Optimal N-Point Configurations on the Sphere: “Magic” Numbers and Smale’s 7th Problem

This paper inquires into the concavity of the map \(N\mapsto v_s(N)\) from the integers \(N\ge 2\) into the minimal average standardized Riesz pair-energies \(v_s(N)\) of \(N\) -point configurations on the sphere \(\mathbb {S}^2\) for various \(s\in \mathbb {R}\) . The standardized Riesz pair-energy of a pair of points on \(\mathbb {S}^2\) a chordal distance \(r\) apart is \(V_s(r)= s^{-1}\left( r^{-s}-1 \right) \) , \(s \ne 0\) , which becomes \(V_0(r) = \ln \frac{1}{r}\) in the limit \(s\rightarrow 0\) . Averaging it over the \(\left( \begin{array}{c} N\\ 2\end{array}\right) \) distinct pairs in a configuration and minimizing over all possible \(N\) -point configurations defines \(v_s(N)\) . It is known that \(N\mapsto v_s(N)\) is strictly increasing for each \(s\in \mathbb {R}\) , and for \(s<2\) also bounded above, thus “overall concave.” It is (easily) proved that \(N\mapsto v_{-2}^{}(N)\) is even locally strictly concave, and that so is the map \(2n\mapsto v_s(2n)\) for \(s<-2\) . By analyzing computer-experimental data of putatively minimal average Riesz pair-energies \(v_s^x(N)\) for \(s\in \{-1,0,1,2,3\}\) and \(N\in \{2,\ldots ,200\}\) , it is found that the map \(N\mapsto {v}_{-1}^x(N)\) is locally strictly concave, while \(N\mapsto {v}_s^x(N)\) is not always locally strictly concave for \(s\in \{0,1,2,3\}\) : concavity defects occur whenever \(N\in {\mathcal {C}}^{x}_+(s)\) (an \(s\) -specific empirical set of integers). It is found that the empirical map \(s\mapsto {\mathcal {C}}^{x}_+(s),\ s\in \{-2,-1,0,1,2,3\}\) , is set-theoretically increasing; moreover, the percentage of odd numbers in \({\mathcal {C}}^{x}_+(s),\ s\in \{0,1,2,3\}\) is found to increase with \(s\) . The integers in \({\mathcal {C}}^{x}_+(0)\) are few and far between, forming a curious sequence of numbers, reminiscent of the “magic numbers” in nuclear physics. It is conjectured that these new “magic numbers” are associated with optimally symmetric optimal-log-energy \(N\) -point configurations on \(\mathbb {S}^2\) . A list of interesting open problems is extracted from the empirical findings, and some rigorous first steps toward their solutions are presented. It is emphasized how concavity can assist in the solution to Smale’s \(7\) th Problem, which asks for an efficient algorithm to find near-optimal \(N\) -point configurations on \(\mathbb {S}^2\) and higher-dimensional spheres.     

The Distribution of the Area Under a Bessel Excursion and its Moments

A Bessel excursion is a Bessel process that begins at the origin and first returns there at some given time \(T\) . We study the distribution of the area under such an excursion, which recently found application in the context of laser cooling. The area \(A\) scales with the time as \(A \sim T^{3/2}\) , independent of the dimension, \(d\) , but the functional form of the distribution does depend on \(d\) . We demonstrate that for \(d=1\) , the distribution reduces as expected to the distribution for the area under a Brownian excursion, known as the Airy distribution, deriving a new expression for the Airy distribution in the process. We show that the distribution is symmetric in \(d-2\) , with nonanalytic behavior at \(d=2\) . We calculate the first and second moments of the distribution, as well as a particular fractional moment. We also analyze the analytic continuation from \(d<2\) to \(d>2\) . In the limit where \(d\rightarrow 4\) from below, this analytically continued distribution is described by a one-sided Lévy \(\alpha \) -stable distribution with index \(2/3\) and a scale factor proportional to \([(4-d)T]^{3/2}\) .     

