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We examine the diffusive dynamics of a lattice random walk subject to resetting in a one-dimensional spatially heterogeneous environment composed of two media separated by an interface. At random times the walker may reset its position to the interface, but only when in the left medium. In addition the spatial heterogeneity results from having unequal diffusivities and biases in the two media. We construct the Master equation for the dynamics of the walker occupation probability in unbounded space, solve it exactly in terms of generating functions, and analyse the dynamics of the first and second moment. Making use of the closed form solution in the unbounded case, we build the analytic solution of the Master equation in finite and semi-infinite domains. By bounding the space on the right with a reflecting boundary we study the first-passage dynamics to a single fully absorbing target placed in the left medium away from the interface. As reset strongly increases the time to reach the target, we find that the first-passage dynamics enter the motion-limited regime even for relative small resetting probability. We also identify a surprising non-monotonic dependence of the first-passage probability mode as a function of the bias. By deriving an analytic expression for the mean first-passage time, we show when its value is independent of the diffusivity and bias in the left medium, uncovering another example of the so-called mean disorder indifference phenomenon. [ABSTRACT FROM AUTHOR]

We study the T -system of type A ∞ , also known as the octahedron recurrence/equation, viewed as a 2 + 1 -dimensional discrete evolution equation. Generalizing earlier work on arctic curves for the Aztec Diamond obtained from solutions of the octahedron recurrence with 'flat' initial data, we consider initial data along parallel 'slanted' planes perpendicular to an arbitrary admissible direction (r , s , t) ∈ Z + 3 . The corresponding solutions of the T -system are interpreted as partition functions of dimer models on some suitable 'pinecone' graphs introduced by Bousquet–Mélou, Propp, and West in 2009. The T -system formulation and some exact solutions in uniform or periodic cases allow us to explore the thermodynamic limit of the corresponding dimer models and to derive exact arctic curves separating the various phases of the system. This direct approach bypasses the standard general theory of dimers using the Kasteleyn matrix approach and uses instead the theory of Analytic Combinatorics in Several Variables, by focusing on a linear system obeyed by the dimer density generating function. [ABSTRACT FROM AUTHOR]

Adsorption of a single AB random copolymer (RC) chain onto homogeneous and inhomogeneous ab surfaces with a regular periodic pattern is studied theoretically. For the averaging over disorder in the RC sequence, the constrained annealed approximation, known as the Morita approximation is employed. A general scheme for constructing the Morita approximation of an arbitrary order m is proposed; it is based on representation of the RC monomer sequence as a Markov chain of overlapping (m –1)-ads. The problem is solved within the framework of the generating functions approach for the two-dimensional partially directed walk model of the polymer on a square lattice. Temperature dependences of various adsorption characteristics are obtained. The accuracy of the Morita approximation is assessed by comparison with the numerical results for RC with quenched sequences obtained by the transfer matrix approach. It is shown that for RC adsorption on a homogeneous surface for AB -copolymers with adsorbing A and neutral B blocks, it is sufficient to use the Morita approximation of second or third order. If the non-adsorbing B block is repelled from the surface, then a higher order of the approximation (5th–6th) is required. An important indicator of validity of the Morita approximation is the entropy: if the order of the approximation is not high enough, the entropy becomes negative in the low-temperature regime which means that the Morita approximation is wrong in this order. In the case when the surface is periodically inhomogeneous, the Morita approximation works worse and very high orders are required, which can be beyond the computational capabilities. Moreover, the Morita approximations become degenerate: approximations of two or more consecutive orders give the same result. [ABSTRACT FROM AUTHOR]

Relativistic fluids of divergence type constitute a special class of fluids whose states are determined from the knowledge of a scalar generating function χ and a dissipation tensor I μ ν i.e. a second rank, symmetric and traceless tensor. Both (χ , I μ ν) depend upon fourteen variables, known in the literature as Lagrange multipliers, and these multipliers satisfy a symmetric, quasilinear, first order system determined by (χ , I μ ν). For particular choices of the generating function χ, this system can be symmetric-hyperbolic and causal. We show in this work that this characteristic property of fluids of divergence type originates in the overdetermined nature of their dynamical equations. Combining their overdetermined nature with the work of Friedrichs on overdetermined system of conservation laws with more recent work by Boillat, Ruggeri and coworkers, we prove that fluids of divergence type are locally determined by a vector potential depending upon the Lagrange multipliers. For the case where the dynamical variables describing fluid states contain a symmetric energy momentum tensor, this vector potential is the gradient of a scalar field and this field is precisely the generating function χ introduced by Pennisi and independently by Geroch and Lindblom. Examples of scalar generating functions χ are discussed where this system is a symmetric hyperbolic and causal system in an open vicinity of equilibrium states. [ABSTRACT FROM AUTHOR]

