Hitting Matrix and Domino Tiling with Diagonal Impurities

As a continuation to our previous work (Nakano and Sadahiro in Fundam. Inform. 117:249–264, 2012; Nakano and Sadahiro in J. Stat. Phys. 139(4):565–597, 2010), we consider the domino tiling problem with impurities. (1) If we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the hitting matrix method (Fomin in Trans. Am. Math. Soc. 353(9):3563–3583, 2001). (2) We have an alternative proof of the main result in Nakano and Sadahiro (Fundam. Inform. 117:249–264, 2012) and result in (1) above using the formula by Kenyon and Wilson (Trans. Am. Math. Soc. 363(3):1325–1364, 2011; Electron. J. Comb. 16(1):112, 2009) of counting the number of groves on circular planar graphs. (3) We study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity, as well as the scaling limit of those.     

Quantum Random Walks with General Particle States

A convergence theorem is obtained for quantum random walks with particles in an arbitrary normal state. This unifies and extends previous work on repeated-interactions models, including that of Attal and Pautrat (Ann Henri Poincaré 7:59–104 2006) and Belton (J Lond Math Soc 81:412–434, 2010; Commun Math Phys 300:317–329, 2010). When the random-walk generator acts by ampliation and either multiplication or conjugation by a unitary operator, it is shown that the quantum stochastic cocycle which arises in the limit is driven by a unitary process.     

Non-Equilibrium Statistical Mechanics of Turbulence

The macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow for a suitable mechanical system (Ruelle, PNAS 109:20344–20346, 2012). Turbulent fluctuations (intermittency) then correspond to thermal fluctuations, and this allows to estimate the exponents \(\tau _p\) and \(\zeta _p\) associated with moments of dissipation fluctuations and velocity fluctuations. This approach, initiated in an earlier note (Ruelle, 2012), is pursued here more carefully. In particular we derive probability distributions at finite Reynolds number for the dissipation and velocity fluctuations, and the latter permit an interpretation of numerical experiments (Schumacher, Preprint, 2014). Specifically, if \(p(z)dz\) is the probability distribution of the radial velocity gradient we can explain why, when the Reynolds number \(\mathcal{R}\) increases, \(\ln p(z)\) passes from a concave to a linear then to a convex profile for large \(z\) as observed in (Schumacher, 2014). We show that the central limit theorem applies to the dissipation and velocity distribution functions, so that a logical relation with the lognormal theory of Kolmogorov (J. Fluid Mech. 13:82–85, 1962) and Obukhov is established. We find however that the lognormal behavior of the distribution functions fails at large value of the argument, so that a lognormal theory cannot correctly predict the exponents \(\tau _p\) and \(\zeta _p\) .     

