Liouville theorems and Harnack inequalities for Allen-Cahn type equation

We first give a logarithmic gradient estimate for positive solutions of Allen-Cahn equation on Riemannian manifolds with Ricci curvature bounded below. As its natural corallary, Harnack inequality and a Liouville theorem for classical positive solutions are obtained. Later, we consider similar estimate under integral curvature condition and generalize previous results to a class nonlinear equations which contain some classical elliptic equations such as Lane-Emden equation, static Whitehead-Newell equation and static Fisher-KPP equation. Last, we briefly generalize them to equation with gradient item under Bakry-\'{E}mery curvature condition

Differential Harnack inequalities for semilinear parabolic equations on Riemannian manifolds I: Bakry-\'{E}mery curvature bounded below

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation \begin{equation*} \partial_t u=\Delta_V u+H(u) \end{equation*} on complete Riemannian manifolds with Bakry-\'Emery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As an application of this method, we obtain new and improved estimates for logarithmic-type equations and Yamabe-type equations. Moreover, under the non-negative Bakry-\'Emery curvature condition, we obtain complete sharp estimates for these equations. As a natural consequence of these results, we also establish sharp Harnack inequalities and Liouville-type theorems for these equations

The Value of Information Concealment

We consider a revenue optimizing seller selling a single item to a buyer, on whose private value the seller has a noisy signal. We show that, when the signal is kept private, arbitrarily more revenue could potentially be extracted than if the signal is leaked or revealed. We then show that, if the seller is not allowed to make payments to the buyer, the gap between the two is bounded by a multiplicative factor of 3, if the value distribution conditioning on each signal is regular. We give examples showing that both conditions are necessary for a constant bound to hold. We connect this scenario to multi-bidder single-item auctions where bidders' values are correlated. Similarly to the setting above, we show that the revenue of a Bayesian incentive compatible, ex post individually rational auction can be arbitrarily larger than that of a dominant strategy incentive compatible auction, whereas the two are no more than a factor of 5 apart if the auctioneer never pays the bidders and if each bidder's value conditioning on the others' is drawn according to a regular distribution. The upper bounds in both settings degrade gracefully when the distribution is a mixture of a small number of regular distributions

Determining Optimal Traffic Opening Time Through Concrete Strength Monitoring: Wireless Sensing

Construction and concrete production are time-sensitive and fast-paced; as such, it is crucial to monitor the in-place strength development of concrete structures in real-time. Existing concrete strength testing methods, such as the traditional hydraulic compression method specified by ASTM C 39 and the maturity method specified by ASTM C 1074, are labor-intensive, time consuming, and difficult to implement in the field. INDOT’s previous research (SPR-4210) on the electromechanical impedance (EMI) technique has established its feasibility for monitoring in-situ concrete strength to determine the optimal traffic opening time. However, limitations of the data acquisition and communication systems have significantly hindered the technology’s adoption for practical applications. Furthermore, the packaging of piezoelectric sensor needs to be improved to enable robust performance and better signal quality. In this project, a wireless concrete sensor with a data transmission system was developed. It was comprised of an innovated EMI sensor and miniaturized datalogger with both wireless transmission and USB module. A cloud-based platform for data storage and computation was established, which provides the real time data visualization access to general users and data access to machine learning and data mining developers. Furthermore, field implementations were performed to prove the functionality of the innovated EMI sensor and wireless sensing system for real-time and in-place concrete strength monitoring. This project will benefit the DOTs in areas like construction, operation, and maintenance scheduling and asset management by delivering applicable concrete strength monitoring solutions

Field Implementation of Concrete Strength Sensor to Determine Optimal Traffic Opening Time

In the fast-paced and time-sensitive fields of construction and concrete production, real-time monitoring of concrete strength is crucial. Traditional testing methods, such as hydraulic compression (ASTM C 39) and maturity methods (ASTM C 1074), are often laborious and challenging to implement on-site. Building on prior research (SPR 4210 and SPR 4513), we have advanced the electromechanical impedance (EMI) technique for in-situ concrete strength monitoring, crucial for determining safe traffic opening times. These projects have made significant strides in technology, including the development of an IoT-based hardware system for wireless data collection and a cloud-based platform for efficient data processing. A key innovation is the integration of machine learning tools, which not only enhance immediate strength predictions but also facilitate long-term projections vital for maintenance and asset management. To bring this technology to practical use, we collaborated with third-party manufacturers to set up a production line for the sensor and datalogger assembly. The system was extensively tested in various field scenarios, including pavements, patches, and bridge decks. Our refined signal processing algorithms, benchmarked against a mean absolute percentage error (MAPE) of 16%, which is comparable to the ASTM C39 interlaboratory variance of 14%, demonstrate reliable accuracy. Additionally, we have developed a comprehensive user manual to aid field engineers in deploying, connecting, and maintaining the sensing system, paving the way for broader implementation in real-world construction settings

Emergence of multifractality through cascade-like transitions in a mosaic interpolating Aubry-Andr\'{e}-Fibonacci chain

In this paper, we explore the localization features of wave functions in a family of mosaic quasiperiodic chains obtained by continuously interpolating between two limits: the mosaic Aubry-Andr\'{e} (AA) model, known for its exact mobility edges with extended states in the band-center region, and localized ones in the band-edge regions for a large enough modulation amplitude, and the mosaic Fibonacci chain, which exhibits its multifractal nature for all the states except for the extended one with $E=0$ for an arbitrary finite modulation amplitude. We discover that the mosaic AA limit for the states in the band-edge regions evolves into multifractal ones through a cascade of delocalization transitions. This cascade shows lobes of lower fractal dimension values separated by maxima of fractal dimension. In contrast, the states in the band-center region (except for the $E=0$ state) display an anomalous cascading process, where it emerges lobes of higher fractal dimension values are separated by the regions with lower fractal dimensions. Our findings offer insight into understanding the multifractality of quasiperiodic chains.Comment: 12 pages, 11 figure