Logarithmic Ln Research Articles

Respiratory muscle training in individuals with spinal cord injury: effect of training intensity and -volume on improvements in respiratory muscle strength.

Retrospective cohort study. To investigate the effect of training intensity and -volume on improvements in respiratory muscle strength in individuals with spinal cord injury (SCI). SCI rehabilitation center in Switzerland. Inpatients with SCI, lesion level C4-T12, who had at least 10 individualized inspiratory muscle training sessions with respiratory function measurements before and after the training period. Multiple regression analysis with natural logarithmic (ln) transformation of the outcome values was used to examine the effect of training intensity and -volume, lesion level and completeness, and baseline respiratory muscle strength on improvements in respiratory muscle strength. Overall, 67 individuals were analyzed. Variation in PImax was explained by PImax at baseline and training intensity. This adjusted effect size suggested a 7% (95% CI 2.8 to11.6%) increase in PImax per 10 units of increase in training intensity. Controlling for the variation in baseline PEmax, the effect of training intensity on PEmax was conditional on AIS (p < 0.021). While individuals with motor complete lesions showed a 6.8% (95% CI 2.1 to11.7%) increase in PEmax per 10 units of increase in training intensity, the corresponding adjusted effect size in those with motor incomplete lesions was 0.1% (95% CI -4.3 to4.5%). The full models explained 57 and 60% of the variance of lnPImax and lnPEmax, respectively. The intensity of inspiratory muscle training was more relevant than training volume for the improvement of respiratory muscle strength in individuals with SCI. Thus, training intensity should be chosen as high as possible.

Cardiac autonomic modulation post-maximal incremental exercise is not influenced by body mass index in young adult men

The purpose of this study was to analyze whether cardiac autonomic modulation recovery at short-term after maximal incremental exercise test in young adults is influenced by the body mass index (BMI). Forty-eight healthy untrained men (22.26 ± 3.08 years old) were evaluated; 18 were normal weight (NW: 21.89 ± 1.75 kg m−2), 11 were overweight (Ov: 27.00 ± 1.38 kg m−2) and 19 were obese (O: 34.38 ± 3.36 kg m−2). After the maximal incremental exercise test, a 10 min recovery (5 min actively, followed by 5 min passively) was performed to analyze cardiac autonomic modulation recovery. The root mean square successive differences between adjacent RR intervals (RMSSD) and standard deviation of all normal RR intervals (SDNN) indexes were determined in successive non-overlapped 30 s segments transformed in natural logarithmic—Ln. LnRMSSD and LnSDNN indexes were reduced at the end of the incremental exercise test and remained reduced throughout 5 min of active recovery, and throughout 5 min after this time compared to baseline resting (p 0.05) among the groups. However, throughout 5 min (330–600 s) after active recovery, the LnSDNN values were higher than the end of exercise (NW: 0.56 ± 0.19; Ov: 0.56 ± 0.25; O: 0.57 ± 0.16 ms; p < 0.05). Although there was no vagal reactivation at short-term after a maximal incremental exercise test, there were alterations in the overall heart rate variability regardless of BMI in apparently healthy young adult men.

Scaling and Similarity of the Anisotropic Coherent Eddies in Near-Surface Atmospheric Turbulence

Abstract The low-wavenumber regime of the spectrum of turbulence commensurate with Townsend’s “attached” eddies is investigated here for the near-neutral atmospheric surface layer (ASL) and the roughness sublayer (RSL) above vegetation canopies. The central thesis corroborates the significance of the imbalance between local production and dissipation of turbulence kinetic energy (TKE) and canopy shear in challenging the classical distance-from-the-wall scaling of canonical turbulent boundary layers. Using five experimental datasets (two vegetation canopy RSL flows, two ASL flows, and one open-channel experiment), this paper explores (i) the existence of a low-wavenumber k−1 scaling law in the (wind) velocity spectra or, equivalently, a logarithmic scaling ln(r) in the velocity structure functions; (ii) phenomenological aspects of these anisotropic scales as a departure from homogeneous and isotropic scales; and (iii) the collapse of experimental data when plotted with different similarity coordinates. The results show that the extent of the k−1 and/or ln(r) scaling for the longitudinal velocity is shorter in the RSL above canopies than in the ASL because of smaller scale separation in the former. Conversely, these scaling laws are absent in the vertical velocity spectra except at large distances from the wall. The analysis reveals that the statistics of the velocity differences Δu and Δw approach a Gaussian-like behavior at large scales and that these eddies are responsible for momentum/energy production corroborated by large positive (negative) excursions in Δu accompanied by negative (positive) ones in Δw. A length scale based on TKE dissipation collapses the velocity structure functions at different heights better than the inertial length scale.

