Finite Constant Research Articles

BPS boojums in ${\cal N}=2$ supersymmetric gauge theories I

We study 1/4 Bogomol'nyi-Prasad-Sommerfield (BPS) composite solitons of vortex strings, domain walls and boojums in ${\cal N}=2$ supersymmetric Abelian gauge theories in four dimensions. We obtain solutions to the 1/4 BPS equations with the finite gauge coupling constant. To obtain numerical solutions for generic coupling constants, we construct globally correct approximate functions which allow us to easily find fixed points of gradient flow equations. We analytically/numerically confirm that the negative mass of a single boojum appearingat the endpoint of the vortex string on the logarithmically bent domain wall is equal to the half-mass of the 't Hooft-Polyakov monopole. We examine various configurations and clarify how the shape of the boojum depends on the coupling constants and moduli parameters. We also find analytic solutions to the 1/4 BPS equations for specific values of the coupling constants.

Models of interacting pairs of thin, quasi-geostrophic vortices: steady-state solutions and nonlinear stability

We study pairwise interactions of elliptical quasi-geostrophic (QG) vortices as the limiting case of vanishingly thin uniform potential vorticity ellipsoids. In this limit, the product of the vertical extent of the ellipsoid and the potential vorticity within it is held fixed to a finite non-zero constant. Such elliptical “lenses” inherit the property that, in isolation, they steadily rotate without changing shape. Here, we use this property to extend both standard moment models and Hamiltonian ellipsoidal models to approximate the dynamical interaction of such elliptical lenses. By neglecting non-elliptical deformations, the simplified models reduce the dynamics to just four degrees of freedom per vortex. For simplicity, we focus on pairwise interactions between identical elliptical vortices initially separated in both the horizontal and vertical directions. The dynamics of the simplified models are compared with the full QG dynamics of the system, and show good agreement as expected for sufficiently distant lenses. The results reveal the existence of families of steadily rotating equilibria in the initial horizontal and vertical separation parameter space. For sufficiently large vertical separations, equilibria with varying shape exist for all horizontal separations. Below a critical vertical separation (stretched by the constant ratio of buoyancy to Coriolis frequencies N / f), comparable to the mean radius of either vortex, a gap opens in horizontal separation where no equilibria are possible. Solutions near the edge of this gap are unstable. In the full QG system, equilibria at the edge of the gap exhibit corners (infinite curvature) along their boundaries. Comparisons of the model results with the full nonlinear QG evolution show that the early stages of the instability are captured by the Hamiltonian elliptical model but not by the moment model that inaccurately estimates shorter-range interactions.

Resetting of fluctuating interfaces at power-law times

What happens when the time evolution of a fluctuating interface is interrupted by resetting to a given initial configuration after random time intervals τ distributed as a power-law ? For an interface of length L in one dimension, and an initial flat configuration, we show that depending on α, the dynamics in the limit exhibit a spectrum of rich long-time behavior. It is known that without resetting, the interface width grows unbounded with time as in this limit, where β is the so-called growth exponent characteristic of the universality class for a given interface dynamics. We show that introducing resetting leads to fluctuations that are bounded at large times for . Corresponding to such a reset-induced stationary state is a distribution of fluctuations that is strongly non-Gaussian, with tails decaying as a power-law. The distribution exhibits a distinctive cuspy behavior for a small argument, implying that the stationary state is out of equilibrium. For , on the contrary, resetting to the flat configuration is unable to counter the otherwise unbounded growth of fluctuations in time, so that the distribution of fluctuations remains time dependent with an ever-increasing width, even at long times. Although stationary for , the width of the interface grows forever with time as a power-law for , and converges to a finite constant only for larger α, thereby exhibiting a crossover at . The time-dependent distribution of fluctuations exhibits for and for small arguments further interesting crossover behavior from cusp to divergence across . We demonstrate these results by exact analytical results for the paradigmatic Edwards–Wilkinson (EW) dynamical evolution of the interface, and further corroborate our findings by extensive numerical simulations of interface models in the EW and Kardar–Parisi–Zhang universality class.

The chirally rotated Schrödinger functional: theoretical expectations and perturbative tests

AbstractThe chirally rotated Schrödinger functional (χSF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schrödinger functional (SF), with different lattice symmetries and a common continuum limit expected from universality. The explicit breaking of flavour and parity symmetries needs to be repaired by tuning the bare fermion mass and the coefficient of a dimension 3 boundary counterterm. Once this is achieved one expects the mechanism of automatic O(a) improvement to be operational in the χSF, in contrast to the standard formulation of the SF. This is expected to significantly improve the attainable precision for step-scaling functions of some composite operators. Furthermore, the χSF offers new strategies to determine finite renormalization constants which are traditionally obtained from chiral Ward identities. In this paper we consider a complete set of fermion bilinear operators, define corresponding correlation functions and explain the relation to their standard SF counterparts. We discuss renormalization and O(a) improvement and then use this set-up to formulate the theoretical expectations which follow from universality. Expanding the correlation functions to one-loop order of perturbation theory we then perform a number of non-trivial checks. In the process we obtain the action counterterm coefficients to one-loop order and reproduce some known perturbative results for renormalization constants of fermion bilinears. By confirming the theoretical expectations, this perturbative study lends further support to the soundness of the χSF framework and prepares the ground for non-perturbative applications.

