Range Of Exponents Research Articles

Genetic diversity reduces competition and increases tree growth on a Norway spruce (Picea abies [L.] Karst.) provenance mixing experiment

Many recent studies address that diversification of tree species and stand structure can increase the level and stability of growth and other forest functions and services to mitigate natural and human disturbances. Most studies so far focussed on the diversification of tree species mixing and stand structure. The potential of intra-specific genetic diversification in terms of provenance mixture was hardly explored so far.Here we analyse how a mixing of species provenances of Norway spruce (Picea abies [L.] Karst.) affects the competition between neighboring trees and their growth. We based our study on the 40-years-old provenance-mixing experiment Vohenstrauß 622 in the Upper Palatinate Forest in South Germany. Here 21 different provenances from clonal propagation of Norway spruce across Germany were combined in individual tree mixture, replicated in different spacing, and thinning. Our main objective was to analyze how the individual tree competition, tree growth, and stand productivity were affected by neighboring trees of different provenances.First we show that at an tree age of 43 years the different provenances strongly vary between 10 and 27% in their crown diameter, cd, stem diameter, d, crown length, cl, and tree height, h, with a ranking cvcd > cvd > cvcl > cvh in the coefficient of variation, cv. We found significant differences of the stem and crown allometry between the provenances; e.g. the range of the allometric exponents of h∝dα was α = 0.12–0.58 and of cd∝dα α = 0.47–0.84.Second we found that different provenances in the neighborhood reduced the inter-individual competition and increased the stem diameter growth. Having 10 different provenances in the neighborhood increases the diameter growth of a tree on average by 28 % compared with having only 5 provenances in the neighborhood. Stem diameter followed a saturation curve with increasing number of provenances.Third we showed that an increasing number of provenances also increased the productivity at the stand level. The effects on stand growth are not only determined by the number of different provenances per area but also by their spatial mixing pattern.The inter-provenance complementarity of tree allometry is discussed as main cause of the revealed reduction of competition and increase of growth. We further discuss how spatially explicit individual tree models may incorporate tree structure as proxy of provenance characteristics. We conclude that genetic diversification in terms of provenance mixture can increase and stabilize tree and stand growth analogously to tree species mixing and may be further explored and considered as seminal silvicultural option.

지수의 범위를 정수에서 유리수로 확장할 때 교과서 기술에 나타나는 논리적 비약에 대한 분석

In this study, the patterns of description adopted in high school Mathematics I textbooks in accordance with the 2015-revised curriculum while expanding the range of the exponent by applying the principle of the permanence of equivalent forms are classified into two categories; hypothetical (analytic) description and synthetic description. While expanding the range of exponents from integers to rational numbers, the logical leap that intervenes in description of the process of discovering the definition of rational exponent in the textbooks was analyzed. At first, the logical leap was identified when I replied to the question raised by a high school mathematics teacher through e-mail. From my analysis, it was found that it originate from the necessity of a textbook to describe the content as efficiently and concisely as possible. The textbooks describe the way to arrive at the definition of the rational exponent by reorganizing it rather than presenting it step by step along the discovery process, which is found to weaken heuristic power of the analysis and synthesis and hinder the teacher’s understanding of the process of discovering the definition. In this respect, it is necessary to revise the relevant part of the current high school Mathematics I textbooks and supplement the precise and detailed explanation in the teachers’ guide books.

Small cap decoupling inequalities: Bilinear methods

We obtain sharp small cap decoupling inequalities associated to the moment curve for certain range of exponents p . Our method is based on the bi-linearization argument due to Bourgain and Demeter. Our result generalizes theirs to all higher dimensions.

Fourier restriction in low fractal dimensions

AbstractLet$S \subset \mathbb {R}^{n}$be a smooth compact hypersurface with a strictly positive second fundamental form,$E$be the Fourier extension operator on$S$, and$X$be a Lebesgue measurable subset of$\mathbb {R}^{n}$. If$X$contains a ball of each radius, then the problem of determining the range of exponents$(p,q)$for which the estimate$\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set$X$: there is a number$0 < \alpha \leq n$such that$|X \cap B_R| \lesssim R^{\alpha }$for all balls$B_R$in$\mathbb {R}^{n}$of radius$R \geq 1$. On the left-hand side of this estimate, we are integrating the function$|Ef(x)|^{q}$against the measure$\chi _X \,{\textrm {d}}x$. Our approach consists of replacing the characteristic function$\chi _X$of$X$by an appropriate weight function$H$, and studying the resulting estimate in three different regimes: small values of$\alpha$, intermediate values of$\alpha$, and large values of$\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on$\mathbb {R}^{n}$and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range$0 < \alpha < n/2$.