The application of a sodium laser guide star is the key difference between a modern adaptive optics system and traditional adaptive optics system. Especially in system such as multiconjugate adaptive optics, a sodium laser guide star asterism that is formed by several laser guide stars in a certain pattern is required to probe more atmospheric turbulence in different directions. To achieve this, a sodium laser guide star asterism launching platform is required. In this paper, we introduce the sodium laser guide star asterism launching platform built and tested on the 1.8 m telescope of Gaomeigu Observatory. The platform has two functions: one function is to compare the performance of sodium laser guide stars generated by different lasers at the same place, and the other function is to generate a sodium laser guide star asterism with an adjustable shape. The field test results at the beginning of 2021 verified the important role of the platform, which is also the first time that a sodium laser guide star asterism was realized in China. [ABSTRACT FROM AUTHOR]

In this paper, we study a double scaling limit of two multi-matrix models: the U (N) 2 × O (D) -invariant model with all quartic interactions and the bipartite U (N) × O (D) -invariant model with tetrahedral interaction (D being here the number of matrices and N being the size of each matrix). Those models admit a double, large N and large D expansion. While N tracks the genus of the Feynman graphs, D tracks another quantity called the grade. In both models, we rewrite the sum over Feynman graphs at fixed genus and grade as a finite sum over combinatorial objects called schemes. This is a result of combinatorial nature which remains true in the quantum mechanical setting and in quantum field theory. Then we proceed to the double scaling limit at large D, i.e. for vanishing grade. In particular, we find that the most singular schemes, in both models, are the same as those found in Benedetti et al for the U (N) 2 × O (D) -invariant model restricted to its tetrahedral interaction. This is a different universality class than in the 1-matrix model whose double scaling is not summable. [ABSTRACT FROM AUTHOR]

We study a Langevin-like model which describes an inertial particle in a one-dimensional harmonic potential and subjected to two heat baths and one athermal environment. The thermal noises are white and Gaussian, and the temperatures of heat reservoirs are different. The athermal medium act through an external non-Gaussian noise of Poisson type. We calculate exactly the time-dependent cumulant-generating function of position and velocity of the particle, as well as an expression of this generating function for stationary states. We discuss the long-time behavior of first cumulants of the energy injected by the athermal reservoir and the heat exchanged with thermal baths. In particular, we find that the covariance of stochastic heat due to distinct thermal reservoirs exhibits a complex dependence on properties of athermal noise. [ABSTRACT FROM AUTHOR]

Adsorption of a single homopolymer chain with bending stiffness onto a heterogeneous regular stripe-patterned surface consisting of adsorbing and non-adsorbing stripes is studied theoretically in the framework of the lattice model and the generating functions approach. The stiffness is introduced by assigning a statistical weight to a trans -isomer (a straight segment) with respect to a gauche -isomer (a kink). The temperature is taken as the main control parameter since it affects both the strength of the monomer units’ attraction to the adsorbing stripes and the chain stiffness. It is shown that the adsorption transition temperature is a non-monotonic function on the bending energy having a minimum. The position of this minimum depends on the stripes’ width and only slightly deviates from zero bending energy. Temperature dependences of the main conformational and thermodynamic characteristics of the adsorbed chain are obtained. It is demonstrated that in most of the studied cases the adsorption is accompanied by the chain localization on a single adsorbing unit stripe and the chain stiffness enhances this effect. [ABSTRACT FROM AUTHOR]

This paper is concerned with construction of quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters. The bosonâ€"fermion correspondence for these symmetric functions have been presented. In virtue of quantum fields, we derive a series of infinite order nonlinear integrable equations, namely, universal character hierarchy, symplectic KP hierarchy and symplectic universal character hierarchy, respectively. In addition, the solutions of these integrable systems have been discussed. [ABSTRACT FROM AUTHOR]

We consider discrete-time Markov chains and study large deviations of the pair empirical occupation measure, which is useful to compute fluctuations of pure-additive and jump-type observables. We provide an exact expression for the finite-time moment generating function, which is split in cycles and paths contributions, and scaled cumulant generating function of the pair empirical occupation measure via a graph-combinatorial approach. The expression obtained allows us to give a physical interpretation of interaction and entropic terms, and of the Lagrange multipliers, and may serve as a starting point for sub-leading asymptotics. We illustrate the use of the method for a simple two-state Markov chain. [ABSTRACT FROM AUTHOR]

