Exact Dependence Research Articles

On exact correlation functions in SU(N) N = 2 $$ \mathcal{N}=2 $$ superconformal QCD

We consider the exact coupling constant dependence of extremal correlation functions of ${\cal N} = 2$ chiral primary operators in 4d ${\cal N} = 2$ superconformal gauge theories with gauge group SU(N) and N_f=2N massless fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt* equations. In the case at hand, the tt* equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU(2) gauge group. This ansatz requires a surprising new non-renormalization theorem in ${\cal N} = 2$ superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the ${\cal N} = 2$ chiral ring of the SU(N) theory, and in all explicitly computed examples we find agreement with the tt* equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the ${\cal N} = 2$ chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU(N) theory using information solely from the S^4 partition function of the theory obtained by supersymmetric localization.

Resummation and matching of b-quark mass effects in b b ¯ H $$ b\overline{b}H $$ production

We use a systematic effective field theory setup to derive the $b\bar{b}H$ production cross section. Our result combines the merits of both fixed 4-flavor and 5-flavor schemes. It contains the full 4-flavor result, including the exact dependence on the $b$-quark mass, and improves it with a resummation of collinear logarithms of $m_b/m_H$. In the massless limit, it corresponds to a reorganized 5-flavor result. While we focus on $b\bar{b}H$ production, our method applies to generic heavy-quark initiated processes at hadron colliders. Our setup resembles the variable flavor number schemes known from heavy-flavor production in deep-inelastic scattering, but also differs in some key aspects. Most importantly, the effective $b$-quark PDF appears as part of the perturbative expansion of the final result where it effectively counts as an $O(\alpha_s)$ object. The transition between the fixed-order (4-flavor) and resummation (5-flavor) regimes is governed by the low matching scale at which the $b$-quark is integrated out. Varying this scale provides a systematic way to assess the perturbative uncertainties associated with the resummation and matching procedure and reduces by going to higher orders. We discuss the practical implementation and present numerical results for the $b\bar{b}H$ production cross section at NLO+NLL. We also provide a comparison to the corresponding predictions in the fixed 4-flavor and 5-flavor results and the Santander matching prescription. Compared to the latter, we find a slightly reduced uncertainty and a larger central value, with its central value lying at the lower edge of our uncertainty band.

Dynamics of noisy quantum systems in the Heisenberg picture: Application to the stability of fractional charge

Based on the Heisenberg-picture analog of the master equation, we develop a method for computing the exact time dependence of noise-averaged observables for general noninteracting fermionic systems with noisy fluctuations. Upon noise averaging, these fluctuations generate effective interactions, limiting analytical approaches. While the short-time dynamics can be studied with Langevin-type numerical simulations, the long-time limit is not amenable to such simulations. Our results provide access to this long-time limit. As a simple example, we examine the fate of the fractional charge in cold-atom emulations of polyacetylene after stochastic driving. We find that in a quantum quench to a fluctuating hopping Hamiltonian, the fractional charge remains robust for hopping between different sublattices, while it becomes unstable in the presence of noisy hopping on the same sublattice.

Solid volume fraction estimation of bone:marrow replica models using ultrasound transit time spectroscopy

The acceptance of broadband ultrasound attenuation (BUA) for the assessment of osteoporosis suffers from a limited understanding of both ultrasound wave propagation through cancellous bone and its exact dependence upon the material and structural properties. It has recently been proposed that ultrasound wave propagation in cancellous bone may be described by a concept of parallel sonic rays; the transit time of each ray defined by the proportion of bone and marrow propagated. A Transit Time Spectrum (TTS) describes the proportion of sonic rays having a particular transit time, effectively describing the lateral inhomogeneity of transit times over the surface aperture of the receive ultrasound transducer. The aim of this study was to test the hypothesis that the solid volume fraction (SVF) of simplified bone:marrow replica models may be reliably estimated from the corresponding ultrasound transit time spectrum. Transit time spectra were derived via digital deconvolution of the experimentally measured input and output ultrasonic signals, and compared to predicted TTS based on the parallel sonic ray concept, demonstrating agreement in both position and amplitude of spectral peaks. Solid volume fraction was calculated from the TTS; agreement between true (geometric calculation) with predicted (computer simulation) and experimentally-derived values were R2=99.9% and R2=97.3% respectively. It is therefore envisaged that ultrasound transit time spectroscopy (UTTS) offers the potential to reliably estimate bone mineral density and hence the established T-score parameter for clinical osteoporosis assessment.

