Equilateral Hyperbola Research Articles

Good correlation between necessary remifentanil concentrations in individual patients determined from sedative-analgesic interactions using pharmacokinetic simulations and the remifentanil concentrations used at anesthesiologists’ discretion

Background: Analgesic and sedative agents interact with each other, and their relationship is explained in a curve convex below. In the automated control anesthesia system based on this relationship that we developed, the dose of analgesic is adjusted with the necessary remifentanil concentration, determined with propofol-remifentanil interactions, as an estimated maximal individual concentration (esMIC). With the system, a study in patients under anesthesia management by an anesthesiologist was conducted to compare the effect-site concentration (ESC) of remifentanil administered at the anesthesiologist’s discretion and the esMIC calculated with drug-drug interactions for their relationship and to assess whether the analgesic dose administered based on esMIC is appropriate.
 Methodology: In the present study involving 20 patients, anesthesiologists changed the propofol (P) infusion rate and remifentanil (R) infusion rate to maintain BIS value of 45. The estimated target-effect-site concentration of propofol for maintaining BIS 45 and the ESC of remifentanil (ESC_R) based on the model by Minto et al. were calculated. Moreover, with these data sets, the isodynamic curve of ESC of propofol (ESC_P) for maintaining BIS 45 (Y) and ESC_R (X) was determined to be an equilateral hyperbola (Y = c/(X-a) + b). With the ESC_R at which even raising ESC_R would result in small decreases in ESC_P considered the esMIC, the ESC_R at the point at which the slope of this curve is apos;1 (neutral point) and at the point of the curve where the y-component deviation of the asymptote (y = b) is 20% (esMIC20) calculated every 6 sec. The ESC_R at the time point of an adequate analgesic state as deemed by the anesthesiologist after anesthesia had been started for 15 min was compared with the esMIC20 determined by objective calculations.
 
 Results: The ESC_R and esMIC were 11.9 ± 2.4 and 11.6 ± 2.0 ng/ml, respectively. Moreover, the median ESC_R and esMIC in the patients showed a very good correlation (correlation coefficient R² = 0.88, p < 0.01).
 Conclusion: The necessary analgesic concentrations estimated with drug-drug interactions did not contradict those determined at anesthesiologists’ discretion and are considered to assure a reasonable analgesic state.
 Key words: Analgesia; Sedation; Remifentanil; Effect-Site Concentrations
 Citation: Matsuki Y, Nagata O, Shigemi K. Good correlation between necessary remifentanil concentrations in individual patients determined from sedative-analgesic interactions using pharmacokinetic simulations and the remifentanil concentrations used at anesthesiologists’ discretion. Anaesth. pain intensive care 2023;27(5):491−494; DOI: 10.35975/apic.v27i5.2300
 Received: April 04, 2023; Reviewed: May 13, 2023; Accepted: June 16, 2023

Direct proof through an equilateral hyperbola to find the infinite Pythagorean triples and a direct proof to find the sum of all odd numbers through a telescopic series

In this article a prove new properties like a connection with Pythagorean triples and a hiperbolic equation. I prove a direct proof about a serie, a telescopic series about the even sum and odd sum. I had an article that use this properties but this results is mine. I discovered this theorem and prove in this article.

Quaternionic Product of Equilateral Hyperbolas and Some Extensions

This note concerns a product of equilateral hyperbolas induced by the quaternionic product considered in a projective manner. Several properties of this composition law are derived and, in this way, we arrive at some special numbers as roots or powers of unit. Using the algebra of octonions, we extend this product to oriented equilateral hyperbolas and to pairs of equilateral hyperbolas. Using an inversion we extend this product to Bernoulli lemniscates and q-lemniscates. Finally, we extend this product to a set of conics. Three applications of the given products are proposed.

GENERALIZATION OF CIRCUITS FOR DETERMINING THE ACCOUNTING PARAMETER OF THE DISTANCE FROM THE PHYSICAL FIELD POINT TO THE POINT ENERGY SOURCES.