High harmonic generation in a gas-filled hollow-core photonic crystal fiber

High harmonic generation (HHG) of intense infrared laser radiation (Ferray et al., J. Phys. B: At. Mol. Opt. Phys. 21:L31, 1988; McPherson et al., J. Opt. Soc. Am. B 4:595, 1987) enables coherent vacuum-UV (VUV) to soft-X-ray sources. In the usual setup, energetic femtosecond laser pulses are strongly focused into a gas jet, restricting the interaction length to the Rayleigh range of the focus. The average photon flux is limited by the low conversion efficiency and the low average power of the complex laser amplifier systems (Keller, Nature 424:831, 2003; Südmeyer et al., Nat. Photonics 2:599, 2008; Röser et al., Opt. Lett. 30:2754, 2005; Eidam et al., IEEE J. Sel. Top. Quantum Electron. 15:187, 2009) which typically operate at kilohertz repetition rates. This represents a severe limitation for many experiments using the harmonic radiation in fields such as metrology or high-resolution imaging. Driving HHG with novel high-power diode-pumped multi-megahertz laser systems has the potential to significantly increase the average photon flux. However, the higher average power comes at the expense of lower pulse energies because the repetition rate is increased by more than a thousand times, and efficient HHG is not possible in the usual geometry. So far, two promising techniques for HHG at lower pulse energies were developed: external build-up cavities (Gohle et al., Nature 436:234, 2005; Jones et al., Phys. Rev. Lett. 94:193, 2005) and resonant field enhancement in nanostructured targets (Kim et al., Nature 453:757, 2008). Here we present a third technique, which has advantages in terms of ease of HHG light extraction, transverse beam quality, and the possibility to substantially increase conversion efficiency by phase-matching (Paul et al., Nature 421:51, 2003; Ren et al., Opt. Express 16:17052, 2008; Serebryannikov et al., Phys. Rev. E (Stat. Nonlinear Soft Matter Phys.) 70:66611, 2004; Serebryannikov et al., Opt. Lett. 33:977, 2008; Zhang et al., Nat. Phys. 3:270, 2007). The interaction between the laser pulses and the gas occurs in a Kagome-type Hollow-Core Photonic Crystal Fiber (HC-PCF) (Benabid et al., Science 298:399, 2002), which reduces the detection threshold for HHG to only 200 nJ. This novel type of fiber guides nearly all of the light in the hollow core (Couny et al., Science 318:1118, 2007), preventing damage even at intensities required for HHG. Our fiber guided 30-fs pulses with a pulse energy of more than 10 μJ, which is more than five times higher than for any other photonic crystal fiber (Hensley et al., Conference on Lasers and Electro-Optics (CLEO), IEEE Press, New York, 2008).     

Thermodynamics and holography of tachyon cosmology

Recently, we have investigated the dynamics of the universe in tachyon cosmology with non-minimal coupling to matter (Farajollahi et al. in Mod Phys Lett A 26(15):1125–1135, 2011; Phys Lett B 711(3–4)15:225–231,2012; Phys Rev D 83:124042, 2011; JCAP 10:014, 20112011; JCAP 05:017, 2011). In particular, for the interacting holographic dark energy (IHDE), the model is studied in Farajollahi et al. (Astrophys Space Sci 336(2):461–467, 2011). In the current work, a significant observational program has been conducted to unveil the model’s thermodynamic properties. Our result shows that the IHDE version of our model better fits the observational data than $\Lambda $ CDM model. The first and generalized second thermodynamics laws for the universe enveloped by cosmological apparent and event horizon are revisited. From the results, both first and generalized second laws, constrained by the observational data, are satisfied on cosmological apparent horizon.In addition, the total entropy is verified with the observation only if the horizon of the universe is taken as apparent horizon. Then, due to validity of generalized second law, the current cosmic acceleration is also predicted.     

The Vlasov-Poisson-Boltzmann System without Angular Cutoff

This paper is concerned with the Vlasov-Poisson-Boltzmann system for plasma particles of two species in three space dimensions. The Boltzmann collision kernel is assumed to be angular non-cutoff with 3 < γ < ?2s and 1/2 ≤ s < 1, where γ , s are two parameters describing the kinetic and angular singularities, respectively. We establish the global existence and convergence rates of classical solutions to the Cauchy problem when initial data is near Maxwellians. This extends the results in Duan et al. (J Diff Eqs 252(12):6356–6386, 2012, Math Models Methods Appl Sci 23(6):927, 2013) for the cutoff kernel with ?2 ≤ γ ≤ 1 to the case ?3 < γ < ?2 as long as the angular singularity exists instead and is strong enough, i.e., s is close to 1. The proof is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in Gressman and Strain (J Amer Math Soc 24(3):771–847, 2011) and the Vlasov-Poisson-Landau system in Guo (J Amer Math Soc 25:759–812, 2012).     