Notes on [formula omitted]-weak tractability: A refined classification of problems with (sub)exponential information complexity

In the last 20 years a whole hierarchy of notions of tractability was proposed and analyzed by several authors. These notions are used to classify the computational hardness of continuous numerical problems S=(Sd)d∈N in terms of the behavior of their information complexity n(ε,Sd) as a function of the accuracy ε and the dimension d. By now a lot of effort was spent on either proving quantitative positive results (such as, e.g., the concrete dependence on ε and d within the well-established framework of polynomial tractability), or on qualitative negative results (which, e.g., state that a given problem suffers from the so-called curse of dimensionality). Although several weaker types of tractability were introduced recently, the theory of information-based complexity still lacks a notion which allows to quantify the exact (sub-/super-) exponential dependence of n(ε,Sd) on both parameters ε and d. In this paper we present the notion of (s,t)-weak tractability which attempts to fill this gap. Within this new framework the parameters s and t are used to quantitatively refine the huge class of polynomially intractable problems. For linear, compact operators between Hilbert spaces we provide characterizations of (s,t)-weak tractability w.r.t. the worst case setting in terms of singular values. In addition, our new notion is illustrated by classical examples which recently attracted some attention. In detail, we study approximation problems between periodic Sobolev spaces and integration problems for classes of smooth functions.

Bistability in inhomogeneity—Effects of flow coherent structures on the fate of a bistable reaction

We present a numerical study on the mixing process between two stable states of a chemical reaction model. The two stable states of the reactions are found in practice not to coexist, and a single stable state of homogeneous scalar concentration is achieved over long time. With all other parameters fixed, we find the dependence of the final state on the rate of reaction. Interestingly, with the existence of coherent structures, at a range of intermediate rate of reaction, we find that the final state also depends on the initial locations of the scalar impurity. The exact dependence on initial condition is explored in detail. These results lead to the fundamental understanding on the variability of biogeochemical tracers in the environment induced by nonlinear fluid stirring.

Resonant frequency of re-entrant klystron cavity

Resonant frequency of re-entrant klystron cavities is difficult to predict through closed form expressions, since the accurate field profile and their solutions in the resonator volume are not known. Often conflicting requirements need a judicious trade-off between cavity parameters by designer. In absence of knowledge of exact dependence of frequency on geometrical parameters the graphical data is still relied upon. Barroso et al proposed a mathematically simple formulation for calculating the resonant frequency of a cylindrical re-entrant cavity. The frequency equation suggested by Barroso et al is valid for a particular range of cavity dimensions of short cavities. This equation is modified and a closed form expression for resonating frequency of the re-entrant resonator is derived which is applicable to both long and short cavities. The analytical results are compared with earlier reported results and simulation results obtained using Computer Simulation Technology microwave studio.

Smoluchowski diffusion equation for active Brownian swimmers.

We study the free diffusion in two dimensions of active Brownian swimmers subject to passive fluctuations on the translational motion and to active fluctuations on the rotational one. The Smoluchowski equation is derived from a Langevin-like model of active swimmers and analytically solved in the long-time regime for arbitrary values of the Péclet number; this allows us to analyze the out-of-equilibrium evolution of the positions distribution of active particles at all time regimes. Explicit expressions for the mean-square displacement and for the kurtosis of the probability distribution function are presented and the effects of persistence discussed. We show through Brownian dynamics simulations that our prescription for the mean-square displacement gives the exact time dependence at all times. The departure of the probability distribution from a Gaussian, measured by the kurtosis, is also analyzed both analytically and computationally. We find that for the inverse of Péclet numbers ≲0.1, the distance from Gaussian increases as ∼t(-2) at short times, while it diminishes as ∼t(-1) in the asymptotic limit.

Atomic-size tip enhanced cooling of field emission from the n-type silicon semiconductor

The Nottingham effect of an n-type silicon semiconductor tip was theoretically investigated for use as a practical solid state cooler. The vacuum potential was obtained in the form which explicitly included the semiconductor cathode geometry. This leads to a relatively exact dependence of the energy exchange Δε and the cooling power density Γ on the geometry of the device. By systematic calculations of the field emission cooling, Γ was obtained as a function of the bias V, the tip radius R, and temperature T. The current density j increased with decreasing R at fixed V and T. The Δε increased with decreasing R at fixed j and T. As T increased, both Δε and Γ increased considerably. When an atomic-size silicon tip was taken, a meaningful cooling was obtained at V as small as several volts. At V = 6.8 V, a sharp tip of R = 0.5 nm yielded the maximum Γ = 250, 1941, and 6148 W/cm2 at T = 300, 600, and 900 K, respectively. This implies that an optimized configuration of an n-Si cathode produces a useful field emission cooling for micro-electronic devices with a low bias.

