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Two‐dimensional (2D) images of light beams reflected off the objects in space impinge on the retinal photoreceptors of our two laterally separated eyes. Nevertheless, we experience our visual percept as a single 3D entity—our visual world that we tend to identify with the physical world. However, experiments point to different geometries in these two worlds. Using the binocular system with asymmetric eyes (AEs), this article studies the global geometric aspects of visual space in the Riemannian geometry framework. The constant‐depth curves in the horizontal field of binocular fixations consist of families of arcs of ellipses or hyperbolas depending on the AE parameters and the eyes' fixation point. For a single set of AE parameters, there is a unique symmetric fixation at the abathic distance such that the constant‐depth conics are straight frontal lines. Critically, the distribution of these constant‐depth lines is independent of such fixations. In these cases, a two‐parameter family of the Riemann metrics is proposed to account for the retinotopy of the brain's visual pathways and simulated constant‐depth lines. The obtained geodesics for a subset of metric parameters include incomplete geodesics that give finite distances to the horizon. The Gaussian curvature of the phenomenal horizontal field is analyzed for all the metric parameters. The sign of the curvature can be inferred from the global behavior of the constant‐depth ellipses and hyperbolas only when for the metric parameters for which the constant‐depth frontal lines at the abathic distance fixations are geodesics. [ABSTRACT FROM AUTHOR]

The paper describes how the attitude of a sounding rocket, launch vehicle or satellite with respect to the Earth can be estimated from camera images of the Earth horizon. Details about detecting the horizon in the camera image, fitting hyperbolae or ellipses to the detected horizon curve and deriving the Earth nadir vector and the corresponding error covariance from the fitted conic section are given. The presented method works at low heights, where the projected horizon mostly appears to be hyperbolic, as well as at large heights, where the projected horizon mostly appears to be elliptic and it is irrelevant if the Earth is fully or only partially in the field of view of the camera. The method can be universally used to estimate the direction vectors and attitude with respect to any spherical celestial body such as the Sun or Moon. Using the example of a sounding rocket mission with two cameras aboard, it is illustrated how the estimates of the Earth nadir and the Sun direction vectors are fused with the measurements of a strapdown inertial measurement unit and a GPS receiver to obtain an accurate and continuous estimate of the three-dimensional orientation of the sounding rocket with respect to the Earth. [ABSTRACT FROM AUTHOR]

Sets of bilinear constraints are important in various machine learning models. Mathematically, they are hyperbolas in a product space. In this paper, we give a complete formula for projections onto sets of bilinear constraints or hyperbolas in a general Hilbert space. [ABSTRACT FROM AUTHOR]

Copyright of Revista Latinoamericana de Filosofía is the property of Centro de Investigaciones Filosoficas and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Among the triangle congruence axioms, the side-side-angle (SsA) axiom states that two triangles are congruent if and only if two pairs of corresponding sides and the angles opposite the longer sides are equal. We construct two triangle sequences in which the items satisfy a modified condition. We require that the opposite angles of the shorter sides be equal. The locus of the intersection points of other sides of triangles is derived to be a hyperbola, and in a generalized form defined by a complete quadrilateral, it is a conic section. [ABSTRACT FROM AUTHOR]

Inspired by the reality that the collection of fixed/common fixed points can embrace any symmetrical geometric shape comparable to a disc, a circle, an elliptic disc, an ellipse, or a hyperbola, we investigate the subsistence of a fixed point and a common fixed point and study their geometry in a partial metric space by introducing some novel contractions and notions of a fixed ellipse-like curve and a common fixed ellipse-like curve which is symmetrical in shape but entirely different than that of an ellipse in a Euclidean space. We look at new hypotheses essential for the collection of nonunique fixed/common fixed points of some mathematical operators to incorporate an ellipse-like curve keeping in view the symmetry in fixed/common fixed points approaches. Appropriate nontrivial examples verify established conclusions. We conclude our work by applying our results to construct the mathematical model and solve the Production–Consumption Equilibrium problem of economics. [ABSTRACT FROM AUTHOR]

This paper's core objective is to introduce a novel notion called hyperbolic fuzzy set (HFS) where, the grades follow the stipulation that the product of optimistic and pessimistic degree must be less than or equal to one (1), rather than their sum not exceeding one (1) as in case of IFSs. The concept of HFS originates from a hyperbola, which provides extreme flexibility to the decision makers in the representation of vague and imprecise information. It is observed that IFSs, Pythagorean fuzzy sets (PFSs), and q-rung orthopair fuzzy sets (Q-ROFSs) often failed to express the uncertain information properly under some specific situations, while HFS tends to overcome such limitations by being applicable under those perplexed situations too. In this paper, we first define some basic operational laws and few desirable properties of HFSs. Second, we define a novel score function, accuracy function, and also establish some of their properties. Third, a novel similarity and distance measure is proposed for HFSs that are capable of distinguishing between different physical objects or alternatives based on the grounds of "similitude degree" and "farness coefficient", respectively. Later, the advantages of all of these newly defined measures have been showcased by performing a meticulous comparative analysis. Finally, these measures have been successfully applied in various COVID-19 associated problems such as medical decision-making, antivirus face-mask selection, efficient sanitizer selections, and effective medicine selection for COVID-19. The final results obtained with our newly defined measures comply with several other existing methods that we considered and the decision strategy adopted is simple, logical, and efficient. The significant findings of this study are certain to aid the healthcare department and other frontline workers to take necessary measures to reduce the intensity of the coronavirus transmission, so that we can hopefully progress toward the end of this ruthless pandemic. [ABSTRACT FROM AUTHOR]

