Register automata with linear arithmetic

We propose a novel automata model over the alphabet of rational numbers, which we call register automata over the rationals (RA-Q). It reads a sequence of rational numbers and outputs another rational number. RA-Q is an extension of the well-known register automata (RA) over infinite alphabets, which are finite automata equipped with a finite number of registers/variables for storing values. Like in the standard RA, the RA-Q model allows both equality and ordering tests between values. It, moreover, allows to perform linear arithmetic between certain variables. The model is quite expressive: in addition to the standard RA, it also generalizes other well-known models such as affine programs and arithmetic circuits. The main feature of RA-Q is that despite the use of linear arithmetic, the so-called invariant problem---a generalization of the standard non-emptiness problem---is decidable. We also investigate other natural decision problems, namely, commutativity, equivalence, and reachability. For deterministic RA-Q, commutativity and equivalence are polynomial-time inter-reducible with the invariant problem

Double-exciton component of the cyclotron spin-flip mode in a quantum Hall ferromagnet

We report on the calculation of the cyclotron spin-flip excitation (CSFE) in a spin-polarized quantum Hall system at unit filling. This mode has a double-exciton component which contributes to the CSFE correlation energy but can not be found by means of a mean field approach. The result is compared with available experimental data.Comment: 9 pages, 2 figure

Learning Optimal Deterministic Auctions with Correlated Valuation Distributions

In mechanism design, it is challenging to design the optimal auction with correlated values in general settings. Although value distribution can be further exploited to improve revenue, the complex correlation structure makes it hard to acquire in practice. Data-driven auction mechanisms, powered by machine learning, enable to design auctions directly from historical auction data, without relying on specific value distributions. In this work, we design a learning-based auction, which can encode the correlation of values into the rank score of each bidder, and further adjust the ranking rule to approach the optimal revenue. We strictly guarantee the property of strategy-proofness by encoding game theoretical conditions into the neural network structure. Furthermore, all operations in the designed auctions are differentiable to enable an end-to-end training paradigm. Experimental results demonstrate that the proposed auction mechanism can represent almost any strategy-proof auction mechanism, and outperforms the auction mechanisms wildly used in the correlated value settings

Existence of Positive Solutions For A Fourth-Order p-Laplacian Boundary Value Problem *

Abstract This article is concerned with the existence of positive solutions of a fourth

Truthful Auctions for Automated Bidding in Online Advertising

Automated bidding, an emerging intelligent decision making paradigm powered by machine learning, has become popular in online advertising. Advertisers in automated bidding evaluate the cumulative utilities and have private financial constraints over multiple ad auctions in a long-term period. Based on these distinct features, we consider a new ad auction model for automated bidding: the values of advertisers are public while the financial constraints, such as budget and return on investment (ROI) rate, are private types. We derive the truthfulness conditions with respect to private constraints for this multi-dimensional setting, and demonstrate any feasible allocation rule could be equivalently reduced to a series of non-decreasing functions on budget. However, the resulted allocation mapped from these non-decreasing functions generally follows an irregular shape, making it difficult to obtain a closed-form expression for the auction objective. To overcome this design difficulty, we propose a family of truthful automated bidding auction with personalized rank scores, similar to the Generalized Second-Price (GSP) auction. The intuition behind our design is to leverage personalized rank scores as the criteria to allocate items, and compute a critical ROI to transform the constraints on budget to the same dimension as ROI. The experimental results demonstrate that the proposed auction mechanism outperforms the widely used ad auctions, such as first-price auction and second-price auction, in various automated bidding environments