Logarithmic Exponent Research Articles

Graph distances in scale-free percolation: the logarithmic case

AbstractScale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on independent and identically distributed vertex weights and the Euclidean distance $|x-y|$ . Depending on the various parameters involved, we get a rich phase diagram. We study graph distance and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.

Logarithmic finite-size scaling of the self-avoiding walk at four dimensions.

The n-vector spin model, which includes the self-avoiding walk (SAW) as a special case for the n→0 limit, has an upper critical dimensionality at four spatial dimensions (4D). We simulate the SAW on 4D hypercubic lattices with periodic boundary conditions by an irreversible Berretti-Sokal algorithm up to linear size L=768. From an unwrapped end-to-end distance, we obtain the critical fugacity as z_{c}=0.147622380(2), improving over the existing result z_{c}=0.1476223(1) by 50 times. Such a precisely estimated critical point enables us to perform a systematic study of the finite-size scaling of 4D SAW for various quantities. Our data indicate that near z_{c}, the scaling behavior of the free energy simultaneously contains a scaling term from the Gaussian fixed point and the other accounting for multiplicative logarithmic corrections. In particular, it is clearly observed that the critical magnetic susceptibility and the specific heat logarithmically diverge as χ∼L^{2}(lnL)^{2y[over ̂]_{h}} and C∼(lnL)^{2y[over ̂]_{t}}, and the logarithmic exponents are determined as y[over ̂]_{h}=0.251(2) and y[over ̂]_{t}=0.25(3), in excellent agreement with the field theoretical prediction y[over ̂]_{h}=y[over ̂]_{t}=1/4. Our results provide a strong support for the recently conjectured finite-size scaling form for the O(n) universality classes at 4D.

Log-Lipschitz embeddings of homogeneous sets with sharp logarithmic exponents and slicing products of balls

If X X is a compact subset of a Banach space with X − X X-X homogeneous (equivalently ‘doubling’ or with finite Assouad dimension), then X X can be embedded into some R n \mathbb {R}^n (with n n sufficiently large) using a linear map L L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c , α > 0 c,\alpha >0 such that \[ c ‖ x − y ‖ | log ⁡ ‖ x − y ‖ | α ≤ | L x − L y | ≤ c ‖ x − y ‖ for all x , y ∈ X , ‖ x − y ‖ > δ , c\ \frac {\|x-y\|}{|\,\log \|x-y\|\,|^\alpha }\le |Lx-Ly|\le c\|x-y\|\quad \mbox {for all}\quad x,y\in X,\ \|x-y\|>\delta , \] for some δ \delta sufficiently small. It is known that one must have α > 1 \alpha >1 in the case of a general Banach space and α > 1 / 2 \alpha >1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k k -fold product of unit volume N N -balls is bounded independent of k k (this provides a ‘qualitative’ generalisation of a result on slices of the unit cube due to Hensley (Proc. AMS 73 (1979), 95–100) and Ball (Proc. AMS 97 (1986), 465–473)).

The two-dimensional 4-state Potts model in a magnetic field

We present a solution of the nonlinear renormalization group equations leading to the dominant and subdominant singular behaviors of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non-vanishing magnetic field. The solutions (i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and (ii) prove recently proposed relations among logarithmic exponents.

Conditional mutual information-based feature selection for congestive heart failure recognition using heart rate variability

Feature selection plays an important role in pattern recognition systems. In this study, we explored the problem of selecting effective heart rate variability (HRV) features for recognizing congestive heart failure (CHF) based on mutual information (MI). The MI-based greedy feature selection approach proposed by Battiti was adopted in the study. The mutual information conditioned by the first-selected feature was used as a criterion for feature selection. The uniform distribution assumption was used to reduce the computational load. And, a logarithmic exponent weighting was added to model the relative importance of the MI with respect to the number of the already-selected features. The CHF recognition system contained a feature extractor that generated four categories, totally 50, features from the input HRV sequences. The proposed feature selector, termed UCMIFS, proceeded to select the most effective features for the succeeding support vector machine (SVM) classifier. Prior to feature selection, the 50 features produced a high accuracy of 96.38%, which confirmed the representativeness of the original feature set. The performance of the UCMIFS selector was demonstrated to be superior to the other MI-based feature selectors including MIFS-U, CMIFS, and mRMR. When compared to the other outstanding selectors published in the literature, the proposed UCMIFS outperformed them with as high as 97.59% accuracy in recognizing CHF using only 15 features. The results demonstrated the advantage of using the recruited features in characterizing HRV sequences for CHF recognition. The UCMIFS selector further improved the efficiency of the recognition system with substantially lowered feature dimensions and elevated recognition rate.

