Scour-control and scour-resistant design for hydraulic structures

There now exists in theoretical analyses, in laboratory findings, and in conventional engineering-design practice, a skeleton of facts on which a rational scour-control procedure can be based. The writer has assembled these facts and stated them in as straightforward a manner as he believes, in consistent with current knowledge. Examples of the manner in which the analysis is taken into account in ordinary engineering design are given in support, thereof

The Fr\"ohlich Polaron at Strong Coupling -- Part II: Energy-Momentum Relation and Effective Mass

We study the Fr\"ohlich polaron model in $\mathbb{R}^3$, and prove a lower bound on its ground state energy as a function of the total momentum. The bound is asymptotically sharp at large coupling. In combination with a corresponding upper bound proved earlier, it shows that the energy is approximately parabolic below the continuum threshold, and that the polaron's effective mass (defined as the semi-latus rectum of the parabola) is given by the celebrated Landau--Pekar formula. In particular, it diverges as $\alpha^4$ for large coupling constant $\alpha$

Characterisation of gradient flows for a given functional

Let $X$ be a vector field and $Y$ be a co-vector field on a smooth manifold $M$. Does there exist a smooth Riemannian metric $g_{\alpha \beta}$ on $M$ such that $Y_\beta = g_{\alpha \beta} X^\alpha$? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.Comment: 17 page

Asymptotic series for low-energy excitations of the Fr\"ohlich Polaron at strong coupling

We consider the confined Fr\"ohlich polaron and establish an asymptotic series for the low-energy eigenvalues in negative powers of the coupling constant. The coefficients of the series are derived through a two-fold perturbation approach, involving expansions around the electron Pekar minimizer and the excitations of the quantum field.Comment: 45 page

Friedrichs Diagrams -- Bosonic and Fermionic

We give a mathematically precise review of a diagrammatic language introduced by Friedrichs in order to simplify computations with creation and annihilation operator products. In that language, we establish explicit formulas and algorithms for evaluating bosonic and fermionic commutators. Further, as an application, we demonstrate that the non-linear Hartree dynamics can be seen as a subset of the diagrams arising in the time evolution of a Bose gas.Comment: 24 pages, 8 figure

Selling to Intermediaries: Optimal Auction Design in a Common Value Model

We characterize revenue maximizing auctions when the bidders are intermediaries who wish to resell the good. The bidders have differential information about their common resale opportunities: each bidder privately observes an independent draw of a resale opportunity, and the highest signal is a sufficient statistic for the value of winning the good. If the good must be sold, then the optimal mechanism is simply a posted price at which all bidders are willing to purchase the good, and all bidders are equally likely to be allocated the good, irrespective of their signals. If the seller can keep the good, then under the optimal mechanism, all bidders make the same expected payment and have the same expected probability of receiving the good, independent of the signal. Conditional on the good being sold, the allocation discriminates in favor of bidders with lower signals. In some cases, the optimal mechanism again reduces to a posted price. The model provides a foundation for posted prices in multi-agent screening problems

Optimal Auction Design in a Common Value Model

We study auction design when bidders have a pure common value equal to the maximum of their independent signals. In the revenue maximizing mechanism, each bidder makes a payment that is independent of his signal and the allocation discriminates in favor of bidders with lower signals. We provide a necessary and sufficient condition under which the optimal mechanism reduces to a posted price under which all bidders are equally likely to get the good. This model of pure common values can equivalently be interpreted as model of resale: the bidders have independent private values at the auction stage, and the winner of the auction can make a take-it-or-leave-it-offer in the secondary market under complete information

Counterfactuals with Latent Information

We describe a methodology for making counterfactual predictions when the information held by strategic agents is a latent parameter. The analyst observes behavior which is rationalized by a Bayesian model in which agents maximize expected utility given partial and differential information about payoff-relevant states of the world. A counterfactual prediction is desired about behavior in another strategic setting, under the hypothesis that the distribution of and agents’ information about the state are held fixed. When the data and the desired counterfactual prediction pertain to environments with finitely many states, players, and actions, there is a finite dimensional description of the sharp counterfactual prediction, even though the latent parameter, the type space, is infinite dimensional