The distribution of shortest path lengths (DSPL) of random networks provides useful information on their large scale structure. In the special case of random regular graphs (RRGs), which consist of N nodes of degree c â©ľ 3, the DSPL, denoted by P (L = â„"), follows a discrete Gompertz distribution. Using the discrete Laplace transform we derive a closed-form (CF) expression for the moment generating function of the DSPL of RRGs. From the moment generating function we obtain CF expressions for the mean and variance of the DSPL. More specifically, we find that the mean distance between pairs of distinct nodes is given by ⟨ L âź© = ln N ln (c âˆ' 1) + 1 2 âˆ' ln c âˆ' ln (c âˆ' 2) + Îł ln (c âˆ' 1) + O ln N N , where Îł is the Eulerâ€"Mascheroni constant. While the leading term is known, this result includes a novel correction term, which yields very good agreement with the results obtained from direct numerical evaluation of ⟨ L âź© via the tail-sum formula and with the results obtained from computer simulations. However, it does not account for an oscillatory behavior of ⟨ L âź© as a function of c or N. These oscillations are negligible in sparse networks but detectable in dense networks. We also derive an expression for the variance Var(L) of the DSPL, which captures the overall dependence of the variance on c but does not account for the oscillations. The oscillations are due to the discrete nature of the shell structure around a random node. They reflect the profile of the filling of new shells as N is increased. The results for the mean and variance are compared to the corresponding results obtained in other types of random networks. The relation between the mean distance and the diameter is discussed. [ABSTRACT FROM AUTHOR]

Epidemics on complex networks is a widely investigated topic in the last few years, mainly due to the last pandemic events. Usually, real contact networks are dynamic, hence much effort has been invested in studying epidemics on evolving networks. Here we propose and study a model for evolving networks based on varying degrees, where at each time step a node might get, with probability r, a new degree and new neighbors according to a given degree distribution, instead of its former neighbors. We find analytically, using the generating functions framework, the epidemic threshold and the probability for a macroscopic spread of disease depending on the rewiring rate r. Our analytical results are supported by numerical simulations. We find that the impact of the rewiring rate r has qualitative different trends for networks having different degree distributions. That is, in some structures, such as random regular networks the dynamics enhances the epidemic spreading while in others such as scale free (SF) the dynamics reduces the spreading. In addition, we unveil that the extreme vulnerability of static SF networks, expressed by zero epidemic threshold, vanishes for only fully evolving network, r = 1, while for any partial dynamics, i.e. r < 1, this zero threshold exists. Finally, we find the epidemic threshold also for a general distribution of the recovery time. [ABSTRACT FROM AUTHOR]

We study the double scaling limit of the O (N)3-invariant tensor model, initially introduced in Carrozza and Tanasa (2016 Lett. Math. Phys. 106 1531). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the two-point function, we rewrite the sum over Feynman graphs at each order in the 1/ N expansion as a finite sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the 1/ N expansion. This leads to a double scaling limit which picks up contributions from all orders in the 1/ N expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of the Feynman graphs, as well as an analytic analysis of the singularities of the relevant generating series. [ABSTRACT FROM AUTHOR]

We explore the time evolution law of a two-mode squeezed light field (pure state) passing through twin diffusion channels, and we find that the final state is a squeezed chaotic light field (mixed state) with entanglement, which shows that even though the two channels are independent of each other, since the two modes of the initial state are entangled with each other, the final state remains entangled. Nevertheless, although the squeezing (entanglement) between the two modes is weakened after the diffusion, it is not completely removed. We also highlight the law of photon number evolution. In the calculation process used in this paper, we make full use of the summation method within the ordered product of operators and the generating function formula for two-variable Hermite polynomials. [ABSTRACT FROM AUTHOR]

Non-relativistic helium-like ions (âˆ' e, âˆ' e, Ze) with static nucleus in a d -dimensional space R d (d > 1) are considered. Assuming r âˆ'1 Coulomb interactions, a two-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domain Z â©˝ 10. For odd d = 3, 5, the variational energy is given by a rational algebraic function of the variational parameters while for even d = 2, 4 it is shown for the first time that it corresponds to a more complicated non-algebraic expression. This twofold analyticity will hold for any d. It allows us to construct reasonably accurate approximate solutions for the ground state energy E 0(Z, d) in the form of compact analytical expressions. We call them generalized Majorana solutions. They reproduce the first leading terms in the celebrated 1 Z expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge Z c vs d and the Shannon entropy S r (d) vs Z are also calculated within the present variational approach. In the light of these results, for the physically important case d = 3 a more general three-parametric correlated Hylleraas-type trial is used to compute the finite mass effects in the Majorana solution for a three-body Coulomb system with arbitrary charges and masses. It admits a straightforward generalization to any d as well. Concrete results for the systems e âˆ' e âˆ' e +, H 2 + and H âˆ' are indicated explicitly. Comparison of our variational analytical results with corresponding highly accurate numerical values reported in the literature indicates a fair agreement, within the limits of accuracy imposed by the approximate character of our treatment. [ABSTRACT FROM AUTHOR]

