Noether’s theorem for fractional variational problems of variable order

We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether’s theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.     

Stern-Gerlach Experiment’s Interpretations and Noether’s Theorem

In this paper we suggest that theories treating two interacting objects in a different manner (for instance electromagnetic field of a laser classically, and the interacting atom as a quantum object) should be called “mixed”. Mixed theories are not so rare in Physics. One just should look at the whole area of Atomic, Molecular and Optical Physics in which mixed theories are often used, and, also, theories including quantum object interacting with classical surroundings that are the subject of our present discussion: the field of Quantum decoherence, when applied to resolving the dilemma should classical trajectories be used in explaining the Stern-Gerlach experiment or not. Consequently we are proving one improved corollary to Noether’s theorem, stating that mixed theories are not supporting the law of conservation of angular momentum and spin, as they are not based on the isotropy of space-time.     

Noether＇s theorem and one-step corrections method for holonomic system

In this paper, a new computational method for improving the accuracy of numerically computed solutions is introduced. The computational method is based on the one-step method and conserved quantities of holonomic systems are considered as kinematical constraints in this method.     

Noether’s theorem for non-conservative Hamilton system based on El-Nabulsi dynamical model extended by periodic laws

This paper focuses on the Noether symmetries and the conserved quantities for both holonomic and nonholonomic systems based on a new non-conservative dynamical model introduced by El-Nabulsi. First, the El-Nabulsi dynamical model which is based on a fractional integral extended by periodic laws is introduced, and El-Nabulsi–Hamilton’s canonical equations for non-conservative Hamilton system with holonomic or nonholonomic constraints are established. Second,the definitions and criteria of El-Nabulsi–Noether symmetrical transformations and quasi-symmetrical transformations are presented in terms of the invariance of El-Nabulsi–Hamilton action under the infinitesimal transformations of the group. Finally, Noether’s theorems for the non-conservative Hamilton system under the El-Nabulsi dynamical system are established,which reveal the relationship between the Noether symmetry and the conserved quantity of the system.     

Comment on “On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models" by D. Helbing and A.F. Johansson

There has been a debate in the transportation research community over the validity of a class of traffic flow models. These models have the peculiar property that one of its characteristic speeds is faster than that of vehicular traffic. This note attempts to provide an overview of the diverging views on how to interpret this property, and specific comments on the interpretation of Helbing and Johansson [On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models, Eur. Phys. J. B (2009), DOI: 10.1140/epjb/e2009-00182-7]. We showed that having faster-than-traffic characteristics does produce counterintuitive predictions, and they cannot be explained away by a linear stability analysis. As such, the existence of such characteristics must be justified by the physics of traffic, and verified through empirical observations.     

Conservation Laws and Associated Noether Type Vector Fields via Partial Lagrangians and Noether’s Theorem for the Liang Equation

We show how one can construct conservation laws of the Liang equation which is not variational but may be regarded as Euler-Lagrange in part. This first requires the determination of the Noether-type symmetries associated with the partial Lagrangian. The final construction of the conservation laws resort to a formula equivalent to Noether’s theorem. A variety of subclasses are given and, for each, a large number of conserved flows are found—the method is usable for any general choice of the variable speed of sound.     

Large Deviations, Guerra’s and A.S.S. Schemes, and the Parisi Hypothesis

We investigate the problem of computing $$\lim_{N \to \infty}\frac{1}{aN}\log EZ_N^a$$ for any value of a, where Z _N is the partition function of the celebrated Sherrington-Kirkpatrick (SK) model, or of some of its natural generalizations. This is a natural “large deviation” problem. Its study helps to get a fresh look at some of the recent ideas introduced in the area, and raises a number of natural questions. We provide a complete solution for a ≥ 0.