Law Exponent Research Articles

Electrochemical modification of the porosity and zeta potential of montmorillonitic soft rock

The porosity (including the specific surface area and pore volume-diameter distribution) of montmorillonitic soft rock (MSR) was studied experimentally with an electrochemical treatment, based on which the change in porosity was further analyzed from the perspective of its electrokinetic potential (<TEX>${\zeta}$</TEX> potential) and the isoelectric point of the electric double layer on the surface of the soft rock particles. The variation between the <TEX>${\zeta}$</TEX> potential and porosity was summarized, and used to demonstrate that the properties of softening, degradation in water, swelling, and disintegration of MSR can be modified by electrochemical treatment. The following conclusions were drawn. The specific surface area and total pore volume decreased, whereas the average pore diameter increased after electrochemical modification. The reduction in the specific surface area indicates a reduction in the dispersibility and swelling-shrinking of the clay minerals. After modification, the <TEX>${\zeta}$</TEX> potential of the soft rock was positive in the anodic zone, there was no isoelectric point, and the rock had lost its properties of softening, degradation in water, swelling, and disintegration. The <TEX>${\zeta}$</TEX> potential increased in the intermediate and cathodic zones, the isoelectric point was reduced or unchanged, and the rock properties are reduced. When the <TEX>${\zeta}$</TEX> potential is increased, the specific surface area and the total pore volume were reduced according to the negative exponent law, and the average pore diameter increased according to the exponent law.

A 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers

This paper addresses three-dimensional (3-D) free vibration characteristic of thick circular/annular functionally graded (FG) plates with surface-bonded piezoelectric layers on the basis of 3-D Ritz solution. Three displacement components along with electrical potential field of the plate are expressed by a set of Chebyshev polynomials multiplied by geometry boundary functions. Both open-circuit and closed-circuit surface conditions are taken into account. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey either exponent or power law distribution of the volume fraction of the constituents. The effect of thickness-to-radius ratio, inner-to-outer radius ratio, piezo-to-host thickness ratio and gradient index on the natural frequencies of coupled piezoelectric FG circular/annular plates is investigated for different electrical and mechanical boundary conditions. It is observed that, unlike isotropic homogeneous circular/annular plates, frequency parameters of their piezoelectric coupled FG counterparts significantly increase with an enhancement in the host plate thickness to radius ratio. Results also show that the frequency parameters for open-circuit condition are higher than those for closed-circuit condition.

Dissecting the spin distribution of dark matter haloes

(Abridged) We apply a very general statistical theorem introduced by Cramer (1936) to study the origin of the deviations of the halo spin PDF from the reference lognormal shape. We find that these deviations originate from correlations between two quantities entering the definition of spin, namely the ratio $J/M^{5/2}$ (which depends only on mass) and the total gravitational binding energy $E$. To reach this conclusion, we have made usage of the results deduced from two high spatial- and mass resolution simulations. Our simulations cover a relatively small volume and produce a sample of more than 16.000 gravitationally bound halos, each traced by at least 300 particles. We verify that our results are stable to different systematics, by comparing our results with those derived by the GIF2 and by a more recent simulation performed by Maccio' et al. We find that the spin probability distribution function shows systematic deviations from a lognormal, at all redshifts z <= 1. These deviations depend on mass and redshift: at small masses they change little with redshift, and also the best lognormal fits are more stable. The J-M relationship is well described by a power law of exponent $\alpha$ very near to the linear theory prediction (alpha=5/3), but systematically lower than this at z<= 0.3. We argue that the fact that deviations from a lognormal PDF are present only for high-spin halos could point to a role of large-scale tidal fields in the evolution of the spin PDF.