“Wandering” eigenfrequencies of a two-dimensional elastic body with a blunted cusp

At the pointed cusp of a two-dimensional plate, a tip of small length h is broken off. By means of asymptotic analysis, a new effect of “wandering” of eigenfrequencies of longitudinal vibrations of the plate with the blunted cusp is found: as h → +0, the frequencies prove to be almost periodic functions in the logarithmic scale ln h; i.e., when the fragment length decreases, they chaotically move at a high speed O(h−1) along the real semi-axis (κ†, +∞), while κ† > 0 is the cutoff point of the continuous spectrum of the problem with an ideal cusp.

Application of a Modified Arrhenius Equation to Describe the Time-Temperature Equivalence in Relaxation Analysis of Metal Seals

For the application of the time-temperature superposition principle a suitable relation is needed to describe the time-temperature shift factor α. Therefore, the Arrhenius equation is widely used due to its simple form and often leads to suitable results. Where, the Arrhenius equation presents a linear relation for the temperature-dependent shift factor in logarithmic scale ln(α) with the absolute inverse temperature (1/ϑ). However, in cases with a large temperature range which eventually include more complex reaction processes, the functional relation between ln(α) and (1/ϑ) is nonlinear in the ‘Arrhenius plot’. In those cases, the monotone change of the nonlinear range in the ‘Arrhenius plot’ can be interpreted as a transient range between two approximately linear or constant regions. An extended application of the modified Arrhenius equation from Nakamura (1989) is presented in this study for this transient range. The introduced method was applied to describe the time-temperature equivalence in the relaxation analysis of restoring seal force of metal seals, which are used in lid-systems of transport and interim storage casks for radioactive materials. But, the method is widely valid and can be used for different objectives which are characterized by thermorheologically simple behavior with nonlinear sensitivity to inverse temperature.

Comparison of Physiological and Psychological Relaxation Using Measurements of Heart Rate Variability, Prefrontal Cortex Activity, and Subjective Indexes after Completing Tasks with and without Foliage Plants.

The objective of this study was to compare physiological and psychological relaxation by assessing heart rate variability (HRV), prefrontal cortex activity, and subjective indexes while subjects performed a task with and without foliage plants. In a crossover experimental design, 24 university students performed a task transferring pots with and without a foliage plant for 3 min. HRV and oxyhemoglobin (oxy-Hb) concentration in the prefrontal cortex were continuously measured. Immediately thereafter, subjective evaluation of emotions was performed using a modified semantic differential (SD) method and a profile of mood state questionnaire (POMS). Results showed that the natural logarithmic (ln) ratio of low frequency/high frequency, as an estimate of sympathetic nerve activity, was significantly lower while performing the task with foliage plants for the average 3 min measurement interval. Oxy-Hb concentration in the left prefrontal cortex showed a tendency to decrease in the 2–3 min interval in the task with foliage plants compared to the task without plants. Moreover, significant psychological relaxation according to POMS score and SD was demonstrated when the task involved foliage plants. In conclusion, the task involving foliage plants led to more physiological and psychological relaxation compared with the task without foliage plants.

Thermal Reduction of NOx with Recycled Plastics.

