Absolute Minima Research Articles

Asymptotics of Finite Energy Monopoles on AC 3-Manifolds

We study the asymptotic behavior of finite energy \(\mathrm SU(2)\) monopoles, and general critical points of the \(\mathrm SU(2)\) Yang–Mills–Higgs energy, on asymptotically conical 3-manifolds with only one end. Our main results generalize classical results due to Groisser and Taubes in the particular case of the flat 3-dimensional Euclidean space \({\mathbb {R}}^3\). Indeed, we prove the integrality of the monopole number, or charge, of finite energy configurations, and derive the classical energy formula establishing monopoles as absolute minima. Moreover, we prove that the covariant derivative of the Higgs field of a critical point of the energy decays quadratically along the end and that its transverse component with respect to the Higgs field, as well as the corresponding component of the curvature of the underlying connection, actually decay exponentially. Additionally, under the assumption of positive Gaussian curvature on the asymptotic link, we prove that the curvature of any critical point connection decays quadratically. Furthermore, we deduce that any irreducible critical point converges uniformly along the conical end to a limiting configuration at infinity consisting of a reducible Yang–Mills connection and a parallel Higgs field.

Successive minima and asymptotic slopes in Arakelov geometry

Let$X$be a normal and geometrically integral projective variety over a global field$K$and let$\bar {D}$be an adelic${\mathbb {R}}$-Cartier divisor on$X$. We prove a conjecture of Chen, showing that the essential minimum$\zeta _{\mathrm {ess}}(\bar {D})$of$\bar {D}$equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that$\zeta _{\mathrm {ess}}(\bar {D})$can be read on the Okounkov body of the underlying divisor$D$via the Boucksom–Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space$X = {\mathbb {P}}_K^{d}$, our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and Rémond.

The Effect of Attending Physical Rehabilitation After the First Acute Lateral Ankle Sprain on Static Postural Control in Patients With Chronic Ankle Instability.

Static postural control deficits are commonly documented among individuals with chronic ankle instability (CAI). Evidence suggests individuals with CAI who seek medical attention after an ankle sprain report fewer subjective symptoms. It is unknown if seeking medical attention and receiving supervised physical rehabilitation has a similar effect on objective outcomes, such as static postural control. To compare measures of single-limb postural control and center of pressure (COP) location between participants with CAI who did or did not self-report attending supervised rehabilitation at the time of their first lateral ankle sprain. Retrospective cohort. Laboratory. Patients (or Other Participants): Twenty-nine participants with CAI who did (n = 14) or did not (n = 15) self-report attending supervised rehabilitation. Self-reported attendance or not of supervised rehabilitation at the time of initial injury. Participants performed three 20-second trials of single-limb stance on a force plate with eyes open. Main outcome measures included the COP velocities, time-to-boundary (TTB) absolute minima, mean of TTB minima, and SD of TTB minima in the anteroposterior and mediolateral directions. The spatial distribution of the COP data points under the foot was quantified within 4 equally proportional sections labeled anteromedial, anterolateral, posteromedial, and posterolateral. Participants who reported attending supervised rehabilitation after their initial ankle sprain had a lower COP velocity in the anterior-posterior direction (P = .030), and higher TTB anterior-posterior absolute minimum (P = .033) and mean minima (P = .050) compared with those who did not attend supervised rehabilitation. Among individuals with CAI, not attending supervised rehabilitation at the time of initial injury may lead to worse static postural control outcomes. Clinicians should continue advocating for patients recovering from an acute ankle sprain to seek medical attention and provide continued care in the form of physical rehabilitation.

Inverse Problem of the Theory of Reflection for Three-Layer Structures

We propose an original approach to analyzing the exact solution of the direct problem in the theory of reflection that allows an algorithm for solving the inverse problem, including that in media with complex refractive indices, to be constructed. We suggest a complete set of seven observable parameters that is in a one-to-one correspondence with the complete set of structure material parameters. We derive the characteristic equation specifying the positions of the absolute minima of the reflection coefficient and the equation to calculate the layer thickness.