This paper presents a generalization of two schemes for determining the parameter for taking into account the influence of the distance from an arbitrary point of a physical field to a point source of energy. Previously, both of these schemes were based on central projection [1].The first scheme made it possible to take into account the unlimited distance between the point of the physical field and the point source of energy. The analytical description of this scheme is an equilateral hyperbola, which can be used instead of central projection to determine the parameter t taking into account the influence of the distance between the indicated points.The second scheme simplifies the determination of the parameter t by limiting the distance l, which affects the potential of an arbitrary point of the physical field, conditionally assuming that there is a maximum distance lmax at which the energy potential at the point of the physical field is practically absent.In this study, two previously presented circuits were generalized to determine parameter t, which takes into account the influence of the distance from the physical field point to the point energy sources. These schemes were based on central projection, which in this study was replaced by matching the points of the two axes using a hyperbolic constraint.

On quaternionic product on some sets of hyperbolas

The aim of this paper is to introduce some products, induced by the quaternionic product, on the set of equilateral hyperbolas. The study is a mix of elements from Euclidean and projective geometry. Some properties of these products are highlighted and are connected with some special numbers as roots or powers of the unit. Then we extend these products in a natural manner to oriented equilateral hyperbolas and to pairs of equilateral hyperbolas, using the algebra of octonions. Further, using an inversion, we extend these products to Bernoulli lemniscates and to Bernoulli q-lemniscates. Finally, using some isomorphisms, we extend these products to conics, and using an inversion, we extend these products to other curves. In order to highlight the importance of these studied products, the paper ends with some applications.

Геометрия электростатических полей

Electrostatic fields have been most fully studied as special cases of electromagnetic field. They are created by a set of charged bodies that are considered immovable in relation to the observer and unchanged in time [1–3; 19; 20; 27]. Since any field is characterized by basic quantities, then such quantities for electrostatic fields are strength E and potential ϕ. Therefore, geometrically, such fields are characterized by a combination of force and equipotential lines. These fields were considered in the thesis of N.P. Anikeeva [1]. In particular, the author notes that in the case of dissimilar equal charges "... families of force and equipotential lines make up two orthogonal bundles of circles" [1, p. 59]. However, it is necessary to clarify that each "force" circle gj represents by itself not one but two lines of force, which emanate from a positive charge and terminate on a negative one. A similar review on the work [1] can be done with respect to the picture of two equal positive charges’ electrostatic field. Here the author considers a family ui of equipotential lines, which are Cassini ovals. It is truly said that these ovals belong to the fourth-order bicircular curves of genre 1 (have two double imaginary points, which are cyclic). But these ovals’ family includes one curve of zero genre — the Bernoulli lemniscate; it has three double points (two of them are the same cyclic ones, and one is real, which coincides with the origin of coordinates). In addition, it has been noted that "... the lines of current are equilateral hyperbolas gj» [1, p. 63]. However, clarification is also required here. The lines of force exit from each point charge and each line has two opposite directions. One such line of "double direction" forms only one branch of an equilateral hyperbola. A similar set of branches of equilateral hyperbolas also emanates from the second charge.

The Predictability of the Seismicity on Shiveluch Volcano

This paper presents the results from estimating the predictability of the seismicity of Shiveluch Volcano based on the earthquake catalog for the Northern Group of Kamchatka Volcanoes for 1971–1996 and 1999–2013. The mathematical model that we employed is a nonlinear second-order differential equation, while the algorithms of optimization and predictability are ours. The calculations show that seismicity can be successfully predicted for time intervals of a few weeks to a few months during phases of higher activity and for times of between a few months and a few years during phases of lower seismicity. The prediction distances are in excess of the error by factors of 20 to 50 on average. The nonlinearities in both times of higher and lower rates are close to the law of an equilateral hyperbola. We concluded that the predictability of seismicity can possibly be used in an integrated complex to predict extrusive and effusive activity and accompanying explosive activity. The prediction of major bursts of explosive activity related to failure in the existing volcanic edifice requires additional monitoring of the dome structure and the stability of the rocks that make up the dome.

