Gauge Symmetries and Dirac Conjecture

The gauge symmetries of a constrained system can be deduced from the gauge identities with Lagrange method, or the first-class constraints with Hamilton approach. If Dirac conjecture is valid to a dynamic system, in which all the first-class constraints are the generators of the gauge transformations, the gauge transformations deduced from the gauge identities are consistent with these given by the first-class constraints. Once the equivalence vanishes to a constrained system, in which Dirac conjecture would be invalid. By using the equivalence, two counterexamples and one example to Dirac conjecture are discussed to obtain defined results.     

Calogero Conjecture and Hubble Law

Acceptance of the Calogero hypothesis on the cosmic origin of quantizationin the framework of Nelson stochastic mechanics would imply that the age ofuniverse is larger than the standard quantum mechanical interpretation of theredshift measurement implies. This is due to variation of h with the radius of theuniverse and thus with time.     

On the Volumorphism Group, the First Conjugate Point is Always the Hardest

We find a simple local criterion for the existence of conjugate points on the group of volume-preserving diffeomorphisms of a 3-manifold with the Riemannian metric of ideal fluid mechanics, in terms of an ordinary differential equation along each Lagrangian path. Using this criterion, we prove that the first conjugate point along a geodesic in this group is always pathological: the differential of the exponential map always fails to be Fredholm.Much of this work was completed while the author was a Lecturer at the University of Pennsylvania. The author is grateful for their hospitality.     

Quasicrystals and Almost Periodicity

We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let denote the space of uniformly discrete subsets of the Euclidean space. Let denote the elements of that admit an autocorrelation measure. A Patterson set is an element of such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that S is a Patterson set if and only if S is almost periodic in (,), where denotes the Besicovitch topology. Let be an ergodic random element of . We prove that is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum. As an illustration, we study deformed model sets.     

Spouge's Conjecture on Complete and Instantaneous Gelation

We investigate the stochastic counterpart of the Smoluchowski coagulation equation, namely the Marcus–Lushnikov coagulation model. It is believed that for a broad class of kernels, all particles are swept into one huge cluster in an arbitrarily small time, which is known as a complete and instantaneous gelation phenomenon. Indeed, Spouge (also Domilovskii et al. for a special case) conjectured that K(i, j)=(ij) , >1, are such kernels. In this paper, we extend the above conjecture and prove rigorously that if there is a function (i, j), increasing in both i and j such that _j=1 1/(j(i, j))< for all i, and K(i, j)ij(i, j) for all i, j, then complete and instantaneous gelation occurs. Evidently, this implies that any kernels K(i, j)ij(log(i+1)log(j+1)) , >1, exhibit complete instantaneous gelation. Also, we conjuncture the existence of a critical (or metastable) sol state: if lim _i+j ij/K(i, j)=0 and _{i, j=1} 1/K(i, j)=, then gelation time T _g satisfies 0<T _g<. Moreover, the gelation is complete after T _g.     

Modified No-hair Conjecture and the Limiting Process

It is argued that the Schwarzschild black hole solution follows as a unique limit of the Brans-Dicke Class I solutions, provided the correct iterated limit is taken. Such a uniqueness is essential for the validity of a recent version of the no-hair conjecture. A non-trivial modification to this version is proposed in order to exclude Brans-Dicke Class IV solutions which appear to represent scalar hair black holes in general.     

Bipartite Matching and Van der Waerden Conjecture

In this paper I show that the free energy F and the cost C associated to a bipartite matching problem can be explicitly estimated in term of the solution of a suitable system of equations (cavity equations in the following). The proof of these results relies on a well known result in combinatorics: the Van der Waerden conjecture (Egorychev–Falikman Theorem). Cavity equations, derived by a mean field argument by Mèzard and Parisi, can be considered as a smoothed form of the dual formulation for the bipartite matching problem. Moreover cavity equation are the Euler–Lagrange equations of a convex functional G parameterized by the temperature T. In term of their unique solution it is possible to define a free-energy-like function of the temperature g(T). g is a strictly decreasing concave function of T and C=g(0). The convexity of G allows to define an explicit algorithm to find the solution of the cavity equations at a given temperature T. Moreover, once the solution of the cavity equations at a given temperature T is known, the properties of g allow to find exact estimates from below and from above of the cost C.     

Weak Gravity Conjecture and extremal black holes

Motivated by the desire to improve our understanding of the Weak Gravity Conjecture, we compute the one-loop correction of charged particles to the geometry and entropy of extremal black holes in 4D. We use the entropy function formalism to extend previous analysis that dealt with neutral particles, and obtain the corrections to the horizon entropy for different regimes of black hole masses. These corrections are small in general. They are furthermore reduced when supersymmetry is present, and disappear in N = 4 supergravity. We provide some speculative arguments that in a theory with only sub-extremal particles, classical Reissner-Nordstrom black holes actually possess an infinite microcanonical entropy, though only a finite amount is visible to an external observer, as shown by the horizon entropy computation.     