For 3D seismic data processing, azimuth plays an important role, especially in complex subsurface regions. In such regions, elliptical or anelliptical orthorhombic models are commonly used to describe wave propagation. In these models, the behaviour of the slowness surface needs more detailed analysis. Umbilic points defined by the equal principal curvatures exist in a complex 3D model. In the vicinity of the umbilic points, the traveltime surface has the shifted hyperbola form that needs to be considered in processing operations like velocity analysis. Fractured media characterized by the orthorhombic model are more likely to have umbilic points, and it is important to address their positions. If exists, umbilic points can provide additional constraints in inverting for model parameters. Through a defined condition, we examine the position of the umbilic point and derive their explicit formulas. We analyse umbilic points for elliptical and anelliptical orthorhombic models in the numerical example. For the elliptical orthorhombic model, the formulas for the umbilic point position on different symmetry planes are derived and corresponding conditions are also identified. Further, we numerically examine umbilic point positions for anelliptical orthorhombic models and observe that the umbilic points are located out of two vertical symmetry planes. Moreover, caused by interference from two neighbouring umbilic points, a more significant deviation in traveltime is found in the anelliptical orthorhombic model. [ABSTRACT FROM AUTHOR]

The bidirectional fluxes of cadmium and calcium across the plasma membrane were assessed and compared in subapical maize root segments. This homogeneous material provides a simplified system for investigating ion fluxes in whole organs. The kinetic profile of cadmium influx was characterized by a combination of a saturable rectangular hyperbola (Km = 30.15) and a straight line (k = 0.0013 L h−1 g−1 fresh weight), indicating the presence of multiple transport systems. In contrast, the influx of calcium was described by a simple Michaelis–Menten function (Km = 26.57 µM). The addition of calcium to the medium reduced cadmium influx into the root segments, suggesting a competition between the two ions for the same transport system(s). The efflux of calcium from the root segments was found to be significantly higher than that of cadmium, which was extremely low under the experimental conditions used. This was further confirmed by comparing cadmium and calcium fluxes across the plasma membrane of inside-out vesicles purified from maize root cortical cells. The inability of the root cortical cells to extrude cadmium may have driven the evolution of metal chelators for detoxifying intracellular cadmium ions. [ABSTRACT FROM AUTHOR]

Most imaging methods based on ultrasonic Lamb waves in structural health monitoring requires reference signals, recorded in the intact state. This paper focuses on a novel baseline-free method for damage localization using Lamb waves based on a hyperbolic algorithm. This method employs a special array with a relatively small number of transducers and only one branch of the hyperbola. The novel symmetrical array was arranged on plate structures to eliminate the direct waves. The time difference between the received signals at symmetrical sensors was obtained from the damage-scattered waves. The sequence of time difference for constructing the hyperbolic trajectory was calculated by the cross-correlation method. Numerical simulation and experimental measurements were implemented on an aluminum plate with a through-thickness hole in the current state. The imaging results show that both the damages outside and inside the diamond-shaped arrays can be localized, and the positioning error reaches the maximum for the diamond-shaped array with the minimum size. The results indicate that the position of the through-hole in the aluminum plate can be identified and localized by the proposed baseline-free method. [ABSTRACT FROM AUTHOR]

The current study evaluated the canopy cover competition for light and heat in a jujube–cotton intercropping system to measure the growth and yield performance of cotton, and the optimal cotton planting configuration. In this study, a two-year field experiment (2020 and 2021) was studied with different spacing configuration modes designed as follows: two rows of cotton (CM1) planted 1.4 m apart, four rows of cotton (CM2) planted 1.0 m apart, and six rows of cotton (CM3) planted 0.5 m apart, spacing intercropped jujube trees, respectively. The control treatment consisted of monocultured cotton (CK). The light-response curve was plotted using an LI-6400 XT photosynthesis instrument. Based on the modified rectangular hyperbola model, the photosynthetic characteristics were fitted, and the dry matter distribution characteristics and yield were compared. The results showed that with the increase in photosynthetically active radiation, the net photosynthetic rate (Pn) of each growth phase decreased first and then increased rapidly in the range of 0–200 μmol·m−2·s−1 and then decreased slightly after the inflection point (light saturation point). The light-response curves of stomatal conductance and transpiration rate showed a linear relationship. The trend in the intercellular CO2 concentration response curve was opposite to that of Pn. The maximum Pn (Pmax) of intercropped cotton was significantly impacted by configuration modes, of which CM2 treatment generated 1.8% and 22.8% higher Pmax than the CM1 and CM3 treatments. The cotton yield in the two years ranked as CK > CM3 > CM2 > CM1, and the average land equivalent ratio of CM2 was significantly higher than that of CM3 (22.4%) and CM1 (95.9%). The six-row configuration resulted in greater competition with the trees, which affected the accumulation of below-ground dry matter, while the four-row configuration formed a reasonable canopy structure, which ensured that more photosynthetic substances were distributed to the generative organs. The reasonable four-rows configuration mode may improve the photosynthetic efficiency of intercropped cotton economic yield. [ABSTRACT FROM AUTHOR]

The approach developed earlier for describing the chains of conjugated enzymatic reactions of the photosynthesis process for the first time allowed the authors to propose a rigorous mathematical model of a nonrectangular hyperbola, which describes the dependence of the photosynthesis rate on light intensity (I), concentrations of CO2, and the interaction of these two factors. The fundamental parameters of this model are the light (IK) and carbon dioxide (CK) substrate constants as well as the curvature of the hyperbola (γ), pairing parameters (rIP, rCP), and the maximum rate of photosynthesis (Pm). Special rectifying coordinates are proposed in which the entire family of hyperbolas has the form of a straight line. The main parameters can be found only when constructing experimental data in rectifying coordinates. The proposed model of a nonrectangular hyperbola is applicable both for describing the process of photosynthesis in algae and higher plants and for the absorption of vitamin B12 algae and the dependence of the rate of food consumption by zooplankton and fish. [ABSTRACT FROM AUTHOR]