Estimation of pressure distribution for shock wave through the bend of bend laneway

Propagation of the air shock wave caused by explosion via the bend of a bend laneway has obvious nonlinear characteristics, compared with its propagation in a straight laneway. These characteristics are important bases to analyze the accident of gas explosion in underground mines and to estimate the blast resistance of underground structures in mines. In this work, the rule of the shock wave propagation via the laneway bend and the pressure distribution are studied by means of the numerical simulation approach. Theoretical results show that attenuation of the peak overpressure with distance does not obey exponent law when the air shock wave goes through the laneway bend. At some locations within the bend zone, the overpressures are higher than ones around those locations, the front of original plane wave bends in the bend of the laneway and after passing through the bend, it gradually returns to the state of plane wave propagation. There is a span dependent on the cross section dimension of laneway and the bend angle and increasing with the bend angle, in which the peak overpressure of shock wave does not uniformly attenuate with distance. When the bend angle is equal to 135°, this span is about five times as long as the corresponding equivalent diameter of the laneway. Additionally, the impulse of air shock wave attenuates uniformly via the laneway bend. On the end section of the complicated pressure distribution area in the bend, it is 66.65–98.7% of that in the straight laneway at the same scaled distance.

Surface-potential decay of biased-probe contact-charged amorphous polymer films

We have investigated the decay of scanning Kelvin probe force microscopy (KPFM) and electric force microscopy (EFM) signals from biased-probe contact-charged films of three different amorphous polymers representing wide-ranging water absorption capabilities. The surface-potential decay (SPD) has been measured by repeatedly scanning the charge pattern as a function of dissipation time t while varying the relative humidity (RH), the film thickness d, the temperature, the charging voltage, and the load on the scanning probe. Whereas increases in KPFM and EFM peak widths are appreciable only in the long run, the decay in the peak heights is rapid at the beginning and then strongly slowing down with time. Peak heights can be approximated for t&amp;lt;1 hour by power laws of negative exponents (−β), with 0&amp;lt;β&amp;lt;0.5 in dry conditions. β increases for thinner films and when scanning with higher probe loads. Raising the humidity or heating to temperatures well below the glass transition temperature of the polymer considerably increases β, with much stronger impacts for polymers with a higher water uptake capability. From the findings, we conclude that ionic charge carriers are trapped by the charge injection process in the volume of the polymers at low depths. A main contribution to SPD is by drift of the ions in their own space-charge field, mutually repelling each other and being attracted by their mirror charge in the grounded back electrode. Lateral drifts for small t are not resolved, increases in peak widths for t⪢1 h are predominantly due to increased probe—charge carrier distances. We interpret the power law approximation in terms of dispersive transport theory. We approximate trap-controlled apparent mobilities μ from isothermal KPFM peak height data, taken within a few minutes after charging, by a linear and a hyperbolic SPD model. Both models yield μ≈10−14 cm2/(V s) for thin films (d≈50 nm) in dry conditions. For mobilities derived similarly from isohumid measurements series, we find an exponential increase as a function of RH%. We furthermore suggest that two more mechanisms contributing to SPD are: first, by potential shielding of charge carriers by water dipoles, and second, in an indirect manner, by diffusion of injected water.

Exothermic process of cast-in-place pile foundation and its thermal agitation of the frozen ground under a long dry bridge on the Qinghai-Tibet Railway

A number of dry bridges have been built to substitute for the roadbed on the Qinghai-Tibet Railway, China. The aim of this study was to investigate the exothermic process of cast-in-place (CIP) pile foundation of a dry bridge and its harm to the stability of nearby frozen ground. We present 3D heat conduction functions of a concrete pile and of frozen ground with related boundaries. Our analysis is based on the theory of heat conduction and the exponent law describing the adiabatic temperature rise caused by hydration heat. Results under continuous and initial conditions were combined to establish a finite element model of a CIP pile-frozen ground system for a dry bridge under actual field conditions in cold regions. Numerical results indicated that the process could effectively simulate the exothermic process of CIP pile foundation. Thermal disturbance to frozen ground under a long dry bridge caused by the casting temperature and hydration heat of CIP piles was substantial and long-lasting. The simulated thermal analysis results agreed with field measurements and some significant rules relating to the problem were deduced and conclusions reached.

