Power-law Curve Research Articles

High-Temperature Stress-Rupture Properties of a Ferritic Steel for Solid Oxide Fuel Cell Interconnect

The high-temperature stress-rupture properties of a ferritic steel (Crofer 22 H) for use as an interconnect material in planar solid oxide fuel cells are investigated. Constant-load creep tests are carried out at 650 to 800oC. For a given applied stress, the creep rupture time is decreased with an increase in temperature. The creep rupture time is well correlated with the applied stress by a simple power law for each testing temperature. The fitted power-law curves of 650-750oC look somewhat parallel to each other while the curve of 800oC is steeper than the others. A greater reduction of creep strength in the long-life region is present at 800oC. Larson-Miller relationship also does well in correlating the creep rupture time with applied stress and temperature. Fractographic and microstructural observations indicate such a significant reduction of creep strength in the long-life region at 800oC can be partially attributed to a precipitate coarsening effect.

Investigation into the creeping polarization and strain in PZT-855 under combined mechanical and electrical loadings

The time-dependent remnant polarization and remnant strains evolving with time are measured and modeled on initially unpoled ferroceramic PZT-855 under a compressive stress and an electric field. Experiments reveal that the compression stress significantly inhibits the hysteresis loop curves of the electric displacement and the strain since the two hysteresis loop curves against the applied electric field become narrower and narrower as the compression stress becomes larger and larger. Moreover, the compression stress decreases the coercive electric field significantly when compared to that under an applied purely electric field. The variable tendency of the coercive filed in the present case seems like a power-law curve. It is also seen that the compression stress significantly influences the variable features of the remnant polarization and remnant strains against time although the time-dependent feature still remains. In particular, the larger the compression stress is, the smaller the remnant polarization and remnant strains are. For some extreme cases, the remnant strain transfers from a tensile strain to a compressive strain. The power-law formulation proposed by Liu and Huber (J Eur Ceram Soc 29:2799–2806, 2006) is extended to model the present experimental data. However, unlike those under purely electric fields, as the mechanical loading varies from 0 to 100 MPa, the predicted saturation of the remnant polarization appears much earlier than the present experimental data. This limits the application of the power-law model. Further theoretical investigations are needed which will be given in a sequel.

LATE-TIME OBSERVATIONS OF GRB 080319B: JET BREAK, HOST GALAXY, AND ACCOMPANYING SUPERNOVA

The Swift discovered GRB080319B was by far the most distant source ever observed at naked eye brightness, reaching a peak apparent magnitude of 5.3 at a redshift of z=0.937. We present our late-time optical (HST, Gemini & VLT) and X-ray (Chandra) observations, which confirm that an achromatic break occurred in the power-law afterglow light curve at ~11 days post-burst. This most likely indicates that the gamma-ray burst (GRB) outflow was collimated, which for a uniform jet would imply a total energy in the jet E_{jet} \gsim 10^{52} erg. Our observations also show a late-time excess of red light, which is well explained if the GRB was accompanied by a supernova (SN), similar to those seen in some other long-duration GRBs. The latest observations are dominated by light from the host and show that the GRB took place in a faint dwarf galaxy (r(AB)\approx27.0, rest-frame M_B\approx-17.2). This galaxy is small even by the standards of other GRB hosts, which is suggestive of a low metallicity environment. Intriguingly, the properties of this extreme event - a small host and bright supernova - are entirely typical of the very low-luminosity bursts such as GRB980425 and GRB060218.

COSMOLOGICAL EVOLUTION OF MASSIVE BLACK HOLES: EFFECTS OF EDDINGTON RATIO DISTRIBUTION AND QUASAR LIFETIME

A power-law time-dependent light curve for active galactic nuclei (AGNs) is expected by the self-regulated black hole growth scenario, in which the feedback of AGNs expels gas and shut down accretion. This is also supported by the observed power-law Eddington ratio distribution of AGNs. At high redshifts, the AGN life timescale is comparable with (or even shorter than) the age of the universe, which sets a constraint on the minimal Eddington ratio for AGNs on the assumption of a power-law AGN light curve. The black hole mass function (BHMF) of AGN relics is calculated by integrating the continuity equation of massive black hole number density on the assumption of the growth of massive black holes being dominated by mass accretion with a power-law Eddington ratio distribution for AGNs. The derived BHMF of AGN relics at z = 0 can fit the measured local mass function of the massive black holes in galaxies quite well, provided the radiative efficiency ∼0.1 and a suitable power-law index for the Eddington ratio distribution are adopted. In our calculations of the black hole evolution, the duty cycle of AGN should be less than unity, which requires the quasar life timescale τQ ≳ 5 × 108 years.