On equilibrium Metropolis simulations on self-organized urban street networks

Urban street networks of unplanned or self-organized cities typically exhibit astonishing scale-free patterns. This scale-freeness can be shown, within the maximum entropy formalism (MaxEnt), as the manifestation of a fluctuating system that preserves on average some amount of information. Monte Carlo methods that can further this perspective are cruelly missing. Here we adapt to self-organized urban street networks the Metropolis algorithm. The “coming to equilibrium” distribution is established with MaxEnt by taking scale-freeness as prior hypothesis along with symmetry-conservation arguments. The equilibrium parameter is the scaling; its concomitant extensive quantity is, assuming our lack of knowledge, an amount of information. To design an ergodic dynamics, we disentangle the state-of-the-art street generating paradigms based on non-overlapping walks into layout-at-junction dynamics. Our adaptation reminisces the single-spin-flip Metropolis algorithm for Ising models. We thus expect Metropolis simulations to reveal that self-organized urban street networks, besides sustaining scale-freeness over a wide range of scalings, undergo a crossover as scaling varies—literature argues for a small-world crossover. Simulations for Central London are consistent against the state-of-the-art outputs over a realistic range of scaling exponents. Our illustrative Watts–Strogatz phase diagram with scaling as rewiring parameter demonstrates a small-world crossover curving within the realistic window 2–3; it also shows that the state-of-the-art outputs underlie relatively large worlds. Our Metropolis adaptation to self-organized urban street networks thusly appears as a scaling variant of the Watts–Strogatz model. Such insights may ultimately allow the urban profession to anticipate self-organization or unplanned evolution of urban street networks.

Isodiametry, Variance, and Regular Simplices from Particle Interactions

Consider a collection of particles interacting through an attractive-repulsive potential given as a difference of power laws and normalized so that its unique minimum occurs at unit separation. For a range of exponents corresponding to mild repulsion and strong attraction, we show that the minimum energy configuration is uniquely attained—apart from translations and rotations—by equidistributing the particles over the vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in two dimensions and regular tetrahedron in three). If the attraction is not assumed to be strong, we show that these configurations are at least local energy minimizers in the relevant $$d_\infty $$ metric from optimal transportation, as are all of the other uncountably many unbalanced configurations with the same support. We infer the existence of phase transitions. The proof is based in part on a simple isodiametric variance bound which characterizes regular simplices; it shows that among probability measures on $${{\mathbf {R}}}^n$$ whose supports have at most unit diameter, the variance around the mean is maximized precisely by those measures which assign mass $$1/(n+1)$$ to each vertex of a (unit-diameter) regular simplex.

An anisotropic regularity condition for the 3D incompressible Navier–Stokes equations for the entire exponent range

We show that a suitable weak solution to the incompressible Navier–Stokes equations on R3×(−1,1) is regular on R3×(−1,0] if ∂3u belongs to M2p/(2p−3),α((−1,0);Lp(R3)) for any α>1 and p∈(3/2,∞), which is a logarithmic-type variation of a Morrey space in time. For each α>1 this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces Lq((−1,0);Lp(R3)) that are subcritical, that is for which 2/q+3/p<2.

The stochastic Strichartz estimates and stochastic nonlinear Schrödinger equations driven by Lévy noise

We establish a new version of the stochastic Strichartz estimate for the stochastic convolution driven by jump noise which we apply to the stochastic nonlinear Schrödinger equation with nonlinear multiplicative jump noise in the Marcus canonical form. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to stochastic nonlinear Schrödinger equation in L2(Rn) with either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.

Economic Transition and the Evolution of City-Size Distribution of China’s Urban System

The evolution of city size distribution in China has gained a great deal of scholarly attention. However, little is known about the effect of economic transition on the reorganization of city size distribution in China. Using an urban hierarchy with cascade structure model, we decompose Zipf’s law into two exponential functions that provide a new way of examining the dynamic processes of urban system evolution. This study aims to investigate the dominating latent forces that affect China’s city size distribution through mathematical modeling of the hierarchical scaling laws based on census data of 1982, 1990, 2000, and 2010. A number of features of China’s city size distribution are found. First, the size distribution of Chinese cities displayed a clear trend of evolving toward the Zipf distribution, which is the result of economic transition from planned to market. Second, the rank-size pattern still deviates slightly from the standard Zipf distribution, as indicated by the narrow scaling range and departure of the scaling exponent from the theoretically expected value. We argue that the top-down state regulation is a critical cause of deviation of China’s city size distribution from Zipf’s law.