A full understanding of the growth and distribution of tree roots is conducive to guiding precision irrigation, fertilization, and other agricultural work during agricultural production. Detecting tree roots with a ground-penetrating radar is a repeatable detection method that does no harm to the earth surface and tree roots. In this research, a rapid and accurate automatic detection was conducted on hyperbolic waveforms formed by root targets in B-scan images based on YOLOv5s. Following this, the regions of interest containing target hyperbolas were generated. Three or more coordinate points on the hyperbola were selected according to the three-point fixed circle (TPFC) method to locate the root system and estimate the root diameter. The results show that the accuracy of hyperbola detection using YOLOv5s was 96.7%, the recall rate was 86.6%, and the detection time of a single image was only 13 ms. In the simulation image, the TPFC method was used to locate the root system and estimate the root diameter through three different frequency antennas (500 MHz, 750 MHz, and 1000 MHz). A more accurate result was obtained when the antenna frequency was 1000 MHz, with the average distance error of root system positioning being 3.17 cm, and the slope and R 2 of the linear fitting result between the estimated root diameter and the actual one being 1.029 and 0.987, respectively. Verified by the pre-buried root test and wilderness field test, both root localization and root diameter estimation in our research were proved to gain good results and conform to the rules found in simulation experiments. Therefore, we believe that this method can quickly and accurately detect the root system, locate and estimate the root diameter, and provide a new perspective for the non-destructive detection of the root system and the three-dimensional reconstruction of the root system. [ABSTRACT FROM AUTHOR]

The exact determination of settlement and the time required to complete that settlement is very important from the safety point of view. There are several methods available to find the rate of consolidation that is highly dependent on the settlement versus time curve. Here, it has been attempted to identify the suitable method for the determination of the rate of consolidation of coal ash. After analyzing the value of the coefficient of consolidation (Cv) estimated from various methods, the rectangular hyperbola method (RH method) is found to be in best linear correlation between the measured Cv and calculated Cv. The coal ash shows an increment in Cv with the increase in effective pressure. The Cv obtained from the RH method is higher than the Cv obtained from Taylor's and Casagrande method that helps in close approximation with field Cv. The precise determination of Cv helps in better prediction of the time required for the consolidation settlement that will increase the performance of the structure. [ABSTRACT FROM AUTHOR]

Pseudo-Dionysius the Areopagite should be celebrated as the Doctor of Darkness (Doctor Tenebrarum). In a Gothic style that reflects the depth of his many masks, his enduring achievement has been to have conceived of a theological grammar to speak of God beyond the haunting spectre of both ancient and modern nihilism. His Christian theological grammar is distinguished by speaking in the sense of a hyperbola (ύπερβολή) or excess of signification, which signifies beyond yet within the world: first in the positive or cataphatic grammar, a positive judgment speaks of God; second in the negative or apophatic grammar, a higher or hyper-negative judgment annuls the positive, even as it speaks of God 'beyond being', and 'beyond intellect', as the absolutely originary source of any such positive judgment; and, third in the proportionate or analogical grammar, this procession of divine Power reciprocally annuls the infinite repetition of all such negative judgments, even as, from the centre of this cycle, it constitutes the absolutely higher ground from which alone speech of God is warranted. At the centre of this cycle, Christ can be acknowledged by faith to descend into the depths of the negative so as to shine from within the essentially proportioned and hyperbolic grammar of analogy. The elements of the Latin Scholastic analogy of being can, as this commentary will show, be reconstructed from Pseudo-Dionysius's hyperbolic grammar. In these hyperbolic arcs, the infinite repetition of hyper-negative judgments is annulled, and yet, in a reciprocal determination, equally constituted to virtually proceed from a higher ground in the essential proportions of analogy. Like fireflies that carry the torch of the Sun before the doors of night, the theological grammar of Pseudo-Dionysius the Areopagite can be shown to be radically shaped by a Christian and Trinitarian theology, in which Christ the Logos is the originary ground and source, not only of the hyper-negative, but of the essential proportions of the analogia entis. [ABSTRACT FROM AUTHOR]

Improving the detection accuracy of hyperbola in B-scan images has been a considerable challenge for ground penetrating radar (GPR) to detect tree roots. In this paper, a method for data enhancement and target detection, both based on deep learning was proposed to identify hyperbolas in GPR B-scan images. First, the authors used a cyclic consistent adversarial network (CycleGAN) to augment the original data. In this procedure, the hyperbolic features of the images were preserved and created a wider variety of training samples. Then, the authors could apply the enhanced dataset to the YOLOv5 detection model to evaluate the effectiveness of their method. Meanwhile, the detection effects of Yolov3, Yolov5, Faster R-CNN, and CenterNet detection models on the enhanced dataset were compared. The results showed that applying the enhanced dataset to the Yolov5 detection model exhibited better detection accuracy compared to other combinations of datasets and detection models. The authors demonstrate that the proposed method increases data diversity and the number of samples, improving the precision and recall of hyperbolic curves. These results provide a new method for tree root localization with important effects. [ABSTRACT FROM AUTHOR]