DUST ATTENUATION IN DISK-DOMINATED GALAXIES: EVIDENCE FOR THE 2175 Å DUST FEATURE

The attenuation of starlight by interstellar dust is investigated in a sample of low redshift, disk-dominated star-forming galaxies using photometry from GALEX and SDSS. By considering broadband colors as a function of galaxy inclination we are able to confidently separate trends arising from increasing dust opacity from possible differences in stellar populations, since stellar populations do not correlate with inclination. All commonly employed dust attenuation curves (such as the Calzetti curve for starbursts, or a power-law curve) provide poor fits to the ultraviolet colors for moderately and highly inclined galaxies. This conclusion rests on the fact that the average FUV-NUV color varies little from face-on to edge-on galaxies, while other colors such as NUV-u and u-r vary strongly with inclination. After considering a number of model variations, we are led to speculate that the presence of the strong dust extinction feature at 2175A seen in the Milky Way (MW) extinction curve is responsible for the observed trends. Independent of our interpretation, these results imply that the modeling of dust attenuation in the ultraviolet is significantly more complicated than traditionally assumed. These results also imply a very weak dependence of the FUV-NUV color on total FUV attenuation, and we conclude from this that it is extremely difficult to use only the observed UV spectral slope to infer the total UV dust attenuation, as is commonly done. We propose several simple tests that might finally identify the grain population responsible for the 2175A feature.

The Time Fractals

Ralph N. Elliott wrote the wave principle in 1938. In 1975 Benoit B. Mandelbrot coined the term fractal and in 1982 published his ideas in Fractal Geometry of Nature. The book brought fractals into the mainstream of professional and popular mathematics. In February 1999, Benoit Mandelbrot submitted an article to Scientific American called ”A Multifractal Walk down Wall Street.” In the article, he discussed how fractal geometry can be used to model the stock market curves. The enclosed research reworks the Mandelbrot Multifractal from a time cycle rather than trend perspective to prove that time fractal is more proportionate than the price fractal and is the real law of nature, which drives everything in nature. The case is validated by illustrating power law curves in time cycle periodicities. Power law is seen across nature and in diverse social trends. The power law in prices is a subject of extended study, but there has been no research attempt made to prove power law in time cycle periodicities. Testing cycle periodicity needs large historical data. Long term time series are difficult to obtain and many emerging markets have seen stock market trading activity only start a decade back. The continued prosperity after 1980’s was a reason why time fractals did not get researchers attention, unlike price fractal which was actively studied and researched. The fact that what we can see is what we can relate too more also made researchers focus more on price than time, which is less visible. Cycles are not conventionally believed to be patterns. Patterns are understood either conventionally or as Elliott wave fractals. Even few Elliott wave practitioners have admitted the limitation of the Elliott Wave structure as being more sharp on form than on time. These were few reasons why time time fractals remained unproven. This study further connects its findings with the existing research on various economic cycles finally extending the proof to a long – short intermarket strategy on an asset pair.

Quantitative assessment of conjunctival microvascular circulation of the human eye

Accessibility to the bulbar conjunctival microvasculature provides a means to assess blood supply to the cerebral cortex and thus optimize therapeutic interventions designed to prevent or reduce the risk of cerebral vascular disease and stroke. The feasibility of a method for quantitative measurements of conjunctiva blood vessel diameter, blood velocity, and flow in the human eye is reported. The method is based on slit lamp biomicroscope digital imaging coupled with a space time image analysis technique. A sequence of conjunctiva microvasculature images was captured at a rate of 50 Hz. The images were analyzed to determine blood vessel diameter, velocity and flow. Blood vessel diameter measurements ranged between 8.7 and 24.3 microns, with a mean value of 15.5 microns. Blood flow rate ranged between 27.3 and 296.9 pl/s, with a mean value of 111.8 pl/s. The relationship between blood flow and vessel diameter was fit with a power law curve ( R = 0.87). The application of this technique for in vivo quantitative assessment of blood flow dynamics has potential to impact diagnosis and monitoring of various cardiovascular and blood disorders.

