We discuss a space of Young measures in connection with some variational problems. We use it to present a proof of the Theorem of Tonelli on the existence of minimizing curves. We generalize a recent result of Ambrosio, Gigli and Savaré on the decomposition of the weak solutions of the transport equation. We also prove, in the context of Mather theory, the equality between Closed measures and H...

In this study, we develop a physical model of grand piano string involving high-fidelity expression hammer shank dynamics. A is modeled as general beam, and its governing equation analyzed using the finite element method (FEM). addition, rotational motion explicitly expressed incorporated into formulation FEM. The developed also involves more strings, which numerically difference (FDM). propose...

How humans infer causation from covariation has been the subject of a vigorous debate, most recently between the computational causal power account (P. W. Cheng, 1997) and associative learning theorists (e.g., K. Lober & D. R. Shanks, 2000). Whereas most researchers in the subject area agree that causal power as computed by the power PC theory offers a normative account of the inductive process...

A hallmark feature of elemental associative learning theories is that multiple cues compete for associative strength when presented with an outcome. Cue competition effects have been observed in humans, both in forward and in backward blocking procedures (e.g., Shanks, 1985) and are often interpreted as evidence for an associative account of human causal learning (e.g., Shanks & Dickinson, 1987...

The generalization of Shanks' e-transformation to double series is discussed and a class of nonlinear transformations, the [A/S]f£ transformations, for accelerating the convergence of infinite double series is presented. It is constructed so as to sum exactly infinite double series whose terms satisfy certain finite linear double difference equations; in that sense it is a generalization of Sha...

Erdős conjectured that there are x1−o(1) Carmichael numbers up to x, whereas Shanks was skeptical as to whether one might even find an x up to which there are more than √ x Carmichael numbers. Alford, Granville and Pomerance showed that there are more than x2/7 Carmichael numbers up to x, and gave arguments which even convinced Shanks (in person-to-person discussions) that Erdős must be correct...

In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax– Oleinik semigroups. This is equivalent to the solvability of an associated multi– time Hamilton–Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions ...

We discuss a space of Young measures in connection with some variational problems. We use it to present a proof of the Theorem of Tonelli on the existence of minimizing curves. We generalize a recent result of Ambrosio, Gigli and Savaré on the decomposition of the weak solutions of the transport equation. We also prove, in the context of Mather theory, the equality between Closed measures and H...