On a functional of Kobayashi for Higgs bundles

We define a functional [Formula: see text] for the space of Hermitian metrics on an arbitrary Higgs bundle over a compact Kähler manifold, as a natural generalization of the mean curvature energy functional of Kobayashi for holomorphic vector bundles, and study some of its basic properties. We show that [Formula: see text] is bounded from below by a nonnegative constant depending on invariants of the Higgs bundle and the Kähler manifold, and that when achieved, its absolute minima are Hermite–Yang–Mills metrics. We derive a formula relating [Formula: see text] and another functional [Formula: see text], closely related to the Yang–Mills–Higgs functional, which can be thought of as an extension of a formula of Kobayashi for holomorphic vector bundles to the Higgs bundles setting. Finally, using 1-parameter families in the space of Hermitian metrics on a Higgs bundle, we compute the first variation of [Formula: see text], which is expressed as a certain [Formula: see text]-Hermitian inner product. It follows that a Hermitian metric on a Higgs bundle is a critical point of [Formula: see text] if and only if the corresponding Hitchin–Simpson mean curvature is parallel with respect to the Hitchin–Simpson connection.

Анализ изменений приземной температуры в предгорной зоне

Based on the series of surface air temperature (average, absolute maxima, absolute minima) for 1961-2018, obtained from instrumental data from the Kislovodsk station, an analysis of changes in seasonal and annual temperatures is carried out. We obtained averaged values in the base (1961-1990, normal) and modern (1991-2018) periods, the results of the t-test for determining the statistical equality/inequality of temperatures for two sub-periods, as well as characteristics of the distribution shape (asymmetry, kurtosis), the rate of temperature change with the criterion of statistical significance (F-test), the value of the Hurst exponent H for determining the stability of the series. Based on the results of statistical and fractal analysis, it can be concluded about a steady increase in temperatures (with the selection of summer averages, maximum and minimum) in the foothill zone of southern Russia (Kislovodsk).

Higher dimensional Lemniscates: the geometry of r particles in n-space with logarithmic potentials

We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension $ n \geq 3$. Lemniscates are defined as follows. Given m points $w_j $ in $\mathbb R^n$, consider the function $F(x)$ which is the product of the distances $ |x-w_j|$: the singular level sets of the function $F$ are called lemniscates. We show via complex analysis that the critical points of $F$ have Hessian of positivity at least $(n-1)$. This implies that, if $F$ is a Morse function, then $F$ has only local minima and saddle points with negativity 1. The critical points lie in the convex span of the points $|w_j| $ (these are absolute minima): but we made also the discovery that $F$ can also have other local minima, and indeed arbitrarily many. We discuss several explicit examples. We finally prove in the appendix that all critical points are isolated.

Winter positions of Arctic front during periods of cooling and warming

Winter positions of the Arctic front (AF) during the known periods of the climate cooling (1949–1980) and warming (1981–2012) were analyzed within the sector 10° W – 60° E. The AF positios were determined by the following indicators: 1) a surface pressure; 2) horizontal wind divergence; 3) geostrophic vortex; 4) geostrophic heat advection. The main extrema of these four dynamic characteristics coincide and fall on the latitude 72.5° N. This corresponds to the average position of the AF for a given resolution and confirms correctness of our choice of these characteristics as the AF indicators. Relative differences between mean profiles of all values of the above warm and cold periods were calculated using method of normalization of each value for the corresponding latitude by the standard deviation for the entire period (1949–2012). To study variability of the AF position we used mean yearly winter profiles of the variables under investigation together with the statistical analysis of positions of the extrema within the latitude degrees. For pressure and geostrophic advection positions of the absolute minima were determined while for geostrophic vortex and divergence – positions of the absolute maxima. The data show that according to different criteria the AF average positions for the period 1949–2012 lie within the zone 72.4–73.4 N. The interannual variability of the AF positions lies within the 1–2 degrees of latitude and corresponds to the range of the air temperature variability above the zone of maximal changes in the sea ice area. According to the standard deviation values of the divergence and the geostrophic vortex are the most stable in region of the AF passage. Comparison of differences of the studied characteristics between the warm and cold periods shows that the changes in the AF positions are not statistically significant (P(t) < 91% t‑criterion) unlike the changes in positions of isolines which characterize the warming (P(t) = 100%). Thus, despite significant changes in properties of the surface and the temperature regime to the north of 72.5 N (the warming), according to all the criteria the AF climatic position remains quasi‑stationary for 32‑year periods of averaging.