7120 A retrospective review of treatment discontinuation and survival in patients with advanced renal cell carcinoma treated with sunitinib or sorafenib

Accumulation curves are useful tools to estimate species diversity. Here we argue that they can also be used in the study of global parasite species richness. Although this basic idea is not completely new, our approach differs from the previous ones as it treats each host species as an independent sample. We show that randomly resampling host–parasite records from the existing databases makes it possible to empirically model the relationship between the number of investigated host species, and the corresponding number of parasite species retrieved from those hosts. This method was tested on 21 inclusive lists of parasitic worms occurring on vertebrate hosts. All of the obtained models conform well to a power law curve. These curves were then used to estimate global parasite species richness. Results obtained with the new method suggest that current predictions are likely to severely overestimate parasite diversity.

Dipolar magnetic moment of the bodies of the solar system and the Hot Jupiters

Context The planets magnetic field has been explained based on the dynamo theory, which presents as many difficulties in mathematical terms as well as in predictions. It proves to be extremely difficult to calculate the dipolar magnetic moment of the extrasolar planets using the dynamo theory. Objective The aim is to find an empirical relationship (justifying using first principles) between the planetary magnetic moment, the mass of the planet, its rotation period and the electrical conductivity of its most conductive layer. Then this is applied to Hot Jupiters. Method Using all the magnetic planetary bodies of the solar system and tracing a graph of the dipolar magnetic moment versus body mass parameter, the rotation period and electrical conductivity of the internal conductive layer is obtained. An empirical, functional relation was constructed, which was adjusted to a power law curve in order to fit the data. Once this empirical relation has been defined, it is theoretically justified and applied to the calculation of the dipolar magnetic moment of the extra solar planets known as Hot Jupiters. Results Almost all data calculated is interpolated, bestowing confidence in terms of their validity. The value for the dipolar magnetic moment, obtained for the exoplanet Osiris (HD209458b), helps understand the way in which the atmosphere of a planet with an intense magnetic field can be eroded by stellar wind. The relationship observed also helps understand why Venus and Mars do not present any magnetic field.

A probabilistic neural network for earthquake magnitude prediction

A probabilistic neural network (PNN) is presented for predicting the magnitude of the largest earthquake in a pre-defined future time period in a seismic region using eight mathematically computed parameters known as seismicity indicators. The indicators considered are the time elapsed during a particular number (n) of significant seismic events before the month in question, the slope of the Gutenberg-Richter inverse power law curve for the n events, the mean square deviation about the regression line based on the Gutenberg-Richter inverse power law for the n events, the average magnitude of the last n events, the difference between the observed maximum magnitude among the last n events and that expected through the Gutenberg-Richter relationship known as the magnitude deficit, the rate of square root of seismic energy released during the n events, the mean time or period between characteristic events, and the coefficient of variation of the mean time. Prediction accuracies of the model are evaluated using three different statistical measures: the probability of detection, the false alarm ratio, and the true skill score or R score. The PNN model is trained and tested using data for the Southern California region. The model yields good prediction accuracies for earthquakes of magnitude between 4.5 and 6.0. The PNN model presented in this paper complements the recurrent neural network model developed by the authors previously, where good results were reported for predicting earthquakes with magnitude greater than 6.0.

Percolation in nanoporous gold and the principle of universality for two-dimensional to hyperdimensional networks

Percolation in nanoporous gold can be achieved with as little as 8% by volume of gold. Samples of nanoporous gold of various morphologies are analyzed with a combination of electrical and optical data. Growing thin films and complex multiply connected three-dimensional networks both display nonuniversal character. Growing films have two-dimensional morphology but a three-dimensional percolation threshold and nonuniversal critical coefficients, yet similar silver films percolate as expected with universal coefficients. Growing gold however regresses to two-dimensional resistive behavior between 65% to 100% gold, and this regime lies along a single power-law curve shared by the hyperdimensional networks of gold, suggesting underlying symmetry governed by diffusion-limited aggregation. Models of data imply either hyperdimensionality or major internal property changes as density shifts. The distinctive flat spectral signature found near the percolation threshold is common to all highly porous samples and is explained quantitatively in terms of effective plasmonic response. Parameters from fits of effective medium models to optical and resistivity data are in close agreement, especially at the highest porosities. They imply an effective dimension which increases continuously as porosity grows via the increased branching needed for structural integrity.

