Noninjectivity of the monodromy of certain equicritical strata

The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group. We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on CP1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{C}\\mathbb{P}^1$$\\end{document} induced by the logarithmic derivative df/f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{d}f/f$$\\end{document} of a polynomial in the stratum.

An ellipsometric procedure developed for noncontact in situ measurement of the CdTe buffer-layer temperature is presented. The procedure is based on the temperature dependence of the energy position of CdTe critical points and is intended for determining the initial temperature of the growth surface before epitaxy of the cadmium–mercury–telluride compound. An express method for determining the position of critical points by the spectra of ellipsometric parameter Ψ is proposed. A series of calibrated experiments is performed. They result in determination of the temperature dependences of the position of critical points. Estimations and the experiment show that the temperature-measurement accuracy is ±3°C.

An ellipsometric procedure developed for noncontact in situ measurement of the CdTe buffer-layer temperature is presented. The procedure is based on the temperature dependence of the energy position of CdTe critical points and is intended for determining the initial temperature of the growth surface before epitaxy of the cadmium–mercury–telluride compound. An express method for determining the position of critical points by the spectra of ellipsometric parameter Ψ is proposed. A series of calibrated experiments is performed. They result in determination of the temperature dependences of the position of critical points. Estimations and the experiment show that the temperature-measurement accuracy is ±3°C.

Purpose. Conduct a study on the kinetics of decomposition of undeformed supercooled austenite during continuous cooling of boron-containing steel. Establish the patterns of structure formation in boron-containing steel during continuous cooling at various rates from heating temperatures. Methodology. The study of phase transformations and the kinetics of decomposition of undeformed supercooled austenite was conducted using differential thermal analysis on samples of boron-containing steel, accompanied by investigations of microstructure and hardness. Findings. The kinetics of transformations were studied, and thermokinetic diagrams of the decomposition of undeformed supercooled austenite in boron-containing steel were constructed. Critical points were examined, and the peculiarities of the influence of the cooling rate of undeformed austenite on the volume fraction of structural components in boron-containing steels were identified. Originality. The positions of critical temperature points have been demonstrated, and both qualitative and quantitative depictions of the structure formation of the investigated boron-containing steel have been obtained. The influence of cooling rate on the nature of austenite transformations and the mechanical properties of the studied steel has been revealed. Practical value. The conducted research (knowledge of the positions of critical points) enables the targeted selection of heating, cooling, and isothermal holding temperatures during the development of annealing regimes both in the subcritical temperature range and with partial or complete phase recrystallization. The obtained results can be utilized in designing various thermal treatment regimes applied in industrial technologies for the production of coiled rolled products made from boron-containing steel for cold forging.

In the preceding paper, basic thermodynamic functions have been derived for a solution of macromolecules associating by means of functional groups (stickers), thus forming junctions, each of which consists of m stickers and bm molecules of solvent. In the present communication, phase stability of the systems under consideration is analyzed. For several values of parameters m and b, spinodal curves are calculated and their minima (i.e. critical solution points) are localized. At higher junction multiplicity values (e.g. at m = 10), a local maximum appears on the spinodal, dividing it into two lobes, each displaying a different critical point. The effect of the stoichiometric parameter b on splitting the spinodal and on positions of the critical solution points is described.

Electrostatic preorganization is thought to be a principle factor responsible for the impressive catalytic capabilities of enzymes. The full protein structure is believed to facilitate catalysis by exerting a highly specific electrostatic field on the active site. Computationally determining the extent of electrostatic preorganization is a challenging process. We propose using the topology and geometry of the electron charge density in the enzyme's active site to asses the effects of electrostatic preorganization. In support of this approach we study the convergence of features of the charge density as the size of the active site model increases in Histone Deacetylase 8. The magnitude of charge density at critical points and most Bader atomic charges are found to converge quickly as more of the protein is included in the simulation. The exact position of critical points however, is found to converge more slowly and be strongly influenced by the protein residues that are further away from the active site. We conjecture that the positions of critical points are affected through perturbations to the wavefunctions in the active site caused by dipole moments from amino acid residues throughout the protein. We further hypothesize that electrostatic preorganization, from the point of view of charge density, can not be easily understood through the charges on atoms or the nature of the bonding interactions, but through the relative positions of critical points that are known to correlate with reactivity and reaction barriers.

