Hamilton-Jacobi approach to thermodynamic transformations

In this note, we formulate and study a Hamilton-Jacobi approach for describing thermodynamic transformations. The thermodynamic phase space assumes the structure of a contact manifold with the points representing equilibrium states being restricted to certain submanifolds of this phase space. We demonstrate that Hamilton-Jacobi theory consistently describes thermodynamic transformations on the space of externally controllable parameters or equivalently the space of equilibrium states. It turns out that in the Hamilton-Jacobi description, the choice of the principal function is not unique but, the resultant dynamical description for a given transformation remains the same irrespective of this choice. Some examples involving thermodynamic transformations of the ideal gas are discussed where the characteristic curves on the space of equilibrium states completely describe the dynamics. The geometric Hamilton-Jacobi formulation which has emerged recently is also discussed in the context of thermodynamics.Comment: v1: Preliminary version; v2: Revised versio

Generalized virial theorem for contact Hamiltonian systems

We formulate and study a generalized virial theorem for contact Hamiltonian systems. Such systems describe mechanical systems in the presence of simple dissipative forces such as Rayleigh friction, or the vertical motion of a particle falling in a fluid (quadratic drag) under the action of constant gravity. We find a generalized virial theorem for contact Hamiltonian systems which is distinct from that obtained earlier for the symplectic case. The `contact' generalized virial theorem is shown to reduce to the earlier result on symplectic manifolds as a special case. Various examples of dissipative mechanical systems are discussed. We also formulate a generalized virial theorem in the contact Lagrangian framework.Comment: Preliminary version. Comments would be very much appreciate

Daylighting and Thermo-Electrical performance of an Autonomous Suspended Particle Device Evacuated Glazing

Suspended particle device (SPD) glazing is an AC powered switchable glazing. PV powered SPD evacuated (vacuum) glazing was proposed with the potential of reducing the heating demand, cooling demand and artificial lighting demand of a building. To achieve an autonomous SPD vacuum glazing, semi empirical simulation and outdoor characterisation was explored in this thesis. Transmission of SPD glazing (area 0.058 m2) varied from 5% when opaque to 55% when transparent in the presence of 110 V, 0.07 W AC supply was characterised in outdoor test cell in Dublin. The SPD glazing has variable spectral transmission in the presence of variable applied voltage, with high transmission in the near infrared between 700 to 1100 nm. 30% transparent SPD glazing in a particular room configuration provided a constant 4% daylight factor with acceptable glare. Use of a 0.34 m2 vertical photovoltaic (PV) panel was investigated to self-power (autonomous) an SPD glazing system. The dynamic behaviour of the PV-powered SPD glazing gave good switching times that would maintain occupant comfort. It was observed that SPD material inside a glazing unit absorbs solar radiation giving a high glazing surface temperature. For this SPD glazing alone, the overall heat transfer coefficient (U-value) was found to be 5.9 W/m2K typical of a single glazing. A SPD switchable double-glazing, was found to have a U-value of 1.99 W/m2K. A vacuum glazing was attached to the SPD glazing was found to have a U-value of 1.14 W/m2K

Subconvexity for GL(1)GL(1) twists of Rankin-Selberg LL-functions

Let $f$ and $g$ be two Hecke-Maass or holomorphic primitive cusp forms for $SL(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of modulus $p$, a prime. A subconvex bound for the central values of the Rankin-Selberg L-functions is $L(s, f \otimes g \otimes \chi)$ is give by $$L(\frac{1}{2}, f \otimes g \otimes \chi) \ll_{f,g,\epsilon}p^{\frac{19}{20}+ \frac{202}{100}\theta +\epsilon} ,$$ for any $\epsilon > 0$, where the implied constant depends only on the forms $f,g$ and $\epsilon$.Comment: First Draft. arXiv admin note: text overlap with arXiv:2111.0069

Statistical ensembles and logarithmic corrections to black hole entropy

In this paper, we consider general statistical ensembles and compute logarithmic corrections to the microcanonical entropy resulting due to thermodynamic fluctuations which are controlled by the boundary conditions, i.e. due to choice of ensemble. The framework is applied to the case of non-extremal black holes to give certain logarithmic corrections to the Bekenstein-Hawking entropy. We argue that within the framework of black hole chemistry, where the cosmological constant is identified with bulk pressure, the isoenthalpic-isobaric entropy rather than microcanonical entropy carries a more natural and consistent thermodynamic interpretation as black hole entropy. Logarithmic corrections to both microcanonical and isoenthalpic-isobaric entropies of black holes are computed, and we show that the latter set of corrections in black hole chemistry are of the same form as corrections to the microcanonical entropy in theories where the cosmological constant is not interpreted as a thermodynamic pressure. Finally, we compute logarithmic corrections to entropy in the framework of holographic black hole chemistry. We emphasize upon the choice of statistical ensemble, both in the bulk and on the boundary, in order to have a consistent comparison between them. The corrections studied in this paper are distinct from those obtained from Euclidean quantum gravity

Making Risk Minimization Tolerant to Label Noise

In many applications, the training data, from which one needs to learn a classifier, is corrupted with label noise. Many standard algorithms such as SVM perform poorly in presence of label noise. In this paper we investigate the robustness of risk minimization to label noise. We prove a sufficient condition on a loss function for the risk minimization under that loss to be tolerant to uniform label noise. We show that the $0-1$ loss, sigmoid loss, ramp loss and probit loss satisfy this condition though none of the standard convex loss functions satisfy it. We also prove that, by choosing a sufficiently large value of a parameter in the loss function, the sigmoid loss, ramp loss and probit loss can be made tolerant to non-uniform label noise also if we can assume the classes to be separable under noise-free data distribution. Through extensive empirical studies, we show that risk minimization under the $0-1$ loss, the sigmoid loss and the ramp loss has much better robustness to label noise when compared to the SVM algorithm

Robust Loss Functions under Label Noise for Deep Neural Networks

In many applications of classifier learning, training data suffers from label noise. Deep networks are learned using huge training data where the problem of noisy labels is particularly relevant. The current techniques proposed for learning deep networks under label noise focus on modifying the network architecture and on algorithms for estimating true labels from noisy labels. An alternate approach would be to look for loss functions that are inherently noise-tolerant. For binary classification there exist theoretical results on loss functions that are robust to label noise. In this paper, we provide some sufficient conditions on a loss function so that risk minimization under that loss function would be inherently tolerant to label noise for multiclass classification problems. These results generalize the existing results on noise-tolerant loss functions for binary classification. We study some of the widely used loss functions in deep networks and show that the loss function based on mean absolute value of error is inherently robust to label noise. Thus standard back propagation is enough to learn the true classifier even under label noise. Through experiments, we illustrate the robustness of risk minimization with such loss functions for learning neural networks.Comment: Appeared in AAAI 201

Optimization of PV powered SPD switchable glazing to minimise probability of loss of power supply

Suspended particle device (SPD) glazing is an electrically actuated switchable glazing. It requires alternate current (AC) power supply to switch from opaque to transparent state. To power this glazing using PV device requires inverter. Optimization of AC powered switchable SPD glazing using photovoltaic (PV) device has been evaluated using loss of power supply probability (LPSP). Electrically switchable direct current (DC) powered electrochromic glazing was also considered in this investigation as it doesn\u27t need any inverter to couple with PV. It is concluded that behaviour of these glazings is the dominant factor in performance optimization outweighting than azimuthal orientation and inclination of PV