180 Different Time Course of PDS Versus EPS Functional Dyspepsia Symptoms After Ingestion of a Meal

Background: According to the Rome III consensus, functional dyspepsia (FD), is subcategorized into Postprandial Distress Syndrome (PDS) and Epigastric Pain Syndrome (EPS). Relationship to ingestion of a meal has been proposed as a distinguishing feature of PDS (comprising early satiation and postprandial fullness) and not of EPS (comprising epigastric pain and burning). The aim of the present study was to study the time-course of different FD symptoms after ingestion of a standardized meal. FD patients underwent a 13C-octanoic acid gastric emptying breath test during which they were asked to score the severity (0-4) of 6 symptoms (pain, fullness, bloating, nausea, epigastric burning and belching) at each sampling point (every 15 min for 4 hrs postprandially). For each symptom, a meal-related severity score was obtained by adding scores at all time-points. Different curve models were evaluated for goodness of fit of the time after the meal vs. symptom intensity based on the lowest sum of squared errors (SSE). Results: 218 consecutive FD patients (149 women, age 39±1) participated in the study. Meal ingestion was associated with a significant increase in intensity of each symptom at 15 min, and this remained significant throughout the 4 study hours. Meal-related symptom score was highest for fullness (23.2±1.3) and bloating (18.8±1.2), followed by belching (14.8±1.0) and epigastric pain (14.4±1.2) and lowest for nausea (13.0±1.1) and epigastric burning (9.7±1.0). Curve fitting revealed two different patterns. For postprandial fullness, a power law curve with exponential cutoff (y=C*x(T)*e(-x/K)+d) provided the best fit (SSE 0.005, C=0.383; T=-0.336; K=139.937; d=0.647). The same curve type also provided the best fit for bloating (SSE 0.002, C=0.217; T=-0.455; K=112.917; d=0.535), for nausea (SSE 0.008, C=0.042; T=-0.685; K=99.842; d=0.497), and for belching (SSE 0.005, C=0.233; T=-0.404; K=108.262; d=0.403). For epigastric pain, a sigmoid curve (y=a/(1.0+e(-(x-b)/c))+d) provided the best fit (SSE 0.037, a=-0.846; b= 7.029; c=-8.730; d=1.040). The same curve type also provided the best fit for epigastric burning (SSE 0.026, a=-0.646; b=2.938, c=-15.215, d=0.613). Conclusion: In FD patients, epigastric pain and burning follow a different pattern and time course after a standardized meal compared to fullness, bloating, nausea and belching. The different types of curve that best approximate these respective time courses do not depend on symptom intensity, and suggest different underlying pathophysiological mechanisms. The observations and curve fitting support the Rome III subdivision of FD into EPS versus PDS.

Measuring the vertical albedo profile of a subarctic boreal forest canopy

The validation of airborne and satellite-derived albedo measurements suffers from the fact that the surface albedo of forest is difficult to measure in-situ over large areas. The goal of this study is to examine whether or not the estimation of the surface albedo of a forest stand from ground level measurements is possible. In addition, knowledge about the vertical behavior of albedo, and therefore transmitted solar radiation, is important in the modeling of interactions of sunlight with the forest canopy. Such modeling is useful for forest growth estimations among other things. To achieve these goals, an albedometer set-up capable of vertical albedo profiling has been used to obtain data from a boreal forest stand in Northern Finland during periods in summer 2006 and winter 2007. The results show a strong relationship between the data and fitted power-law regression curves. Power-law regression fits best likely because of the radiation transmission characteristics of boreal forest.

Low‐Luminosity GRB 060218: A Collapsar Jet from a Neutron Star, Leaving a Magnetar as a Remnant?