In the present work, we study the fidelity susceptibility and the entanglement entropy in an antiferromagnetic spin-1 chain with additional next-nearest neighbor interactions and three-site interactions, which are given by H=(J1SiSi+1+ J2SiSi+2)+[J3(SiSi+1)(Si+1Si+2)+ h.c.]. By using the density matrix renormalization group method, the ground-state properties of the system are calculated with very high accuracy. We investigate the effect of the three-site interaction J3 on the fidelity susceptibility numerically, and then analyze its relation with the quantum phase transition (QPT). The fidelity measures the similarity between two states, and the fidelity susceptibility describes the associated changing rate. The QPT is intuitively accompanied by an abrupt change in the structure of the ground-state wave function, so generally a peak of the fidelity susceptibility indicates a QPT and the location of the peak denotes the critical point. For the case of J2=0, a peak of the fidelity susceptibility is found by varying J3, and the height of the peak grows as the system size increases. The location of the peak shifts to a slightly lower J3 up to a particular value as the system size increases. Through a finite size scaling, the critical point J3c=0.111 of the QPT from the Haldane spin liquid to the dimerized phase is identified. We also study the effect of the three-site interaction on the entanglement entropy between the right half part and the rest. It is noted that the peak of the entanglement entropy does not coincide with the critical point. Instead, the critical point is determined by the position at which the first-order derivative of the entanglement entropy takes its minimum, since a second-order QPT is signaled by the first derivative of density matrix element. Moreover, the entanglement entropy disappears when J3=1/6, which corresponds to the size-independent Majumdar-Ghosh point. The positions of quantum critical points extracted from these two quantum information observables agree well with those obtained by the string order parameters, which characterizes the topological order in the Haldane phase. Secondly, we also study the case of J20, and obtain the critical points by both the fidelity susceptibility and the entanglement entropy. Finally we provide a ground-state phase diagram of the system. To sum up, the quantum information observables are effective tools for detecting diverse QPTs in spin-1 models.

While the phenomenon of stereoscopy has been to a limited extent developed in a practical manner, there appears to have been but little actual research work done within recent years, the impression being a general one that the particular field had been already thoroughly explored. If an attempt be made to reproduce the experiments described by Wheatstone in his original paper, in which his great discovery of stereoscopy was published, it will be found that the results are somewhat different in important respects from those described. From experiments recently carried out by the author, it appears that stereoscopic vision is possible only within certain limits. For certain pairs of objects, whether on the same or different horizons, there are generally two extreme critical points beyond which stereoscopic vision breaks down. For certain pairs of objects on the same horizon there are two inner critical points. Objects of dissimilar form but approximately the same average angular dimensions can frequently be combined stereoscopically: thus a circle can be combined with a triangle if their average angular dimensions are about equal. When the angular dimensions are very different, combination is generally impossible: thus, for example, a thin line cannot be combined with a thick line or triangle or other form of considerably larger size. When the pairs of objects are dissimilar in size there is only one pair of outer and inner critical points if the objects are on the same horizon and one outer critical point if they are on different horizons. For pairs of objects any of which can be combined together, there are two pairs of such critical points. Data regarding the positions of the critical points under various conditions are contained in the paper, which also deals generally with the effects of variation of the conditions of observation and particularly variations of the illumination.

To evaluate the effect of tibial tuberosity advancement (TTA) on tibiofemoral shear force as reflected by measurement of cranial tibial subluxation (CTS) and patella tendon angle (PTA) in the canine cranial cruciate ligament (CrCL) deficient stifle joint. In vitro cadaver study. Canine cadaveric hind limbs (n=10). CTS and PTA were evaluated from lateral radiographic projections in axially loaded intact CrCL stifle joints, after transection of the CrCL, at a maximally advanced tibial tuberosity position, and at a critical point position. A custom-designed hinge plate allowed alteration of the tibia to tibial tuberosity distance (Ti-TT) under axial load. Digitized radiographic images were used to quantify CTS, PTA, and Ti-TT. Comparisons within groups were made using 1-way repeated measures ANOVA. A post hoc Tukey's HSD test was used to determine post-ANOVA pair-wise comparison within these groups. Significance was set at a value of P<.05. CTS occurred after CrCL transection, which was significantly different from the intact position (P<.01). Subsequent stability of the stifle joint was obtained by advancing the tibial tuberosity. In the maximally advanced tibial tuberosity position, caudal tibial thrust was generated resulting in caudal tibial subluxation that was significantly different from the transected CrCL position (P<.01) and from the intact CrCL position (P<.01). Despite a stable joint, there was slight CTS at the critical point position, which was significantly different from the intact CrCL position (P<.05). The PTA at the maximally advanced position was significantly different from the intact, critical point and reference 90 degrees PTAs (P<.01). The PTA at the critical point position was significantly different from the intact and maximally advanced tibial tuberosity PTAs (P<.01), but not different (P>.05) from the reference 90 degrees PTA. We demonstrated that advancement of the tibial tuberosity neutralized cranial tibial thrust, and converted cranial tibial thrust into caudal tibial thrust. Neutralization of tibiofemoral shear forces occurred at a PTA of 90.3+/-9.0 degrees. TTA can effectively change the magnitude and direction of the tibiofemoral shear force, and thus may be used to prevent craniotibial translation in a CrCL deficient stifle joint.