Ab-initio Calculations of Electronic, Transport, and Bulk Properties of cubic Boron Nitride (zb-BN)

We present results from ab-initio, self consistent, local density approximation (LDA) calculations of electronic and related properties of cubic boron nitride (zb-BN). We employed the Ceperley and Alder LDA potential and the linear combination of atomic orbitals (LCAO) formalism in our non-relativistic computations. We solved the system of LDA equations self-consistently, through the implementation of the LCAO formalism, following the Bagayoko, Zhao, and Williams (BZW) method as enhanced by Ekuma and Franklin (BZW-EF). The BZW-EF method includes a methodical search for the optimal basis set that yields the absolute minima of the occupied energies. This search entails increasing the size of the basis set and the related modifications of angular symmetry and of radial orbitals. Our calculated, indirect band gap of 6.48 eV, from the Γ to the Χ points, and bulk modulus of 375 GPa are in excellent agreement with corresponding experimental values at room temperature (RT). The calculated widths of the lowest valance band and that of the entire valence bands of 5.65 eV and 20.26 eV are in excellent agreement with the measured values of 5.5 eV and 20 eV, respectively. We have also calculated electron and hole effective masses for zb-BN, and the total (DOS) and partial (pDOS) densities of states.  Â

Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions

This paper, which is the follow-up to part I, concerns the equation ( − Δ ) s v + G ′ ( v ) = 0 (-\Delta )^{s} v+G’(v)=0 in R n \mathbb {R}^{n} , with s ∈ ( 0 , 1 ) s\in (0,1) , where ( − Δ ) s (-\Delta )^{s} stands for the fractional Laplacian—the infinitesimal generator of a Lévy process. When n = 1 n=1 , we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits ± 1 \pm 1 at ± ∞ \pm \infty ) if and only if the potential G G has only two absolute minima in [ − 1 , 1 ] [-1,1] , located at ± 1 \pm 1 and satisfying G ′ ( − 1 ) = G ′ ( 1 ) = 0 G’(-1)=G’(1)=0 . Under the additional hypotheses G ( − 1 ) > 0 G(-1)>0 and G ( 1 ) > 0 G(1)>0 , we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution. For n ≥ 1 n\geq 1 , we prove some results related to the one-dimensional symmetry of certain solutions—in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.

Entorhinal cortex stellate cell synchronization

Theta-frequency oscillations (8-12 Hz) of large groups of synaptically coupled cells are commonly seen throughout the mammalian brain. The theta oscillations of entorhinal cortex grid cells and hippocampal place cells have received special attention recently due to their role they apparently play in the encoding of positional information. In a previous study [1], we found that biologically realistic computational models of two or three synaptically coupled entorhinal cortex stellate cells [2] generally synchronize their oscillations much more quickly from an initially random phase configuration if their intrinsic firing frequencies are near the theta range (with the absolute minima generally at about 15-20 Hz) than at significantly either higher or lower ranges. However, our model looked only at identical neurons with the same intrinsic frequencies and lacking any channel noise. Here we present results from further studies in which realistic channel noise and frequency heterogeneity are included. We found that channel noise could either increase or decrease the time for coupled stellate cells to synchronize by a significant amount, depending on which channel the noise was applied to. We also found that the synchronization time appears to be largely independent of the underlying cell parameters that determine the intrinsic frequencies (the persistent sodium and hyperpolarization-activated cation conductances) and depend only on the frequencies themselves, potentially simplifying the process of parameter sweeps. We found evidence that synchronization is favored for a mean firing frequency of 18 Hz even when the individual intrinsic frequencies of the neurons differ by up to about 0.5 Hz (see Figure ​Figure1).1). Finally, we found evidence of a bifurcation in the stability of synchronization states of two coupled identical stellate cells, with in-phase synchronization suddenly switching to out-of-phase synchronization as the synaptic coupling strength is increased, suggesting a complex picture of entorhinal cortex synchronization patterns similar to previous studies of Hodgkin-Huxley neurons [3]. Figure 1 Long-term synchronization of two coupled entorhinal cortex stellate cell neurons with slightly different intrinsic firing frequencies. The mean of their intrinsic frequencies is plotted against a parameter (“Delta Coll 1”) relating the ...

Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term

A gradient-like property is established for second order semilinear conservative systems in presence of a linear damping term which is asymptotically weak for large times. The result is obtained under the condition that the only critical points of the potential are absolute minima. The damping term may vanish on large intervals for arbitrarily large times and tends to $0$ at infinity, but not too fast (in a non-integrable way). When the potential satisfies an adapted, uniform, Łojasiewicz gradient inequality, convergence to equilibrium of all bounded solutions is shown, with examples in both analytic and non-analytic cases.