A tenant‐mix model for shopping malls

PurposeThe purpose of this paper is to develop a novel tenant mix model for shopping malls based on an analogy from ecological theories.Design/methodology/approachThis study empirically investigates the tenant species‐area relationship and tenant species‐abundance distribution in shopping malls. In this study, the tests on species‐area relationship and species‐abundance distribution in shopping malls are derived from ecological theories. Empirical tests by a sample of 18 shopping malls for the species‐area relationship and of five malls for the species‐abundance distribution are carried out in Hong KongFindingsIt shows that, in line with the findings of biogeography, the tenant species‐area relationship follows a power law of exponent of about 0.20. Furthermore, the species‐abundance distributions of the five large‐scale malls are found to be closely in track with a geometric distribution as commonly found in ecology. These results imply that tenant mix strategies are governed by two principles: the number of tenant species is related to the mall size; and the shop area allocation follows a geometric distribution.Research limitations/implicationsThe study provides the first quantitative tenant mix model on the number of tenant species in a particular mall size, and on the tenant species abundance distribution pattern. These results provide far‐reaching implications for research and practice, including a quantitative benchmarking of tenant mix strategy and an optimal design of shopping malls.Practical implicationsThe model is the first tenant mix model for practitioners to formulate quantitative tenant mix strategy, and evaluate the effects of tenant mix on the performance of a shopping mall.Originality/valueIt is the first quantitative model for tenant mix, and would open up a novel agenda for research on tenant mix strategies.

A Tenant Mix Model for Shopping Malls

The purpose of this paper is to develop a novel tenant mix model for shopping malls from an analogy of ecological theories. This study empirically investigated the tenant species-area relationship and tenant species-abundance distribution in shopping malls. By a sample of 25 shopping malls are carried out in Hong Kong, it is showed that, in line with the findings of biogeography, the tenant species-area relationship follows a power law of exponent of 0.30. And the analysis results suggest that different developers or mall operators adopt similar tenant-mix strategies. Furthermore, the species abundance distributions of several large-scale malls are found to be closely in track with a geometric distribution commonly found in ecology. The study provides a very good prediction model on the number of tenant species in a particular mall size, and on the tenant species abundance distribution pattern. These results provide very far-reaching implications for research and practice, including a quantitative benchmarking of tenant mix strategy and an optimal design of shopping malls.

Gateless Gating Model vs. Gated Pore Model: Phase Pinning of Guanidinium Toxins in Sodium Channels

Unlike the gated pore model, which assumes the existence of a movable gate within the molecule, the gateless gating model (H. R. Leuchtag, Voltage-Sensitive Ion Channels, Springer 2008) (VSIC) explains ion-channel gating as a phase transition. Under this model, the closed channel conformation imposed by the resting electric field is a compact, ordered phase with behavior, such as critical temperature, hysteresis, fractional dispersion exponent (constant phase angle) and Curie-Weiss law, similar to that of phases seen in ferroelectric liquid crystals (VSIC, pp. 355-383). In the absence of a toxin molecule, threshold membrane depolarization indirectly brings about a stochastic phase transition to a less ordered phase in which S4 segments expand by the mutual repulsion of their positively charged residues. The resulting wider pitch of permeation pathway α helices elastically linked to the S4s permits ion replacement in the interloop H bonds and the subsequent percolation of permeant ions through the channel (VSIC, 477f, 506f). With an externally applied tetrodotoxin (TTX) or analog molecule complexed in the channel, however, the ordered phase is pinned by the toxin, inhibiting the transition to the ion-conducting phase (VSIC, 76, 382f). Phase pinning by impurities is an established effect in ferroelectric liquid crystals. In the toxin, a guanidinium group, H2N+=C(NH2)2, a highly resonant, planar, positive ion, is active in pinning the closed phase. The fact that guanidinium is also found in ferroelectric crystals such as guanidinium aluminum sulfate hexahydrate suggests that TTX enhances the spontaneous polarization of the resting phase. This explanation by the gateless gating model of specific toxin action is based on physical principles; in contrast, the phrase “TTX blocks the pore” offered by the gated pore model is vague and at the macroscopic rather than molecular scale.