Signal-to-noise ratio improvements in laser flow diagnostics using time-resolved image averaging and high dynamic range imaging

Two alternative image readout approaches are demonstrated to improve the signal-to-noise ratio (SNR) in temporally resolved laser-based imaging experiments of turbulent phenomena. The first method exploits the temporal decay characteristics of the phosphor screens of image intensifiers when coupled to an interline-transfer CCD camera operated in double-frame mode. Specifically, the light emitted by the phosphor screen, which has a finite decay constant, is equally distributed and recorded over the two sequential frames of the detector so that an averaged image can be reconstructed. The characterization of both detector and image intensifier showed that the technique preserves the correct quantitative information, and its applicability to reactive flows was verified using planar Rayleigh scattering and tested with the acquisition of images of both steady and turbulent partially premixed methane/air flames. The comparison between conventional Rayleigh results and the averaged ones showed that the SNR of the averaged image is higher than the conventional one; with the setup used in this work, the gain in SNR was seen to approach 30 %, for both the steady and turbulent cases. The second technique uses the two-frame readout of an interline-transfer CCD to increase the image SNR based on high dynamic range imaging, and it was tested in an unsteady non-reactive flow of Freon-12 injected in air. The result showed a 15 % increase in the SNR of the low-pixel-count regions of an image, when compared to the pixels of a conventionally averaged one.

A Cluster-Size Averaging Model for Strongly Discontinuous Percolation

We propose a network percolation model, called cut-off model, which exhibits strongly discontinuous transition by an averaging effect of cluster sizes. In this model, a randomly selected bond is added if the size of the cluster formed by the bond is less than a times of the mean cluster size. It is shown that the model is strongly discontinuous when a is a finite constant.

Remodeling by fibroblasts alters the rate-dependent mechanical properties of collagen

The ways that fibroblasts remodel their environment are central to wound healing, development of musculoskeletal tissues, and progression of pathologies such as fibrosis. However, the changes that fibroblasts make to the material around them and the mechanical consequences of these changes have proven difficult to quantify, especially in realistic, viscoelastic three-dimensional culture environments, leaving a critical need for quantitative data. Here, we observed the mechanisms and quantified the mechanical effects of fibroblast remodeling in engineered tissue constructs (ETCs) comprised of reconstituted rat tail (type I) collagen and human fibroblast cells. To study the effects of remodeling on tissue mechanics, stress-relaxation tests were performed on ETCs cultured for 24, 48, and 72h. ETCs were treated with deoxycholate and tested again to assess the ECM response. Viscoelastic relaxation spectra were obtained using the generalized Maxwell model. Cells exhibited viscoelastic damping at two finite time constants over which the ECM showed little damping, approximately 0.2s and 10–30s. Different finite time constants in the range of 1–7000s were attributed to ECM relaxation. Cells remodeled the ECM to produce a relaxation time constant on the order of 7000s, and to merge relaxation finite time constants in the 0.5–2s range into a single time content in the 1s range. Results shed light on hierarchical deformation mechanisms in tissues, and on pathologies related to collagen relaxation such as diastolic dysfunction. Statement of SignificanceAs fibroblasts proliferate within and remodel a tissue, they change the tissue mechanically. Quantifying these changes is critical for understanding wound healing and the development of pathologies such as cardiac fibrosis. Here, we characterize for the first time the spectrum of viscoelastic (rate-dependent) changes arising from the remodeling of reconstituted collagen by fibroblasts. The method also provides estimates of the viscoelastic spectra of fibroblasts within a three-dimensional culture environment. Results are of particular interest because of the ways that fibroblasts alter the mechanical response of collagen at loading frequencies associated with cardiac contraction in humans.

Charge transfer energies of benzene physisorbed on a graphene sheet from constrained density functional theory

Constrained density functional theory (CDFT) is used to evaluate the energy level alignment of a benzene molecule as it approaches a graphene sheet. Within CDFT the problem is conveniently mapped onto evaluating total energy differences between different charge-separated states, and it does not consist in determining a quasi-particle spectrum. We demonstrate that the simple local density approximation provides a good description of the level aligmnent along the entire binding curve, with excellent agreement to experiments at an infinite separation and to GW calculations close to the bonding distance. The method also allows us to explore the effects due to the presence of graphene structural defects and of multiple molecules. In general all our results can be reproduced by a classical image charge model taking into account the finite dielectric constant of graphene.