Spatial and temporal variation of extremely low minimum temperatures in Hungary during the period between 1951 and 2010

For this study, data of 16 meteorological stations have been processed over a period of 60 years with the purpose to reveal the spatial and temporal structure of the frequencies of absolute extreme minima in Hungary with special reference to the surmised global rise of temperatures on a worldwide scale. In the main areas of fruit growing, the monthly or seasonal absolute temperature maxima and minima are presented during the 60 year-long period and projected on the geographical map. For the main fruit- and vegetable growing regions the probability of winter- and late frosts is of prime interest. The time series of extreme temperatures though did not prove signifi cant changes over the period observed, but the information upon changes and their tendencies is a precious tool being utilised in planning, choice of adequate varieties for a longer period of time in the future. The deleterious winter frosts experienced in fruit production is not a consequence of a sole drop to a minimum temperature, but of an earlier period of mild temperatures during the winter, which sensibilised the trees. Frequent and extreme variations of temperatures may cause troubles at any time during the year and reduce the yields conspicuously. We ought to get familiar with the hazards of our climate and fi nd optimal solutions to mitigate the damages expected. The seasonal and monthly distribution projected on the geographical map we can follow up also the spatial relations and the signifi cance of their occurrence. Coeffi cients of variation between meteorological happenings at different localities facilitate the calculation of the probability of risks on the surrounding areas.

Large gallanes and the PSEPT theory: a theoretical study of Ga n H n+2 clusters (n = 7–9)

Do alanes Al(n)H(n+2) and gallanes Ga(n)H(n+2) satisfy the polyhedral skeletal electron pair theory (PSEPT)? Taking into account previous work on this question, this paper provides a convincing answer and proposes the reformulation of the (n + 1) electron pairs rule of Wade and Mingos (W-M) for such systems. Following recent studies of tetra-, penta-, hexa-, hepta-, octa-, and nonaalanes as well as valence-isoelectronic/related gallanes, in this paper we present an analysis of the hydrides of aluminum and gallium A(n)H(n+2) (A = Al, Ga and n = 7-9). The aim is still to examine the applicability of PSEPT, especially the W-M rule, to these clusters. Exploration of the total potential energy surfaces (PESs) of hepta-, octa-, and nonagallanes shows that the absolute minima have a nido-like polyhedron arrangement. Unlike the smaller Ga(n)H(n+2) (n = 4, 5, 6), it seems that the size of the cluster largely dictates its preferred geometry. Although none of them have closed (totally triangular) cages, these clusters exhibit significant compactness, comparable to borane double anions, B(n)H(n) (2-), which are the archetypes for the PSEPT theory.

Curvature statistics of streamlines in various turbulent flows

Geometrical features of turbulent flows are analyzed by studying the curvature field of streamlines based on the instantaneous fluctuating velocity field of four different direct numerical simulations (DNS) at Taylor-based Reynolds numbers in the range Re λ = 50–300. Among the flows are two decaying homogeneous isotropic flows, one isotropic forced flow and the other homogeneous shear flow. Two different measures of curvature are extracted, the first one is the geometrical curvature of streamlines based on Frenet’s formulas, and the second one is the Gaussian curvature describing the divergence of streamlines. The two curvature fields are found to be related to each other by the divergence of the directional tensor of streamlines, which is also related to the total acceleration along streamlines. The latter identity allows the identification of the convective term in the Navier–Stokes equations, as the term determining the curvature-related geometrical features of streamlines. Based on the random sweeping hypothesis, a scaling of the standard deviation of the two curvatures of streamlines in turbulent flows with the inverse of the Taylor microscale is obtained. The curvature fields are in a first step analyzed via their probability density functions (pdfs) of the fluctuating fields (including their sign), and a normalization with the standard deviation yields a good collapse of the curves for different Reynolds numbers and flow types. Both, the positive and negative tails of the pdf display a pronounced algebraic tail with a decay exponent of −4. To analyze the curvatures as geometrical features of streamlines, the pdf of their absolute values normalized with their respective mean values are examined, and the curves are found to collapse well for the different Reynolds numbers. All pdfs again display an algebraic tail for large values of the curvature with a scaling exponent of −4. This result is theoretically explained based on the expression for the Gaussian curvature, and turns out to be related to the regions in the flow close to stagnation points where streamlines get strongly diverted. It turns out that at the origin the pdf of the Gaussian curvature is dominated by regions where the gradient of the absolute value in streamline direction us vanishes. us =0, However, it defines an isosurface in which all local extreme points of the field of the absolute value of the velocity u as well as those of the turbulent kinetic energy k lie. As stagnation points are absolute minima of the kinetic energy field, these points too lie in the isosurface. The topology of the isosurface is analyzed in the vicinity of stagnation points, and it is found that independent of the type of stagnation point, the isosurface is locally always a degenerated quadric surface of cone type, a conjecture that is verified based on the comparison of the analytical expansion and the DNS data.