We assess the accuracy of Bayesian polynomial extrapolations from small parameter values, x, to large values of x. We consider a set of polynomials of fixed order, intended as a proxy for a fixed-order effective field theory (EFT) description of data. We employ Bayesian model averaging (BMA) to combine results from different order polynomials (EFT orders). Our study considers two 'toy problems' where the underlying function used to generate data sets is known. We use Bayesian parameter estimation to extract the polynomial coefficients that describe these data at low x. A 'naturalness' prior is imposed on the coefficients, so that they are O (1) . We BMA different polynomial degrees by weighting each according to its Bayesian evidence and compare the predictive performance of this BMA with that of the individual polynomials. The credibility intervals on the BMA forecast have the stated coverage properties more consistently than does the highest evidence polynomial, though BMA does not necessarily outperform every polynomial. [ABSTRACT FROM AUTHOR]

We try to develop an atlas-guided automatic planning (AGAP) approach and evaluate its feasibility and performance in rectal cancer intensity-modulated radiotherapy. The developed AGAP approach consisted of four independent modules: patient atlas, similar patient retrieval, beam morphing (BM), and plan fine-tuning (PFT) modules. The atlas was setup using anatomy and plan data from Pinnacle auto-planning (P-auto) plans. Given a new patient, the retrieval function searched the top similar patient by a generic Fourier descriptor algorithm and retrieved its plan information. The BM function generated an initial plan for the new patient by morphing the beam aperture from the top similar patient plan. The beam aperture and calculated dose of the initial plan were used to guide the new plan optimization in the PFT function. The AGAP approach was tested on 96 patients by the leave-one-out validation and plan quality was compared with the P-auto plans. The AGAP and P-auto plans had no statistical difference for target coverage and dose homogeneity in terms of V100% (p = 0.76) and homogeneity index (p = 0.073), respectively. The CI index showed they had a statistically significant difference. But the ΔCI was both 0.02 compared to the perfect CI index of 1. The AGAP approach reduced the bladder mean dose by 152.1 cGy (p < 0.05) and V50 by 0.9% (p < 0.05), and slightly increased the left and right femoral head mean dose by 70.1 cGy (p < 0.05) and 69.7 cGy (p < 0.05), respectively. This work developed an efficient and automatic approach that could fully automate the IMRT planning process in rectal cancer radiotherapy. It reduced the plan quality dependence on the planner experience and maintained the comparable plan quality with P-auto plans. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the distribution of the total angular momentum in a relativistic configuration. Using cumulants and generating function formalism this analysis can be reduced to the study of individual subshells with N equivalent electrons of momentum j. An expression as a nth-derivative is provided for the generating function of the J distribution and efficient recurrence relations are established. It is shown that this distribution may be represented by a Gram–Charlier-like series which is derived from the corresponding series for the magnetic quantum number distribution. The numerical efficiency of this expansion is fair when the configuration consists of several subshells, while the accuracy is less good when only one subshell is involved. An analytical expression is given for the odd-order momenta while the even-order ones are expressed as a series which provides an acceptable accuracy though being not convergent. Such expressions may be used to obtain approximate values for the number of transitions in a spin–orbit split array: it is shown that the approximation is often efficient when few terms are kept, while some complex cases require to include a large number of terms. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of the distribution of the total magnetic quantum number M in a relativistic subshell with N equivalent electrons of momentum j. This distribution is analyzed through its cumulants and through their generating function, for which an analytical expression is provided. This function also allows us to get the values of the cumulants at any order. Such values are useful to obtain the moments at various orders. Since the cumulants of the distinct subshells are additive this study directly applies to any relativistic configuration. Recursion relations on the generating function are given. It is shown that the generating function of the magnetic quantum number distribution may be expressed as an nth derivative of a polynomial. This leads to recurrence relations for this distribution which are very efficient even in the case of large j or N. The magnetic quantum number distribution is numerically studied using the Gram–Charlier and Edgeworth expansions. The inclusion of high-order terms may improve the accuracy of the Gram–Charlier representation for instance when a small and a large angular momenta coexist in the same configuration. However such series does not exhibit convergence when high orders are considered and the account for the first two terms often provides a fair approximation of the magnetic quantum number distribution. The Edgeworth series offers an interesting alternative though this expansion is also divergent and of asymptotic nature. [ABSTRACT FROM AUTHOR]

Trajectory-based methods are well-developed to approximate steady-state probability distributions for stochastic processes in large-system limits. The trajectories are solutions to equations of motion of Hamiltonian dynamical systems, and are known as eikonals. They also express the leading flow lines along which probability currents balance. The existing eikonal methods for discrete-state processes including chemical reaction networks are based on the Liouville operator that evolves generating functions of the underlying probability distribution. We have previously derived [1, 2] a representation for the generators of such processes that acts directly on the hierarchy of moments of the distribution, rather than on the distribution itself or on its generating function. We show here how in the large-system limit the steady-state condition for that generator reduces to a mapping from eikonals to the ratios of neighboring factorial moments, as a function of the order k of these moments. The construction shows that the boundary values for the moment hierarchy, and thus its whole solution, are anchored in the interior fixed points of the Hamiltonian system, a result familiar from Freidlin–Wenztell theory. The direct derivation of eikonals from the moment representation further illustrates the relation between coherent-state and number fields in Doi–Peliti theory, clarifying the role of canonical transformations in that theory. [ABSTRACT FROM AUTHOR]