Frame-dragging fields and spin 1 gravitomagnetic radiation

Experimental results published in 2004 (Ciufolini and Pavlis in Nature 431:958–960, 2004) and 2011 (Everitt et al. in Phys Rev Lett 106:221101, 1–5, 2011) have confirmed the frame-dragging phenomenon for a spinning earth predicted by Einstein’s field equations. Since this is observed as a precession caused by the gravitomagnetic (GM) field of the rotating body, these experiments may be viewed as measurements of a GM field. The effect is encapsulated in the classic steady state solution for the vector potential field $\zeta $ of a spinning sphere–a solution applying to a sphere with angular momentum J and describing a field filling space for all time (Weinberg in Gravitation and Cosmology, Wiley, New York, 1972). In a laboratory setting one may visualise the case of a sphere at rest $(\zeta =0, \text{ t}<0)$ , being spun up by an external torque at $\text{ t}=0$ to the angular momentum J: the $\zeta $ field of the textbook solution cannot establish itself instantaneously over all space at $\text{ t}=0$ , but must propagate with the velocity c, implying the existence of a travelling GM wave field yielding the textbook $\zeta $ field for large enough t (Tolstoy in Int J Theor Phys 40(5):1021–1031, 2001). The linearized GM field equations of the post-Newtonian approximation being isomorphic with Maxwell’s equations (Braginsky et al. in Phys Rev D 15(6):2047–2060, 1977), such GM waves are dipole waves of spin 1. It is well known that in purely gravitating systems conservation of angular momentum forbids the existence of dipole radiation (Misner et al. in Gravitation, Freeman & Co., New York, 1997); but this rule does not prohibit the insertion of angular momentum into the system from an external source–e.g., by applying a torque to our laboratory sphere.     

An Alternative Approach to Understanding the Observed Positron Fraction

Space-based observations by PAMELA (Adriani et al., Nature 458, 607, 2009), Fermi-LAT (Ackerman et al., Phys. Rev. Lett. 105, 01103, 2012), and AMS (Aguilar et al., Phys. Rev. Lett. 110, 141102, 2013) have demonstrated that the positron fraction (e+/total-e) increases with increasing energy above about 10 GeV. According to the propagation model for Galactic cosmic rays in widespread use (Moskalenko & Strong, Astrophys. J. 493, 693, 1998), the production of secondary positrons from interaction of cosmic-ray protons and heavier nuclei with the interstellar medium gives a generally falling positron fraction between 10 and 100 GeV, with secondary positrons accounting for only ～20 % of the observed positron fraction at 100 GeV; so some other physical phenomena have been proposed to explain the data. An alternative approach to interpreting the positron observations is to consider these data as presenting an opportunity for re-examining models of Galactic cosmic-ray propagation. Following release of the PAMELA data, three groups published propagation models (Shaviv, et al., Phys. Rev. Lett. 103, 111302, 2009, Cowsik and Burch, Phys. Rev. D. 82, 023009, 2010, Katz et al., Mon. Not. R. Aston. Soc. 405, 1458 2010) in which the observed positron fraction is explained entirely by secondary positrons produced in the interstellar medium. In May of this year, stimulated by the AMS extension of the positron data to higher energy with excellent statistics, two of those groups presented further development of their calculations (Cowsik et al. 2013, Blum et al. 2013), again concluding that the observed positrons can be understood as secondaries. None of the authors of these five papers was registered for the 33rd International Cosmic Ray Conference (ICRC). Although I am not an author of any of these papers, I have some close familiarity with one of these recent papers, so the conference organizers invited me to bring this alternative approach to the attention of the conference. The present paper is a summary of the material I presented, along with a brief comment about reaction at the conference to this approach.     

Asymptotic Expansion of β Matrix Models in the One-cut Regime

We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and Maurel-Segala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N.     

Note on Solutions of the Strominger System from Unitary Representations of Cocompact Lattices of {SL(2,\mathbb{C})}

This note is motivated by a recently published paper (Biswas and Mukherjee in Commun Math Phys 322(2):373–384, 2013). We prove a no-go result for the existence of suitable solutions of the Strominger system in a compact complex parallelizable manifold \({M = G/\Gamma}\) . For this, we assume G to be non-abelian, the Hermitian metric to be induced from a right invariant metric on G, the Bianchi identity to be satisfied using the Chern connection and furthermore the gauge field to be flat. In Biswas and Mukherjee (Commun Math Phys 322(2):373–384, 2013) it is claimed that one such solution exists on \({SL(2, \mathbb{C})/\Gamma}\) . Our result contradicts the main result in Biswas and Mukherjee (Commun Math Phys 322(2):373–384, 2013).     

