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Bayes' theorem plays an increasingly prominent role in statistical applications but remains controversial among statisticians. The term \"controversial theorem\" sounds like an oxymoron, but Bayes' theorem has played this part for two-and-a-half centuries. Twice it has soared to scientific celebrity, twice it has crashed, and it is currently enjoying another boom. The theorem itself is a landmark of logical reasoning and the first serious triumph of statistical inference, yet is still treated with suspicion by most statisticians. There are reasons to believe in the staying power of its current popularity, but also some signs of trouble ahead.

We study the problem of estimating the ridges of a density function. Ridge estimation is an extension of mode finding and is useful for understanding the structure of a density. It can also be used to find hidden structure in point cloud data. We show that, under mild regularity conditions, the ridges of the kernel density estimator consistently estimate the ridges of the true density. When the data are noisy measurements of a manifold, we show that the ridges are close and topologically similar to the hidden manifold. To find the estimated ridges in practice, we adapt the modified mean-shift algorithm proposed by Ozertem and Erdogmus [J. Mach. Learn. Res. 12 (2011) 1249-1286]. Some numerical experiments verify that the algorithm is accurate.

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group GG, we show that a finite subset XX with |XX−1X|/|X||X X ^{-1}X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of GG. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.

We prove that the 2-primary π₆₁ is zero. As a consequence, the Kervaire invariant element θ₅ is contained in the strictly defined 4-fold Toda bracket 〈2, θ₄, θ₄, 2〉. Our result has a geometric corollary: the 61-sphere has a unique smooth structure, and it is the last odd dimensional case — the only ones are S¹, S³, S⁵ and S⁶¹. Our proof is a computation of homotopy groups of spheres. A major part of this paper is to prove an Adams differential d₃(D₃) = B₃. We prove this differential by introducing a new technique based on the algebraic and geometric Kahn-Priddy theorems. The success of this technique suggests a theoretical way to prove Adams differentials in the sphere spectrum inductively by use of differentials in truncated projective spectra.

We answer in the affirmative the following question of Jörg Brendle: If there is a $\\Sigma _2^1$ mad family, is there then a $\\Pi _1^1$ mad family?

We show that ℤ is definable in ℚ by a universal first-order formula in the language of rings. We also present an ∀∃-formula for ℤ in ℚ with just one universal quantifier. We exhibit new diophantine subsets of ℚ like the complement of the image of the norm map under a quadratic extension, and we give an elementary proof for the fact that the set of nonsquares is diophantine.

We establish virtual surjection to pairs (VSP) as a general criterion for the finite presentability of subdirect products of groups: if Γ 1 ,...,Γ n are finitely presented and S < Γ 1 × ··· × Γ n projects to a subgroup of finite index in each Γ i × Γ j , then S is finitely presentable, indeed there is an algorithm that will construct a finite presentation for S. We use the VSP criterion to characterize the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither FP ∞ nor of Stallings-Bieri type.

This paper studies a model of long-term contracting for experimentation. We consider a principalagent relationship with adverse selection on the agent's ability, dynamic moral hazard, and private learning about project quality. We find that each of these elements plays an essential role in structuring dynamic incentives, and it is only their interaction that generally precludes efficiency. Our model permits an explicit characterization of optimal contracts.

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair {x + y,xy}. We answer this question affirmatively in a strong sense by exhibiting a large new class of nonlinear patterns that can be found in a single cell of any finite partition of ℕ. Our proof involves a correspondence principle that transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as x² – y² = z and x² + 2y² – 3z² = w.

I provide a truthmaker semantics for Angell's system of analytic implication and establish completeness.