Estimating the predictability of the seismicity rate: The 1964 eruption of Shiveluch Volcano

This paper describes a method for estimating the predictability of the seismicity rate for volcanic edifices. The mathematical model is a nonlinear second-order differential equation, with our own algorithms for optimization and for estimating predictability. The method is illustrated by the 1964 eruption on Shiveluch Volcano. Calculations showed that the seismicity rate can be predicted with success for a time interval ranging between a few days and a few tens of days during the active phase and between a few tens of days and a few hundreds of days during the decay phase. The level of predictability is rather high, with the prediction distance exceeding the mean error by two to four orders of magnitude. The nonlinear character of both the active phase of the process and of its decay phase is close to the law of an equilateral hyperbola. The method, as developed in this paper, allows a realistic approach to the prediction of that part of the volcanic process which involves brittle deformation and/or rapid displacements of material along the newly arisen cracks.

Quantum-Wave Dualism for Description of the Dynamic Systems

On the basis of the new formulation of the problem of three bodies were obtained: 1) a new definition of the locus of the points of an equilateral hyperbola; 2) a new "dynamic formula of the golden section" to the representative point of a scalar field displacement of the center; 3) a new interpretation of the deleted point as the point preceding the transition to another branch of the rectangular hyperbola. It is shown that the author introduced the space of possible states of a dynamical system described by a scalar field as a hypersphere with a displaced center - quantized punctures offset center. That is in addition to the known quantization of time and the level of quantization is proposed a new method based on the quantization introduced by the author of the concept of the quantum motion, which is in contrast to the "photon" of theoretical physics is filled with rigorous mathematical content. Thus, the overall dimension of the state space and the space of possible states is (3n + 2). It is proved that the curvature of the space of possible states (phase space + hyperspace displacement of the center), as predicted by Einstein, is a geometric property of the scalar field. The space of possible states is a pulsating wave hyperspherical variable curvature center coordinates which are the algebraic sum of scalar waves of torsion on the branch of an equilateral hyperbola and scalar waves exponential motion. The proof is based on the quantum analogue of the Pythagorean theorem.

Static Aeroelasticity Solution for Divergence Dynamic Pressure of Wing

This paper introduces the solution for calculating the divergence dynamic pressure of wing under subsonic speed based on static aeroelasticity. Several groups of results (q ~ θ values) are calculated and the divergence dynamic pressure can be obtained through two methods, i.e. making q ~ θ value curves or deducing divergence dynamic pressure formula based on the characteristics of equilateral hyperbola. According to several groups of calculated data, the error caused by the two methods is minor and can be ignored basically.

Prediction of Vertical Ultimate Bearing Capacity in Piles Based on the Improved Hyperbolic Model

Vertical static load test is widely used in the determination of pile bearing capacity, the mathematical model used to fit test pile data in determining the bearing capacity is essential. From the perspective of analytic geometry, the paper analyzes the traditional method of hyperbola, of which the asymptotic line of equilateral hyperbola was used to determine the ultimate bearing capacity. By extending the equal-axed conditions, a more general form of hyperbolic equation is derived and feasibility of such method is also analyzed, which indicates that the maximum point of curvature in such hyperbolic curve can determine the ultimate bearing capacity and such method is proved to be reasonable in practical projects.

Curves with rational chord-length parametrization

It has been recently proved that rational quadratic circles in standard Bézier form are parameterized by chord-length. If we consider that standard circles coincide with the isoparametric curves in a system of bipolar coordinates, this property comes as a straightforward consequence. General curves with chord-length parametrization are simply the analogue in bipolar coordinates of nonparametric curves. This interpretation furnishes a compact explicit expression for all planar curves with rational chord-length parametrization. In addition to straight lines and circles in standard form, they include remarkable curves, such as the equilateral hyperbola, Lemniscate of Bernoulli and Limaçon of Pascal. The extension to 3D rational curves is also tackled.