CCSD(T)-F12a Study of Reactions of Interstellar Anions C2nH−(n = 2−4) with HCCH

Explicitly correlated coupled cluster theory at the CCSD(T)-F12a level (Adler, T. B.; Knizia, G.; Werner, H.-J. J. Chem. Phys. 2007, 127, 221106) was employed to study the energetics of reactions of interstellar anions C(2n)H(-) with HCCH, which may formally be regarded as proton-transfer reactions. The gas-phase acidities at 0 K for the polyynes HC(2n)H with n = 2-4 are predicted to be 1516.5, 1483.2, and 1462.6 kJ mol(-1). The energy profiles of the collinear reactions are characterized by absolute minima corresponding to the hydrogen-bonded species HC(2n)(-)...HCCH, saddle points at which the proton is shared between the two anion moieties, and a second minimum in the exit channel of the reaction, which becomes less pronounced with increasing n.

Hamiltonian systems as selfdual equations

Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative action functionals obtained by a generalization of Bogomolnyi’s trick of ‘dcompleting squares’. Reminiscent of the selfdual Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the corresponding Euler-Lagrange equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study certain Lagrangian intersections.

DFT-Based Studies on the Jahn−Teller Effect in 3d Hexacyanometalates with Orbitally Degenerate Ground States

The topology of the ground-state potential energy surface of M(CN)(6) with orbitally degenerate (2)T(2g) (M = Ti(III) (t(2g)(1)), Fe(III) and Mn(II) (both low-spin t(2g)(5))) and (3)T(1g) ground states (M = V(III) (t(2g)(2)), Mn(III) and Cr(II) (both low-spin t(2g)(4))) has been studied with linear and quadratic Jahn-Teller coupling models in the five-dimensional space of the epsilon(g) and tau(2g) octahedral vibrations (Tg[symbol: see text](epsilon(g)+tau(2g)) Jahn-Teller coupling problem (T(g) = (2)T(2g), (3)T(1g))). A procedure is proposed to give access to all vibronic coupling parameters from geometry optimization with density functional theory (DFT) and the energies of a restricted number of Slater determinants, derived from electron replacements within the t(2g)(1,5) or t(2g)(2,4) ground-state electronic configurations. The results show that coupling to the tau(2g) bending mode is dominant and leads to a stabilization of D(3d) structures (absolute minima on the ground-state potential energy surface) for all complexes considered, except for [Ti(CN)(6)](3-), where the minimum is of D(4h) symmetry. The Jahn-Teller stabilization energies for the D3d minima are found to increase in the order of increasing CN-M pi back-donation (Ti(III) < V(III) < Mn(III) < Fe(III) < Mn(II) < Cr(II)). With the angular overlap model and bonding parameters derived from angular distortions, which correspond to the stable D(3d) minima, the effect of configuration interaction and spin-orbit coupling on the ground-state potential energy surface is explored. This approach is used to correlate Jahn-Teller distortion parameters with structures from X-ray diffraction data. Jahn-Teller coupling to trigonal modes is also used to reinterpret the anisotropy of magnetic susceptibilities and g tensors of [Fe(CN)(6)](3-), and the (3)T(1g) ground-state splitting of [Mn(CN)(6)](3-), deduced from near-IR spectra. The implications of the pseudo Jahn-Teller coupling due to t(2g)-e(g) orbital mixing via the trigonal modes (tau(2g)) and the effect of the dynamic Jahn-Teller coupling on the magnetic susceptibilities and g tensors of [Fe(CN)(6)](3-) are also addressed.

CRITICALITY OF CHARACTERISTIC VECTOR FIELDS ON ALMOST COSYMPLECTIC MANIFOLDS

Main interest of the present paper is to investigate the criticality of characteristic vector fields on almost cosymplectic manifolds. Killing critical characteristic vector fields are absolute minima. This paper contains some examples of non-Killing critical characteristic vector fields.