A Note on Permutation Twist Defects in Topological Bilayer Phases

We present a mathematical derivation of some of the most important physical quantities arising in topological bilayer systems with permutation twist defects as introduced by Barkeshli et al. (Phys Rev B 87:045130_1-23, 2013). A crucial tool is the theory of permutation equivariant modular functors developed by Barmeier et al. (Int Math Res Notices 2010:3067–3100, 2010; Transform Groups 16:287–337, 2011).     

n-Orthodistributivity in Orthomodular Lattices

A useful generalization of distributivity in lattices n-distributivity, \(n \in \mathbb{N}\) , was introduced in Huhn (Acta Sci. Math. 33:297–305, 1972). In Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987), ‘orthogonalized’ versions, n-orthodistributivity, \(n \in \mathbb{N}\) , of these equations were introduced and discussed. The discussion and results of Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987) centered on the class of modular ortholattices. In this paper we discuss and present some preliminary results for these conditions in orthomodular lattices. In particular, we completely classify the n-(ortho)distributive orthomodular lattices arising from Greechie’s classical 1971 construction, and we prove that a certain simple atomless orthomodular lattice, presented in Roddy (Algebra Univers. 29:564–597, 1992), is 4-orthodistributive. It is not 3-orthodistributive.     

Optimal escape from circular orbits around black holes

An obvious strategy to escape from a stable circular orbit in the Schwarzschild spacetime is to employ a tangential instantaneous acceleration. Using the theory of optimal rocket trajectories in general relativity, recently developed in Henriques and Natário (J Optim Theory Appl 154:500–552; 2011), we show that this manoeuvre satisfies the optimality conditions for maximizing the rocket’s final energy (given a fixed amount of fuel) if and only if the magnitude of the acceleration is smaller than a certain bound. This is the general relativistic version of a result by Lawden (J Brit Interplan Soc 12:68–71; 1953).     

First principles electronic structure investigation of order of singlet and triplet states of oxyhemoglobin and analysis of possible influence of muon trapping

Interest in the possibility of magnetic character for oxyhemoglobin (OxyHb) has been recently stimulated by the observations of muon spin-lattice relaxation effects studied (Nagamine et al., Proc Jpn Acad Ser B Phys Biol Sci 83:120–126, 2007) with the muon-spin rotation (μSR) technique. In view of this, we have carried out first-principles electronic structure investigations involving Hartree–Fock theory combined with many body perturbation effects for the singlet and triplet states of OxyHb. Our results indicate that using two recent x-ray structural data (Paoli et al., J Mol Biol 256:775, 1996; Park et al., J Mol Biol 360:690, 2006) for OxyHb, for only Hartree–Fock theory without many-body effects included, the singlet state lies above the triplet state by energies of about 0.08 and 0.13 a.u. for the two structures in Paoli et al. (J Mol Biol 256:775, 1996) and Park et al. (J Mol Biol 360:690, 2006). Incorporation of many-body effects by the perturbation method reverses the order, with the triplet state located 0.18 and 0.14 a.u. above the singlet state for the structures in Paoli et al. (J Mol Biol 256:775, 1996) and Park et al. (J Mol Biol 360:690, 2006). Physical reasons for these relative orderings of the singlet and triplet states will be discussed. It is clear that OxyHb by itself would be in a singlet state at room temperature or below, since from our calculation, the triplet state lies about KT above the singlet state with T having the value of 44,098 K and 56,449 K for the two structural data in Paoli et al. (J Mol Biol 256:775, 1996) and Park et al. (J Mol Biol 360:690, 2006). As regards the muon spin-lattice relaxation effects obtained by recent μSR measurements (by Nagamine et al., Proc Jpn Acad Ser B Phys Biol Sci 83:120–126, 2007) at room temperature, the sensitive dependence of the singlet-triplet separation on many-body effects in our investigation suggests that it is possible that the singlet-triplet separation could be reversed or reduced significantly when a muon is trapped near an oxygen atom of the oxygen molecule, allowing the triplet to be occupied at room temperature and lead to significant muon spin-lattice relaxation.     