Dissecting localization phenomena of dynamical processes on networks

Localization phenomena permeate many branches of physics playing a fundamental role on dynamical processes evolving on heterogeneous networks. These localization analyses are frequently grounded, for example, on eigenvectors of adjacency or non-backtracking matrices which emerge in theories of dynamic processes near to an active to inactive phase transition. We advance in this problem gauging nodal activity to quantify the localization in dynamical processes on networks whether they are near to a transition or not. The method is generic and applicable to theory, stochastic simulations, and real data. We investigate spreading processes on a wide spectrum of networks, both analytically and numerically, showing that nodal activity can present complex patterns depending on the network structure. Using annealed networks we show that a localized state at the transition and an endemic phase just above it are not incompatible features of a spreading process. We also report that epidemic prevalence near to the transition is determined by the delocalized component of the network even when the analysis of the inverse participation ratio indicates a localized activity. Also, dynamical processes with distinct critical exponents can be described by the same localization pattern. Turning to quenched networks, a more complex picture, depending on the type of activation and on the range of degree exponent, is observed and discussed. Our work paves an important path for investigation of localized activity in spreading and other processes on networks.

Enhanced robustness of single-layer networks with redundant dependencies.

Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak variant of this model: Nodes with dependency neighbors may continue to function if at least one of their dependency neighbors is active. We show that this relaxation of the dependency rule allows for more robust structures and a rich variety of critical phenomena, as percolation is not determined strictly by finite dependency clusters. We study Erdős-Rényi and random scale-free networks with an underlying Erdős-Rényi network of dependency links. We identify a special "cusp" point above which the system is always stable, irrespective of the density of dependency links. We find continuous and discontinuous hybrid percolation transitions, separated by a tricritical point for Erdős-Rényi networks. For scale-free networks with a finite degree cutoff we observe the appearance of a critical point and corresponding double transitions in a certain range of the degree distribution exponent. We show that at a special point in the parameter space, where the critical point emerges, the giant viable cluster has the unusual critical singularity S-S_{c}∝(p-p_{c})^{1/4}. We study the robustness of networks where connectivity degrees and dependency degrees are correlated and find that scale-free networks are able to retain their high resilience for strong enough positive correlation, i.e., when hubs are protected by greater redundancy.

The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space

We study a natural question that, apparently, has not been well addressed in the literature. Given functions u with support in the unit ball B1⊂Rn and with gradient in the Morrey space Mp,λ(B1), where 1<p<λ<n, what is the largest range of exponents q for which necessarily u∈Lq(B1)? While David R. Adams proved in 1975 that this embedding holds for q≤λp/(λ−p), an article from 2011 claimed the embedding in the larger range q<np/(λ−p). Here we disprove this last statement by constructing a function that provides a counterexample for q>λp/(λ−p). The function is basically a negative power of the distance to a set of Hausdorff dimension n−λ. When λ∉Z, this set is a fractal. We also make a detailed study of the radially symmetric case, a situation in which the exponent q can go up to np/(λ−p).

Coppersmith's lattices and “focus groups”: An attack on small-exponent RSA

We present a principled technique for reducing the lattice and matrix size in some applications of Coppersmith's lattice method for finding roots of modular polynomial equations. It relies on extrapolating patterns from the actual behavior of Coppersmith's attack for smaller parameter sizes, which can be thought of as “focus group” testing. When applied to the small-exponent RSA problem, our technique reduces lattice dimensions and consequently running times, and hence can be applied to a wider range of exponents. Moreover, in many difficult examples our attack is not only faster but also more successful in recovering the RSA secret key. We include a discussion of subtleties concerning whether or not existing metrics (such as enabling condition bounds) are decisive in predicting the true efficacy of attacks based on Coppersmith's method. Finally, indications are given which suggest certain lattice basis reduction algorithms (such as Nguyen-Stehlé's L2) may be particularly well-suited for Coppersmith's method.

A Method for Fans’ Potential Malfunction Detection of ONAF Transformer Using Top-Oil Temperature Monitoring

The fan is one of the key components of the power transformer cooling system. The operating condition of fans determines transformers’ internal temperature rise and long-term reliability. However, at present, the fans’ condition monitoring only includes switch status (online) and regular maintenance (offline), online direct monitoring of the fans’ operating condition is lacking due to economic costs. In view of the above-mentioned problem, this paper proposes a transformer fan early fault detection method based on the oil exponent, which is monitored by the existing transformer top-oil temperature data, thereby detecting the abnormality of the fans. In this method, the oil exponent was chosen as the characteristic criterion. First, to obtain the range of oil exponent in different cooling modes, a set of physical models describing global oil flow and its interaction with air was established based on fluid dynamics and heat transfer principle. Then, regarding the constantly changing top-oil temperature, ambient temperature and load current, an oil exponent tracking algorithm using particle swarm optimization (PSO) was proposed within an improved IEC dynamic thermal model. The operation data from an oil-immersed transformer with a rated capacity of 120-MVA and rated voltage of 220-kV was selected to verify the above methods under two different scenarios.