Container vessel fleet deployment for liner shipping with stochastic dependencies in shipping demand

The problem of optimal container vessels deployment is one of great significance for the liner shipping industry. Although the pioneering work on this problem dates back to the early 1990s, only until recently have researchers started to acknowledge and account for the significant amount of uncertainty present in shipping demand in real world container shipping. In this paper, new analytical results are presented to further relax the input requirements for this problem. Specifically, only the mean and variance of the maximum shipping demand are required to be known. An optional symmetry assumption is shown to further reduce the feasible region and deployment cost for typical confidence levels. Moreover, unlike previous work that tends to ignore stochastic dependencies between the shipping demands on the various routes (that are known to exist in the real world), our models account for such dependencies in the most general setting to date. A salient feature of our modeling approach is that the exact dependence structure does not need to be specified, something that is hard, if not simply impossible, to determine in practice. A numerical case study is provided to illustrate the proposed models.

Influences of p-type layer structure and doping profile on the temperature dependence of the foward voltage characteristic of GaInN light-emitting diode

Many GaInN light-emitting diodes (LEDs) are subjected to a great temperature variation during their serving. In these applications, it is advantageous that GaInN LEDs have a weak temperature dependence of forward voltage. However, the factors determining the exact temperature dependence of the forward voltage characteristics are not fully understood. In this paper, two series of GaInN LEDs are prepared for investigating the correlation between the epitaxial structural and the temperature dependence of the forward voltage characteristics. The forward voltage characteristics of samples are studied in a temperature range from 100 K to 350 K. The curves of forward voltage versus temperature (dV/dT) are compared and analyzed. For the three samples in series I, according to the barrier thickness and emitting wavelength, they are designated as blue multiquantum well (MQW) with thin barrier (sample A), blue MQW with thick barrier (sample B), and green barrier with thick barrier (sample C) respectively. Their structures of active region including the insertion layer between n-GaN and MQW, the MQW, and the emitting wavelength are different from each other. However, the same slopes of dV/dT at room temperature (300 K± 50 K) are observed in the samples. Moreover, samples B and C with the same p-type layer design also have the same slopes of dV/dT at cryogenic temperatures. Sample A with a much thinner p-type layer shows a lower slope than samples B and C. Based on the these experimental data, it is deduced that the intrinsic physic properties of active region such as structure and emission wavelength have a little influence on the variation of the slope of dV/dT either at room temperature or at cryogenic temperatures. Moreover, the Mg concentration of the p-GaN main region determines the slope of dV/dT at cryogenic temperatures. Low doping concentration leads to a high slope of dV/dT.#br#In order to find the decisive factor determining the slope of dV/dT at room temperature, three samples in series II are grown. For sample E, at the MQW-EBL (electron blocking layer) interface, the Mg concentration increases very slowly while an abruptly varying doping profile is observed for samples D and F. The slopes of samples D and F are both -1.3 mV·K-1. This is very close to the calculation value of the lower bond for the change in forward voltage (-1.2 mV·K-1). Meanwhile, the slope of sample E is -2.5 mV·K-1, which is much higher than those of samples D and F. Thus, it is suggested that the major factor influencing the slope of dV/dT at room temperature is the Mg doping profile of the initial growth stage of the p-AlGaN electron blocking layer. These phenomena are mainly attributed to the changes of the activation energy of p-AlGaN and p-GaN, since it relies on the doping concentration and temperature. Our findings clarify the roles of active region, p-AlGaN and p-GaN in the temperature dependence of the forward voltage characteristic. More importantly, the results obtained in this study are helpful for optimizing the growth parameters to achieve LED devices with forward voltage that has a low sensitivity to temperature.