Revisiting the evolutionary origin of allometric metabolic scaling in biology

1 In 1997 West, Brown & Enquist published a theoretical explanation for the long-known empirical observation of 3/4-power scaling of organismal metabolic rates with body mass, using an attractive combination of general physical and physiological principles with evolutionary optimization. This model generated hundreds of studies, exploring its implications for physiology, populations, biodiversity, whole-ecosystem functioning, and even medical, engineering and social sciences. 2 However, the evolutionary optimization part of the model has never been carefully scrutinized. 3 In this article we perform the evolutionary optimization as proposed by the authors rigorously and show that it actually leads to a biologically irrelevant network, i.e. a single vessel. 4 Moreover, we find that by relaxing some assumptions in a realistic way we obtain a feasible network but with isometric scaling of metabolism, which is in conflict with Kleiber's empirical law of an allometric exponent of around 3/4. 5 Hence, we conclude that the West-Brown-Enquist model cannot account for the observed universal metabolic scaling relation. 6 As possible solutions to this paradox, we discuss (weak) body-size dependence of capillary properties (leading to different predictions of intra- and inter-specific metabolic scaling), and expanding the West-Brown-Enquist model to an integrated network model (including body shape and oxygen transport to and from the arterial system). Alternatively, the ultimate explanation for the observed allometric patterns may need to be found outside the framework of flow-limited network theory. 7 We conclude that Kleiber's law is still as theoretically unexplained as ecologically important.

Some Generic Properties of Level Spacing Distributions of 2D Real Random Matrices

We study the level spacing distribution P(S) of 2D real random matrices both symmetric as well as general, non-symmetric. In the general case we restrict ourselves to Gaussian distributed matrix elements, but different widths of the various matrix elements are admitted. The following results are obtained: An explicit exact formula for P(S) is derived and its behaviour close to S = 0 is studied analytically, showing that there is linear level repulsion, unless there are additional constraints for the probability distribution of the matrix elements. The constraint of having only positive or only negative but otherwise arbitrary non-diagonal elements leads to quadratic level repulsion with logarithmic corrections. These findings detail and extend our previous results already published in a preceding paper. For symmetric real 2D matrices also other, non-Gaussian statistical distributions are considered. In this case we show for arbitrary statistical distribution of the diagonal and non-diagonal elements that the level repulsion exponent ρ is always ρ = 1, provided the distribution function of the matrix elements is regular at zero value. If the distribution function of the matrix elements is a singular (but still integrable) power law near zero value of S, the level spacing distribution P(S) is a fractional exponent power law at small S. The tail of P(S) depends on further details of the matrix element statistics. We explicitly work out four cases: the uniform (box) distribution, the Cauchy-Lorentz distribution, the exponential distribution and, as an example for a singular distribution, the power law distribution for P(S) near zero value times an exponential tail.

Linear Stability of Fractional Reaction - Diffusion Systems

Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on anomaly exponents of various species. In addition to monotonous instability, known from normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants' diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modes become stable, whereas oscillatory instability persists until the anomalous critical value is also exceeded. An exact formula for the anomalous critical value is obtained. It is shown that in the short wave limit the growth rate is a power law of the wave number. When the anomaly exponents differ, disturbance modes are governed by power laws of the distinct exponents. If the difference between the diffusion anomaly exponents is small, the splitting of the power law exponents is absent at the leading order and emerges only as a next-order effect. In the short wave limit the onset of instability is governed by the anomaly exponents, whereas the ratio of diffusion coefficients influences the complex growth rates. When the exponent of the inhibitor is greater than that of the activator, the system is always unstable due to the inhibitor enhanced diffusion relatively to the activator. If the exponent of the activator is greater, the system is always stable. Existence of oscillatory unstable modes is also possible for waves of moderate length.

Fisher waves and front roughening in a two-species invasion model with preemptive competition

We study front propagation when an invading species competes with a resident; we assume nearest-neighbor preemptive competition for resources in an individual-based, two-dimensional lattice model. The asymptotic front velocity exhibits an effective power-law dependence on the difference between the two species' clonal propagation rates (key ecological parameters). The mean-field approximation behaves similarly, but the power law's exponent slightly differs from the individual-based model's result. We also study roughening of the front, using the framework of nonequilibrium interface growth. Our analysis indicates that initially flat, linear invading fronts exhibit Kardar-Parisi-Zhang (KPZ) roughening in one transverse dimension. Further, this finding implies, and is also confirmed by simulations, that the temporal correction to the asymptotic front velocity is of O(t(-2/3)).