• Investigated test-line response curve for a qualitative serum hCG lateral flow assay. • Image analysis was performed to obtain the Test line integral (S) vs [hCG]. • S vs [hCG] was well-characterized by a hyperbola: S/Smax = [hCG]/([hCG] + K). • K was 202 mIU/mL; interval between 10% and 90% Smax was a factor of 81 in [hCG]. • At claimed sensitivity of [hCG] = 10 mIU/mL, S/N was 2.7. As an adjunct to verification of performance characteristics of a qualitative serum hCG lateral flow immunoassay (LFI), we performed image analysis to characterize the dose vs response curve (visibility of the test line), as a means of understanding the transition from negative to positive as a function of increasing [hCG]. Using serum samples of known [hCG], device images were obtained using a scanner at the prescribed reading time (5 min). Image analysis (using Python and R) was used to obtain the integral (S) of the test-line color as a function of [hCG]. Data for S as a function of [hCG] were well characterized by a simple hyperbola: S = Smax [hCG]/([hCG] + K), where K = 202 mIU/ml (r = 0.997). Replicates of S at K had CV of 7.3 %. By eye, uncertainty of test results among users occurred only below the assay's stated sensitivity of 10 mIU/ml, in region of S < 3 % of Smax, and signal:noise ratio < 3. By image analysis, the dose vs response (Test line integral) for this qualitative serum hCG LFI was a simple hyperbola. Characterization of the dose vs response curve was useful in verification of the assay's performance characteristics. [ABSTRACT FROM AUTHOR]

In this work, the method of drawing of nine celebrated Minoan Late Bronze Age wall-paintings, is studied. Four of these frescoes were unearthed at Akrotiri, Thera, while the other five were excavated in Crete; these frescoes have never been thoroughly studied before, concerning their method of drawing. The authors demonstrate that practically all actually drawn contours in these frescoes, optimally fit six geometric stencilsguides, namely, four hyperbolae and two Archimedes' (linear) spirals. It is important to note that the hyperbola and the linear spiral do not exist in nature, definitely not with the precision encountered here. We would like to stress that all wall-paintings the authors have studied so far, i.e. more than twenty-five (25), had been drawn via the use of the same stencils, a fact that renders the eventuality that this was indeed the method of drawing of these frescoes, almost certain. The considered wall-paintings cover a period of at least two centuries, located both before and after the gigantic eruption of Thera volcano. Moreover, here, possible methods of construction of the determined stencils are proposed, as well as the most probable one, according to our opinion. Methodologically, the authors determine couples of the longest contour parts, CP, appearing in the studied frescoes and of a proper sub-curve GPopt of one of the aforementioned stencils; these couples are chosen so as they fit with exceptionally low approximation errors. The latter terms are rigorously defined here, for the first time. Special care is taken, mathematically and programmatically, to ensure the maximum possible smoothness between successive optimally fit stencils' parts GPopt. Next, the authors employed these modern prerequisites, to introduce the more probable sequence of actions taken by the prehistoric artist(s) for drawing the specific frescoes. Finally, crucial queries and conjectures are set, together with a number of plausible answers. [ABSTRACT FROM AUTHOR]

A test of the evolutionary theory was conducted by replicating Bradshaw et al.'s (1977, 1978, 1979) experiments in which human participants worked on single‐alternative variable‐interval (VI) schedules of reinforcement under three punishment conditions: no punishment, superimposed VI punishment, and superimposed variable‐ratio (VR) punishment. Artificial organisms (AOs) animated by the theory worked in the same environments. Four principal findings were reported for the human participants: (1) their behavior was well described by an hyperbola in all conditions, (2) the asymptote of the hyperbola under VI punishment was equal to the asymptote in the absence of punishment, but the asymptote under VR punishment was lower than the asymptote in the absence of punishment, (3) the parameter in the denominator of the hyperbola was larger under both VI and VR punishment than in the absence of punishment, and (4) response suppression under punishment was greater at lower than at higher reinforcement frequencies. These four outcomes were also observed in the behavior of the AOs working in the same environments, thereby confirming the theory's first‐order predictions about the effects of punishment on single‐alternative responding. [ABSTRACT FROM AUTHOR]

Disc infiltrometer can be used to characterize near saturated, soil near surface hydraulic conductivity, which is mandatory for hydrological modelling of runoff, irrigation management and artificial recharge. The hydraulic conductivity determination from the disc infiltrometer measurements requires soil-specific water retention characteristic parameters. This study proposed an alternate procedure for determining near surface near saturated soil hydraulic conductivity (K) and saturated hydraulic conductivity (Ks) for 14 soils comprising 4 different soil textures in the field. The study demonstrated the usefulness of rectangular hyperbola method (RHM) for predicting long-term final infiltration rate (if) from 50 min short-term mini disc infiltrometer (MDI) measurements. The estimated if was utilized for determining K directly. For this purpose, a multiplication factor R was proposed for four USDA surface soil textures. In addition, an empirical relationship was proposed for determining Ks from if estimated by RHM. The advantage of the proposed empirical procedure is that if, K and Ks can be determined from 50 min short-term MDI measurements without the knowledge of soil specific water retention parameters. The proposed methodology can be used to determine K and Ks of other soil textures, vegetated and organic soils. [ABSTRACT FROM AUTHOR]

Plants have the capacity to fix CO2 through photosynthesis. To reveal the photosynthetic processes of Quercus acutissima Carruth. and Quercus variabilis Bl., their net photosynthetic rates were quantified during the early and peak growing seasons. To evaluate forest photosynthetic efficiencies, the photosynthetic light response curves of Q. acutissima and Q. variabilis were fitted by the rectangular hyperbola model (RHM), non-rectangular hyperbola model (NHM), and modified rectangular hyperbola model (Ye model). The results revealed the following: (1) All daily variation curves of the net photosynthetic rate, stomatal conductivity, and transpiration rate were single-peaked. The peak times of the Q. acutissima and Q. variabilis' net photosynthetic rates appeared at 12:00 am during the early growing season and 10:00 am during the peak growing season. (2) The photosynthetic capacities of both Q. acutissima and Q. variabilis during peak growing seasons were higher than during the early growing season. (3) The net photosynthetic rate was found to be positively correlated with stomatal conductivity, the transpiration rate, and photosynthetically active radiation, and it was negatively correlated with the intercellular CO2 concentration. (4) The Ye model provided the best fit for the light response curves of Q. acutissima and Q. variabilis when compared with the rectangular hyperbola and nonrectangular hyperbola models. The photosynthetic performance of Q. acutissima was superior to that of Q. variabilis; thus, it can be employed as a priority tree species in carbon sink forests. [ABSTRACT FROM AUTHOR]