The gamma-ray burst (GRB) 060218 has ~ 10^5 times lower luminosity than typical long GRBs, and is associated with a supernova (SN). The radio afterglow displays no jet break, so that this burst might arise from a mildly-relativistic spherical outflow produced by the SN shock sweeping the stellar surface. Since this model is energetically difficult, we propose that the radio afterglow is produced by a non-relativistic phase of an initially collimated outflow (jet). Our jet model is supported by the detection of optical linear polarization in the SN component. We also show analytically that the jet can penetrate a progenitor star. We analyzed the observational data of the prompt emission of this burst and obtained a smooth power-law light curve which might last longer than 10^6s. This behavior contrasts with the long intermittent activities with the X-ray flares of typical GRBs, implying that the central engine of this burst is different from those of typical GRBs. This argument is consistent with the analysis of the SN component of this burst, which suggests that the progenitor star was less massive and collapsed to a neutron star instead of a black hole. The collimation-corrected event rate of such low-luminosity GRBs is estimated to be ~ 10 times higher than that of typical long GRBs, and they might form a different GRB population: low-luminosity GRBs are produced by mildly-relativistic jets from neutron stars at the collapses of massive stars, while typical long GRBs by highly-relativistic jets from black holes. We suggest that the central engine of GRB 060218 is a pulsar (or a magnetar) with the initial rotation period P_0 \~ 10ms and the magnetic field B ~ 10^16G. A giant flare from the magnetar might be observed in future.

Bayesian power-law regression with a location parameter, with applications for construction of discharge rating curves

This paper presents a Bayesian approach for fitting the standard power-law rating curve model to a set of stage-discharge measurements. Methods for eliciting both regional and at-site prior information, and issues concerning the determination of prior forms, are discussed. An efficient MCMC algorithm for the specific problem is derived. The appropriateness of the proposed method is demonstrated by applying the model to both simulated and real-life data. However, some problems came to light in the applications, and these are discussed.

Root tensile strength relationships and their slope stability implications of three shrub species in the Northern Apennines (Italy)

The role of root strength is important in stabilising steep hillslopes which are seasonally affected by storm-induced shallow landslides. In the Italian Apennines, steep (25–40°) slopes underlain by mudstone are generally stable if they are covered by shrubs whose roots anchor into the soil mantle. To quantify the mechanical reinforcement of roots to soil, the root tensile breaking force and the root tensile strength of three autochthonous shrub species commonly growing on stiff clay soils of the Northern Italian Apennines, Rosa canina (L.), Inula viscosa (L.) and Spartium junceum (L.), were measured by means of field and laboratory tests. For each test approximately 150 root specimens were used. The tensile force increases with increasing root diameter following a second-order polynomial regression curve. The tensile strength decreases with increasing root diameter following a power law curve. The field in situ tensile force required to break a root is always smaller than that obtained from laboratory tests for the same root diameter, although their difference becomes negligible if the root diameter is smaller than 5 mm. The influence of root tensile strength on soil shear strength was verified based on the infinite slope stability model. The root reinforcement was calculated using the number and mean diameter of roots. The factor of safety was calculated for three different soil thickness values (0.1, 0.3, and 0.6 m) and topographic slopes between 10° and 45°. The factor of safety for the combination of 0.6 m soil thickness, slopes smaller than 30°, and vegetation of I. viscosa (L.) or S. junceum (L.) is always larger than 1. If a slope is steeper, the factor of safety may be smaller than 1 for I. viscosa (L.), although it is still larger than 1 for S. junceum (L.). In the stiff clayey areas of the Northern Italian Apennines, I. viscosa (L.) mainly colonizes fan/cone/taluses and stabilises these zones up to a topographic gradient < 30° for a soil 0.6 m thick. S. junceum (L.) colonizes not only fan/cone/taluses but also headwalls and cliffs and, for a 0.6 m thick soil, it stabilises these areas up to 45°. The effectiveness of this reinforcement, however, depends strongly on the frequency of soil and seasonal grass vegetation removal due to shallow landsliding before the entrance of the shrub species.

Global structures in a composite system of two scale-free discs with a coplanar magnetic field