The Normality of Meromorphic Functions with Multiple Zeros and Poles Concerning Sharing Values

In this paper we study the problem of normal families of meromorphic functions concerning shared values. Let F be a family of meromorphic functions in the plane domain and n, k be two positive integers such that , and let a, b be two finite complex constants such that . Suppose that (1) and share b in D for every pair of functions f, ; (2) All zeros of f have multiplicity at least k + 2 and all poles of f have multiplicity at least 2 for each in D; (3) Zeros of are not the b points of f(z) for each in D. Then F is normal in D. And some examples are provided to show the result is sharp.

Inference after truncated one-sided sequential test

ABSTRACTAssume i.i.d. random variables {X1, …, Xn, …} follow the standard exponential family {dFθ(x) = exp(θx − c(θ))dF0(x)}. For the one-sided hypothesis test H0: θ = θ0 < 0 and Ha: θ ⩾ θ1 > 0 where c(θ0) = c(θ1), the truncated sequential probability ratio test stops at min (τ, T) where , and H0 is rejected if τ < T. Inference problems based on asymptotic pivots are considered given τ < T by assuming T/b → finite constant larger than 1/c′(θ1).

Microscopic identification of dissipative modes in relativistic field theories

We present an argument to support the existence of dissipative modes in relativistic field theories. In an O(N) $\varphi^4$ theory in spatial dimension $d\le 3$, a relaxation constant $\Gamma$ of a two-point function in an infrared region is shown to be finite within the two-particle irreducible (2PI) framework at the next-leading order (NLO) of 1/N expansion. This immediately implies that a slow dissipative mode with a dispersion $p_0\sim i\Gamma \p^2$ is microscopically identified in the two-point function. Contrary, NLO calculation in the one-particle irreducible (1PI) framework fails to yield a finite relaxation constant. Comparing the results in 1PI and 2PI frameworks, one concludes that dissipation emerges from multiple scattering of a particle with a heat bath, which is appropriately treated in the 2PI-NLO calculation through the resummation of secular terms to improve long-time behavior of the two-point function. Assuming that this slow dissipative mode survives at the critical point, one can identify the dynamic critical exponent $z$ for the two-point function as $z=2-\eta$. We also discuss possible improvement of the result.

Coherent quantum transport in disordered systems: A unified polaron treatment of hopping and band-like transport.

Quantum transport in disordered systems is studied using a polaron-based master equation. The polaron approach is capable of bridging the results from the coherent band-like transport regime governed by the Redfield equation to incoherent hopping transport in the classical regime. A non-monotonic dependence of the diffusion coefficient is observed both as a function of temperature and system-phonon coupling strength. In the band-like transport regime, the diffusion coefficient is shown to be linearly proportional to the system-phonon coupling strength and vanishes at zero coupling due to Anderson localization. In the opposite classical hopping regime, we correctly recover the dynamics described by the Fermi's Golden Rule and establish that the scaling of the diffusion coefficient depends on the phonon bath relaxation time. In both the hopping and band-like transport regimes, it is demonstrated that at low temperature, the zero-point fluctuations of the bath lead to non-zero transport rates and hence a finite diffusion constant. Application to rubrene and other organic semiconductor materials shows a good agreement with experimental mobility data.

Singular Poisson equations on Finsler–Hadamard manifolds

In the first part of the paper we study the reflexivity of Sobolev spaces on non-compact and not necessarily reversible Finsler manifolds. Then, by using direct methods in the calculus of variations, we establish uniqueness, location and rigidity results for singular Poisson equations involving the Finsler-Laplace operator on Finsler-Hadamard manifolds having finite reversibility constant.

A new coupled reduced alternating group explicit method for nonlinear singular two-point boundary value problems on a variable mesh

In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve the nonlinear singular two-point boundary value problems u″ = f(r, u, u′), 0 < r < 1 subject to given natural boundary conditions u(0) = A1, u(1) = A2 where A1 and A2 are finite constants, along with a third-order numerical method on a geometric mesh. The proposed method is applicable to singular and nonsingular problems. We have discussed the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two-parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.

Exact differentiable penalty for a problem of quadratic programming with the use of a gradient-projective method

AbstractIt is known that the solution obtained for a problem of quadratic programming (QP) is not exact in the case of reduction of the original problem to a sequence of problems of unconditional minimization by a method of penalty functions with a differentiable penalty. In this paper we obtain an exact solution to a QP problem for a finite penalty constant and a smooth penalty with the use of a gradient-projective method.