Formability improvement is extremely crucial in incremental sheet forming (ISF) of low-ductility material and high wall angle 3-dimensional parts. The contact-induced vibration from the tool rotation provides an alternative route for optimizing formability; however, the underlying mechanism induced by the contact-induced vibration effect has not yet been fully elucidated. This study designed a contact-induced vibration tool (V-Tool) with a flat elliptical shape tip and applied it into the ISF of a 0.5 mm Al5052 sheet. Hyperbola shapes were manufactured by the ISF with V-Tool and conventional tool (C-Tool) with different rotation speeds to experimentally investigate the forming limit, temperature, hardness, fracture morphology, and material deformation. The main results show that the vibration effect and the resultant shear deformation by V-Tool result in the lower friction heat verified by the peak temperature from 180 °C to 80 °C and higher formability from the thinning rate of 54%–84% as compared to those of the conventional tool (C-Tool). Further, the deformation observation with the drilled holes on initial sheets indicates that the contact-induced vibration effect in the ISF with V-Tool may produce more shear deformation on the inner surface to delay the continuous enlargement of the drilled hole along the meridional direction.

In this article, we study the weak-star density of the linear span of the trigonometric functions { e m , n (x , y) = e π i (m x + n y) , e m , n < β > (x , y) = e π i β (m / x + n / y) ; m , n ∈ Z } for a positive real β. We aim to extend the results of Hedenmalm and Montes-Rodríguez (2011) [18] and Canto-Martín, Hedenmalm, and Montes-Rodríguez (2014) [8] in the plane. They have extensively studied the weak-star completeness of the hyperbolic trigonometric system in L ∞ (R). This is the dual formulation of the Heisenberg uniqueness pair for (hyperbola, certain lattice-cross). As in their work, β = 1 turns out to be the critical value. In particular, one of our main results asserts that the space spanned by the aforesaid trigonometric functions is weak-star dense in L ∞ of the set Θ 1 , β = (− 1 , 1 ] 2 ∪ (R ∖ (− β , β ]) 2 if and only if 0 < β ≤ 1 , and the corresponding pre-annihilator space has finite dimension whenever β > 1. However, for β > 1 , the pre-annihilator space can be made infinite-dimensional by allowing functions with slightly bigger support than Θ 1 , β. To be precise, let Θ β ″ ⊆ R 2 ∖ Θ 1 , β be such that (− β , β ] 2 ∩ Θ β ″ has positive Lebesgue measure. We prove that the weak-star closure of the linear span of e m , n and e m , n < β > as m , n varies over Z , has infinite codimension in L ∞ (Θ 1 , β ∪ Θ β ″) whenever β > 1. Our proof goes via the analysis of a two-dimensional Gauss-type map and its corresponding Perron-Frobenius operator. [ABSTRACT FROM AUTHOR]

We proposed a simple hyperbolic positioning method that does not require solving simultaneous quadratic equations. Moreover, we introduced the mathematical concept of a "pencil" into analytical calculations in the hyperbolic positioning method for a better understanding. In many recent studies using positioning biotelemetry, the specific procedure for intersection calculation of hyperbolas has rarely been described. This might be one of two major obstacles, with the other being clock synchronisation among receivers, for positioning biotelemetry users, including potential users. We focus only on the intersection calculation in this paper. Therefore, we propose a novel method and introduce the mathematical concept into analytical calculations. The computing performances of the novel method, an analytical method applying the concept of a pencil, and an approximating method using the Newton-Raphson method were compared regarding positioning correctness, accuracy, and calculation speed. In the novel method, hyperbolas were represented using the parameter θ, which was treated as a discrete variant. The finer the tick-width of the parameter θ, the more accurate was its positioning, but it took slightly longer to calculate. By setting the tick-width to 0.01°, a simulated trajectory was correctly and accurately localised, as in the analytical method which always correctly returned the accurate solution. The approximating method has a major limitation concerning correctness. It returns a single solution regardless of two intersections of hyperbolas; however, the positioning is accurate when the hyperbolas intersect at a single point. This study approached one major difficulty in positioning biotelemetry and will help biotelemetry users overcome this drawback with a simple and intuitive understanding of hyperbolic positioning. [ABSTRACT FROM AUTHOR]

The modern definition of enzymology is synonymous with the Michaelis–Menten equation instituted by Leonor Michaelis and Maud Menten. Most textbooks, or chapters within, discussing enzymology start with the derivation of the equation under the assumption of rapid equilibrium (as done by Michaelis–Menten) or steady state (as modified by Briggs and Haldane) conditions to highlight the importance of this equation as the bedrock on which interpretation of enzyme kinetic results is dependent. However, few textbooks or monographs take the effort of placing the equation within its right historical context and discuss the assumptions that have gone into its institution. This guide will dwell on these in substantial detail. Further, this guide will attempt to instil a sense of appreciation for the mathematical curve rectangular hyperbola, its unique attributes and how ubiquitous the curve is in biological systems. To conclude, this guide will discuss the limitations of the equation, and the method it embodies, and trace the journey of how investigators are attempting to move beyond the steady‐state approach and the Michaelis–Menten equation into full progress curve, pre–steady state and single‐turnover kinetic analysis to obtain greater insights into enzyme kinetics and catalysis. [ABSTRACT FROM AUTHOR]

