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We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g. with Cα metric). These coordinates are bi-Lipschitz on embedded balls of the domain or manifold, with distortion constants that depend only on natural geometric properties of the… (More)

- Peter W. Jones, Mauro Maggioni, Raanan Schul
- Proceedings of the National Academy of Sciences…
- 2008

We use heat kernels or eigenfunctions of the Laplacian to construct local coordinates on large classes of Euclidean domains and Riemannian manifolds (not necessarily smooth, e.g., with (alpha) metric). These coordinates are bi-Lipschitz on large neighborhoods of the domain or manifold, with constants controlling the distortion and the size of the… (More)

- Raanan Schul
- 2008

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in Rd . Their results formed the basis of quantitative rectifiability in Rd . We… (More)

- Raanan Schul, Burton M. Slotnick, Yadin Dudai
- Behavioral neuroscience
- 1996

Rats were trained to discriminate an aqueous compound of an odor and taste (amyl acetate and NaCl) from the components of the compound before removal of one olfactory bulb and the contralateral ventrolateral frontal cortex. In postoperative tests, experimental rats performed much more poorly than nonlesioned controls or controls which had all lesions made… (More)

- Kobi Rosenblum, Raanan Schul, Noga Meiri, Yaron R. Hadari, Yehiel Zick, Yadin Dudai
- Proceedings of the National Academy of Sciences…
- 1995

Protein tyrosine phosphorylation is a major signal transduction pathway involved in cellular metabolism, growth, and differentiation. Recent data indicate that tyrosine phosphorylation also plays a role in neuronal plasticity. We are using conditioned taste aversion, a fast and robust associative learning paradigm subserved among other brain areas by the… (More)

The purpose of this essay is to present a partial survey of a family of theorems that are usually referred to as analyst’s traveling salesman theorems (also referred to as geometric traveling salesman theorems). There have been several new theorems recently added to this family of theorems which we feel should be collected together to give some bigger… (More)

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R, n ≥ 2. To each locally finite Borel measure μ, we associate a function J̃2(μ, x) which uses a weighted sum to record how closely the mass of μ is concentrated near a line in the triples of dyadic cubes containing x. We show that J̃2(μ,… (More)

- Raanan Schul
- 2008

We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such sets are ‘flat’ on most scales and in most locations. Our result is quantitative, and when combined with work of I. Hahlomaa, gives a characterization of 1-Ahlfors regular subsets of 1Ahlfors-regular curves in metric spaces. Our result is a generalization to the metric… (More)

For d ≥ 2, we construct a doubling measure ν on Rd and a rectifiable curve Γ such that ν(Γ) > 0.

We identify two sufficient conditions for locally finite Borel measures on R to give full mass to a countable family of Lipschitz images of R. The first condition, extending a prior result of Pajot, is a sufficient test in terms of L affine approximability for a locally finite Borel measure μ on R satisfying the global regularity hypothesis lim sup r↓0… (More)