Large deviation theory and instanton calculus for stochastic systems are widely used to gain insight into the evolution and probability of rare events. At its core lies the fact that rare events are, under the right circumstances, dominated by their least unlikely realization. Their computation through a saddle-point approximation of the path integral for the corresponding stochastic field theory then reduces an inefficient stochastic sampling problem into a deterministic optimization problem: finding the path of smallest action, the instanton. In the presence of heavy tails, though, standard algorithms to compute the instanton critically fail to converge. The reason for this failure is the divergence of the scaled cumulant generating function (CGF) due to a non-convex large deviation rate function. We propose a solution to this problem by 'convexifying' the rate function through a nonlinear reparametrization of the observable, which allows us to compute instantons even in the presence of super-exponential or algebraic tail decay. The approach is generalizable to other situations where the existence of the CGF is required, such as exponential tilting in importance sampling for Monte-Carlo algorithms. We demonstrate the proposed formalism by applying it to rare events in several stochastic systems with heavy tails, including extreme power spikes in fiber optics induced by soliton formation. [ABSTRACT FROM AUTHOR]

Let N ⩾ 2 and ρ ∈ (0, 1/N2]. The homogenous Cantor set E is the self-similar set generated by the iterated function system Let s = dimH E be the Hausdorff dimension of E, and let be the s-dimensional Hausdorff measure restricted to E. In this paper we describe, for each x ∈ E, the pointwise lower s-density and upper s-density Θ∗s(μ, x) of μ at x. This extends some early results of Feng et al (2000 J. Math. Anal. Appl. 250 692–705). Furthermore, we determine two critical values ac and bc for the sets respectively, such that dimH E*(a) > 0 if and only if a < ac, and that dimH E*(b) > 0 if and only if b > bc. We emphasize that both values ac and bc are related to the Thue–Morse type sequences, and our strategy to find them relies on ideas from open dynamical systems and techniques from combinatorics on words. [ABSTRACT FROM AUTHOR]

We study the SYK2 model of N Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling the model to two- and four-point sources. In particular, we define the generalized spectral form factor (SFF) and level spacing distribution function by generalizing from the partition function to the generating function. For N = 2, we obtain an exact solution of the generalized SFF. It exhibits qualitatively similar behavior to the higher N case with a source term. The exact solution helps understand the behavior of the generalized SFF. We calculate the generalized level spacing distribution function and the mean value of the adjacent gap ratio defined by the generating function. For the SYK2 model with a four-point source term, we find a Gaussian unitary ensemble behavior in the near-integrable region of the theory, which indicates a transition to chaos. This behavior is confirmed by the connected part of the generalized SFF with an unfolded spectrum. The departure from this Gaussian random matrix behavior as the relative strength of the source term is increased is consistent with the observation that the four-point source term alone, without the SYK2 couplings turned on, exhibits an integrable spectrum from the SFF and level spacing distribution function in the large N limit. [ABSTRACT FROM AUTHOR]

This paper proposes a solution method for identification problems in the context of contact mechanics when overabundant data are available on a part Γm of the domain boundary while data are missing from another part of this boundary. The first step is then to find a solution to a Cauchy problem. The method used by the authors for solving Cauchy problems consists of expanding the displacement field known on Γm toward the inside of the solid via the minimization of a function that measures the gap between solutions of two well-posed problems, each one exploiting only one of the superabundant data. The key question is then to build an appropriate gap functional in strongly nonlinear contexts. The proposed approach exploits a generalization of the Bregman divergence, using the thermodynamic potentials as generating functions within the framework of generalized standard materials (GSMs), but also implicit GSMs in order to address Coulomb friction. The robustness and efficiency of the proposed method are demonstrated by a numerical bi-dimensional application dealing with a cracked elastic solid with unilateral contact and friction effects on the crack's lips. [ABSTRACT FROM AUTHOR]

We study spectral gaps of the one-dimensional totally asymmetric simple exclusion process (TASEP) with open boundaries in the maximal current phase. Earlier results for the model with periodic boundaries suggest that the gaps contributing to the universal KPZ regime may be understood as points on an infinite genus Riemann surface built from a parametric representation of the cumulant generating function of the current. We perform explicit analytic continuations from the known large deviations of the current for open TASEP, and confirm the results for the gaps by an exact Bethe ansatz calculation, with additional checks using high precision extrapolation numerics. [ABSTRACT FROM AUTHOR]

We study the large deviations of the distribution P(Wτ) of the work associated with the propulsion of individual active Brownian particles in a time interval τ, in the region of the phase diagram where macroscopic phase separation takes place. P(Wτ) is characterised by two peaks, associated to particles in the gaseous and in the clusterised phases, and two separate non-convex branches. Accordingly, the generating function of Wτ's cumulants displays a double singularity. We discuss the origin of such non-convex branches in terms of the peculiar dynamics of the system phases, and the relation between the observation time τ and the typical persistence times of the particles in the two phases. [ABSTRACT FROM AUTHOR]