Laurent series based RBF-FD method to avoid ill-conditioning

We propose a new approach to avoid the inherent ill-condition in the computation of RBF-FD weights, which is due to the fact that the RBF interpolation matrix is nearly singular. The new approach is based on the semi-analytical computation of the Laurent series of the inverse of the RBF interpolation matrix. Once the Laurent series is obtained, it can be used to compute the RBF-FD weights of any differential operator exactly without extra cost. The proposed method also provides analytical formulas for the RBF-FD weights in terms of the parameters involved in the problem. These formulas can be used to derive the exact dependence of the truncation error in the approximation of any differential operator of a given function. Furthermore, from the analysis presented here one can derive the values of the parameters involved in the problem for which the RBF interpolation matrix becomes ill-conditioned and, hence, for which the weights cannot be obtained numerically.

Top-quark mass effects in double and triple Higgs production in gluon-gluon fusion at NLO

The observation of double and triple scalar boson production at hadron colliders could provide key information on the Higgs self couplings and the potential. As for single Higgs production the largest rates for multiple Higgs production come from gluon-gluon fusion processes mediated by a top-quark loop. However, at variance with single Higgs production, top-quark mass and width effects from the loops cannot be neglected. Computations including the exact top-quark mass dependence are only available at the leading order, and currently predictions at higher orders are obtained by means of approximations based on the Higgs-gluon effective field theory (HEFT). In this work we present a reweighting technique that, starting from events obtained via the MC@NLO method in the HEFT, allows to exactly include the top-quark mass and width effects coming from one- and two-loop amplitudes. We describe our approach and apply it to double Higgs production at NLO in QCD, computing the needed one-loop amplitudes and using approximations for the unknown two-loop ones. The results are compared to other approaches used in the literature, arguing that they provide more accurate predictions for distributions and for total rates as well. As a novel application of our procedure we present predictions at NLO in QCD for triple Higgs production at hadron colliders.

Exact analytical solution for the Kissinger equation: Determination of the peak temperature and general properties of thermally activated transformations

A key parameter of many transformations when heated at a constant rate is the peak temperature, i.e., the temperature at which the transformation rate is at its maximum. The most universal approach to determine the peak temperature for thermally activated transformations is the Kissinger equation. In this paper, we solve Kissinger equation to deduce the exact dependence of the peak temperature on the heating rate. This analytical solution is based on the Lambert W-function. In addition, an approximate solution is derived that is used to infer general properties of thermally activated processes and to obtain a test to check the validity of Kissinger method.

High temperature dielectric relaxation anomaly of Y3+ and Mn2+ doped barium strontium titanate ceramics

Relaxation like dielectric anomaly is observed in Y3+ and Mn2+ doped barium strontium titanate ceramics when the temperature is over 450 K. Apart from the conventional dielectric relaxation analysis method with Debye or modified Debye equations, which is hard to give exact temperature dependence of the relaxation process, dielectric response in the form of complex impedance, assisted with Cole-Cole impedance model corrected equivalent circuits, is adopted to solve this problem and chase the polarization mechanism in this paper. Through this method, an excellent description to temperature dependence of the dielectric relaxation anomaly and its dominated factors are achieved. Further analysis reveals that the exponential decay of the Cole distribution parameter n with temperature is confirmed to be induced by the microscopic lattice distortion due to ions doping and the interaction between the defects. At last, a clear sight to polarization mechanism containing both the intrinsic dipolar polarization and extrinsic distributed oxygen vacancies hopping response under different temperature is obtained.

Investigation of photon detection probability dependence of SPADnet-I digital photon counter as a function of angle of incidence, wavelength and polarization

SPADnet-I is a prototype, fully digital, high spatial and temporal resolution silicon photon counter, based on standard CMOS imaging technology, developed by the SPADnet consortium. Being a novel device, the exact dependence of photon detection probability (PDP) of SPADnet-I was not known as a function of angle of incidence, wavelength and polarization of the incident light. Our targeted application area of this sensor is next generation PET detector modules, where they will be used along with LYSO:Ce scintillators. Hence, we performed an extended investigation of PDP in a wide range of angle of incidence (0° to 80°), concentrating onto a 60nm broad wavelength interval around the characteristic emission peak (λ=420nm) of the scintillator. In the case where the sensor was optically coupled to a scintillator, our experiments showed a notable dependence of PDP on angle, polarization and wavelength. The sensor has an average PDP of approximately 30% from 0° to 60° angle of incidence, where it starts to drop rapidly. The PDP turned out not to be polarization dependent below 30°. If the sensor is used without a scintillator (i.e. the light source is in air), the polarization dependence is much less expressed, it begins only from 50°.