The results of experimental investigation of near—critical point states of liquid‐vapour phase transition of of lithium, sodium and aluminium are presented. The metal foil samples were launched by explosively driven steel plate in Helium atmosphere; Li and Na—by direct impact and Al—by impact through the layer of helium. The heating of the Li and Na foils were performed by heat exchange with shocked He layer from the free side of sample; Al—by heating by multiple‐shocked He from the back side of the foil. The temperature of sample surface was measured by fast multi‐channel optical pyrometer. For Li and Na experiments the pressure was obtained from measured shock velocity in helium using base length technique; the 1‐D simulation of the process of launching was performed to obtain pressure for Al experiments. The obtained experimental information allowed to evaluate liquid spinodal line, and the position of critical points on pressure—temperature plane for investigated metals.

The stability of critical/stagnation points and streamline topologies along with their local and global bifurcations for peristaltic flow of an incompressible power-law model have been investigated. The flow is analyzed in an axisymmetric tube subject to the constraints of long wavelength and low Reynolds number, and an exact analytic solution is found in a moving frame of reference. Nonlinear autonomous differential equations are developed to unveil the particle paths and position of critical points in the flow. Stability of obtained critical points and streamline patterns along with their local bifurcations are inspected by using the theory of dynamical system. Three different flow distributions are found in the flow: backward flow, trapping and augmented flow. The transitions between these flow situations correspond to bifurcations in the flow field where non-hyperbolic degenerate points change their behavior to form heteroclinic connection between saddles. The nature and stability of these bifurcations and corresponding topological changes are explored graphically. A global bifurcation diagram is used to epitomize these bifurcations. The fluid behavior index has a significant impact on the stability of critical points and the ranges of flow regions. The backward flow region and trapping region are observed to shrink and expand, respectively, with an increase in the fluid behavior index. The findings of present article are verified by comparing them with the results already existing in the literature.

We develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared. In all, seven different bootstrap methods are examined, each being usable in both parametric and nonparametric contexts. Emphasis is on the way in which the methods cope with first- and second-order departures from normality. Percentile-$t$ and accelerated bias-correction emerge as the most promising of existing techniques. Certain other methods are shown to lead to serious errors in coverage and position of critical point. An alternative approach, based on "shortest" bootstrap confidence intervals, is developed. We also make several more technical contributions. In particular, we confirm Efron's conjecture that accelerated bias-correction is second-order correct in a variety of multivariate circumstances, and give a simple interpretation of the acceleration constant.

Iterated bootstrap procedures may be used to reduce error in many statistical problems. We discuss their use in constructing confidence intervals with accurate coverage and show that bootstrap coverage correction produces improvements in coverage accuracy of order n −1/2 in one-sided intervals, but of order n −1 in two-sided intervals. Explicit formulas are provided for the dominant term in coverage error after iteration in each case. These results are used to compare various iterated bootstrap intervals and to assess the effect of bootstrap iteration on other indicators of interval performance, such as position of critical points and length of interval. We show that, for one-sided intervals, the coverage-correction algorithm yields critical points that are second-order correct. This is not the case for two-sided intervals, where second-order correctness is not crucial in obtaining high-order coverage accuracy. We also show that the asymptotic mean increase in length between the original and coverage-corrected intervals is proportional to the coverage error of the original interval. The arguments are illustrated via several examples, including the application of the bootstrap coverage-correction algorithm to reduce coverage error of Bartlett-corrected likelihood-based confidence regions from order n −2 to order n −3.

In this paper, we perform the dynamical system analysis of the cosmological models framed in the extended teleparallel gravity, the f(T, B) gravity. We use the mapping, f(T, B)rightarrow -T+tilde{f}(T, B), and define the dynamical variables to form the autonomous dynamical system. The critical points are obtained in two well-motivated forms of f(T, B), one that involves the logarithmic form of the boundary term B, and the other one is the non-linear form of the boundary term. The position of critical points is shown in the different evolutionary phases of the Universe such as radiation, matter, and de-Sitter phase. The stability condition of each of the critical points of both the models is derived and the behavior of each point has been obtained mathematically and through the phase portrait. The evolution of standard density parameters such as radiation (Omega _{r}), matter (Omega _{m}), and dark energy (Omega _{DE}) are also analyzed. Further to connect with the present cosmological scenario, the behavior of deceleration and equation of state parameter both in the dark energy phase (omega _{DE}) and total (omega _{tot}) are shown from the initial condition of the dynamical variables. The accelerating behaviour has been obtained for both models.

The visualization of vector fields is still based on piecewise linear approximation. This is fast and good enough in large areas but has drawbacks if the non-linear behavior of a field has local topological implications like close simple critical points or higher order singularities. This article introduces the concept of Clifford algebra into the visualization of vector fields to deal with these difficulties. It derives a close relationship between the description of some polynomial 2D vector fields in Clifford algebra and their topology, especially the index and the position of critical points. This is used to develop an algorithm for vector field visualization without the problems of conventional methods.