Copyright of Ideas y Valores is the property of Universidad Nacional de Colombia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Hyperbolic diffractions in Ground Penetrating Radar (GPR) data are caused by a variety of subsurface objects such as pipes, stones, or archaeological artifacts. Supplementary to their location, the propagation velocity of electromagnetic waves in the subsurface can be derived. In recent years, it was shown that deep learning tools can automatically detect hyperbola in radargrams using data measured over urban infrastructure, which are relatively clear. In contrast, in this study, we used an archaeological dataset with diverse underground structures. In the first step we used the deep learning network RetinaNet to detect hyperbola automatically and achieved an average precision of 0.58. In the next step, 10 different approaches for hyperbola fitting and thus velocity determination were applied. The derived information was validated with manually determined velocities and apex points. It was shown that hyperbola extraction by using a threshold and a column connection clustering (C3) algorithm followed by simple hyperbola fitting is the best method, which had a mean velocity error of 0.021 m/ns compared to manual determination. The average 1D velocity-depth distribution derived in 10 ns intervals was in shape comparable to the manually determined one, but had a systematic shift of about 0.01 m/ns towards higher velocities. [ABSTRACT FROM AUTHOR]

Estimating the velocity of electromagnetic waves in the medium is essential for ground penetrating radar (GPR) to reconstruct target scenes. However, the existing velocity estimation methods based on imaging algorithms, e.g. the back-projection (BP)- and the Stolt migration-based methods, are susceptible to clutter and cannot strike a balance between depth independence and computational efficiency. Therefore, a velocity estimation method is proposed based on the vertex extraction-guided local BP algorithm. Initially, the high-order moment (HOM) is introduced into GPR data by analysing the distributions of target hyperbolas and clutter. Designed to avoid clutter, the proposed HOM sliding window extracts the hyperbola vertex at a high signal-to-clutter ratio region. By virtue of the extracted vertex, the local hyperbola region is subsequently extracted to reduce computation. Additionally, this paper adopts the depth-independent BP algorithm and further develops a local BP algorithm utilizing its imaging flexibility. Guided by the extracted vertex, the proposed local BP algorithm is iteratively performed on the local hyperbola region to estimate the velocity by a focus metric. Comprehensive experiments demonstrate the effectiveness and robustness of the proposed velocity estimation method. The proposed method outperforms the existing methods in clutter robustness, computational efficiency, and depth independence. [ABSTRACT FROM AUTHOR]

In this paper, we give a thorough account of Brianchon and Poncelet's joint memoir on equilateral hyperbolas subject to four given conditions, focusing on the most significant theorems expounded therein, and the determination of the "nine-point circle". We also discuss about the origin of this very rare example of collaborative work for the time, and the general challenge of finding the nature of the loci described by the centres of the conic sections required to pass through m points and to be tangent to n straight lines given in position, m + n = 4, which was posed at the end of their work. In the case m = 4, i.e. when the conic sections have to pass through the vertices of a quadrilateral, the locus of centres is another conic section passing through the intersection points of the opposite sides and the two diagonals of the quadrilateral, respectively, and, as Gergonne showed analytically shortly after, through other significant points connected with the quadrilateral; this curve was later given the name of the "nine-point conic", being a natural generalization of the above mentioned circle. [ABSTRACT FROM AUTHOR]

We propose and analyze an exponentially convergent numerical method for solving differential equations with right-hand fractional Riemann–Liouville derivative and unbounded operator coefficient in Banach spaces. We use a representation of the solution in the form of the Danford–Cauchy integral on a hyperbola that covers the spectrum of the operator coefficient with subsequent application of the exponentially convergent quadrature. To this end, we choose the parameters of the hyperbola in order to guarantee the possibility of analytic extension of the integrand in a strip containing the real axis and then apply the Sinc-quadrature. We prove the exponential accuracy of the method and present a numerical example that confirms the obtained a priori estimate. [ABSTRACT FROM AUTHOR]

In this paper, we provide the exact expressions (not found in the literature) for the curved surface area of revolution (about the x-axis and y-axis) of horizontal and oblique hyperbolas. Closed-form expressions for the curved surface area are obtained in terms of Gauss function, Clausen function, and Appell's hypergeometric function of the first kind. [ABSTRACT FROM AUTHOR]

In this paper we consider special hyperbolas circumscribed to the symmetral triangle of a given triangle. Considering some special cases of these hyperbolas we get equations of some special conics related to the triangles associated to a given triangle as e.g., the Jeřabek hyperbola of the tangential triangle of the given triangle and the circumscribed circle of the symmetral triangle which is also the polar circle of the given triangle. We investigate some other interesting properties of special hyperbolas circumscribed to the symmetral triangle of a given triangle. [ABSTRACT FROM AUTHOR]

Let us define a Fibonacci-Lucas hyperbola as a hyperbola passing through an infinite number of points of the form (Fm, Ln), where the Fm are distinct Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, 13, 21,..., where F0 = 0), and the Ln are distinct Lucas numbers (2, 1, 3, 4, 7, 11, 18, 29,...,. where L0 = 2). The simplest examples are 5x2 - y2 = 4, which contains the points (Fk, Lk) with odd subscripts, e.g. (1, 1), (2, 4), (5, 11), and 5x2 - y2 = -4, which contains the points with even subscripts, e.g. (0, 2), (1, 3), (3, 7); (see [1, 2]). These follow immediately from the identity (1) $$L_n^2 - 5F_n^2 = 4{(- 1)^n}.$$ Our goal is to find more of these Fibonacci-Lucas hyperbolas. [ABSTRACT FROM AUTHOR]