Quantum Measures on Finite Effect Algebras with the Riesz Decomposition Properties

One kind of generalized measures called quantum measures on finite effect algebras, which fulfil the grade-2 additive sum rule, is considered. One basis of vector space of quantum measures on a finite effect algebra with the Riesz decomposition property (RDP for short) is given. It is proved that any diagonally positive symmetric signed measure \(\lambda \) on the tensor product \(E\otimes E\) can determine a quantum measure \(\mu \) on a finite effect algebra \(E\) with the RDP such that \(\mu (x)=\lambda (x\otimes x)\) for any \(x\in E\) . Furthermore, some conditions for a grade-2 additive measure \(\mu \) on a finite effect algebra \(E\) are provided to guarantee that there exists a unique diagonally positive symmetric signed measure \(\lambda \) on \(E\otimes E\) such that \(\mu (x)=\lambda (x\otimes x)\) for any \(x\in E\) . At last, it is showed that any grade- \(t\) quantum measure on a finite effect algebra \(E\) with the RDP is essentially established by the values on a subset of \(E\) .     

Parton distribution functions at LO,NLO and NNLO with correlated uncertainties between orders

Sets of parton distribution functions (PDFs) of the proton are reported for the leading (LO), next-to-leading (NLO) and next-to-next-to-leading-order (NNLO) QCD calculations. The parton distribution functions are determined with the HERAFitter program using the data from the HERA experiments and preserving correlations between uncertainties for the LO, NLO and NNLO PDF sets. The sets are used to study cross-section ratios and their uncertainties when calculated at different orders in QCD. A reduction of the overall theoretical uncertainty is observed if correlations between the PDF sets are taken into account for the ratio of \(WW\) di-boson to \(Z\) boson production cross sections at the LHC.     

Maximal Distance Travelled by N Vicious Walkers Till Their Survival

We consider N Brownian particles moving on a line starting from initial positions \(\mathbf{{u}}\equiv \{u_1,u_2,\ldots u_N\}\) such that \(0 . Their motion gets stopped at time \(t_s\) when either two of them collide or when the particle closest to the origin hits the origin for the first time. For \(N=2\) , we study the probability distribution function \(p_1(m|\mathbf{{u}})\) and \(p_2(m|\mathbf{{u}})\) of the maximal distance travelled by the \(1^{\text {st}}\) and \(2^{\text {nd}}\) walker till \(t_s\) . For general N particles with identical diffusion constants \(D\) , we show that the probability distribution \(p_N(m|\mathbf{u})\) of the global maximum \(m_N\) , has a power law tail \(p_i(m|\mathbf{{u}}) \sim {N^2B_N\mathcal {F}_{N}(\mathbf{u})}/{m^{\nu _N}}\) with exponent \(\nu _N =N^2+1\) . We obtain explicit expressions of the function \(\mathcal {F}_{N}(\mathbf{u})\) and of the N dependent amplitude \(B_N\) which we also analyze for large N using techniques from random matrix theory. We verify our analytical results through direct numerical simulations.     

