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"Mathematical constants"
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Gamma : exploring Euler's constant
\"Among the many constants that appear in mathematics, [pi], e, and i are the most familiar. Following closely behind is [gamma] or gamma, a constant that arises in many mathematical areas yet remains profoundly mysterious. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + ... up to 1/n , minus the natural logarithm of n -- and the numerical value is 0.5772156 ... But unlike its more celebrated colleagues [pi] and e, the exact nature of gamma remains a mystery. In fact, we don't even know if gamma is a fraction. In this tantalizing blend of history and mathematics, Julian Havil takes readers on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.\"--Back cover.
Book
Solving the KPZ equation
We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a \"universal\" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class. As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the Fokker-Planck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.
Journal Article
Nodal sets of Laplace eigenfunctions: proof of Nadirashvili's conjecture and of the lower bound in Yau's conjecture
Let u be a harmonic function in the unit ball B(0,1) ⊂ ℝⁿ, n ≥ 3, such that u(0) = 0. Nadirashvili conjectured that there exists a positive constant c, depending on the dimension n only, such that Hⁿ⁻¹({u = 0} ⋂ B) ≥ c. We prove Nadirashvili's conjecture as well as its counterpart on C∞-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact C∞-smooth Riemannian manifold M (without boundary) of dimension n, there exists c > 0 such that for any Laplace eigenfunction 𝜑λ on M, which corresponds to the eigenvalue λ, the following inequality holds: $\\mathrm{c}\\sqrt{\\mathrm{\\lambda }}\\le {\\mathrm{H}}^{\\mathrm{n}-1}\\left(\\{{\\mathrm{\\phi }}_{\\mathrm{\\lambda }}=0\\}\\right)$.
Journal Article
Learning to Optimize via Posterior Sampling
This paper considers the use of a simple posterior sampling algorithm to balance between exploration and exploitation when learning to optimize actions such as in multiarmed bandit problems. The algorithm, also known as
Thompson Sampling
and as
probability matching
, offers significant advantages over the popular upper confidence bound (UCB) approach, and can be applied to problems with finite or infinite action spaces and complicated relationships among action rewards. We make two theoretical contributions. The first establishes a connection between posterior sampling and UCB algorithms. This result lets us convert regret bounds developed for UCB algorithms into Bayesian regret bounds for posterior sampling. Our second theoretical contribution is a Bayesian regret bound for posterior sampling that applies broadly and can be specialized to many model classes. This bound depends on a new notion we refer to as the
eluder dimension
, which measures the degree of dependence among action rewards. Compared to UCB algorithm Bayesian regret bounds for specific model classes, our general bound matches the best available for linear models and is stronger than the best available for generalized linear models. Further, our analysis provides insight into performance advantages of posterior sampling, which are highlighted through simulation results that demonstrate performance surpassing recently proposed UCB algorithms.
Journal Article
CONFORMAL WELDINGS OF RANDOM SURFACES: SLE AND THE QUANTUM GRAVITY ZIPPER
We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path-decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.
Journal Article
Sharp constants in several inequalities on the Heisenberg group
We derive the sharp constants for the inequalities on the Heisenberg group ℍ n whose analogues on Euclidean space ℝ n are the well known Hardy-Littlewood-Sobolev inequalities. Only one special case had been known previously, due to Jerison-Lee more than twenty years ago. From these inequalities we obtain the sharp constants for their duals, which are the Sobolev inequalities for the Laplacian and conformally invariant fractional Laplacians. By considering limiting cases of these inequalities sharp constants for the analogues of the Onofri and log-Sobolev inequalities on ℍ n are obtained. The methodology is completely different from that used to obtain the ℝ n inequalities and can be (and has been) used to give a new, rearrangement free, proof of the HLS inequalities.
Journal Article
The connective constant of the honeycomb lattice equals √2 + √2
We provide the first mathematical proof that the connective constant of the hexagonal lattice is equal to √2 + √2. This value has been derived nonrigorously by B. Nienhuis in 1982, using Coulomb gas approach from theoretical physics. Our proof uses a parafermionic observable for the self-avoiding walk, which satisfies a half of the discrete Cauchy-Riemann relations. Establishing the other half of the relations (which conjecturally holds in the scaling limit) would also imply convergence of the self-avoiding walk to SLE(8/3).
Journal Article
Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality
We study the convergence properties of an alternating proximal minimization algorithm for nonconvex structured functions of the type: L(x, y) = f(x) + Q(x, v) + g(y), where f and g are proper lower semicontinuous functions, defined on Euclidean spaces, and Q is a smooth function that couples the variables x and y. The algorithm can be viewed as a proximal regularization of the usual Gauss-Seidel method to minimize L. We work in a nonconvex setting, just assuming that the function L satisfies the Kurdyka-Łojasiewicz inequality. An entire section illustrates the relevancy of such an assumption by giving examples ranging from semialgebraic geometry to \"metrically regular\" problems. Our main result can be stated as follows: If L has the Kurdyka-Łojasiewicz property, then each bounded sequence generated by the algorithm converges to a critical point of L. This result is completed by the study of the convergence rate of the algorithm, which depends on the geometrical properties of the function L around its critical points. When specialized to Q(x, y) = || JC — y||² and to f, g indicator functions, the algorithm is an alternating projection mehod (a variant of von Neumann's) that converges for a wide class of sets including semialgebraic and tame sets, transverse smooth manifolds or sets with \"regular\" intersection. To illustrate our results with concrete problems, we provide a convergent proximal reweighted l¹ algorithm for compressive sensing and an application to rank reduction problems.
Journal Article
ESTIMATING AND UNDERSTANDING EXPONENTIAL RANDOM GRAPH MODELS
We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000—1017]. The theory explains a host of difficulties encountered by applied workers: many distinct models have essentially the same MLE, rendering the problems \"practically\" ill-posed. We give the first rigorous proofs of \"degeneracy\" observed in these models. Here, almost all graphs have essentially no edges or are essentially complete. We supplement recent work of Bhamidi, Bresler and Sly [2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2008) 803—812 IEEE] showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erdős—Rényi model. We also find classes of models where the limiting graphs differ from Erdős—Rényi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.
Journal Article
VALID POST-SELECTION INFERENCE
It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid \"post-selection inference\" by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals. Simultaneity is required for all linear functions that arise as coefficient estimates in all submodels. By purchasing \"simultaneity insurance\" for all possible submodels, the resulting post-selection inference is rendered universally valid under all possible model selection procedures. This inference is therefore generally conservative for particular selection procedures, but it is always less conservative than full Scheffé protection. Importantly it does not depend on the truth of the selected submodel, and hence it produces valid inference even in wrong models. We describe the structure of the simultaneous inference problem and give some asymptotic results.
Journal Article