In this article, we revisit to the THINC/QQ (tangent of hyperbola interface capturing method with quadratic surface representation and Gaussian quadrature) scheme to make thorough investigations on its numerical property. Furthermore, it presents some recent progress made extensively on THINC/QQ algorithm to improve numerical accuracy and robustness which include (1) construct the interface equation on the global coordinate so as to apply THINC/QQ on polygonal/polyhedral elements, which ensures the boundedness of numerical solutions as well; (2) determine the coefficients of reconstruction polynomial from the unit normal vector of interface to reduce computational cost and alleviate numerical diffusion; (3) represent the interface with a cubic polynomial rather than quadratic representation used in THINC/QQ to obtain better geometric fidelity; (4) provide an appropriate choice considering different combinations, such as the number of quadrature points and temporal discretization methods. The verification benchmark tests are carried out on different types of grids, which convinces that THINC/QQ scheme with normalization on global coordinate delivers higher accuracy and robustness than the original one, and demonstrates effective competition compared with other existing interface capturing methods. [ABSTRACT FROM AUTHOR]

A sender designs an information structure to persuade a receiver to take an action. The sender is ignorant about the receiver's prior, and evaluates each information structure using the receiver's prior that is the worst for the sender. I characterize the optimal information structures in this environment. I show that there exists an optimal signal with two realizations, characterize the support of the signal realization recommending approval,, and show that the optimal signal is a hyperbola. The lack of knowledge of the receiver's prior causes the sender to hedge her bets: the optimal signal induces the high action in more states than in the standard model, albeit with a lower probability. Increasing the sender's ignorance can hurt both the sender and the receiver. [ABSTRACT FROM AUTHOR]

How to improve the dressing method has become a key technology for high-precision flex and circular rigid splines with involute tooth profile in gear form grinding, where the machining precision mainly depends on dressing accuracy of the grinding wheel. Based on the geometric principle (i.e., a spatial straight line changes to quadric conical surface by continuously rotating around its non-coplanar coordinate axis), a method of generating hyperbola by a single degree of freedom linear dressing motion is proposed to fit the involute tooth profile. With the design of a novel diamond dresser, mechanism model and different compensation models are established to demonstrate the feasibility of the fitting principle. Meanwhile, the error models are also established according to the mapping relationship of the device, which can reveal the error transmission law and the accurate adjustment method of geometric-profiling deviations, and enhance grinding accuracy. Especially, through complementary numerical simulations and experimental method, a specific example is presented to verify that the accuracy of the straight-line feed motion can meet its precision of the fitting and dressing. Eventually, it is hoped that this systematic method can be extended to manufacture high-precision involute gears for small-module precision gear transmission system and establish a theoretical foundation for setting standard on evaluating the tooth profile deviation, which is influenced by the fitting principle errors in gear form grinding. [ABSTRACT FROM AUTHOR]

A Minkowski plane is Euclidean if and only if at least one hyperbola is a quadric. We discuss the higher dimensional consequences too. [ABSTRACT FROM AUTHOR]

This research introduces an innovative algorithm using Ground Penetrating Radar (GPR) for inspecting underground infrastructure non-destructively, surpassing R-CNN methods by combining signal processing with computer vision techniques including hyperbola correlation, point cloud segmentation and noise reduction based on linearity, density and echo patterns. Utilizing on-vehicle multi-channel GPR and without training, it maps pipes in 3D with high precision on both experimental fields and real roads. The method demonstrated remarkable accuracy, with an average error of 0.1175 m in locating pipes, outperforming deep learning methods requiring numerous training data. The ablation study and sensitivity analysis on hyperparameters further prove the improvement from the proposed method and its robustness. These outcomes offer notable improvements in infrastructure monitoring, highlighting the potential for automated underground utility detection. • Introduce a novel method bridging GPR volumetric images to 3D underground pipe map. • Avoid pretraining by hyperbola correlation and geometric point cloud segmentation. • Extract pipes with noise reduction based on linearity, density and echoes. [ABSTRACT FROM AUTHOR]

We study the minimum number of distinct distances between point sets on two curves in R 3. Assume that one curve contains m points and the other n points. Our main results: (a) When the curves are conic sections, we characterize all cases where the number of distances is O (m + n). This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is Ω (min { m 2 / 3 n 2 / 3 , m 2 , n 2 }). (b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is O (m + n). This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is Ω (min { m 2 / 3 n 2 / 3 , m 2 , n 2 }). [ABSTRACT FROM AUTHOR]

In this paper, a high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) scheme is proposed for simulating one-dimensional compressible multi-material flows. The system is discretized by the DG method, and a kind of Taylor's expansion basis functions in the general cell is used to construct the interpolation polynomials of the variables. The mesh velocity at the node recognized as the material interface is set to approximate fluid velocity, which makes our scheme can capture the material interface accurately and clearly. For controlling the oscillations and reducing the numerical dissipation, a reconstruction based on the boundary variation diminishing (BVD) algorithm and Tangent of Hyperbola for INterface Capturing (THINC) function is employed to minimize the jump between the values at the left and right sides of cell boundaries. Due to the essentially monotone and bounded properties of THINC function, the difficulties caused by solving discontinuous solutions and complexities of designing limiters can be avoided. Finally, several examples are presented to demonstrate the accuracy and good performance of our scheme such as the essentially non-oscillatory property and so on for simulating the compressible multi-material flows. [ABSTRACT FROM AUTHOR]

