Features of photon diffusion in a dispersed medium

Energy transfer by thermal radiation in a dispersed medium with a variable refractive index is discussed. This transfer can be described by a surprisingly simple diffusion equation. The process is naturally to interpret as the photon diffusion. The diffusion equation is free from strict conditions of applicability of the radiation transfer equation, which are usually not satisfied in disperse media with densely packed inhomogeneities. Quantum constraints on the value of the photon diffusion coefficient are derived. These restrictions turn out to be similar to the conditions for the applicability of geometric optics. The lower limit of the thermal conductivity coefficient is obtained, which is easier to verify in the experiment. An independent derivation of this limitation is given from considerations of symmetry and dimension.

Should California’s Film Script Cases Be Merged into Trade Secret Law?

In legal disputes where one party claims that it submitted an idea to another party and alleges that the latter used that idea without permission or compensation, two categories of California intellectual property law have increasingly come to resemble one another: (1) trade secret law, most often applied in business or technical contexts; and (2) idea submission law, primarily applied in cases involving film scripts and other media productions. Over the decades, these regimes have developed separately, within distinct business and legal cultures. But recent developments in California trade secret law have brought the two closer together; in some areas, they may even be approaching a unified body of law. This Article explores that possibility. It concludes that although a partial merger is inevitable, the two core causes of action—for asserted trade secrets, a misappropriation claim; for idea submissions, a so-called Desny claim for breach of implied-in-fact contract—will and should remain distinct. A partial merger, however, would lead to beneficial exchanges in areas where their doctrines already overlap: (1) idea submission’s “independent development” and trade secret’s “independent derivation” defenses; and (2) statutory preemption under California’s Uniform Trade Secrets Act (CUTSA). California’s idea submission cases have developed sophisticated and robust means to adjudicate the concept of “independent development”—that is, a defendant’s assertion that despite receiving the plaintiff’s idea, it nonetheless came up with the disputed film, television show, or other concept on its own. By contrast, the important and analogous defense of “independent derivation” in California trade secret law remains underdeveloped. This Article argues that the idea submission cases offer a far more rigorous analysis of the defense and could inform similar decisions under trade secret law. In particular, it proposes a methodology that courts can use to adjudicate the independent derivation defense, inspired by the idea submission cases. The idea submission cases largely survived copyright preemption challenges in the 1990s and 2000s after Ninth Circuit rulings preserved the viability of some idea submission causes of action under state law. But surviving copyright preemption is not the same thing as surviving CUTSA trade secret preemption. This more recent form of IP preemption is broad, and it subsumes tort claims seeking to protect information said to be confidential. This Article argues that the CUTSA preempts peripheral idea submission tort claims such as breach of confidence, but it does not preempt the core claim at the heart of California’s idea submission regime—the Desny claim for breach of implied-in-fact contract. The proposed partial merger recognizes the public policy ends of each regime: protecting weaker parties who submit ideas to film and media studios (in narrowlydefined circumstances), and ensuring that litigants cannot use tort claims to subvert the protections the CUTSA and related employee mobility rules provide for the free use of publicly available information that does not meet the statutory definition of a trade secret.

A Boltzmann Approach to Percolation on Random Triangulations

We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length$n$decays exponentially with$n$except at a particular value$p_{c}$of the percolation parameter$p$for which the decay is polynomial (of order$n^{-10/3}$). Moreover, the probability that the origin cluster has size$n$decays exponentially if$p<p_{c}$and polynomially if$p\geqslant p_{c}$.The critical percolation value is$p_{c}=1/2$for site percolation, and$p_{c}=(2\sqrt{3}-1)/11$for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at$p_{c}$, the percolation clusters conditioned to have size$n$should converge toward the stable map of parameter$\frac{7}{6}$introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

The Form of Even Perfect Numbers-- an Independent Derivation

Here, I present my own, independently pursued derivation of the general form of even perfect numbers.

Spinning bodies in general relativity

A covariant Hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of Hamiltonians accounting for specific properties and interactions of spinning bodies. The dynamics for a minimal and a specific non-minimal Hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy–momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

Potential vorticity in magnetohydrodynamics

A version of Noether's second theorem using Lagrange multipliers is used to investigate fluid relabelling symmetries conservation laws in magnetohydrodynamics (MHD). We obtain a new generalized potential vorticity type conservation equation for MHD which takes into account entropy gradients and the J × B force on the plasma due to the current J and magnetic induction B. This new conservation law for MHD is derived by using Noether's second theorem in conjunction with a class of fluid relabelling symmetries in which the symmetry generator for the Lagrange label transformations is non-parallel to the magnetic field induction in Lagrange label space. This is associated with an Abelian Lie pseudo algebra and a foliated phase space in Lagrange label space. It contains as a special case Ertel's theorem in ideal fluid mechanics. An independent derivation shows that the new conservation law is also valid for more general physical situations.

New construction of algebro-geometric solutions to the Camassa–Holm equation and their numerical evaluation

An independent derivation of solutions to the Camassa–Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay’s identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the limit where the surface degenerates to the Riemann sphere, and where solitons and cuspons appear.

GAUSS' LINKING NUMBER REVISITED

In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism. A brief discussion of a possibly independent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided.

Clivadas sem operador no português brasileiro (Clefts without an operator in Brazilian Portuguese)

O português apresenta um tipo de sentença que tem sido tradicionalmente analisado como uma pseudo-clivada reduzida, as semi-clivadas, obtidas através do apagamento do operador-Q. Construcões semelhantes são também encontradas no Espanhol Caribenho (EsC) e analisadas como tendo um Operador nulo. Percebendo que nem todas as pseudo-clivadas têm uma reduzida correspondente, Bosque (1999) e Camacho (2006), tratando do EsC, e Mioto (2008), tratando do português brasileiro (PB), propoÃÉem uma derivação independente para as semi-clivadas. O presente trabalho se constitui numa tentativa diferente de explicar as semi-clivadas, independentemente das pseudo-clivadas, para dar conta de construções não permitidas em EsC. A diferença proposta é que, enquanto nas pseudo- clivadas o foco é o argumento, o adjunto ou o VP, nas semi- clivadas o foco é o resíduo de VP (depois da subida do verbo) ou Adverbiais adjuntos a VP.PALAVRAS-CHAVE: Construções de Foco Estreito e Contrastivo. Espanhol Caribenho. Movimento Residual. Português Brasileiro.ABSTRACT Portuguese exhibits a type of sentence which has been traditionally analyzed as the reduction of a pseudo-cleft, built up through the deletion/ erasure of the wh-operator. Similar constructions are also found in the Caribbean dialects of Spanish (CS) and analyzed as containing a null Operator. Realizing that not all pseudo-clefts with an overt wh-operator have a corresponding reduced cleft, or semi-cleft, Bosque (1999) and Camacho (2006), for CS, and Mioto (2008), for Brazilian Portuguese, propose an independent derivation for semi-clefts. This paper is another attempt at deriving reduced or semi-clefts independently of pseudo-clefts, an analysis that can account for constructions found out in BP, which are inexistent in CS. I claim that, while pseudo-clefts focalize arguments, VP or adjuncts, reduced clefts focalize only remnant VPs and VP-adjuncts.KEYWORDS: Narrow and Contrastive Focus Construction. Caribbean Spanish. Remnant Movement. Brazilian Portuguese.