New Types of Thermodynamics from (1+1)(1+1)-Dimensional Black Holes

For normal thermodynamic systems superadditivity $\S$, homogeneity \H and concavity \C of the entropy hold, whereas for $(3+1)$-dimensional black holes the latter two properties are violated. We show that $(1+1)$-dimensional black holes exhibit qualitatively new types of thermodynamic behaviour, discussed here for the first time, in which \C always holds, \H is always violated and $\S$ may or may not be violated, depending of the magnitude of the black hole mass. Hence it is now seen that neither superadditivity nor concavity encapsulate the meaning of the second law in all situations.Comment: WATPHYS-TH93/05, Latex, 10 pgs. 1 figure (available on request), to appear in Class. Quant. Gra

A Fresh Look at Entropy and the Second Law of Thermodynamics

This paper is a non-technical, informal presentation of our theory of the second law of thermodynamics as a law that is independent of statistical mechanics and that is derivable solely from certain simple assumptions about adiabatic processes for macroscopic systems. It is not necessary to assume a-priori concepts such as "heat", "hot and cold", "temperature". These are derivable from entropy, whose existence we derive from the basic assumptions. See cond-mat/9708200 and math-ph/9805005.Comment: LaTex file. To appear in the April 2000 issue of PHYSICS TODA

Construction and Analysis of Random Networks with Explosive Percolation

The existence of explosive phase transitions in random (Erdös Rényi-type) networks has been recently documented by Achlioptas, D’Souza, and Spencer via simulations. In this Letter we describe the underlying mechanism behind these first-order phase transitions and develop tools that allow us to identify (and predict) when a random network will exhibit an explosive transition. Several interesting new models displaying explosive transitions are also presented

Hierarchical networks, power laws, and neuronal avalanches

We show that in networks with a hierarchical architecture, critical dynamical behaviors can emerge even when the underlying dynamical processes are not critical. This finding provides explicit insight into current studies of the brain\u27s neuronal network showing power-lawavalanches in neural recordings, and provides a theoretical justification of recent numerical findings. Our analysis shows how the hierarchical organization of a network can itself lead to power-law distributions of avalanche sizes and durations, scaling laws between anomalous exponents, and universal functions—even in the absence of self-organized criticality or critical points. This hierarchy-induced phenomenon is independent of, though can potentially operate in conjunction with, standard dynamical mechanisms for generating power laws

Quasi-Homogeneous Thermodynamics and Black Holes

We propose a generalized thermodynamics in which quasi-homogeneity of the thermodynamic potentials plays a fundamental role. This thermodynamic formalism arises from a generalization of the approach presented in paper [1], and it is based on the requirement that quasi-homogeneity is a non-trivial symmetry for the Pfaffian form $\delta Q_{rev}$. It is shown that quasi-homogeneous thermodynamics fits the thermodynamic features of at least some self-gravitating systems. We analyze how quasi-homogeneous thermodynamics is suggested by black hole thermodynamics. Then, some existing results involving self-gravitating systems are also shortly discussed in the light of this thermodynamic framework. The consequences of the lack of extensivity are also recalled. We show that generalized Gibbs-Duhem equations arise as a consequence of quasi-homogeneity of the thermodynamic potentials. An heuristic link between this generalized thermodynamic formalism and the thermodynamic limit is also discussed.Comment: 39 pages, uses RevteX. Published version (minor changes w.r.t. the original one

Combinatorial Games with a Pass: A dynamical systems approach

By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game.Comment: 39 pages, 13 figures, published versio

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

Discovering New Physics in the Decays of Black Holes

If the scale of quantum gravity is near a TeV, the LHC will be producing one black hole (BH) about every second, thus qualifying as a BH factory. With the Hawking temperature of a few hundred GeV, these rapidly evaporating BHs may produce new, undiscovered particles with masses ~100 GeV. The probability of producing a heavy particle in the decay depends on its mass only weakly, in contrast with the exponentially suppressed direct production. Furthemore, BH decays with at least one prompt charged lepton or photon correspond to the final states with low background. Using the Higgs boson as an example, we show that it may be found at the LHC on the first day of its operation, even with incomplete detectors.Comment: 4 pages, 3 figure