Antineutrino science in KamLAND

The primary goal of KamLAND is a search for the oscillation of \({\bar{\nu }}_\mathrm{e}\) ’s emitted from distant power reactors. The long baseline, typically 180 km, enables KamLAND to address the oscillation solution of the “solar neutrino problem” with \({\bar{\nu }}_{e} \) ’s under laboratory conditions. KamLAND found fewer reactor \({\bar{\nu }}_{e} \) events than expected from standard assumptions about \(\overline{\nu }_e\) propagation at more than 9 \(\sigma \) confidence level (C.L.). The observed energy spectrum disagrees with the expected spectral shape at more than 5 \(\sigma \) C.L., and prefers the distortion from neutrino oscillation effects. A three-flavor oscillation analysis of the data from KamLAND and KamLAND + solar neutrino experiments with CPT invariance, yields \(\Delta m_{21}^2 \) = [ \(7.54_{-0.18}^{+0.19} \times \) 10 \(^{-5}\) eV \(^{2}\) , \(7.53_{-0.18}^{+0.19} \times \) 10 \(^{-5}\) eV \(^{2}\) ], tan \(^{2}\theta _{12}\) = [ \(0.481_{-0.080}^{+0.092} \) , \(0.437_{-0.026}^{+0.029} \) ], and sin \(^{2}\theta _{13}\) = [ \(0.010_{-0.034}^{+0.033} \) , \(0.023_{-0.015}^{+0.015} \) ]. All solutions to the solar neutrino problem except for the large mixing angle region are excluded. KamLAND also demonstrated almost two cycles of the periodic feature expected from neutrino oscillation effects. KamLAND performed the first experimental study of antineutrinos from the Earth’s interior so-called geoneutrinos (geo \({\bar{\nu }}_{e} \) ’s), and succeeded in detecting geo \({\bar{\nu }}_{e} \) ’s produced by the decays of \(^{238}\) U and \(^{232}\) Th within the Earth. Assuming a chondritic Th/U mass ratio, we obtain \(116_{-27}^{+28} {\bar{\nu }}_{e}\) events from \(^{238}\) U and \(^{232}\) Th, corresponding a geo \({\bar{\nu }}_{e}\) flux of \(3.4_{-0.8}^{+0.8}\times \) 10 \(^{6}\) cm \(^{-2}\)  s \(^{-1}\) at the KamLAND location. We evaluate various bulk silicate Earth composition models using the observed geo \({\bar{\nu }}_{e} \) rate.     

Refractive index changes and nuclear activation upon irradiation of lithium niobate crystals with different low-mass,high-energy ions

Refractive index changes \(\Delta n\) in lithium niobate crystals upon irradiation with high-energy protons, deuterons, \(^3\) He, and \(^4\alpha \) particles (up to 14 MeV/nucleon) are created, and the accompanying, unwanted nuclear activation is investigated. The measurements give answers to the question which ion is the best choice depending on the requirements: largest values of \(\Delta n\) are achieved with \(^4\alpha \) particles, low nuclear activation with deuterons, or the best tradeoff between \(\Delta n\) and activation with \(^3\) He, respectively.     

Local Central Limit Theorem for Determinantal Point Processes

We prove a local central limit theorem (LCLT) for the number of points \(N(J)\) in a region \(J\) in \(\mathbb R^d\) specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of \(N(J)\) tends to infinity as \(|J| \rightarrow \infty \) . This extends a previous result giving a weaker central limit theorem for these systems. Our result relies on the fact that the Lee–Yang zeros of the generating function for \(\{E(k;J)\}\) —the probabilities of there being exactly \(k\) points in \(J\) —all lie on the negative real \(z\) -axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble. A LCLT is also established for the probability density function of the \(k\) -th largest eigenvalue at the soft edge, and of the spacing between \(k\) -th neighbors in the bulk.     

Polymeric N-stage serial-cascaded four-port optical router with scalable 3N channel wavelengths for wideband signal routing application

Device architecture and design scheme of a universal \(N\) -stage cascaded polymer four-port optical router with scalable 3 \(N\) channel wavelengths are proposed. Basic cross-coupling two-microring resonator routing element based on polymer materials is optimized for single-mode transmission, low optical loss and phase-match between microring waveguide and channel waveguide. Then, a one-stage four-port optical router is constructed using four-group basic routing elements, which has 12 possible I/O routing paths and 3 channel wavelengths. The insertion losses of each channel wavelength along every routing path are within the range of 0.04–0.63 dB, the maximum crosstalk between the on-port along each routing path and other off-ports is less than \(-39\)  dB, and the device footprint size is \(\sim \) 0.13 mm \(^{2}\) . Compared with the previously reported four-port silicon optical routers, this device possesses similar ring radius ( \(\sim \) 10  \(\upmu \) m) and device size ( \(<\) 1 mm \(^{2})\) . Aiming at wideband signal routing applications, we then construct a universal \(N\) -stage cascaded polymer four-port optical router possessing scalable 3 \(N\) channel wavelengths. The proposed routing structure has potential application in photonic networks-on-chip, because of low insertion loss, low crosstalk, small footprint size, and scalable wideband 3 \(N\) routing wavelengths.     

