Nontrivial Function Research Articles

On the stability of continuously stratified quasi-geostrophic hetons

In this paper we examine the stability of quasi-geostrophic hetons in a stably, continuously stratified fluid. To this purpose we first determinate numerically equilibrium states. Equilibrium hetons consist of two vortices of equal and opposite strength lying at different depths that are steadily translating without deforming. The situation is studied through a parameter space comprising of the vertical offset between the vortices, their horizontal separation distance and their aspect ratio. The study first shows that the equilibrium vortices are not only strongly deformed in the vertical but that their instability modes are also varying within the height of the structures. The main purpose of the present contribution is to study families of equilibria which stem from the case of two vertically aligned cylindrical vortices. It is however shown that other branches of solutions exist with different properties. The paper concludes that hetons may be sensitive to baroclinic instabilities provided the separation distance between the poles of the hetons is moderate both in the horizontal and in the vertical directions. The hetons become stable and efficient ways to transport properties as far as the poles are distant from one another. The critical separation distance in a non-trivial function of the radius-to-height aspect ratio of the poles.

Spatial extent of branching Brownian motion.

We study the one-dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost (X(max)≥0) and leftmost (X(min)≤0) visited sites up to time t. At each time step the existing particles in the system either diffuse (with diffusion constant D), die (with rate a), or split into two particles (with rate b). We focus on the regime b≤a where these two extreme values X(max) and X(min) are strongly correlated. We show that at large time t, the joint probability distribution function (PDF) of the two extreme points becomes stationary P(X,Y,t→∞)→p(X,Y). Our exact results for p(X,Y) demonstrate that the correlation between X(max) and X(min) is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF p(ζ) of the (dimensionless) span ζ=(X(max)-X(min))/√[D/b], which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior p(ζ)∼1/2(1+Δ)ζ for small spans, with Δ=(a/b-1). In the critical case (Δ=0) this distribution has a nontrivial power law tail p(ζ)∼8π√[3]/ζ(3) for large spans. On the other hand, in the subcritical case (Δ>0), we show that the span distribution decays exponentially as p(ζ)∼(A(2)/2)ζexp(-√[Δ]ζ) for large spans, where A is a nontrivial function of Δ, which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between X(max) and X(min). Finally we verify our results via direct Monte Carlo simulations.

The efficiency of primitive recursive functions: A programmer's view

Using denotational semantics tools, Colson and others studied primitive recursive (p.r.) algorithms, proving the “ultimate obstinacy” property, which has as a consequence that many computations cannot be efficiently implemented in a language that faithfully expresses the classical definition of p.r. functions.As shown by Ritchie and Meyer, the class of p.r. functions can also be characterised by the programming language Loop. Our purpose is to show that the informal, but precise, operational description of Loop (and its variants) is sufficient to prove non-trivial time lower bounds of p.r. algorithms. In particular, we present a simple proof of a property which is similar to ultimate obstinacy, namely that every p.r. program that implements a non-trivial function must execute xi+c times the body of some loop instruction, where xi is the initial contents of some fixed input register, and c does not depend on the input values. This and other lower bounds were obtained without using the lambda calculus or systems with functional parameters (such as Gödel system T).If the conditional break and the decrement instructions are included in the Loop language, the ultimate obstinacy property does not hold. In this case, we use another approach for obtaining lower bounds. The unconditional break instruction does not avoid the obstinacy property; it is equivalent (in function and efficiency) to the conditional instruction if-then-else. The efficient implementation of other functions like step(x) (step(0)=0; step(x)=1 for x≥1) and min⁡(x,y) is also studied.

