability one, a pure global initial state into a pure global final state. The present work considers the generalization of Nielsen's characterization to n-party protocols. It presents a sweeping generalization of the only if part of Nielsen's result. The result presented here pertains also to protocols that do not generate a final state for sure, it considers arbitrary mixed initial states instead of pure states and n-party protocols for an arbitrary n. In this very general setting, local operations and classical communication can never decrease the expected spectra of the local mixed states in the majorization ordering. In other terms, the local states can only become purer (weakly) in expectation. The proof also provides an improvement on Nielsen's. The if part of Nielsen's characterization does not generalize. This is shown by studying the entanglement of three qubits. It is shown that one can find pure states of a system of three qubits that are not equivalent under unitary local operations but define local mixed states on all subparts of the system that have the same spectra. Neither equivalence of pure states under local unitary operations nor accessibility under LOCC operations among a system of three qubits can be characterized by properties of the spectra of the local mixed states.

quant-ph↗

arXiv2012

Measurements and majorization

Majorization is an outstanding tool to compare the purity of mixed states or the amount of information they contain and also the degrees of entanglement presented by such states in tensor products. States are compared by their spectra and majorization defines a partial order on those. This paper studies the effect of measurements on the majorization relation among states. It, then, proceeds to study the effect of local measurements on the agents sharing an entangled global state. If the result of the measurement is recorded, Nielsen and Vidal showed that the expected spectrum after any P.O.V.M. measurement majorizes the initial spectrum, i.e., a P.O.V.M. measurement cannot, in expectation, reduce the information of the observer. A new proof of this result is presented and, as a consequence, the only if part of Nielsen's characterization of LOCC transformations is generalized to n-party entanglement. If the result of a bi-stochastic measurement is not recorded, the initial state majorizes the final state, i.e., no information may be gained by such a measurement. This strengthens a result of A. Peres. In the n-party setting, no local trace preserving measurement by Alice can change the local state of another agent.

quant-ph↗

arXiv2010

Concrete Foundations for Categorical Quantum Physics

An original presentation of Categorical Quantum Physics, in the line of Abramsky and Coecke, tries to introduce only objects and assumptions that are clearly relevant to Physics and does not assume compact closure. Adjoint arrows, tensor products and biproducts are the ingredients of this presentation. Tensor products are defined, up to a unitary arrow, by a universal property related to transformations of composite systems, not by assuming a monoidal structure. Entangled states of a tensor product define mixed states on the components of the tensor product. Coproducts that fit the adjoint structure are shown to be defined up to a unitary arrow and to provide biproducts. An abstract no-cloning result is proved.

quant-ph↗

arXiv2008

Ordinary holomorphic webs of codimension one

The main change with respect to the previous version is a change of terminology : we call "ordinary" the webs previously called "regular". A holomorphic $d$-web of codimension one in dimension $n$ is "ordinary", if it satisfies to some condition of genericity. In dimension at least 3, any such web has a rank bounded from above by a number $π'(n,d)$ strictly smaller than the bound $π(n,d)$ of castelnuovo. This bound $π'(n,d)$ is optimal. Moreover, for some $d$'s, the abelian relations are sections with vanishing covariant derivative of some bundle with a connection, the curvature of which generalizes the Blaschke curvature. In dimension 2, we recover results of Hénaut and Pantazi

math.DS↗

arXiv2008

Similarity-Projection structures: the logical geometry of Quantum Physics

Similarity-Projection structures abstract the numerical properties of real scalar product of rays and projections in Hilbert spaces to provide a more general framework for Quantum Physics. They are characterized by properties that possess direct physical meaning. They provide a formal framework that subsumes both classical boolean logic concerned with sets and subsets and quantum logic concerned with Hilbert space, closed subspaces and projections. They shed light on the role of the phase factors that are central to Quantum Physics. The generalization of the notion of a self-adjoint operator to SP-structures provides a novel notion that is free of linear algebra.

quant-ph↗

arXiv2008

Global stucture of webs in codimension one

We describe the global structure of holomorphic webs in codimension one, and in particular their singularity (caustic). Various concepts are introduced, which have no interest locally near a regular point, such as the type, the reducibility, the quasi-smoothness, the CI property (complete intersection), the dicriticity... We prove for instance that the algebraicity of a web globally defined on a complex projective space may be readen on its caustic (dicriticity), at least if each irreducible component is CI, and the web quasi-smooth. .

math.DS↗

arXiv2007

Quantic Superpositions and the Geometry of Complex Hilbert Spaces

The concept of a superposition is a revolutionary novelty introduced by Quantum Mechanics. If a system may be in any one of two pure states x and y, we must consider that it may also be in any one of many superpositions of x and y. An in-depth analysis of superpositions is proposed, in which states are represented by one-dimensional subspaces, not by unit vectors as in Dirac's notation. Superpositions must be considered when one cannot distinguish between possible paths, i.e., histories, leading to the current state of the system. In such a case the resulting state is some compound of the states that result from each of the possible paths. States can be compounded, i.e., superposed in such a way only if they are not orthogonal. Since different classical states are orthogonal, the claim implies no non-trivial superpositions can be observed in classical systems. The parameter that defines such compounds is a proportion defining the mix of the different states entering the compound. Two quantities, p and theta, both geometrical in nature, relate one-dimensional subspaces in complex Hilbert spaces: the first one is a measure of proximity relating two rays, the second one is an angle relating three rays. The properties of the operation of superposition are very different from those that govern linear combination of vectors.