Fully Developed Turbulence and the Multifractal Conjecture

We review the Parisi-Frisch (Proc. Int. School of Physics “E. Fermi”, pp. 84–87, North-Holland, Amsterdam, 1985) MultiFractal formalism for Navier-Stokes turbulence with particular emphasis on the issue of statistical fluctuations of the dissipative scale. We do it for both Eulerian and Lagrangian Turbulence. We also show new results concerning the application of the formalism to the case of Shell Models for turbulence. The latter case will allow us to discuss the issue of Reynolds number dependence and the role played by vorticity and vortex filaments in real turbulent flows.     

Correlated Fractal Percolation and the Palis Conjecture

Let F ₁ and F ₂ be independent copies of one-dimensional correlated fractal percolation, with almost sure Hausdorff dimensions dim?_H(F ₁) and dim?_H(F ₂). Consider the following question: does dim?_H(F ₁)+dim?_H(F ₂)>1 imply that their algebraic difference F ₁?F ₂ will contain an interval? The well known Palis conjecture states that ‘generically’ this should be true. Recent work by Kuijvenhoven and the first author (Dekking and Kuijvenhoven in J. Eur. Math. Soc., to appear) on random Cantor sets cannot answer this question as their condition on the joint survival distributions of the generating process is not satisfied by correlated fractal percolation. We develop a new condition which permits us to solve the problem, and we prove that the condition of Dekking and Kuijvenhoven (J. Eur. Math. Soc., to appear) implies our condition. Independently of this we give a solution to the critical case, yielding that a strong version of the Palis conjecture holds for fractal percolation and correlated fractal percolation: the algebraic difference contains an interval almost surely if and only if the sum of the Hausdorff dimensions of the random Cantor sets exceeds one.     

Proof of a Conjecture by Lewandowski and Thiemann

It is proven that for compact, connected and semisimple structure groups every degenerate labelled web is strongly degenerate. This conjecture by Lewandowski and Thiemann implies that diffeomorphism invariant operators in the category of piecewise smooth immersive paths preserve the decomposition of the space of integrable functions w.r.t. the degeneracy and symmetry of the underlying labelled webs. This property is necessary for lifting these operators to well-defined operators on the space of diffeomorphism invariant states.     

Sums of Hermitian Squares and the BMV Conjecture

We show that all the coefficients of the polynomial

物理教学中的科学猜想与直觉思维

直觉思维是在早已获得的经验、知识的基础上,凭思维的"感觉"直观地把握事物本质和规律的心理过程 是对客观现象的详细内容或所遇问题没有经过充分逻辑推理和系统论证而作出的一种迅速而"径直"猜度的认识活动.比如,当学生遇到难题百思不得其解时,有时却忽然灵机一动,豁然开朗,从而导致认识活动的飞跃和突破,这种思维活动便属直觉思维,也叫灵感思维、顿悟思维.当然,直觉的猜想和假说需要逻辑的论证和事实的检验.在创造性的思维活动中,逻辑思维和直觉思维是相互补充、相互渗透的,但直觉思维具有独特的作用.正如爱因斯坦所说,在科学研究和创造发明中"真正可贵的因素是直觉".     

Spectra and Transport in Almost Periodic Dimers

We study spectral properties of discrete Schrödinger operators with potentials obtained via dimerization of a class of aperiodic sequences. It is shown that both the nature of the autocorrelation measure of a regular sequence and the presence of generic (full probability) singular continuous spectrum in the hull of primitive and palindromic (four block substitution) potentials are robust under dimerization. Generic results also hold for circle potentials. We illustrate these results with numerical studies of the quantum mean square displacement as a function of time. The numerical techniques provide a very fast algorithm for the time evolution of wave packets.     

跷跷板模型中三阶中微子混合矩阵的幺正性破坏

对标准模型的一种简单扩充就是引入n个重的右手中微子且保持其SU(2)_L×U(1)_Y规范对称性. 通过对角化(3+n)×(3+n)阶中微子质量矩阵, 得到关于ν_e, ν_μ和ν_τ的有效 质量矩阵的精确的解析表达式. 结果表明, 在轻子带电弱流中出现的3×3中微子混合矩阵V必须不是严格幺正的. 如果通过跷跷板机制产生正确的轻的中微子的质量标度, 那么V的幺正性破坏的程度非常小, 几乎可以忽略. 类似的结论同样可以在第二类跷跷板模型中得到.     

Almost every pure state of three qubits is completely determined by its two-particle reduced density matrices

In a system of n quantum particles, we define a measure of the degree of irreducible n-way correlation, by which we mean the correlation that cannot be accounted for by looking at the states of n-1 particles. In the case of almost all pure states of three qubits, we show that there is no such correlation: almost every pure state of three qubits is completely determined by its two-particle reduced density matrices.