Effect of process scale-up on the dissolution of granules with a high content of active pharmaceutical ingredient

The robustness and parametric sensitivity of a high-shear wet granulation process during scale-up were investigated for a formulation containing over 50% by weight of a hydrophobic active pharmaceutical ingredient (API). The sensitivity of granule properties such as particle size distribution, bulk density and dissolution rate with respect to both formulation (binder/solids ratio) and process parameters (agitation rate and wet massing time) was determined at three different scales (0.5, 4.0 and 25.0L). For a fixed impeller speed at a given granulator scale, the dissolution rate was found to be strongly correlated with bulk density, and both parameters could be controlled relatively precisely by the wet massing time. However, during scale-up and for variable agitation rates (expressed by the dimensionless Froude number) dissolution rate was found to be a non-trivial function of both granule size distribution and granule density. A set of process parameters for which the formulation was robust during scale-up was identified.

Gluon fragmentation into quarkonium at next-to-leading order

We present the first calculation at next-to-leading order (NLO) in $\alpha_s$ of a fragmentation function into quarkonium whose form at leading order is a nontrivial function of $z$, namely the fragmentation function for a gluon into a spin-singlet S-wave state at leading order in the relative velocity. To calculate the real NLO corrections, we introduce a new subtraction scheme that allows the phase-space integrals to be evaluated in 4 dimensions. We extract all ultraviolet and infrared divergences in the real NLO corrections analytically by calculating the phase-space integrals of the subtraction terms in $4-2\epsilon$ dimensions. We also extract the divergences in the virtual NLO corrections analytically, and detail the cancellation of all divergences after renormalization. The NLO corrections have a dramatic effect on the shape of the fragmentation function, and they significantly increase the fragmentation probability.

Pulsating strings from two-dimensional CFT on [formula omitted

We propose a state from the two-dimensional conformal field theory on the orbifold (T4)N/S(N) as a dual description for a pulsating string moving in AdS3. We show that, up to first order in the deforming parameter, the energy in both descriptions has the same dependence on the mode number, but with a non-trivial function of the coupling.

FKPP Fronts in Cellular Flows: The Large-Péclet Regime

We investigate the propagation of chemical fronts arising in Fisher--Kolmogorov--Petrovskii--Piskunov-type models in the presence of a steady cellular flow. In the long-time limit, a steadily propagating pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large Peclet number, $\text{Pe} \gg 1$) and arbitrary reaction rate (arbitrary Damkohler number $\text{Da}$). We identify three regimes corresponding to the distinguished limits $\text{Da} = O(\text{Pe}^{-1})$, $\text{Da}=O((\log \text{Pe})^{-1})$, and $\text{Da} = O(\text{Pe})$ and, in each regime, obtain the front speed in terms of a different nontrivial function of the relevant combination of $\text{Pe}$ and $\text{Da}$. Closed-form expressions for the speed, characterized by power-law and logarithmic dependences on $\text{Da}$ and $\text{Pe}$ an...

Exact Correlation Functions inSU(2)N=2Superconformal QCD

We report an exact solution of 2- and 3-point functions of chiral primary fields in SU(2) N=2 super-Yang-Mills theory coupled to four hypermultiplets. It is shown that these correlation functions are non-trivial functions of the gauge coupling, obeying differential equations which take the form of the semi-infinite Toda chain. We solve these equations recursively in terms of the Zamolodchikov metric that can be determined exactly from supersymmetric localization on the four-sphere. Our results are verified independently in perturbation theory with a Feynman diagram computation up to 2-loops. This is a short version of a companion paper that contains detailed technical remarks, additional material and aspects of an extension to SU(N) gauge group.

Interevent time distributions of human multi-level activity in a virtual world

Studying human behavior in virtual environments provides extraordinary opportunities for a quantitative analysis of social phenomena with levels of accuracy that approach those of the natural sciences. In this paper we use records of player activities in the massive multiplayer online game Pardus over 1238 consecutive days, and analyze dynamical features of sequences of actions of players. We build on previous work where temporal structures of human actions of the same type were quantified, and provide an empirical understanding of human actions of different types. This study of multi-level human activity can be seen as a dynamic counterpart of static multiplex network analysis. We show that the interevent time distributions of actions in the Pardus universe follow highly non-trivial distribution functions, from which we extract action-type specific characteristic “decay constants”. We discuss characteristic features of interevent time distributions, including periodic patterns on different time scales, bursty dynamics, and various functional forms on different time scales. We comment on gender differences of players in emotional actions, and find that while males and females act similarly when performing some positive actions, females are slightly faster for negative actions. We also observe effects on the age of players: more experienced players are generally faster in making decisions about engaging in and terminating enmity and friendship, respectively.