We present a derivation of the Feynman–Vernon approach to open quantum systems in the language of super-operators. We show that this gives a new and more direct derivation of the generating function of energy changes in a bath, or baths. As found previously, this generating function is given by a Feynman–Vernon-like influence functional, with only time shifts in the kernels coupling the forward and backward paths. We further show that the new approach extends to an-harmonic and possible non-equilibrium baths, provided that the interactions are bi-linear, and that the baths do not interact between themselves. Such baths are characterized by non-trivial cumulants. Every non-zero cumulant of certain environment correlation functions is thus a kernel in a higher-order term in the Feynman–Vernon action. [ABSTRACT FROM AUTHOR]

We derive an exact expression for the probability density of work done on a particle that diffuses in a parabolic potential with a stiffness varying by an arbitrary piecewise constant protocol. Based on this result, the work distribution for time-continuous protocols of the stiffness can be determined up to any degree of accuracy. This is achieved by replacing the continuous driving by a piecewise constant one with a number n of positive or negative steps of increasing or decreasing stiffness. With increasing n, the work distributions for the piecewise protocols approach that for the continuous protocol. The moment generating function of the work is given by the inverse square root of a polynomial of degree n, whose coefficients are efficiently calculated from a recurrence relation. The roots of the polynomials are real and positive (negative) steps of the protocol are associated with negative (positive) roots. Using these properties the inverse Laplace transform of the moment generating function is carried out explicitly. Fluctuation theorems are used to derive further properties of the polynomials and their roots. [ABSTRACT FROM AUTHOR]

The generation of photon-pairs with controllable spectral correlations is crucial in quantum photonics. Here we present the design of a photonic crystal fiber to generate widely-spaced four-wave mixing bands with spectral correlations that can be tuned through the thermo-optic effect after being infiltrated with heavy water. We present a theoretical study of the purity of the signal and idler photons generated as a function of temperature, pump spectral linewidth and the length of the fiber. [ABSTRACT FROM AUTHOR]

We completely generalize previous results related to the counting of connected Feynman diagrams. We use a generating function approach, which encodes the Wick contraction combinatorics of the respective connected diagrams. Exact solutions are found for an arbitrary number of external legs, and a general algorithm is implemented for this calculus. From these solutions, we calculate many asymptotics expansion terms for a simple analytical tool (Taylor expansion theorem). Our approach offers new perspectives in the realm of Feynman diagrams enumeration (or zero-dimensional quantum field theory). [ABSTRACT FROM AUTHOR]

For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed states. We show that the normalization is related to Jacobi polynomial, so the cumulant expansion in turn represents the new generating function of Jacobi polynomial. [ABSTRACT FROM AUTHOR]

We consider the photon statistical properties emitted from a two-level quantum system under pump–probe driving. The line shapes and Mandel's Q parameters are studied for a single quantum system based on the generating function method developed recently. The control of photon emission from a single quantum system driven simultaneously by a driven field and a tunable probe field is investigated. Our theoretical results and the experimental results are in good agreement. [ABSTRACT FROM AUTHOR]

In this paper the totally asymmetric simple exclusion process (TASEP) with parallel update on an open lattice of size L is considered in the maximumcurrent region. A formal expression for the generating function for the weight of configurations with N particles is given. Further an interpretation in terms of (u, l, d)-colored weighted Motzkin paths is presented. Using previous results (Woelki and Schreckenberg 2009 J. Stat. Mech. P05014,Woelki 2010 Cellular Automata pp 637-45) the generating function is compared with the one for a possible second-class particle dynamics for the parallel TASEP. It is shown that both become physically equivalent in the thermodynamic limit. [ABSTRACT FROM AUTHOR]

We study generating functions for the scalar products of SU(2) coherent intertwiners, which can be interpreted as coherent spin network evaluations on a 2-vertex graph. We show that these generating functions are exactly summable for different choices of combinatorialweights. Moreover, we identify one choice of weight distinguished thanks to its geometric interpretation. As an example of dynamics, we consider the simple case of SU(2) flatness and describe the corresponding Hamiltonian constraint whose quantization on coherent intertwiners leads to partial differential equations that we solve. Furthermore, we generalize explicitly these Wheeler-DeWitt equations for SU(2) flatness on coherent spin networks for arbitrary graphs. [ABSTRACT FROM AUTHOR]

The generating function for the number of non-crossingwalk configurations of n walks between the roots of a two-rooted directed plane graph is introduced. This is shown to be a rational function and the structure and symmetry property of its numerator are discussed. The walk configurations correspond to flows and the equivalent dual generating function for potentials is investigated independently. Also equivalences are drawn with the partially order sets that can be constructed from the walk configurations. Finally, the general results developed here are applied to the directed square and honeycomb lattices. [ABSTRACT FROM AUTHOR]