Investigation of conductivity of adhesive layer including indium tin oxide particles for multi-junction solar cells

We report connection conductivity ( \(C_{\rm c}\) ) of adhesive which including \(\hbox {In}_2\hbox {O}_3\) – \(\hbox {SnO}_2\) (ITO) particles developed for fabrication of stacked-type-multi-junction solar cells. The commercial 20- \(\upmu \) m sized ITO particles were heated in vacuum at temperature ranging from 800 to 1,300  \(^{\circ }{\rm C}\) for 10 min to increase \(C_{\rm c}\) . 6.2 wt% ITO particles were dispersed in commercial Cemedine adhesive gel to form 100 samples structured with n-type Si/adhesive/n-type Si (n-Si sample) and p-type Si/adhesive/p-type Si (p-Si sample). Current density as a function of voltage (J–V) characteristics gave \(C_{\rm c}\) . It ranged from 4.3 to 1.0 S/cm \(^2\) for the n-Si sample with 800 \(^{\circ }{\rm C}\) heat-treated ITO particles. Its standard deviation was 0.59 S/cm \(^2\) . On the other hand, it ranged from 2.0 to 0.6 S/cm \(^2\) for the p-Si sample with 800  \(^{\circ }{\rm C}\) heat-treated ITO particles. Its standard deviation was 0.22 S/cm \(^2\) . The distribution of \(C_{\rm c}\) mainly resulted from contact efficiency of ITO particles to substrate. We theoretically estimated that present \(C_{\rm c}\) achieved a low loss of the power conversion efficiency ( \(E_{\rm ff}\) ) lower than 0.3 % in the application of fabrication of multi-junction solar cell with an intrinsic \(E_{\rm ff}\) of 30 % and an open circuit voltage above 1.9 V.     

Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions

We extend the method of Balister, Bollobás and Walters (Phys. Rev. E 76:011110, 2007) for determining rigorous confidence intervals for the critical threshold of two dimensional lattices to three (and higher) dimensional lattices. We describe a method for determining a full confidence interval and apply it to show that the critical threshold for bond percolation on the simple cubic lattice is between \(0.2485\) and \(0.2490\) with \(99.9999\,\%\) confidence, and the critical threshold for site percolation on the same lattice is between \(0.3110\) and \(0.3118\) with \(99.9999\,\%\) confidence.     

NMR in Magnetic Field of the Earth: Pre-Polarization of Nuclei with Alternating Magnetic Field

The polarization of nuclei in the low static magnetic field \(B_0\) with an alternating magnetic field \(B^{*} (B^{*} \gg B_0)\) at a very low frequency \(f_m\) (but \(f_m\gg 1\) / \({T_1}\) , where \(T_1\) is the spin-lattice relaxation time) has been investigated. The question of the optimization of the energy consumption during the pre-polarization is also considered. The possibilities of the method are illustrated by the observation of nuclear magnetic resonance signals from a few liquids.     

Static Magnetic Susceptibility of the Systems with Anisotropic Kondo Interaction

We theoretically investigated the static magnetic susceptibility in the heavy fermion compounds YbRh \(_2\) Si \(_2\) and YbIr \(_2\) Si \(_2\) . The molecular field approximation together with the renormalization of the Kondo interaction by the high-energy conduction electron excitations results in the Curie–Weiss law and Van Vleck susceptibility with temperature-dependent Curie and Weiss parameters.     