Some Remarks on the Rule of the Middle Third

The so-called ‘rule of the middle third’ states that an inclined force applied at the top of a vertical pillar with rectangular cross-section must intersect the bottom within the middle third of the height of the rectangle in order that the normal stress on the base is one-signed. This rule has been extended by Michell (1900) to the case of a plane elastic wedge loaded at its vertex. We here study plane elastic pillars with other profiles, like a trapezoid, the plane region bounded by two branches of an equilateral hyperbola, a blunt pillar. The result is that the rule is only partially valid.

Detachment folding, fold amplification, and diapirism in thrust wedge experiments

The relations between detachment folding, fold amplification, and salt diapirism in contractional settings have been investigated by means of scaled analogue models. The viscosity of the silicone layer simulating salt in nature and the shortening rates were combined in order to reproduce weak (type 1 models) and strong (type 2 models) décollements. Deformation patterns in the roof sequence exhibited two contrasting styles, (1) outward propagation of detachment folding along the décollement (OFP mode) and (2) passive roof duplex (PRD mode). In type 2 models, detachment folding propagated away from the most external thrust in the floor sequence, while in type 1 models, long‐lived detachment folds almost invariably localized amplified above a floor thrust tip as a result of strain localization. A silicone wall intruded occasionally into the crestal graben of detachment folds in type 1 and OFP models. Best fitting of transition models data points indicates nonlinear relations with regression curves close to the equilateral hyperbola equation for both OFP‐PRD and amplified detachment folds‐box folds transitions. A quantitative comparison of model results with nature has been attempted by plotting salt‐based fold‐and‐thrust belts data points on the scaled transition curves obtained from the modeling. Such a comparison relates shear stress products and ratios to the conditions favoring the amplification of detachment folds and the potential emplacement of ductile diapirs in their core. By reducing the roof sequence strength, pore fluid pressure λb is inferred to shift the equilibrium of fold‐and‐thrust belts toward the field of OFP and diapirism.

Effect of manufacturing error in fillet radii on the reliability of stepped shafts

Stress concentration has been recognized as a major factor in the fatigue life of machine elements. Stress concentration plots for a variety of cases are widely available in machine design books. In this paper the stress concentration curve K t = ψ(α,η,β,r/d) is modelled as an equilateral hyperbola, where parameters α, η, β are constants of the equation, d is a deterministic variable defining the characteristic dimension of the component determining its overall strength, and r is the random variable defining the radius of the surface of an angular junction. The randomness in the fillet radius r is due to the machining error introduced while manufacturing the shaft. The variability in r is modelled by a normal distribution. Using these mathematical models of K t and r, in conjunction with the Weibull equation of stress versus number of cycles to failure (S-N) curve, probability model f(N) to characterize the fatigue life of the stepped shaft is developed. This probability model can be used to calculate (1) the reliability, (2) the mean life, and (3) the scatter in the fatigue life of the shaft. The results are presented in a set of plots which illustrate the effect of the variability of the fillet radius, on the mean life and the scatter in the life. The results can also be used to assess the fatigue life when the stress concentration is caused by notches, holes or keyways in shafts and plates.

A statistical analysis of upper and lower limits of earthquake magnitude

The relationship between frequency and magnitude in homogeneous series of earthquakes is studied. The linear relation of Gutenberg and Richter between the logarithm of frequency and the magnitude is found to be applicable only in a restricted range of magnitudes. The plots of the cumulative number of earthquakes versus magnitude for 20 different series (the total number of events in a single series ranging from about 50 to more than 3000) clearly indicate the presence of vertical asymptotes. This fact suggests that upper and lower limits could exist for the magnitude of a single event, i.e. for the energy that can be released in an earthquake. From the position of the vertical asymptotes it is possible to obtain the upper and lower bounds of magnitude for each series of earthquakes. The empirical shape of the cumulative number-magnitude distribution can be well represented by equilateral hyperbolas. A ‘least χ2’ method has been used to obtain the best-fitting hyperbolas and corresponding the minimum and maximum magnitudes for each series. The shape of the frequency-magnitude relationship is discussed in lesser detail. It is found, however, that after suitable rescaling the earthquakes in each series follow much the same law, with a comparatively small scatter.