Sparse tensor finite elements for elliptic multiple scale problems

Locally periodic, elliptic multiscale problems in a bounded Lipschitz domain D ⊂ R n with K ⩾ 2 separated scales are reduced to an elliptic system of K coupled, anisotropic elliptic one-scale problems in a cartesian product domain of total dimension Kn (e.g. [2,3,11,26]). In [23,31], it has been shown how these coupled elliptic problems could be solved by sparse tensor wavelet Finite Element Methods in log-linear complexity with respect to the number N of degrees of freedom required by multilevel solvers for elliptic one-scale problems in D with the same convergence rate. In the present paper, the high dimensional one-scale limiting problems are discretized by a sparse tensor product Finite Element Method (FEM) with standard, one-scale FE basis functions as used in engineering FE codes. Sparse tensorization and multilevel preconditioning is achieved by a BPX multilevel iteration. We show that the resulting sparse tensor multilevel FEM resolves all physical length scales throughout the domain, with efficiency (i.e., accuracy versus work and memory) comparable to that of multigrid solvers for elliptic one-scale problems in the physical domain D. In particular, our sparse tensor FEM gives numerical approximations of the correct homogenized limit as well as compressed numerical representations of all first order correctors, throughout the physical domain with performance independent of the physical problem’s scale parameters. Numerical examples with standard FE shape functions and BPX multilevel preconditioners for elliptic problems with K = 2 separated, physical scales in spatial dimension n = 2 confirm the theoretical results. In particular, the present approach allows to avoid the construction of wavelet FE bases necessary in previous work [23,31] while achieving resolution of all scales throughout the physical domain in log-linear complexity, with the logarithmic exponent behaving linearly in the number K of scales.

An integral representation formula for logarithmic potentials and embeddings of Bessel-potential spaces

We give an integral representation formula for logarithmic Riesz potentials. This plays an essential role in proving the sharpness of the embeddings of Bessel-potential spaces, which have logarithmic exponents both in the smoothness and in the underlying Lorentz—Zygmund spaces. These results are natural extensions of those obtained by Edmunds, Gurka, Opic and Trebels.

Investigation of contact interactions of (001) MgO: molecular dynamics simulations studies

Molecular dynamics simulations have been performed to investigate surface contact interactions between small MgO probes (blocks of cross-section n × n atoms, where n = 2, 4, 6, 8 and 10), and the MgO (001) surface (periodic MgO slabs) at 300 K. The contact behaviour was monitored by measuring the force on the probe and the interplanar spacings at different vertical distances between the probe and slab. A mechanistic description of the probe's approach has thus been obtained. In all cases, a characteristic instability ‘jump’ was observed, where the probe was attracted towards the surface, signalled by large atomic displacements and an increased attractive force. It was found that the normal force per atom experienced by a probe decreases as the area increases and could be characterised by a logarithmic exponent, q = 0.12 (corresponding simulations for NaCl have yielded a similar value). Hysteresis occurred in every case after the jump and a rich variety of phenomena were observed, such as ‘neck’ formation, accompanied by atomic dislocations, and systematic removal of atoms from the surface, as the probe was withdrawn.

New method of analysing self-avoiding walks in four dimensions

A new method based on the concept of fractal dimensionality is used to study the problem of self-avoiding walks in a four-dimensional lattice. The authors find from Monte Carlo simulations that the confluent logarithmic exponent related to the end-to-end distance is 1/8+or-0.01, in excellent agreement with the prediction derived from the n to 0 vector model.