This study develops technology for mitigation of NOx formed in thermal processes using recycled plastics such as polyethylene (PE). Experiments involve sample characterization, and thermogravimetric decomposition of PE under controlled atmospheres, with NOx concentration relevant to industrial applications. TGA-Fourier transform infrared (FTIR) spectroscopy and NOx chemiluminescence serve to obtain the removal efficiency of NOx by fragments of pyrolyzing PE. Typical NOx removal efficiency amounts to 80%. We apply the isoconversional method to derive the kinetic parameters, and observe an increasing dependency of activation energy on the reaction progress. The activation energies of the process span 135 kJ/mol to 226 kJ/mol, and 188 kJ/mol to 268 kJ/mol, for neat and recycled PE, respectively, and the so-called compensation effect accounts for the natural logarithmic pre-exponential ln (A/min-1) factors of ca. 19-35 and 28-41, in the same order, depending on the PE conversion in the experimental interval of between 5 and 95%. The observed delay in thermal events of recycled PE reflects different types of PE in the plastic, as measurements of intrinsic viscosity indicate that, the recycled PE comprises longer linear chains. The present evaluation of isoconversional activation energies affords accurate kinetic modeling of both isothermal and nonisothermal decomposition of PE in NOx-doped atmosphere. Subsequent investigations will focus on the effect of mass transfer and the presence of oxygen, as reburning of NOx in large-scale combustors take place at higher temperatures than those included in the current study.

New EWMA S2 Control Charts for Monitoring Process Dispersion

The Exponentially Weighted Moving Average (EWMA) control chart is an effective tool for the detection of small shifts in the process variability. This research studied the properties of EWMA charts based on unbiased sample variance, S-2, for monitoring of changes in the process dispersion. However, since an increase in process variance could lead to an increased number of defective products, we only considered upward shifts in the process variance. The proposed schemes were based on simple random sampling and extreme variations of ranked set sampling technique for efficient monitoring. Using Monte Carlo simulations, we compared the relative performance of EWMA charts based on unbiased sample variance, S-2, and its logarithmic transformation ln(S-2) as well as some existing schemes for monitoring the increases in variability of a normal process. It is found that the proposed schemes significantly outperform several other procedures for detecting increases in the process dispersion. Numerical example is given to illustrate the practical application of the proposed schemes using real industrial data.

On uniqueness of numerical solution of boundary integral equations with 3-times monotone radial kernels

The uniqueness of solution of boundary integral equations (BIEs) is studied here when geometry of boundary and unknown functions are assumed piecewise constant. In fact we will show BIEs with 3-times monotone radial kernels have unique piecewise constant solution. In this paper nonnegative radial function Fδ3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if δ3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than Fδ3/2. Note that there is a smart relation between δ3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln(c0r) as fundamental solution of Laplace equation. As a result, an upper bound for condition number of constant Galerkin BEMs system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Also non-singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non-singular BIEs are used. Finally some numerical examples are presented which confirm the analytical results.

The effects of the variations of turbulence parameters on the observable turbulence statistics

In order to study how the statistical properties of the observational quantities of the turbulent molecular cloud models vary with different turbulence parameters, I use the artificially generated random numerical models based on Gaussian fields created by the code PYFC with different values of the 1D density Fourier spectral index beta(n), the turbulence driving parameter, i.e. the b parameter and the 1D Fourier velocity power spectral index beta(upsilon). Then I calculate the line profiles of (CO)-C-13 emission on every sight line. I derive the statistical properties of column density N, line intensity W, peak intensity T-peak and velocity dispersion V-rms on every sight line. I discover that as beta(n) increases, (1) the standard deviations sigma(ln(N/ )) and sigma(ln(W/ )), of the logarithmic column density ln (N / ) and the logarithmic integrated intensity ln (W/ ) increase; (2) the number of the low T-peak values becomes larger; (3) the probability distribution function (PDF) of v(rms) inclines to left a little. As the b parameter increases, (1) the values of both sigma(ln(N/ )) and sigma(ln(W/ )) increase; (2) the numbers of both the low T-peak values and the saturated T-peak values become larger; (3) the PDF of v(rms) inclines to left obviously. As beta(upsilon) increases, (1) the PDF of ln (W/ ) remains generally unchanged; (2) the PDF of T-peak remains generally unchanged; (3) the PDF of V-rms inclines to left obviously. I also discuss the relationship among sigma(N/ Mach number, the b parameter and beta(n).

Patterns of heart rate variability and cardiac autonomic modulations in controlled and uncontrolled asthmatic patients.