Semiclassical strings in supergravity PFT

In this paper, we introduce the bulk viscosity in the formalism of modified gravity theory in which the gravitational action contains a general function \(f(R,T)\) , where \(R\) and \(T\) denote the curvature scalar and the trace of the energy–momentum tensor, respectively, within the framework of a flat Friedmann–Robertson–Walker model. As an equation of state for a prefect fluid, we take \(p=(\gamma -1)\rho \) , where \(0 \le \gamma \le 2\) and a viscous term as a bulk viscosity due to the isotropic model, of the form \(\zeta =\zeta _{0}+\zeta _{1}H\) , where \(\zeta _{0}\) and \(\zeta _{1}\) are constants, and \(H\) is the Hubble parameter. The exact non-singular solutions to the corresponding field equations are obtained with non-viscous and viscous fluids, respectively, by assuming a simplest particular model of the form of \(f(R,T) = R+2f(T)\) , where \(f(T)=\alpha T\) ( \(\alpha \) is a constant). A big-rip singularity is also observed for \(\gamma <0\) at a finite value of cosmic time under certain constraints. We study all possible scenarios with the possible positive and negative ranges of \(\alpha \) to analyze the expansion history of the universe. It is observed that the universe accelerates or exhibits a transition from a decelerated phase to an accelerated phase under certain constraints of \(\zeta _0\) and \(\zeta _1\) . We compare the viscous models with the non-viscous one through the graph plotted between the scale factor and cosmic time and find that the bulk viscosity plays a major role in the expansion of the universe. A similar graph is plotted for the deceleration parameter with non-viscous and viscous fluids and we find a transition from decelerated to accelerated phase with some form of bulk viscosity.     

Fluctuation–Dissipation Relation for Systems with Spatially Varying Friction

When a particle diffuses in a medium with spatially dependent friction coefficient \(\alpha (r)\) at constant temperature \(T\) , it drifts toward the low friction end of the system even in the absence of any real physical force \(f\) . This phenomenon, which has been previously studied in the context of non-inertial Brownian dynamics, is termed “spurious drift”, although the drift is real and stems from an inertial effect taking place at the short temporal scales. Here, we study the diffusion of particles in inhomogeneous media within the framework of the inertial Langevin equation. We demonstrate that the quantity which characterizes the dynamics with non-uniform \(\alpha (r)\) is not the displacement of the particle \(\Delta r=r-r^0\) (where \(r^0\) is the initial position), but rather \(\Delta A(r)=A(r)-A(r^0)\) , where \(A(r)\) is the primitive function of \(\alpha (r)\) . We derive expressions relating the mean and variance of \(\Delta A\) to \(f\) , \(T\) , and the duration of the dynamics \(\Delta t\) . For a constant friction coefficient \(\alpha (r)=\alpha \) , these expressions reduce to the well known forms of the force-drift and fluctuation–dissipation relations. We introduce a very accurate method for Langevin dynamics simulations in systems with spatially varying \(\alpha (r)\) , and use the method to validate the newly derived expressions.     

Pesin’s Entropy Formula for Systems Between C^1 and C^{1+\alpha }

In this article we give a new observation of Pesin’s entropy formula, motivated from Mañé’s proof of (Ergod Theory Dyn Sys 1:95–102, 1981). Let \(M\) be a compact Riemann manifold and \(f:\,M\rightarrow M\) be a \(C^1\) diffeomorphism on \(M\) . If \(\mu \) is an \(f\) -invariant probability measure which is absolutely continuous relative to Lebesgue measure and nonuniformly-H \(\ddot{\text {o}}\) lder-continuous(see Definition 1.1), then we have Pesin’s entropy formula, i.e., the metric entropy \(h_\mu (f)\) satisfies $$\begin{aligned} h_{\mu }(f)=\int \sum _{\lambda _i(x)> 0}\lambda _i(x)d\mu , \end{aligned}$$ where \(\lambda _1(x)\ge \lambda _2(x)\ge \cdots \ge \lambda _{dim\,M}(x)\) are the Lyapunov exponents at \(x\) with respect to \(\mu .\) Nonuniformly-H \(\ddot{\text {o}}\) lder-continuous is a new notion from probabilistic perspective weaker than \(C^{1+\alpha }.\)      

Scaling Limits and Critical Behaviour of the 4-Dimensional n-Component |\varphi |^4 Spin Model

We consider the \(n\) -component \(|\varphi |^4\) spin model on \({\mathbb {Z}}^4\) , for all \(n \ge 1\) , with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent \(\frac{n+2}{n+8}\) for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for \(n =1,2,3\) ; double logarithmic scaling for \(n=4\) ; and is bounded when \(n>4\) . In addition, for the model defined on the \(4\) -dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the \(|\varphi |^4\) model.     