Eine kinematische Kennzeichnung der gleichseitigen Paraboloide

For a given quadric exist (in general) ∞ generating methods by means of cones. In the case of the equilateral hyperbolic paraboloid we know, due toH. Brauner, an explicit generation by an equilateral hyperbola. More general we show: The only quadrics which can be generated with cones by a cylindrical vibration are the equilateral paraboloids. The generating cones are equilateral hyperbolas or parabolas, resp. Some special properties are demonstrated.

A Review of Criticality Safety Models Used in Evaluating Arrays of Fissile Materials

The methods commonly used in this country to evaluate the criticality safety of fissile material arrays include density analog, NB/sup 2//sub N/, surface density, equilateral hyperbola, albedo, and solid-angle techniques. These can be divided into array unit interaction and semiempirical methods. The albedo and solid-angle techniques fall into the former class; the rest fall into the latter class. A study reveals that interaction methods are useful in treating arrays of arbitrary mesh patterns (e.g., triangular or hexagonal) having only a few units, while the semiempirical techniques are more applicable to arrays with a large number of units. The density analog and surface density approaches are easy to apply but typically require more auxiliary calculations, while other methods are more difficult to use, but more broadly applicable. None of the methods satisfactorily handle nonuniformly spaced arrays, arrays with arbitrary amounts of internal moderator, or ''clumped'' arrays, i.e., arrays of arrays. Most of the methods have some provision for treating mixed arrays, but these provisions often do not apply to arrays of arbitrarily arranged fast (e.g., metal) and slow (e.g., solution) units.

Comparative Calculational Evaluation of Array Criticality Models

The NB/sup 2//sub N/, albedo, interaction potential, weighted interaction Oak Ridge Gaseous Diffusion Plant solid-angle, s/v surface density, fraction critical surface density, and equilateral hyperbola methods are compared by attempting to apply them to 14 different example arrays. The example arrays are distinguished by the type of fissile material in the array units and by the number of units in each array. Arrays with units of Pu(95) metal, U(93) metal, damp U(93)O/sub 2/, dry Pu(95)O/sub 2/, U(93)O/sub 2/(NO/sub 3/)/sub 2/ solution, Pu(95)(NO/sub 3/)/sub 4/ solution, or U(5)O/sub 2/F/sub 2/ solution are considered. It is concluded that if the user has access to a Monte Carlo code such as KENO, he is better off using it than any of the other models for criticality evaluations of specific problems. The remainder of the models are most useful in establishing possible designs for a fissile storage or processing facility. The albedo method is found to be the preferred solid-angle technique, while the s/v surface density method and the NB/sup 2//sub N/ method are the most comprehensive and useful of the semiempirical techniques. The s/v method is easier to apply and covers more possible contingencies (e.g., array flooding or persons in arrays), while the more » NB/sup 2//sub N/ method is more difficult to use but has added flexibility, broader applicability, and yields more analytical scaling relations between array parameters. « less

Results of a study of limestones at the Inguri hydroelectric power plant

1. The character of the (E1, σ1) curve, which is an equilateral hyperbola, makes it possible to determine: a) the integral value of the natural stresses in the rock mass over the foundation of the whole area beneath a plate; b) the relation between the deformation modulus, determined by the static method, and the dynamic modulus of elasticity. 2. Repeated loading of the foundation enables us to predict by an indirect method the approximate effect obtainable by cementing a particular sector of ledge rock.