BackgroundPrevious heart rate variability (HRV) studies in asthmatic subjects (AS) demonstrate predominance of parasympathetic drive concomitant with low HRV, which is against the general belief that enhanced parasympathetic modulation improves HRV. The aim of this study was to compare patterns of HRV and cardiac autonomic modulations of AS to healthy control subjects (HS).MethodsEighty AS and forty HS were enrolled in the study. Asthma control test and spirometry were used to discriminate uncontrolled (UA) from controlled (CA) asthmatic patients. Natural logarithmic (Ln) scale of total power (TP), very low frequency (VLF), low frequency (LF) and high frequency (HF) were used to evaluate HRV. Normalized low frequency (LF Norm) and high frequency (HF Norm) were used to determine sympathetic and parasympathetic autonomic modulations respectively.ResultsCA patients achieved significantly higher LnTP, LnLF, LnHF and HF Norm but lower LF Norm and LnLF/HF compared with UA patients (p < 0.05). Although CA patients showed increased HRV and augmented vagal modulation compared with HS, these findings were no longer significant following adjustment for mean heart rates and anti-asthma treatment. All measured HRV parameters were not significantly different in UA patients compared with the HS (p > 0.05).ConclusionsCA is associated with enhanced parasympathetic modulations and higher HRV compared with UA. However, neither CA nor UA patients had different autonomic modulations and/or HRV compared with HS.

Leaf Area Index Derived from Hemispherical Photograph and its Correlation with Aboveground Forest Biomass

Leaf area index (LAI) is one of the key physical factors in the energy exchange between terrestrial ecosystem and atmosphere. It determines the photosynthesis process to produce biomass and plays an important role in performing forest stand reflectance. Therefore building relationship between LAI and biomass from field measurements can be used to develop allometric equations for biomass estimation. This paper studies the relationship between diameter at breast height (DBH) and leaves biomass, DBH and crown biomass (sum up of leaves, twigs and branches) as well as between LAI and leaves biomass; LAI and crown biomass; LAI and Total Above-ground Biomass (TAGB) in East Kalimantan Province. Destructive sampling was conducted to develop allometric equations. The DBH measurements from 52 sample plots were used as training data for model development (35 plots) and for validation (17 plots). A hemispherical photograph was used to record LAI. The result shows that strong corelation (r) exists between natural logarithmic ( ln ) DBH and crown biomass ranging from 0.88 to 0.98. The correlation (r) between LAI and biomass of leaves; leaves + twigs + branches; TAGB were 0.742, 0.768 and 0.772, respectively. Improvement of (r) between LAI and biomass can be conducted by proper time of LAI measurement, when the sky is uniformly overcast.

Charged particles in higher dimensional homogeneous gravitational field: self-energy and self-force

A problem of self-energy and self-force for a charged point-like particle in a higher dimensional homogeneous gravitational field is considered. We study two cases, when a particle has usual electric charge and a case when it has a scalar charge, which is a source of a scalar massless minimally coupled field. We assume that a particle is at rest in the gravitational field, so that its motion is not geodesic and it has an acceleration a directed from the horizon. The self-energy of a point charge is divergent and the strength of the divergence grows with the number of dimensions. In order to obtain a finite contribution to the self- energy we use a covariant regularization method which is a modification of the proper time cut-off and other covariant regularizations. We analyze a relation between the self-energy and self-force and obtain explicit expressions for the self-forces for the electric and scalar charge in the spacetimes with the number of dimensions up to eight. General expressions for the case of higher dimensions are also obtained. We discuss special logarithmic factors ln(a), which are present both in the self-energy and self-force in odd dimensions. Finally, we compare the obtained results with the earlier known results both for the homogeneous gravitational field and for particles near black holes.