Split supersymmetry under GUT and current dark matter constraints

There are four types of two-Higgs doublet models under a discrete \(Z_2\) symmetry imposed to avoid tree-level flavor-changing neutral current, i.e. type-I, type-II, type-X, and type-Y models. We investigate the possibility to discriminate the four models in the light of the flavor physics data, including \(B_s\) – \(\bar{B}_s\) mixing, \(B_{s,d} \rightarrow \mu ^+ \mu ^-\) , \(B\rightarrow \tau \nu \) and \(\bar{B} \rightarrow X_s \gamma \) decays, the recent LHC Higgs data, the direct search for charged Higgs at LEP, and the constraints from perturbative unitarity and vacuum stability. After deriving the combined constraints on the Yukawa interaction parameters, we have shown that the correlation between the mass eigenstate rate asymmetry \(A_{\Delta \Gamma }\) of \(B_{s} \rightarrow \mu ^+ \mu ^-\) and the ratio \(R=\mathcal{B}(B_{s} \rightarrow \mu ^+ \mu ^-)_\mathrm{exp}/ \mathcal{B}(B_{s} \rightarrow \mu ^+ \mu ^-)_\mathrm{SM}\) could be a sensitive probe to discriminate the four models with future precise measurements of the observables in the \(B_{s} \rightarrow \mu ^+ \mu ^-\) decay at LHCb.     

How bosonic is a pair of fermions?

Composite particles made of two fermions can be treated as ideal elementary bosons as long as the constituent fermions are sufficiently entangled. In that case, the Pauli principle acting on the parts does not jeopardise the bosonic behaviour of the whole. An indicator for bosonic quality is the composite boson normalisation ratio \(\chi _{N+1}/\chi _{N}\) of a state of \(N\) composites. This quantity is prohibitively complicated to compute exactly for realistic two-fermion wavefunctions and large composite numbers \(N\) . Here, we provide an efficient characterisation in terms of the purity \(P\) and the largest eigenvalue \(\lambda _1\) of the reduced single-fermion state. We find the states that extremise \(\chi _N\) for given \(P\) and \(\lambda _1\) , and we provide easily evaluable, saturable upper and lower bounds for the normalisation ratio. Our results strengthen the relationship between the bosonic quality of a composite particle and the entanglement of its constituents.     

Efficient Algorithms for the Two-Dimensional Ising Model with a Surface Field

The bond propagation and site propagation algorithms are extended to the two-dimensional (2D) Ising model with a surface field. With these algorithms we can calculate the free energy, internal energy, specific heat, magnetization, correlation functions, surface magnetization, surface susceptibility and surface correlations. The method can handle continuous and discrete bond and surface-field disorder and is especially efficient in the case of bond or site dilution. To test these algorithms, we study the wetting transition of the 2D Ising model, which was solved exactly by Abraham. We can locate the transition point accurately with a relative error of \(10^{-8}\) . We carry out the calculation of the specific heat and surface susceptibility on lattices with sizes up to \(200^2 \times 200\) . The results show that a finite jump develops in the specific heat and surface susceptibility at the transition point as the lattice size increases. For lattice size \(320^2 \times 320\) the parallel correlation length exponent is \(1.86\) , while Abraham’s exact result is \(2.0\) . The perpendicular correlation length exponent for lattice size \(160^2\times 160\) is \(1.05\) , whereas its exact value is \(1.0\) .