Daniel Gottlieb

Daniel Gottlieb
University of California, Los Angeles | UCLA · Department of Mathematics

About

32
Publications
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413
Citations

Publications

Publications (32)
Article
This paper characterizes incentive-compatible allocations when types are mul- tidimensional and the single-crossing condition may not hold. This character- ization is used to obtain the optimal contracts in multidimensional screening as well as the equilibria in multidimensional signaling models. Then, we de- termine the implications of signaling a...
Article
We develop a job-market signalling model where signals convey two pieces of information. This model is employed to study countersignalling (signals nonmonotonic in ability) and the GED exam. A result of the model is that countersignalling is more likely to occur in jobs that require a combination of skills that differs from the combination used in...
Article
For M and N closed oriented connected smooth manifolds of the same dimension, we consider the mapping space Map(M,N; f) of continuous maps homotopic to f : M ® Nf : M \longrightarrow N . We will show that the evaluation map from the space of maps to the manifold N induces a nontrivial homomorphism on the fundamental group only if the self-coincide...
Article
Full-text available
This paper concerns a formula which relates the Lefschetz number L(f) for a map f:M→M′ to the fixed point index I(f) summed with the fixed point index of a derived map on part of the boundary of ∂M. Here M is a compact manifold and M′ is M with a collar attached.
Article
Bo Ju Jiang applied Neilsen theory to the study of periodic orbits of a homeomorphism. His method employs a certain loop in the mapping torus of the homeomorphism. Our interest concerns the persistence of periodic orbits in parameterized families of homeomorphisms. This leads us to consider fibre bundles and equivariant maps, which gives us a nice...
Article
In this paper, we show that when the government is able to transfer wealth between generations, regressive policies are no longer optimal. The optimal educational policy can be decentralized through appropriate Pigouvian taxes and credit provision, is not regressive, and provides equality of opportunities in education (in the sense of irrelevance o...
Article
Given a parameterized space of square matrices, the associated set of eigenvectors forms some kind of a structure over the parameter space. When is that structure a vector bundle? When is there a vector field of eigenvectors? We answer those questions in terms of three obstructions, using a Homotopy Theory approach. We illustrate our obstructions w...
Article
Most of the work being done to unify General Relativity and Quantum Mechanics tries to represent General Relativity in the Quantum Mechanics language. We propose here an approach to represent Quantum Mechanics in the language of Relativity. In order to introduce discretness into the language of Relativity we consider the classical invarients of hom...
Article
This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R
Article
. Fields of Lorentz transformations on a space--time M are related to tangent bundle self isometries. In other words, a gauge transformation with respect to the Gamma + ++ Minkowski metric on each fibre. Any such isometry L : T (M) ! T (M) can be expressed, at least locally, as L = e F where F : T (M) ! T (M) is antisymmetric with respect to the me...
Article
. We study the "Lie Algebra" of the group of Gauge Transformations of Space-time. We obtain topological invariants arising from this Lie Algebra. Our methods give us fresh mathematical points of view on Lorentz Transformations, orientation conventions, the Doppler shift, Pauli matrices, Electro-Magnetic Duality Rotation, Poynting vectors, and the E...
Article
We introduce a new definition of intersection number by means of umkehr maps. This definition agrees with an obvious definition up to sign. It allows us to extend the concept parametrically to fibre bundles. For fibre bundles we can then define intersection number transfers. Fibre bundles arise naturally in equivariant situations, so the trace of t...
Article
. Vector fields defined only over a part of a manifold give rise to indexes and to transfers. These local vector fields form a topological space whose relation to configuration spaces was studied by Dusa McDuff, and whose higher dimensional homotopy and homology promise invariants of parametrized families of local vector fields. We show that the as...
Article
these two concepts to get two algebraic structures which generalize rings. Why haven't we heard of these concepts? b) Consider the function f(z) = ke z for k a constant. This is a function from C to C which is a fixed point of the complex derivative. Show e x has no fixed point for real x. Show that e z has two fixed points. What are they? What are...
Article