Structural and thermoelectric properties of Cu6Fe4Sn12Se32 single crystal

We present structural and thermoelectric properties of quaternary selenide Cu6Fe4Sn12Se32 single crystal. Single crystal X-ray structural analysis revealed that Cu6Fe4Sn12Se32 has the cubic thiospinel structure. Low-temperature logarithmic electrical resistivity ln ρ exhibits Mott’s T−1/4 dependence, indicating that three-dimensional variable-range hopping (VRH) is the dominant electrical conduction mechanism. Furthermore, Efros’T−1/2 dependence of ln ρ appears below 30K. Thermopower S varies as T1/2 at low temperatures, as expected for the VRH conduction, and reaches 150μV/K at 300K. Result of Hall measurement indicates that rather low ρ for a VRH conductor and the large S are attributed to high carrier density and large slope of electron density of states, respectively.

Mathematical model on surface reaction diffusion in the presence of front chemical reaction

The article discusses a mathematical model of solid-phase diffusion over substance surface accompanied a frontal chemical reaction. The purpose of our article is to describe the concentration distribution and surface reacted layer growth. The model is a system parabolic equations, complicated with the presence of mobile front. It takes account of diffusive fluxes redistribution, sublimation from the surface, chemical reaction reversibility. The asymptotic approximation of the obtained nonlinear problem is constructed. Numerical solution was also carried out. Both numerical and analytical solutions conform to each other in a wide range of parameter changes, whereas observed differences are explained. It was obtained that the reaction front at the substrate surface grows as the fourth root of time in the assumed absence of evaporation and reaction reversibility. In the presence of evaporation the logarithmic distribution law ln(t) is obtained. The theoretical possibility of sharp deceleration and stop of reaction product layer growth is obtained.

Abstract 2643: Interactions between polycyclic aromatic hydrocarbon exposure and genetic polymorphisms on benzo(a)pyrene-DNA adducts in a Polish cohort of mothers and newborns

Abstract Background Polycyclic aromatic hydrocarbons (PAHs) are a class of chemicals common in the environment. Major sources of PAHs include fossil fuel combustion and cigarette smoking. Certain PAHs are carcinogenic, though the degree to which genetic variation influences susceptibility to PAH exposure remains unclear. Benzo(a)pyrene (BaP) is a well-studied PAH that is classified as a probable human carcinogen. Here, we evaluated single nucleotide polymorphism (SNP) frequency in genes related to PAH metabolism and detoxification, and explored the interactions between PAH exposure and SNPs on BaP-DNA adducts, an established cancer risk marker, in umbilical cord white blood cells (WBCs). Methods Maternal and umbilical cord blood were collected at delivery from participants in a longitudinal cohort study of Polish mothers and newborns in Krakow, Poland. WBC DNA was isolated from cord blood, and genetic polymorphisms and BaP-DNA cord adducts were then measured in 503 mothers and 445 newborns. Maternal PAH exposure was measured during pregnancy with a personal air exposure monitor. For this study 7 maternal and newborn genes related to PAH metabolism, detoxification and repair were selected for SNP analyses: CYP1A1, CYP1A2, CYP1B1, GSTM3, GSTT2, NQO1 and XRCC1. SNP x PAH interactions on logarithmic (ln)-transformed BaP-DNA adduct levels were explored. Results We observed a number of significant interactions between high PAH exposure and SNPs in both mothers and newborns on BaP-DNA adducts in cord blood. For example, the interaction between a CYP1A2 SNP (rs2069514) and high PAH exposure on cord BaP-DNA adducts in newborns was significant (α=−1.34, p&amp;lt;0.01). We will present significant interaction results from analyses performed on both maternal and newborn CYP1A1, CYP1A2, GSTM3, GSTT2, NQO1 and XRCC1 SNPs. Discussion It is biologically plausible that an individual's abilities to metabolically activate and detoxify BaP and repair DNA damage play a role in susceptibility to DNA adduct formation following prenatal PAH exposure. Ultimately, this susceptibility could correlate with future risk of cancer development. Our findings of several SNP x PAH exposure interactions on cord BaP-DNA adducts support the hypothesis that genetic variability can contribute to susceptibility and potential cancer risk from prenatal PAH exposures. Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 103rd Annual Meeting of the American Association for Cancer Research; 2012 Mar 31-Apr 4; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2012;72(8 Suppl):Abstract nr 2643. doi:1538-7445.AM2012-2643

OC26.03: Derivation of new fetal weight estimation models using fractional limb volume