We define the notion of a spanning tree generating function (STGF) ΣanZn, anzn, which gives the spanning tree constant when evaluated at z = 1, and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one threedimensional case. We show for all lattices that these can also be represented as Dirichlet L-series. Making the connection between STGFs and LGFs produces integral identities and hypergeometric connections, some of which appear to be new. [ABSTRACT FROM AUTHOR]

We solve exactly a two-dimensional partially directed walk model of a semiflexible polymer that has one end tethered to a sticky wall, while a pulling force away from the adsorbing surface acts on the free end of the walk. This model generalizes a number of previously considered adsorption models by incorporating individual horizontal and vertical stiffness effects, in competition with a variable pulling angle. A solution to the corresponding generating function is found by means of the kernel method. While the phases and related phase transitions are similar in nature to those found previously the analysis of the model in terms of its physical variables highlights various novel structures in the shapes of the phase diagrams and related behaviour of the polymer. We review the results of previously considered sub-cases, augmenting these findings to include analysis with respect to the model's physical variables--namely, temperature, pulling force, pulling angle away from the surface, stiffness strength and the ratio of vertical to horizontal stiffness potentials, with our subsequent analysis for the general model focusing on the effect that stiffness has on this pulling angle range. In analysing the model with stiffness we also pay special attention to the case where only vertical stiffness is included. The physical analysis of this case reveals behaviour more closely resembling that of an upward pulling force acting on a polymer than it does of a model where horizontal stiffness acts. The stiffness-temperature phase diagram exhibits re-entrance for low temperatures, previously only seen for three-dimensional or co-polymer models. For the most general model we delineate the shift in the physical behaviour as we change the ratio of vertical to horizontal stiffness between the horizontal-only and the vertical-only stiffness regimes. We find that a number of distinct physical characteristics will only be observed for a model where the vertical stiffness dominates the horizontal stiffness. [ABSTRACT FROM AUTHOR]

Adsorption of an (AB) random copolymer (RC) onto an (ab) random heterogeneous surface (RS), where both the RC and the RS have correlations, is studied using a two-dimensional partially directed walk model of the polymer. The problem is solved by using a combination of the annealed approximation for averaging over polymer and surface randomness with the generating functions (GFs) approach. A compact determinant equation for finding the smallest singularity of the GF of the adsorbed RC chain is derived. Dependences of inverse temperature of the adsorption transition on the RC and the RS correlation parameters are calculated for the chosen sets of monomer- site interaction parameters. We analyze the influence of interplay between correlations in the RC and the RS on the transition temperature, observe and discuss the effect of insensitivity of Bernoullian (uncorrelated) RC (or RS) to correlations in its 'adsorption partner'. An exact solution for the regular alternating copolymer adsorption onto the regular alternating surface is obtained. [ABSTRACT FROM AUTHOR]

We find, and analyse, the exact solution of two friendly directed walks, modelling polymers, which interact with a wall via contact interactions. We specifically consider two walks that begin and end together so as to imitate a polygon. We examine a general model in which a separate interaction parameter is assigned to configurations where both polymers touch the wall simultaneously, and investigate the effect this parameter has on the integrability of the problem. We find an exact solution of the generating function of the model, and provide a full analysis of the phase diagram that admits three phases with one first-order and two second-order transition lines between these phases. We argue that one physically realizable model would see two phase transitions as the temperature is lowered. [ABSTRACT FROM AUTHOR]

We prove that for … ≤ 1 and n ≥ …, where hn(x|q) and hn(x|t, q) are respectively the so-called q-Hermite and the big q-Hermite polynomials, and (q)n denotes the so-called q-Pochhammer symbol. We prove similar equalities involving big q-Hermite and Al-Salam-Chihara polynomials, and Al-Salam-Chihara and the so-called continuous dual q-Hahn polynomials. Moreover, we are able to relate in this way some other 'ordinary' orthogonal polynomials such as, e.g., Hermite, Chebyshev or Laguerre. These equalities give a new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general (i.e. involving more and more parameters) families of orthogonal polynomials. We pose some conjectures concerning Askey-Wilson polynomials and their possible generalizations. We prove that these conjectures are true for the cases q = 1 (classical case) and q = 0 (free case), thus paving the way to generalization of Askey-Wilson polynomials at least in these two cases. [ABSTRACT FROM AUTHOR]

Propagation into the far field changes initialwaves into their Fourier transforms. This implies that the Riemann zeros could be observed experimentally in the radiation pattern generated by an initial wave whose Fourier transform is proportional to the Riemann zeta function on the critical line. Two such waves are examined, generating the Riemann &b.Xi;(t) function (pattern 1) and the function ζ (1/2 + it)/(1/2 + it) (pattern 2). For pattern 1, the radiation side lobes are probably too weak to allow detection of the zeros, but for pattern 2 the lobes are stronger, suggesting a feasible experiment. [ABSTRACT FROM AUTHOR]