A necessary condition for best fitting analytical to simulated density profiles in dark matter haloes

Analytical and geometrical properties of generalized power-law (GPL) density profiles are reviewed, and special effort is devoted to the special cases where GPL density profiles reduce to (i) a double power-law (DPL), and (ii) a single power-law (SPL). Then GPL density profiles are compared with simulated dark haloes (SDH) density profiles, and non-linear least-squares fits are prescribed, involving five parameters (a scaling radius, r 0, a scaling density, ρ 0, and three exponents, α, β, γ), which specify the fitting profile (RFSM5 method). More specifically, the validity of a necessary condition for the occurrence of an extremal point, is related to the existence of an intersection between three surfaces in a three-dimension space. Using the algorithm makes also establish that the extremal point is a fiducial minimum, while the explicit calculation of the Hessian determinant is avoided to gain in simplicity. In absence of a rigorous proof, the fiducial minimum can be considered as nothing but a fiducial absolute minimum. An application is made to a sample of 17 SDHs on the scale of cluster of galaxies, within a flat ΛCDM cosmological model (E. Rasia, G. Tormen, L. Moscardini, MNRAS 351 (2004) 237). In dealing with the averaged SDH density profile (ADP), a virial radius, r vir, equal to the mean over the whole sample, is assigned, which allows the calculation of the remaining parameters. The following results are found. (i) A necessary condition for the occurrence of an extremal point is satisfied for eight sample haloes, and is not for the remaining nine together with ADP. In the former alternative, an extremal minimum point (EMP) may safely exist. In the latter alternative, the occurrence of an EMP cannot be excluded, but only a non-extremal minimum can be determined. (ii) The occurrence of an EMP implies a sum of square residuals which is systematically lower than its counterpart deduced by use of numerical RFSM5 methods. Accordingly, EMPs may safely be thought of as absolute minima. With regard to sample haloes where no EMP is detected, the above result maintains in four cases (including ADP), while the contrary holds for the remaining five cases. (iii) The best fit (with no EMP detected) is provided by DPL density profiles for three sample haloes. In addition, DPL density profiles make a rough, but viable approximation in fitting SDH density profiles. The contrary holds for SPL density profiles. No evident correlation is found between SDH dynamical state (relaxed or merging) and asymptotic inner slope of the logarithmic density profile or (for SDH comparable virial masses) scaled radius. Mean values and standard deviations of some parameters are calculated and, in particular, the decimal logarithm of the scaled radius, ξ vir, reads 〈 log ξ vir 〉 ≈ 0.59 and σ s log ξ vir ≈ 0.59 , the standard deviation exceeding by a factor 3.3–3.4 its counterpart evaluated in an earlier attempt using Navarro et al. (J.F. Navarro, C.S. Frenk, S.D.M. White, MNRAS 275 (1995) 720, J.F. Navarro, C.S. Frenk, S.D.M. White, ApJ 462 (1996) 563, J.F. Navarro, C.S. Frenk, S.D.M. White, ApJ 490 (1997) 493) density profiles (J.S. Bullock, T.S. Kolatt, Y. Sigad, MNRAS 321 (2001) 559). If a large dispersion still maintains for richer samples, in dealing with analytical RSFM5 methods, a low dispersion found in N-body simulations seems to be an artefact, due to the assumption of Navarro et al. (or any equivalent choice) density profile. It provides additional support to the idea, that Navarro et al. density profiles may be considered as a convenient way to parametrize SDH density profiles, without implying that it necessarily produces the best possible fit (J.S. Bullock, T.S. Kolatt, Y. Sigad, MNRAS 321 (2001) 559). With regard to RFSM5 methods formulated in the current paper, the exponents of both the best fitting GPL density profile to ADP, ( α, β, γ) ≈ (0.3, 4.5, 1.5), and related averages calculated over the whole halo sample, ( α ¯ , β ¯ , γ ¯ ) ≈ ( 3.0 , 3.9 , 1.3 ) , are far from their Navarro et al. counterparts, ( α, β, γ) = (1, 3, 1). The last result, together with the large value of the standard deviation, σ s log ξ vir , is interpreted as due to a certain degree of degeneracy in fitting GPL to SDH density profiles. If it is a real feature of the problem, or it could be reduced by the next generation of high-resolution simulations, still remains an open question. Values of asymptotic inner slope of fitting logarithmic density profiles, are consistent with results from recent high-resolution simulations (J. Diemand, B. Moore, J. Stadel, MNRAS 353 (2004) 624; D. Reed, F. Governato, L. Verde, et al., MNRAS 357 (2005) 82).