We investigate a theoretical magnetohydrodynamic (MHD) disc problem involving a composite disc system of gravitationally coupled stellar and gaseous discs with a coplanar magnetic field in the presence of an axisymmetric dark matter halo. The two discs are expediently approximated as razor-thin, with a barotropic equation of state, a power-law surface mass density, a ring-like magnetic field, and a power-law rotation curve of radius r. By imposing the scale-free condition, we construct analytically stationary global MHD perturbation configurations for both aligned and logarithmic spiral patterns, using our composite MHD disc model. MHD perturbation configurations in a composite system of partial discs in the presence of an axisymmetric dark matter halo are also considered. Our study generalizes the previous analyses of Lou & Shen and Shen & Lou on the unmagnetized composite system of two gravitationally coupled isothermal and scale-free discs, of Lou and Shen et al. on the cases of a single coplanarly magnetized isothermal and scale-free disc, and of Lou & Zou on magnetized two coupled singular isothermal discs. We derive analytically the stationary MHD dispersion relations for both aligned and unaligned perturbation structures and analyse the corresponding phase relationships between surface mass densities and the magnetic field. Compared with earlier results, we obtain three solution branches corresponding to superfast MHD density waves (sFMDWs), fast MHD density waves (FMDWs) and slow MHD density waves (SMDWs), respectively. We examine the m = 0 cases for both aligned and unaligned MHD perturbations. By evaluating the unaligned m = 0 case, we determine the marginal stability curves where the two unstable regimes corresponding to Jeans collapse instability and ring fragmentation instability are identified. We find that the aligned m = 0 case is simply the limit of the unaligned m = 0 case with the radial wavenumber ξ → 0 (i.e. the breathing mode) which does not merely represent a rescaling of the equilibrium state. We further show that a composite system of partial discs behaves much differently from a composite system of full discs in certain aspects. We provide numerical examples by varying dimensionless parameters β (rotation velocity index), η (ratio of effective sound speeds of the two discs), δ (ratio of surface mass densities of the two discs), q (a measure of coplanar magnetic field strength), F (gravitational potential ratio), ξ (radial wavenumber). Our formalism provides a useful theoretical framework in the study of stationary global perturbation configurations for MHD disc galaxies with bars, spirals and barred spirals.

Benioff Strain Release Before Earthquakes in China: Accelerating or Not?

We systematically analyzed the Benioff strain release before 65 earthquakes with M S over 6.0 in China from 1978 to 2003 to investigate the generality of the widely discussed accelerating moment release (AMR) phenomenon before strong and intermediate-strength earthquakes. In this approach, a strong or intermediate-strength earthquake is selected as a “target earthquake,” and retrospective analysis of seismic activity before the “target earthquake” is performed. Simple searching area (three circular areas with different radius centered at the epicenter of the “target earthquake”) and unified temporal range (8 years) are taken in the analysis. Justification of whether AMR exists is by both visual inspection and by power-law curve fitting. It is found that more than 3/5 of the earthquakes under consideration exhibit clear pre-shock AMR property, and 1/3 of the events seem to be sensitive to the searching area. AMR behavior shows apparent focal mechanism dependence: 15 out of 17 dip-slip earthquakes with stable moment release characteristics against the changing of searching areas exhibit AMR behavior, while 16 out of 25 strike-slip earthquakes with stable moment release characteristics exhibit AMR behavior.

The effect of variation of microstructure on fracture mechanics parameters of HSLA-100 steel

The variations in the FCGR curve, the J–R curve, the blunting-line slope and the stretch zone width (SZW) for systematic variation of microstructure in Cu-strengthened HSLA-100 steel has been studied. The microstructural variation in the steel has been introduced through ageing at various temperatures after an initial quenching treatment. This has resulted in progressive tempering of the as-quenched martensitic matrix, accompanied by nanoscale precipitation of coherent Cu particles that gradually coarsen and loose coherency with overageing. It was observed that although there was a systematic trend, the FCGR curves were least sensitive to microstructural changes. The variation of fracture toughness, characterized by JQ in most cases and by KQ for the microstructures displaying highest strengths, correlated well with the inverse relationship between fracture toughness and strength. A systematic trend was also observed for the pre-exponent and exponent of the power-law tearing curve (for cases in which brittle fracture was precluded), the blunting-line slope and the SZW. The effect of coherency of precipitates in restricting plastic flow, as implied from the variation of mechanical properties with ageing temperature is thought to be responsible for the effects observed.

Existence of the frequentistic estimate for power-law regression with a location parameter, with applications for making discharge rating curves

Practical application of the power-law regression model with an unknown location parameter can be plagued by non-finite least squares parameter estimates. This presents a serious problem in hydrology, since stream flow data is mainly obtained using an estimated stage–discharge power-law rating curve. This study provides a set of sufficient requirements for the data to ensure the existence of finite least squares parameter estimates for a power-law regression with an unknown location parameter. It is shown that in practice, these requirements act as necessary for having a finite least squares solution, in most cases. Furthermore, it is proved that there is a finite probability for the model to produce data having non-finite least squares parameter estimates. The implications of this result are discussed in the context of asymptotic predictions, inference and experimental design. A Bayesian approach to the actual regression problem is recommended.