Weighted integrability of polyharmonic functions

To address the uniqueness issues associated with the Dirichlet problem for the N-harmonic equation on the unit disk D in the plane, we investigate the Lp integrability of N-harmonic functions with respect to the standard weights (1−|z|2)α. The question at hand is the following. If u solves ΔNu=0 in D, where Δ stands for the Laplacian, and∫D|u(z)|p(1−|z|2)αdA(z)<+∞, must then u(z)≡0? Here, N is a positive integer, α is real, and 0<p<+∞; dA is the usual area element. The answer will, generally speaking, depend on the triple (N,p,α). The most interesting case is 0<p<1. For a given N, we find an explicit critical curve p↦β(N,p) – a piecewise affine function – such that for α>β(N,p) there exist nontrivial functions u with ΔNu=0 of the given integrability, while for α≤β(N,p), only u(z)≡0 is possible. We also investigate the obstruction to uniqueness for the Dirichlet problem, that is, we study the structure of the functions in PHN,αp(D) when this space is nontrivial. We find a new structural decomposition of the polyharmonic functions – the cellular decomposition – which decomposes the polyharmonic weighted Lp space in a canonical fashion. Corresponding to the cellular expansion is a tiling of part of the (p,α) plane into cells.The above uniqueness for the Dirichlet problem may be considered for any elliptic operator of order 2N. However, the above-mentioned critical integrability curve will depend rather strongly on the given elliptic operator, even in the constant coefficient case, for N>1.

Simple Belief Elicitation

We present methods of belief elicitation which are applicable for any non-trivial utility function. Unlike existing techniques that account for deviations from risk-neutrality, these methods are highly transparent to subjects. Rather than identifying beliefs exactly we identify bounds on beliefs, thus trading off precision for generality and simplicity.

DNA walker circuits: computational potential, design, and verification

Unlike their traditional, silicon counterparts, DNA computers have natural interfaces with both chemical and biological systems. These can be used for a number of applications, including the precise arrangement of matter at the nanoscale and the creation of smart biosensors. Like silicon circuits, DNA strand displacement systems (DSD) can evaluate non-trivial functions. However, these systems can be slow and are susceptible to errors. It has been suggested that localised hybridization reactions could overcome some of these challenges. Localised reactions occur in DNA ‘walker’ systems which were recently shown to be capable of navigating a programmable track tethered to an origami tile. We investigate the computational potential of these systems for evaluating Boolean functions and forming composable circuits. We find that systems of multiple walkers have severely limited potential for parallel circuit evaluation. DNA walkers, like DSDs, are also susceptible to errors. We develop a discrete stochastic model of DNA walker ‘circuits’ based on experimental data, and demonstrate the merit of using probabilistic model checking techniques to analyse their reliability, performance and correctness. This analysis aids in the design of reliable and efficient DNA walker circuits.

Minimal functions on the random graph

We show that there is a system of 14 non-trivial finitary functions on the random graph with the following properties: Any non-trivial function on the random graph generates one of the functions of this system by means of composition with automorphisms and by topological closure, and the system is minimal in the sense that no subset of the system has the same property. The theorem is obtained by proving a Ramsey-type theorem for colorings of tuples in finite powers of the random graph, and by applying this to find regular patterns in the behavior of any function on the random graph. As model-theoretic corollaries of our methods we rederive a theorem of Simon Thomas classifying the first-order closed reducts of the random graph, and prove some refinements of this theorem; also, we obtain a classification of the maximal reducts closed under primitive positive definitions, and prove that all reducts of the random graph are model-complete.

Higher spin entanglement entropy from CFT

We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential $\mu$ for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in $\mu$ in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, R) x SL(3, R) Chern-Simons theory. Our result suggests that the order $\mu^2$ correction to the entanglement entropy may be universal for W-algebra CFTs with spin-three chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS_3.