quant-ph↗

arXiv2007

A presentation of Quantum Logic based on an "and then" connective

When a physicist performs a quantic measurement, new information about the system at hand is gathered. This paper studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the "and then" operation that combines old and new information. The "and then" connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the "and then" connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch (1969) satisfies all the properties studied in this paper. All properties satisfied by Finch's operation in modular lattices are valid in Hilbert Space Quantum Logic. It is not known whether all properties of Hilbert Space Quantum Logic are satisfied by Finch's operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.

quant-ph↗

arXiv2005

Algebras of Measurements: the logical structure of Quantum Mechanics

In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute. Keywords: Quantum measurements, Measurement algebras, Quantum Logic. PACS: 02.10.-v.

quant-ph↗

arXiv2002

Combinatorial Auctions with Decreasing Marginal Utilities

In most of microeconomic theory, consumers are assumed to exhibit decreasing marginal utilities. This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valuations is NP-hard, we present an efficient greedy 2-approximation algorithm for this case and generalize it to the case of limited complementarities. No such approximation algorithm exists in a setting allowing for arbitrary complementarities. Some results about strategic aspects of combinatorial auctions among players with decreasing marginal utilities are also presented.

cs.GT↗

arXiv2002

Connectives in Quantum and other Cumulative Logics

Cumulative logics are studied in an abstract setting, i.e., without connectives, very much in the spirit of Makinson's early work. A powerful representation theorem characterizes those logics by choice functions that satisfy a weakening of Sen's property alpha, in the spirit of the author's "Nonmonotonic Logics and Semantics" (JLC). The representation results obtained are surprisingly smooth: in the completeness part the choice function may be defined on any set of worlds, not only definable sets and no definability-preservation property is required in the soundness part. For abstract cumulative logics, proper conjunction and negation may be defined. Contrary to the situation studied in "Nonmonotonic Logics and Semantics" no proper disjunction seems to be definable in general. The cumulative relations of KLM that satisfy some weakening of the consistency preservation property all define cumulative logics with a proper negation. Quantum Logics, as defined by Engesser and Gabbay are such cumulative logics but the negation defined by orthogonal complement does not provide a proper negation.

cs.AI↗

arXiv2002

Nonmonotonic Logics and Semantics

Tarski gave a general semantics for deductive reasoning: a formula a may be deduced from a set A of formulas iff a holds in all models in which each of the elements of A holds. A more liberal semantics has been considered: a formula a may be deduced from a set A of formulas iff a holds in all of the "preferred" models in which all the elements of A hold. Shoham proposed that the notion of "preferred" models be defined by a partial ordering on the models of the underlying language. A more general semantics is described in this paper, based on a set of natural properties of choice functions. This semantics is here shown to be equivalent to a semantics based on comparing the relative "importance" of sets of models, by what amounts to a qualitative probability measure. The consequence operations defined by the equivalent semantics are then characterized by a weakening of Tarski's properties in which the monotonicity requirement is replaced by three weaker conditions. Classical propositional connectives are characterized by natural introduction-elimination rules in a nonmonotonic setting. Even in the nonmonotonic setting, one obtains classical propositional logic, thus showing that monotonicity is not required to justify classical propositional connectives.

cs.AI↗

arXiv2002

Belief Revision and Rational Inference

The (extended) AGM postulates for belief revision seem to deal with the revision of a given theory K by an arbitrary formula, but not to constrain the revisions of two different theories by the same formula. A new postulate is proposed and compared with other similar postulates that have been proposed in the literature. The AGM revisions that satisfy this new postulate stand in one-to-one correspondence with the rational, consistency-preserving relations. This correspondence is described explicitly. Two viewpoints on iterative revisions are distinguished and discussed.

cs.AI↗

arXiv2002

NetNeg: A Connectionist-Agent Integrated System for Representing Musical Knowledge

The system presented here shows the feasibility of modeling the knowledge involved in a complex musical activity by integrating sub-symbolic and symbolic processes. This research focuses on the question of whether there is any advantage in integrating a neural network together with a distributed artificial intelligence approach within the music domain. The primary purpose of our work is to design a model that describes the different aspects a user might be interested in considering when involved in a musical activity. The approach we suggest in this work enables the musician to encode his knowledge, intuitions, and aesthetic taste into different modules. The system captures these aspects by computing and applying three distinct functions: rules, fuzzy concepts, and learning. As a case study, we began experimenting with first species two-part counterpoint melodies. We have developed a hybrid system composed of a connectionist module and an agent-based module to combine the sub-symbolic and symbolic levels to achieve this task. The technique presented here to represent musical knowledge constitutes a new approach for composing polyphonic music.

cs.AI↗