On the existence and stability of a nonlinear wave system with variable exponents

Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two “smart” materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state ( 0 , 0 ) either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings.

Global Bounded Weak Entropy Solutions to the Euler--Vlasov Equations in Fluid-Particle System

In this paper, we consider a fluid-particle system which describes the evolution of a two-phase flow. The system consists of the compressible Euler equations for the fluid (fluid phase) coupled with the Vlasov equation for the particles (disperse phase) through the drag force. We obtain a global bounded weak entropy solution for such one-dimensional Euler--Vlasov equations with arbitrarily large initial data for the whole range of physical adiabatic exponents $\gamma>1$. To achieve this, we apply the vanishing viscosity method and compensated compactness theory. We construct globally defined approximate solutions by adding our novel viscosity terms to the Euler equations, which together with our key observation on relative velocity plays a fundamental role in the hardest part of our proof: the uniform $L^\infty$ estimate. After deriving an entropy dissipation estimate, we prove the convergence of approximate solutions by the compensated compactness argument.

On the Tuning of Fractional Order Resonant Controllers for a Voltage Source Converter in a Weak AC Grid Context

This paper proposes a method for tuning the fractional exponent of different types of fractional order resonant controllers for a voltage source converter in a weak AC grid context. The main objective is to ensure the stability of the controlled system in a weak AC grid environment and to achieve an adequate dynamic response under disturbances. Therefore, six commonly used integer order proportional resonant (PR) control structures are selected from the literature and compared with each other according to their frequency behaviour. Afterwards, a rational approximation for the fractional order term is selected based on continuous fraction expansion technique. The inclusion of a fractional exponent in each integer order PR structure generates the fractional order proportional resonant (FPR) control transfer functions. Once the FPR controllers have been obtained, their closed-loop responses are tested via eigenvalue trajectory analysis. For each FPR control structure, a range of the fractional exponent that ensures stability is obtained. The conclusions of eigenvalue trajectory analysis are tested by implementing the FPR control structures in an specific application consisting in a modular multi-level converter (MMC) connected to a weak AC grid with adjustable short-circuit ratio. By means of time-domain simulations, not only the previous eigenvalue analyses are validated, but also new tuning criteria are given for the fractional exponent in combination with other control parameters, such as the damping frequency and the inductance of the complementary feedback branch. Moreover, a sensitivity analysis of the tuning criteria is carried out for other sizes of the AC filter inductance.

On families between the Hardy–Littlewood and spherical maximal functions

We study a family of maximal operators that provides a continuous link connecting the Hardy-Littlewood maximal function to the spherical maximal function. Our theorems are proved in the multilinear setting but may contain new results even in the linear case. For this family of operators we obtain bounds between Lebesgue spaces in the optimal range of exponents.

Revisiting the relations between Hurst exponent and fractional differencing parameter for long memory

This study aims to verify the efficiency of the estimating methods of long memory process based on the linear relationship between the Hurst exponent (H) and the fractional differencing parameter (d), which are the two approaches used to identify the long memory process. By using the Monte Carlo simulation and empirical examinations, the results show that there is a distinct linear relationship between the Hurst exponent and the fractional differencing parameter. This linear relationship may provide an efficient estimation of the range of Hurst exponent and fractional differencing parameter with each other when the stable index of the stable distribution is close to 2. However, according to the findings, the detailed form of linear equation is impacted by the fat tailed distribution, which becomes more volatile as the stable index of stable distribution is less than 2. Therefore, researchers should be cautious when using the linear relation to estimate H and d them by each other when the distribution has fat tail.

Critical exponentνof the Ising model in three dimensions with long-range correlated site disorder analyzed with Monte Carlo techniques

We study the critical behavior of the Ising model in three dimensions on a lattice with site disorder by using Monte Carlo simulations. The disorder is either uncorrelated or long-range correlated with correlation function that decays according to a power-law $r^{-a}$. We derive the critical exponent of the correlation length $\nu$ and the confluent correction exponent $\omega$ in dependence of $a$ by combining different concentrations of defects $0.05 \leq p_d \leq 0.4$ into one global fit ansatz and applying finite-size scaling techniques. We simulate and study a wide range of different correlation exponents $1.5 \leq a \leq 3.5$ as well as the uncorrelated case $a = \infty$ and are able to provide a global picture not yet known from previous works. Additionally, we perform a dedicated analysis of our long-range correlated disorder ensembles and provide estimates for the critical temperatures of the system in dependence of the correlation exponent $a$ and the concentrations of defects $p_d$. We compare our results to known results from other works and to the conjecture of Weinrib and Halperin: $\nu = 2/a$ and discuss the occurring deviations.