Testing Weak Cross-Sectional Dependence in Large Panels

This article considers testing the hypothesis that errors in a panel data model are weakly cross-sectionally dependent, using the exponent of cross-sectional dependence α, introduced recently in Bailey, Kapetanios, and Pesaran (2012). It is shown that the implicit null of the cross-sectional dependence (CD) test depends on the relative expansion rates of N and T. When T = O(N ε), for some 0 < ε ≤1, then the implicit null of the CD test is given by 0 ≤ α < (2 − ε)/4, which gives 0 ≤ α <1/4, when N and T tend to infinity at the same rate such that T/N → κ, with κ being a finite positive constant. It is argued that in the case of large N panels, the null of weak dependence is more appropriate than the null of independence which could be quite restrictive for large panels. Using Monte Carlo experiments, it is shown that the CD test has the correct size for values of α in the range [0, 1/4], for all combinations of N and T, and irrespective of whether the panel contains lagged values of the dependent variables, so long as there are no major asymmetries in the error distribution.

Dynamics of self-sustained asynchronous-irregular activity in random networks of spiking neurons with strong synapses.

Random networks of integrate-and-fire neurons with strong current-based synapses can, unlike previously believed, assume stable states of sustained asynchronous and irregular firing, even without external random background or pacemaker neurons. We analyze the mechanisms underlying the emergence, lifetime and irregularity of such self-sustained activity states. We first demonstrate how the competition between the mean and the variance of the synaptic input leads to a non-monotonic firing-rate transfer in the network. Thus, by increasing the synaptic coupling strength, the system can become bistable: In addition to the quiescent state, a second stable fixed-point at moderate firing rates can emerge by a saddle-node bifurcation. Inherently generated fluctuations of the population firing rate around this non-trivial fixed-point can trigger transitions into the quiescent state. Hence, the trade-off between the magnitude of the population-rate fluctuations and the size of the basin of attraction of the non-trivial rate fixed-point determines the onset and the lifetime of self-sustained activity states. During self-sustained activity, individual neuronal activity is moreover highly irregular, switching between long periods of low firing rate to short burst-like states. We show that this is an effect of the strong synaptic weights and the finite time constant of synaptic and neuronal integration, and can actually serve to stabilize the self-sustained state.

Bounded gaps between primes with a given primitive root

Fix an integer $g \neq -1$ that is not a perfect square. In 1927, Artin conjectured that there are infinitely many primes for which $g$ is a primitive root. Forty years later, Hooley showed that Artin's conjecture follows from the Generalized Riemann Hypothesis (GRH). We inject Hooley's analysis into the Maynard--Tao work on bounded gaps between primes. This leads to the following GRH-conditional result: Fix an integer $m \geq 2$. If $q_1 < q_2 < q_3 < \dots$ is the sequence of primes possessing $g$ as a primitive root, then $\liminf_{n\to\infty} (q_{n+(m-1)}-q_n) \leq C_m$, where $C_m$ is a finite constant that depends on $m$ but not on $g$. We also show that the primes $q_n, q_{n+1}, \dots, q_{n+m-1}$ in this result may be taken to be consecutive.

Queuing with future information

We study an admissions control problem, where a queue with service rate $1-p$ receives incoming jobs at rate $\lambda\in(1-p,1)$, and the decision maker is allowed to redirect away jobs up to a rate of $p$, with the objective of minimizing the time-average queue length. We show that the amount of information about the future has a significant impact on system performance, in the heavy-traffic regime. When the future is unknown, the optimal average queue length diverges at rate $\sim\log_{1/(1-p)}\frac{1}{1-\lambda}$, as $\lambda\to 1$. In sharp contrast, when all future arrival and service times are revealed beforehand, the optimal average queue length converges to a finite constant, $(1-p)/p$, as $\lambda\to1$. We further show that the finite limit of $(1-p)/p$ can be achieved using only a finite lookahead window starting from the current time frame, whose length scales as $\mathcal{O}(\log\frac{1}{1-\lambda})$, as $\lambda\to1$. This leads to the conjecture of an interesting duality between queuing delay and the amount of information about the future.

Symplectic capacities on surfaces

We classify capacities on the class Symo(2) of connected symplectic surfaces with at most countably many nonplanar ends. To obtain the classification we study diffeomorphism types of surfaces in Symo(2) of infinite genus with nonplanar ends; it turns out that these types are in bijective correspondence with countable successor ordinals of the form ωα · d + 1, where α is an ordinal and d ≥ 0 is an integer. It also turns out that if S1 and S2 are two open surfaces of infinite genera with at most countably many nonplanar ends, then each of the surfaces embeds into the other. Our classification implies that every capacity on the class of symplectic surfaces in Symo(2) of infinite genus differs from the Hofer–Zehnder capacity by a non-negative finite or infinite constant.