In this paper we continue an earlier study (Diao et al 2011 J. Phys. A: Math. Theor. 44 405202) on the generation algorithms of random equilateral polygons confined in a sphere. Here, the equilateral random polygons are rooted at the center of the confining sphere and the confining sphere behaves like an absorbing boundary. One way to generate such a random polygon is the accept/reject method in which an unconditioned equilateral random polygon rooted at origin is generated. The polygon is accepted if it is within the confining sphere, otherwise it is rejected and the process is repeated. The algorithm proposed in this paper offers an alternative to the accept/reject method, yielding a faster generation process when the confining sphere is small. In order to use this algorithm effectively, a large, reusable data set needs to be pre-computed only once. We derive the theoretical distribution of the given random polygon model and demonstrate, with strong numerical evidence, that our implementation of the algorithm follows this distribution. A run time analysis and a numerical error estimate are given at the end of the paper. [ABSTRACT FROM AUTHOR]

We discuss the integrability of the bosonic and Grassmannian massive Thirring models in the presence of defects through the inverse scattering approach. We present a general method to compute the generating functions of modified conserved quantities for any integrable field equation associated with the m×m spectral linear problem. We apply the method to derive in particular the defect contributions for the number of occupation, energy and momentum of the massive Thirring models. [ABSTRACT FROM AUTHOR]

We derive an nth order difference equation as a dual of a very simple periodic equation, and construct &LFLOOR;(n+1)/2&RFOOR; explicit integrals and integrating factors of this equation in terms of multi-sums of products. We also present a generating function for the degrees of its iterates, exhibiting polynomial growth. In conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of Levi and Yamilov. These three facts combine to suggest the integrability of the nth order difference equation. [ABSTRACT FROM AUTHOR]

For the conventional translational shape-invariant potentials (TSIPs), it has demonstrated that the phase contribution devoted by the scattered subwaves in the analytical transfer matrix quantization condition is integrable and independent of n. Based on this fact we propose a novel strategy to generate the whole set of conventional TSIPs and classify them into three types. The generating functions are given explicitly and the Morse potential is taken as an example to illustrate this strategy. [ABSTRACT FROM AUTHOR]

Based on the concept of the cloak generating function, we propose an implicit transformation-independent method for the required parameters of spherical cloaks without knowing the needed coordinate transformation beforehand. A non-ideal discrete model is used to calculate and optimize the total scattering cross-sections of different profiles of the generating function. A bell-shaped quadratic spherical cloak is found to be the best candidate, which is further optimized by controlling the design parameters involved. Such improved invisibility is steady even when the model is highly discretized. [ABSTRACT FROM AUTHOR]

We investigate spin currents and spin-current correlations for double quantum dots in the spin blockade regime. By analysing the time evolution of the density matrix, we obtain the spin resolved currents and derive from a generating function an expression for the fluctuations and correlations. Both the charge current and the spin current turn out to be generally super-Poissonian. Moreover, we study the influence of ac fields acting upon the transported electrons. In particular, we focus on fields that cause spin rotation or photonassisted tunnelling. [ABSTRACT FROM AUTHOR]

In this paper, a scheme for generating various multiatom entangled graph states via resonant interactions is proposed. We investigate the generation of various four-atom graph states first in the ideal case and then in the case in which the cavity decay and atomic spontaneous emission are taken into consideration in the process of interaction. More importantly, we improve the possible distortion of the graph states coming from cavity decay and atomic spontaneous emission by performing appropriate unitary transforms on atoms. The generation of multiatom entangled graph states is very important for constructing quantum one-way computer in a fault-tolerant manner. The resonant interaction time is very short, which is important in the sense of decoherence. Our scheme is easy and feasible within the reach of current experimental technology. [ABSTRACT FROM AUTHOR]

Models of directed paths have been used extensively in the scientific literature to model linear polymers. In this paper, we examine a directed path model of a linear polymer in a confining geometry (a wedge). We are particularly interested in cn, the number of directed lattice paths of length n steps which take steps in the North-East and South-East directions and are confined in the wedge Y = ±X/p, where p is an integer. We examine the case p = 2 in detail and show that the generating function satisfies a functional equation with quartic kernel. We give a formal solution by using the kernel method, show that this model is not D-finite and determine asymptotics to leading order for cn. In particular, the number of paths of length n in the wedge Y = ±X/2 is given by c_n = [0.678 74\ldots]\times 2^{n} + (4/3^{3/4})^{n+o(n)} where the constant 0.67874... can be determined to arbitrary accuracy with little computational effort. [ABSTRACT FROM AUTHOR]

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry. [ABSTRACT FROM AUTHOR]