Quadratic Posylognomials: An Extension of Posynomial Geometric Programming

Abstract The theory of geometric programming is extended to include a new function with logarithmic exponents. The function, defined as a Quadratic Posylognomial (QPL), is a series of nonlinear product terms with positive coefficients and positive variables. A QPL may be created by adding a linear function of the logarithms of the variables to the constant exponents of a posynomial. The logarithm of each nonlinear term is a quadratic form in the logarithms of the primal variables, whereas the logarithm of a general term of a posynomial is linear in the logarithms of the variables. The primal-dual relationships and necessary conditions are developed, and special cases where sufficient conditions exist are derived. The theory is the basis for solution of a machining economics problem using a more accurate QPL tool life equation and has the potential for solution of other engineering problems where a posynomial formulation is inadequate. The extended theory may also prove useful for condensing posynomial geo...

Perception of effort in isometric and dynamic muscular contraction.

The perception of muscular effort was studied using estimation and production methods in the adductor pollicis and quadriceps. A psychometric scale (percentage magnitude) was used. Static contractions were studied in the adductor pollicis, and both dynamic (isokinetic) and static contractions were studied in quadriceps. Linear and logarithmic equations were fitted for the perceived effort as a percentage of the maximum in relation to the produced percentage maximal force or torque. The logarithmic exponent was around or above 1.0. No significant difference was found between mean exponent and intercept values for the adductor pollicis and the quadriceps, or when estimated or produced values for the two muscles were compared. There was no difference in the same subjects between the equations for static and dynamic contractions with low angular velocity of the quadriceps.

Leptoproduction and Drell-Yan processes beyond the leading approximation in chromodynamics

The problem of going beyond the leading logarithmic approximation in QCD for leptoproduction and Drell-Yan processes is considered. All the coefficient functions for leptoproduction are evaluated to order α s Q 2) (apart from two-loop corrections to logarithmic exponents). Existing calculations are thus completed and in part corrected. Particular attention is given to the constraint imposed by the validity at all Q 2 of the Adler sum rule. The question of a convenient definition of effective parton densities appropriate at this level of accuracy is discussed. Phenomenological consequences for leptoproduction are considered with special emphasis on the problem of extraction from the data of the small sea densities which are particularly sensitive to the corrections. The modifications of the Drell-Yan formula relevant for proton-nucleus processes are also explictly calculated to order α s( Q 2).

The [formula omitted] rule with heavy quarks and right-handed currents

The effect of strong interactions on a two-fermion operator is calculated in an asymptotically free theory, thereby completing the calculation of the logarithmic exponents for the operators contributing to non-leptonic amplitudes. The case of models with both left-and right-handed weak currents is discussed and the implications for the ΔI= 1 2 rule and other non-leptonic decays are deduced.

The Effect of Streptococcus Hemolyticus Infection on the Reaction of the Blood of Rabbits

It has been amply demonstrated by many workers using a* wide variety of methods of examination that the reaction of the blood remains constant in health and is subject to slight actual variations in disease. Human blood is faintly alkaline and expressed in a physicochemical sense has a hydrogen-ion concentration only slightly less than that of pure water. As determined by its dissociation into hydrogen and hydroxyl ions, pure water at 20 C. contains approximately 1-10,000,000 gm. of hydrogen ions to the liter and an equivalent amount of hydroxyl ions; in other words, pure water is 1-10,000,000 N acid and also 1-10,000,000 N alkaline. This reaction is conveniently expressed in terms of the hydrogen-ion concentration by the use of logarithmic exponents, such as 10 -7 or as suggested by Sorensen 1 as Ph 7. When human blood is titrated by electrometric methods its reaction is found to be Ph 7.45, and the limits of variation to be well within Ph 7.0 and Ph 8.0. The neutral point Ph 7.0 is reached only in severe acidosis (as a terminal finding) and the limit of alkalinity Ph 8.0 only after the prolonged administration of alkalies. This precise regulation of the reaction of the blood constitutes one of the great constants of the body. The acid radicals produced through the oxidation of carbon, sulphur, and phosphorus in the food under normal conditions and in large part the acid radicals which arise through deranged metabolism in disease are either oxidized or neutralized in the body and then excreted or excreted unchanged. The fixed bases in the food and the ammonia arising from protein metabolism furnish bases for such neutralization, and the excretion of carbon dioxid by the lungs and of acid by the kidneys complete the mechanism. In addition the blood itself is able to take care of considerable quantities of either acid or alkali without change in its reaction through the buffer action