Cappelletti-Montano et al. [6] introduced and studied taut cosymplectic circles/spheres, which are the analogues in the cosymplectic setting of taut contact circles/spheres introduced and studied by Geiges-Gonzalo [11] , [13] , [14]. The main purpose of this paper is to introduce and study, in dimension three, the hyperbolic analogue in the cosymplectic setting. We give a characterization of a taut almost cosymplectic hyperbola, classify compact three-manifolds which admit a cosymplectic hyperbola, and give a complete classification of 3D Lie groups which admit a left invariant taut cosymplectic hyperbola/circle/hyperboloid/sphere. Besides, we introduce the notion of almost bi-contact metric structure. In particular, in dimension three, an almost bi-contact structure defines a taut almost cosymplectic hyperbolas or a taut almost cosymplectic circle. Then we characterize, in dimension three, the existence of an almost bi-contact metric structure, in particular with the associated 1-forms closed, and thus a complete classification of almost bi- α -coKähler three-manifolds with constant extrinsic curvature is established. [ABSTRACT FROM AUTHOR]

Detecting the corrosive environment of reinforced concrete (RC) bridge decks is of critical importance for evaluating the reliability and safety of bridge structures. However, accurately and automatically detecting a corrosive environment with traditional methods is challenging. This paper proposes a method for the automatic corrosive environment detection of bridge decks from ground-penetrating radar (GPR) data based on the single-shot multibox detector (SSD) model. This method can be divided into three steps: data preprocessing, automatic rebar picking, and corrosive environment mapping. First, the GPR data are preprocessed to enhance the contrast of the hyperbolic feature in GPR B-scans. Then the rebars in the B-scan images are automatically picked up by the trained SSD model. Finally, the corrosive environment contour map of the bridge deck is generated with the rebar reflection amplitudes after depth correction. The SSD model was trained with 10,316 B-scan images and tested with 2,578 images. The 300×300-pixel B-scan image typically included three to five hyperbolas. A case study with GPR data from a tested bridge was employed to validate the feasibility of the proposed method. The results show that the accuracy of the automatic corrosive environment detection method can reach 98% and is considerably higher than that of commercial software methods. [ABSTRACT FROM AUTHOR]

In 1626, René Descartes and Claude Mydorge worked closely together on the problem of refraction, apparently discovering what is now known as the sine law of refraction. They constructed a plano-hyperbolic lens in order to test out the truth of this mathematical relationship. In 1637, Descartes finally published the sine method of determining refractions in his Dioptrique, which also demonstrated, on the basis of this relationship, that the hyperbola and ellipse were anaclastic lines (that is, that a lens with their profile would refract rays perfectly to a single point) without mentioning Mydorge. Mydorge himself wrote a short manuscript on the subject of refraction and lenses, which is found in a copy among Mersenne's papers. Scholars have usually assumed that this treatise was written as Mydorge and Descartes worked together in 1626. However, other evidence shows that only the first part of the treatise records the two men's early collaboration. Subsequently, Mydorge and Descartes completed their arguments separately, Descartes with the help of Isaac Beeckman. [ABSTRACT FROM AUTHOR]

Copyright of Contaduría y Administración is the property of Facultad de Contaduria y Administracion-Universidad Nacional Autonoma de Mexico and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of China Sciencepaper is the property of China Sciencepaper and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Sub-bottom profiler (SBP) measurements play a very important role in marine pipeline detection and checking. However, sometimes because of bad sea conditions and complex underwater environments, only poor SBP images can be obtained. Pipeline recognition based on a poor SBP image manually is tedious and has low accuracy. In this article, based on the specific hyperbola shapes caused by pipelines on SBP images, a novel method to detect the marine pipelines automatically and robustly is presented. Two main steps are used. First, a comprehensive edge extraction method is proposed, which includes line-like structure enhancement and preliminary edge extraction, local phase calculation and phase edge extraction, final edge extraction, as well as edge processing. By applying this method, interference such as reverberation can be eliminated, and fine hyperbola edges can be obtained. Second, by referencing the theoretical hyperbola and using a buffer zone and a hyperbola contour consistency (HCC) criterion, the location of the pipeline can be determined robustly. This method was implemented and verified using data collected under real conditions. The results show that the proposed method is practicable. [ABSTRACT FROM AUTHOR]

We present a novel method for deriving Snell's law of refraction without using calculus by instead rotating a certain hyperbola. This approach includes a geometrically motivated equation that turns out to be equivalent to setting the derivative equal to zero. [ABSTRACT FROM AUTHOR]

Hyperbola detection is an important application field of ground-penetrating radar (GPR) systems as underground threats and targets detected by these systems are presented in the form of a hyperbola. However, the low-amplitude hyperbola detection of deeply-buried targets and targets with low metal content has remained a major challenge due to the increase in the attenuation of radar echo with an increase in the detection depth and the features with low resolution extracted by existing methods. In this study, first, a high-level visual feature, namely, phase symmetry, is proposed to effectively improve the feature resolution and the robustness of amplitude change in GPR images. Then, we propose a handcrafted feature descriptor based on phase symmetry, namely, histogram of oriented vector phase symmetry (HOVPS) to improve the detection of hyperbola in GPR images. In constructing HOVPS, we first cite our previous work to enhance the descriptive ability of hyperbola in GPR images. Subsequently, we extend a phase symmetry model to develop a vector feature model [namely, vector phase symmetry (VPS)]. Finally, HOVPS is developed based on the VPS to extract structural information from GPR images. The proposed HOVPS describes the shape features in GPR images using a symmetrical structure, and it is used as the feature input of the classifier for hyperbola detection. The qualitative analysis of the proposed method is performed by comparing the performance of the proposed method for the extraction of features on different GPR data with those of other methods. In addition, we also provide quantitative analysis on different signal-to-noise ratio (SNR) test sets, and the results reveal that the proposed method outperforms various state-of-the-art methods (e.g., GPR histogram of oriented gradient (gprHOG), edge histogram descriptor (EHD), and log-Gabor (LG) feature). [ABSTRACT FROM AUTHOR]