Introduction. Morse's equation, which is the main tool, is equation (7). It is used to prove several different equations, many of which are described briefly in this introduction and more fully in the body of the paper. To motivate the concept of the pullback vector field, which is the first item in the title and the first subject of the paper, we...
Article
For any fibration there is a number we call the trace which measures the best natural transfer that exists. For a group G acting on a space X we get the fibration from the Borel construction and we say the trace for that fibration is the trace of the action, written tr(G,X). The singular set of an action is the subspace of X where G is not acting f...
Article
Introduction. When I was 13, the intellectual world began opening up to me. I remember discussing with my friends the list of Sciences which we were soon to be offered. There was Chemistry, Biology, Geology, Astronomy, and Physics. We knew what Chemistry was. It was the study of Chemicals. Biology was the study of life, Geology studied rocks and As...
Article
F32.47> @ Gamma V = Ø(M) where Ind V is the index of the vector field and Ø(M) is the Euler characteristic of M . ([M], [G 2 -G 5 ], [P]). The Law of Vector Fields can be used to define the index of vector fields, so the whole of index theory follows from (1). The definition of index is not difficult, but proving it is well-defined is a little invo...
Article
this paper is the analogue of the following equation for indices of vector fields, which we call the Law of Vector Fields. (1) Ind V + Ind @ Gamma V = Ø(M) BECKER---GOTTLIEB
Article
this paper I show how the basic concept of angle leads naturally to the basic topological ideas of degree of mapping and of the Euler-Poincar'e Number. My story spans the history of mathematics. It concerns the, perhaps, most widely known non-obvious theorem of mathematics and it contains the same stunning generalization that characterizes the rece...
Article
Given any space-time M without singularities and any event O, there is a natural continuous mapping f of a two dimensional sphere into any space-like slice T not containing O. The set of future null geodesics (or the set of past null geodesics) forms a 2-sphere S 2 and the map f sends a point in S 2 to the point in T which is the intersection of th...
Article
. The concept of the index of a vector field is one of the oldest in Algebraic Topology. First stated by Poincare and then perfected by Heinz Hopf and S. Lefschetz and Marston Morse, it is developed as the sum of local indices of the zeros of the vector field, using the idea of degree of a map and initially isolated zeros. The vector field must be...
Article
. We give a definition of Mathematics. In the context of this definition we investigate the question: Why does Mathematics appear to have an underlying unity? We suggest in large part it is because of the modern notion of function. We give a brief history of the concept of functions. Then we examine the principle that any general concept easily exp...
Article
We give an argument that magnetic monopoles should not exist. It is based on the concept of the index of a vector field. The thrust of the argument is that indices of vector fields are invariants of space-time orientation and of coordinate changes, and thus physical vector fields should preserve indices. The index is defined inductively by means of...
Article
We define the concepts of topological particles and topological radiation. These are nothing more than connected components of defects of a vector field. To each topological particle we assign an index which is an integer which is conserved under interactions with other particles much as electric charge is conserved. For space-like vector fields of...
Article
For a smooth fibre bundle F\mathop ® i E\mathop ® p BF\mathop \to \limits^i E\mathop \to \limits^p B whereF is a compact manifold with or without boundary, a vertical vector fieldV gives rise to a transfer τV as anS-map. Our goal is to show these transfers satisfy an equation analogous to one that the index of vector fields satisfy. This equation...
Article
Two integer invariants of a fibration are defined: the degree, which generalizes the usual notion, and the trace. These numbers represent the smallest transfers for integral homology which can be constructed for the fibrations. Since every action gives rise to a fibration, we have the trace of an action. A list of properties of this trace is develo...
Article
Full-text available
We show that the signaling hypothesis cannot be rejected in general. Formally, any function from the space of types to the space of signals (signaling function) and an increasing wage schedule can be rationalized as an equilibrium profile of many signaling models. In particular, for one-dimensional type models, we may have non-monotonic signals con...

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