Tracy–Widom at High Temperature

We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature \(\beta \) tends to \(0\) . We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy–Widom \(\beta \) law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index \(k\) . Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in Dumaz and Virág (Ann Inst H Poincaré Probab Statist 49(4):915–933, 2013). We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when \(\beta \rightarrow 0\) . As an application, we investigate the maximal eigenvalues statistics of \(\beta _N\) -ensembles when the repulsion parameter \(\beta _N\rightarrow 0\) when \(N\rightarrow +\infty \) . We study the double scaling limit \(N\rightarrow +\infty , \beta _N \rightarrow 0\) and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of Edelman and Sutton (J Stat Phys 127(6):1121–1165, 2007) and Ramírez et al. (J Am Math Soc 24:919–944, 2011) from our later study of the stochastic Airy operator.     

Viability of Noether Symmetry of F(R) Theory of Gravity

Recently, we have explored vices and virtues of $R^{\frac{3}{2}}$ term in the action which has in-built Noether symmetry and anticipated that a linear term might improve the situation (Sarkar et al., arXiv:1201.2987 [astro-ph.CO], 2012). In the absence of a conserved current it is extremely difficult to obtain an analytical solution of the said fourth order theory of gravity in the presence of a linear term. Here, we therefore enlarge the configuration space by including a scalar field in addition and also taking some of the anisotropic models (in the absence of a scalar field) into account. We observe that Noether symmetry remains obscure and it does not even reproduce the one that already exists in the literature (Sanyal, Gen. Relativ. Gravit., 37:407, 2005). However, there exists in general, a conserved current for F(R) theory of gravity in the presence of a non-minimally coupled scalar field (Sanyal, Phys. Lett. B, 624:81, 2005; Mod. Phys. Lett. A, 25:2667, 2010), which simplifies the field equations considerably. Here, we briefly expatiate the non-Noether conserved current and show that indeed the situation is modified.     

Decoherent Histories of Spin Networks

The decoherent histories formalism, developed by Griffiths, Gell-Mann, and Hartle (in Phys. Rev. A 76:022104, 2007; arXiv:1106.0767v3 [quant-ph], 2011; Consistent Quantum Theory, Cambridge University Press, 2003; arXiv:gr-qc/9304006v2, 1992) is a general framework in which to formulate a timeless, ‘generalised’ quantum theory and extract predictions from it. Recent advances in spin foam models allow for loop gravity to be cast in this framework. In this paper, I propose a decoherence functional for loop gravity and interpret existing results (Bianchi et al. in Phys. Rev. D 83:104015, 2011; Phys. Rev. D 82:084035, 2010) as showing that coarse grained histories follow quasiclassical trajectories in the appropriate limit.     

A response to ‘Comments on the effect of liquid layering on the thermal conductivity of nanofluids’, E. Doroodchi, T. M. Evans & B. Moghtaderi, 2009. J Nanopart Res 11(6):1501–1507

This article provides a response to a recent brief communication ‘Comments on the effect of liquid layering on the thermal conductivity of nanofluids’ by Doroodchi et al. in J Nanopart Res 11(6):1501–1507, 2009. It provides an opportunity for us to clarify the fundamental differences between the models of Yu and Choi (2003) and Leong et al. (2006) mentioned in the communication, followed by an explanation of the development of Leong et al.’s model. While we re-affirm that the model of Leong et al. (2006) was developed based on the right methodology, appropriate boundary conditions and mathematical basis and is therefore valid, there are at least three incorrect equations in Doroodchi et al.’s communication which raise serious doubts on their results calculated from the above models. Hence, the comments by Doroodchi et al. (2009) about the model of Leong et al. (2006) are not well-justified.