Develop new fetal weight estimation models that combine fractional limb volume with conventional 2D biometry. A cross-sectional study of 271 fetuses used 3DUS to acquire biparietal diameter (BPD), abdominal circumference (AC), femoral diaphysis length (FDL), fractional arm volume (AVol) and fractional thigh volume (TVol) within 4 days of delivery. Weighted multiple linear regression was used to develop ‘modified Hadlock’ models for comparison to new models using natural logarithmic (Ln) transformed parameters for birth weight (BW). Predicted and observed weights were compared using mean % difference between predicted and actual BW (systematic error) and SD of the % differences (random error). The proportion of newborns with EFW within 5 or 10% of BW, were compared using McNemar's test. More precise fetal weight estimations are possible by combining fractional limb volume with 2D biometry. Substitution of FDL by fractional limb volume, use of Ln transformations with weighted regression analysis, and selective application of squared transformed parameter terms reduced the random error to 6.6%. New fetal weight estimation models, that incorporate soft tissue parameters, may offer new insight regarding the body composition of malnourished fetuses.

Properties and Applications of the Reciprocal Logarithm Numbers

Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln (1+z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers and are denoted by Ak, converge to zero as k→∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling numbers of the first kind. Also appearing here are several applications including a new representation for Euler’s constant known as Hurst’s formula and another for the logarithmic integral. From the properties of the Ak it is found that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst’s formula, thereby indicating that Euler’s constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting the preceding material.

Temporal patterns of growth in larval cohorts of the Japanese sardine Sardinops melanostictus in a coastal nursery area

Growth patterns of larval sardine Sardinops melanostictus were studied in a coastal nursery area, in southern Japan for four monthly hatch cohorts of larvae (November, December, January and February) for the 2003–2004 and 2004–2005 seasons. Laird–Gompertz models were fitted to each cohort using both total length (LT)‐at‐age at capture and mean LT‐at‐age data derived from backcalculations. In both approaches, the absolute daily growth rates (GR) and absolute daily growth rates at the inflection point (GXO) were estimated. In parallel, individual growth rates (GI) were derived from backcalculated LT (LB). Growth showed the following general common patterns irrespective of hatch month, season and methods: (1) significant Laird–Gompertz fits, (2) asymptotic growth, (3) a decrease in GR after the inflexion point, except for February for the 2003–2004 season that showed an apparent constant growth pattern, (4) six in eight cohorts showed GXO ranging from 0·8 to 1·2 mm day−1 and (5) a decreasing tendency of GI from 1·75 to 0·24 mm day−1, from first feeding through the first month of larval life. The contrasting pattern between the 2003–2004 and the 2004–2005 seasons were: (1) allometric v. logarithmic (ln) LT and otolith radius relationships, (2) low GXOv. high GXO, (3) high GRv. low GR when growth turned asymptotic, (4) low GXOv. high GXO when monthly hatch cohorts were combined and (5) LB and GI not differing among monthly hatch cohorts. The differences in growth patterns and growth rates between seasons seemed to be linked to the influx of warmer and oligotrophic waters of the Kuroshio Current that triggered an increase of 3° C in the coastal area for the 2003–2004 seasons. In the overall context, however, the high GXO, within cohorts and seasons reported in the current study, suggests that either sea surface temperature (SST) or food availability, or both are in the optimal range of preferences for S. melanostictus larvae. Consequently, nearshore coastal areas seem to be playing an important role as a nursery area for the larval stage of this species.

Homotopy analysis of unsteady boundary‐layer flow started impulsively from rest along a symmetric wedge

AbstractAn investigation of the velocity boundary‐layer growth resulting from an impulsively started wedge is presented in this paper. By introducing a new set of scaled coordinates, the governing equations describing the impulsively set into motion Falkner‐Skan flows is transformed to a partial differential equation but the number of independent variables are reduced from three to two. Note that the new time scaling avoids the appearance of the singularity resulting from the logarithmic function ln(1‐ξ), thus reducing the difficulties of computations. Analytical approximations for the momentum boundary layer are obtained, which are valid for all time. The results are then matched numerically using the Keller‐Box method.