Nonlinear Elasticity and Yielding of Nanoparticle Glasses

We employ experiment and theory to explore the nonlinear elasticity and yielding of concentrated suspensions of nanoparticles which interact via purely repulsive forces. These glassy suspensions are found to exhibit high exponent power law or simple exponential dependences of the shear elastic modulus and perturbative yield stress on nanoparticle volume fraction, as well as a monotonic decrease of the perturbative yield strain with increasing concentration. Our experimental observations are in good agreement with the predictions of a recently developed microscopic statistical mechanical theory, which describes glassy dynamics based on a nonequilibrium free energy that incorporates local cage correlations and activated barrier hopping processes [(1) Schweizer, K. S.; Saltzman, E. J. J. Chem. Phys. 2003, 119, 1181. (2) Saltzman, E. J.; Schweizer, K. S. J. Chem. Phys. 2003, 119, 1197. (3) Kobelev, V.; Schweizer, K. S. Phy. Rev. E 2005, 71, 021401].

The Seat–Vote Equation in French Legislative Elections (1978–2002)

After outlining a way of calculating the seat–vote equation for the French majority two-ballot majority electoral system, this article shows that the 2002 legislative elections were consistent with the ‘fourth exponent law’ and not to the more general cube law, mainly because of the nationalization of electoral behavior since the 1960s. More generally, the ‘fourth exponent law’ is a good indicator of the transfer of votes into seats in France, as long as the variation in votes between the Left and the Right-wing is not too large. In particular, the ‘fourth exponent law’ makes it possible to highlight bias caused by the drawing of electoral boundaries (specifically, malapportionment) and the presence in the second round of a third party, here the National Front.

Activities for Students: Exponential Growth through Pattern Exploration

Do your students think of arithmetic, algebra, and geometry as separate entities? Do the laws of exponents tend to confuse them? Participants at the summer institute Pattern Exploration: Integrating Math and Science in the Middle Grades developed and used the three activities in this article. The activities have proved to be exceptional tools for providing students in middle school and the early years of high school with models of exponent laws while relating differing fields of mathematics.

Thermodynamics from three-dimensional many-body fragmentation simulations on a cellular automaton model

The thermal equilibrium of many-body systems subject to finite range interactions is investigated numerically, by means of a multipurpose 3D cellular automaton dynamic model developed by the authors. The numerical experiments, carried out at fixed number of bodies, volume and energy, demonstrate the formation of an equilibrium among 3D aggregates of bodies. The distribution of the aggregates against size obeys a power law of (negative) exponent tau approximately 2.2 (against 1.3 in 2D). Our experiments, indicating that the exponent is insensitive to the precise parameter values and the precise parametrization of the interactions, are consistent with the idea of the existence of a universality class corresponding to the thermal equilibrium. The numerical value for the exponent tau is in agreement with the theoretical thermal equilibrium analyses based on various other approaches, numerical and semianalytical, indicating that the cellular automaton approach provides an adequate methodology to investigate thermal equilibria. In this paper, as an illustration of this method, we refer to the problem of formation of clusters of nucleons in heavy ion collisions of nuclei leading on to fragmentation. The theoretical tau value, however, corresponding to the thermal equilibrium among the aggregation clusters, is 15 percent lower than the empirical value ( approximately 2.6 ) , as measured in laboratory nuclear fragmentation experiments induced by collision. There is then only a very approximate correspondence between the experimental and the thermal equilibrium value. On the basis of the results of this paper and of a previous paper of this series, we conjecture that the approximate agreement is due to a partial establishment of a thermodynamic equilibrium during the collision of the nuclei. The thermal equilibrium gives the main contribution to the observed tau value; the deviation from this possibly universal value is largely the consequence of the lack of full thermal equilibrium in actual laboratory experiments. This conjecture is extended to interpret the observed ubiquity of power laws of exponents exceeding 2.2, which refer to the distribution of various types of matter in 3D space.