Lepton mixing parameters from Δ(48) family symmetry and generalised CP

We provide a systematic and thorough exploration of the Δ(48) family symmetry and the consistent generalised CP symmetry. A model-independent analysis of the achievable lepton flavor mixing is performed by considering all the possible remnant symmetries in the neutrino and the charged lepton sectors. We find a new interesting mixing pattern in which both lepton mixing angles and CP phases are nontrivial functions of a single parameter θ. The value of θ can be fixed by the measured reactor mixing angle θ 13, and the excellent agreement with the present data can be achieved. A supersymmetric model based on Δ(48) family symmetry and generalised CP symmetry is constructed, and this new mixing pattern is exactly reproduced.

Confining boundary conditions from dynamical coupling constants

It is shown that it is possible to consistently and gauge invariantly formulate models where the coupling constant is a non-trivial function of a scalar field. In the U(1) case, the coupling to the gauge field contains a term of the form g(ϕ)jμ(Aμ+∂μB) where B is an auxiliary field and jμ is the Dirac current. The scalar field ϕ determines the local value of the coupling of the gauge field to the Dirac particle. The consistency of the equations determines the condition ∂μϕjμ=0 which implies that the Dirac current cannot have a component in the direction of the gradient of the scalar field. As a consequence, if ϕ has a soliton behaviour, like defining a bubble that connects two vacua, we obtain that the Dirac current cannot have a flux through the wall of the bubble, defining a confinement mechanism where the fermions are kept inside those bags. Consistent models with time dependent fine structure constant can be also constructed

Optimally weighted L2 distance for functional data

Many techniques of functional data analysis require choosing a measure of distance between functions, with the most common choice being L2 distance. In this article we show that using a weighted L2 distance, with a judiciously chosen weight function, can improve the performance of various statistical methods for functional data, including k-medoids clustering, nonparametric classification, and permutation testing. Assuming a quadratically penalized (e.g., spline) basis representation for the functional data, we consider three nontrivial weight functions: design density weights, inverse-variance weights, and a new weight function that minimizes the coefficient of variation of the resulting squared distance by means of an efficient iterative procedure. The benefits of weighting, in particular with the proposed weight function, are demonstrated both in simulation studies and in applications to the Berkeley growth data and a functional magnetic resonance imaging data set.

Zero-visibility cops and robber and the pathwidth of a graph

We examine the zero-visibility cops and robber graph searching model, which differs from the classical cops and robber game in one way: the robber is invisible. We show that this model is not monotonic. We show that the zero-visibility copnumber of a graph is bounded above by its pathwidth and cannot be bounded below by any nontrivial function of the pathwidth. As well, we define a monotonic version of this game and show that the monotonic zero-visibility copnumber can be bounded both above and below by positive multiples of the pathwidth.

Modified bosonic gas trapped in a generic 3-dim power law potential

We analyze the consequences caused by an anomalous single-particle dispersion relation suggested in several quantum-gravity models, upon the thermodynamics of a Bose–Einstein condensate trapped in a generic 3-dimensional power-law potential. We prove that the condensation temperature is shifted as a consequence of such deformation and show that this fact could be used to provide bounds on the deformation parameters. Additionally, we show that the shift in the condensation temperature, described as a non-trivial function of the number of particles and the trap parameters, could be used as a criterion to analyze the effects caused by a deformed dispersion relation in weakly interacting systems and also in finite size systems.

On the vanishing rate of smooth CR functions

Let M M be a lineally convex hypersurface of C n \mathbb C^n of finite type, 0 ∈ M 0\in M . Then there exist non-trivial smooth CR functions on M M that are flat at 0 0 , i.e. whose Taylor expansion about 0 0 vanishes identically. Our aim is to characterize the rate at which flat CR functions can decrease without vanishing identically. As it turns out, non-trivial CR functions cannot decay arbitrarily fast, and a possible way of expressing the critical rate is by comparison with a suitable exponential of the modulus of a local peak function.