Quantum quenches and generalized Gibbs ensemble in a Bethe Ansatz solvable lattice model of interacting bosons

We consider quantum quenches in the so-called q-boson lattice model. We argue that the Generalized Eigenstate Thermalization Hypothesis holds in this model, therefore the Generalized Gibbs Ensemble (GGE) gives a valid description of the stationary states in the long time limit. For a special class of initial states (which are the pure Fock states in the local basis) we are able to provide the GGE predictions for the resulting root densities. We also give predictions for the long-time limit of certain local operators. In the q → ∞ limit the calculations simplify considerably, the wave functions are given by Schur polynomials and the overlaps with the initial states can be written as simple determinants. In two cases we prove rigorously that the GGE prediction for the root density is correct. Moreover, we calculate the exact time dependence of a physical observable (the one-site Emptiness Formation Probability) for the quench starting from the state with exactly one particle per site. In the long-time limit the GGE prediction is recovered.

Divergence equation in thin-tube structures

Divergence equation is considered in a thin-tube structure, connected finite union of thin finite cylinders (in the case of two dimensions, respectively, thin rectangles) where the ratio of the diameter and the heights of the cylinders is the small parameter . Various norms of the solution of divergence equation are estimated over the norms of the right-hand side. The exact dependence of constants in these estimates on the small parameter is obtained.

Analysis of higher spin black holes with spin-4 chemical potential

We consider the $AdS_{3}/CFT_{2}$ duality between certain coset WZW theories at large central charge and Vasiliev 3D higher spin gravity with a single complex field. On the gravity side, we discuss a higher spin black hole solution with chemical potential coupled to the spin-4 charge. We compute the perturbative expansion of the higher spin charges and of the partition function at high order in the chemical potential. The result is obtained with its exact dependence on the parameter $\lambda$ characterising the symmetry algebra $\mbox{hs}[\lambda]$. The cases of $\lambda=0,1$ are successfully compared with a CFT calculation. The special point $\lambda=\infty$, the Bergshoeff-Blencowe-Stelle limit, is also solved in terms of the exact generating function for the partition function. The thermodynamics of both the spin-4 and the usual spin-3 black holes is studied in order to discuss the $\lambda$ dependence of the BTZ critical temperature $T_{\rm BTZ}(\lambda)$. In the spin-3 case, it is shown that $T_{\rm BTZ}(\lambda)$ converges for large $\lambda$ to the critical point of the $\lambda=\infty$ known partition function previously found by the authors. In the spin-4 black hole, the picture is qualitatively similar and $T_{\rm BTZ}(\infty)$ is accurately determined by various numerical methods.

Investigation of the Ultimate Strength of Periclase-Carbon Refractory Materials and Analysis of Their High Temperature Strength

A setup for determining the ultimate strength of refractory materials in compression at high temperatures is examined. The value of the ultimate strength of periclase-carbon materials in compression in the temperature range 20 – 500°C are presented. The fracture process and strength are analyzed. High-temperature plants are lined with refractory materials. In addition, the service life of many high-temperature plants is determined by the service life of the lining. Optimization of the factors influencing the stability of the lining makes it possible to increase the working run of a plant several-fold. Even without changing the form of the refractory a significant result can be achieved by simply improving the operating conditions (temperature regimes). Physical factors such as expansion and cracking arise in the case of thermal action on furnace lining. It becomes necessary to operate the plant continually without damaging the integrity of the lining in the working chamber and the technical-economic indices of the process. To prevent the lining from being damaged by the stresses arising during heating it must be operated in a regime where stresses grow at a rate below their relaxation rate. It is important to calculate the values of the thermal stresses when calculating the rate of heating. The stress values calculated using computational relations are compared with the admissible values, and on this basis a conclusion is drawn concerning the rate of heating of the plant. The average heating rate of high-temperature plants is about 60 Kmin [1]. The ultimate strength of the material is used in the calculations as the admissible value of the stresses arising with a change in the temperature field. Knowing the exact temperature dependence of the ultimate strength of the materials used it is possible to determine the maximum rate of heating of a plant (according to the conditions under which stresses arise). The results of tests performed on carbon-containing refractories produced by ‘Kombinat Magnezit’ JSC are presented in [2]. The characteristics and the stability of the parts are indicated and it is noted that the refractories have a high resistance to thermal shearing in the presence of temperature fluctuations. It should be noted that the ultimate strength in compression (ranging from 19 to 61 MPa for different types of refractories) is presented only at temperature 20°C. Thus, the plant data do not show the dynamics of the change in this parameter as a function of temperature.