Rheology and reptation of linear polymers. Ultrahigh molecular weight chain dynamics in the melt

The melt rheology of ultrahigh molecular weight polymeric materials characterized by a narrow molecular weight distribution has been analyzed. Ultrahigh molecular weight polyethylene obtained from a metallocene catalyst shows a well-developed “plateau” modulus in a range of angular frequency of more than 3 decades. The characteristic value of the plateau modulus (∼2 MPa) is in close agreement with those reported for a model high molecular weight monodisperse polyethylene. From this value one can determine a characteristic molecular weight between entanglements of 1200 g mol−1. The molecular weight dependency of different, experimentally based relaxation times obtained from the linear viscoelastic response exhibits an exponent power law close to 3.0 for these materials. This seems to contradict the 3.4 dependence observed in the usual molecular weight range, which is based on the chain contour length fluctuation approach, but is in agreement with the latest reptation-based models. These models predict a crossover from the 3.4 to a 3.0 exponent for very long chains as used here at a constant critical value of the molecular weight Mr close to 100Mc (200Mc when using the well accepted relationship Mc=2Me). This predicted crossover is independent of the polymer’s chemical composition. However, combining results from our experiments with results from literature shows that the experimental values of Mr extend from 15Mc for polystyrene, 25Mc for polyisobutilene, 100Mc for polybutadyene to 220Mc for polyethylene. These results are not predicted by molecular models and demand for new theoretical considerations of chain dynamics, in which the chemical structure is, most probably, a key factor that should be taken into account. It should be noticed that the influence of the molecular weight distribution on the differences observed is not understood. Unfortunately, it is very difficult to obtain monodisperse samples of ultrahigh molecular weight polyethylene and, therefore, the use of the samples studied here the best choice possible, up to now to test and revisit basic and novel aspects of the rheology of polyolefin’s.

Swelling behavior and structural characteristics of wheat gluten polypeptide films.

Wheat gluten films were subjected to controlled thermomechanical treatments to increase the percentage of aggregated sodium dodecyl sulfate (SDS)-insoluble gluten protein, the aggregation reaction being disulfide bonding. The rheological properties of the films were measured under immersion in water, where wheat gluten films are stable and show only slight swelling. The equilibrium swelling of the gluten films in water decreased with the increase of the percentage of SDS-insoluble protein aggregates, and the frequency the independent shear modulus increased sharply with increasing percentage of SDS-insoluble aggregates. Both findings confirm that disulfide bonding between gluten proteins is the predominant cross-linking reaction in the system. A relationship between shear modulus and aggregated protein compatible with a power law (of exponent 3) suggests the existence of a protein network at a molecular scale. However, the classical Flory-Rehner model failed to describe the relationship between the plateau modulus and the gluten volume fraction (a very drastic increase, compatible with a power law of an exponent of about 14). This result shows that gluten cannot be described as an entangled polymer network. The interpretation of both relationships is a network of mesoscale particles which in turn have a fractal inner structure (with a fractal dimension close to 3).

Influence of spatial correlation of fracture centers on the permeability of two‐dimensional fracture networks following a power law length distribution

Observations of outcrops of fractured media show that fractures are correlated at all scales of the medium and that there are also fractures of sizes ranging from the microscale to the macroscale. Recent studies have shown that the correlation pattern is often a fractal characterized by its dimension Dc and that the fracture length distribution is a power law of characteristic exponent −a. We study the influence of these long‐range heterogeneities on the equivalent medium permeability. Fracture correlation and fracture length are present at all scales of the medium and may thus influence the hydraulic properties and their scaling at all scales. We have numerically investigated the effect of the fractal correlation pattern and of the power law length distribution on the equivalent permeability of fracture networks. The correlation and the length distribution have opposite effects on connectivity and permeability. Increasing correlation lets connectivity and permeability decrease and conversely for the length distribution. In most cases, one of the characteristics (i.e., length or correlation) supersedes the other. There is only a small area defined by a &gt; 2 and a ≤ Dc + 1 where both characteristics have a simultaneous influence on permeability. In this area the influence of the correlation remains low, more precisely a scale increase of at least 5 orders of magnitude is required to produce a permeability increase of 1 order of magnitude. Although the correlation pattern does not have much influence on the equivalent permeability, its influence on the flow pattern increases from negligible for networks at threshold to important for dense networks.