Elliptical Curves Research Articles

Vortex-pair dynamics in three-dimensional homogeneous dipolar superfluids

The static and dynamic properties of vortices in dipolar Bose-Einstein condensates (dBECs) can be considerably modified relative to their nondipolar counterparts by the anisotropic and long-ranged nature of the dipole-dipole interaction. Working in a uniform dBEC, we analyze the structure of single vortices and the dynamics of vortex pairs, investigating the deviations from the nondipolar paradigm. For a straight vortex line, we find that the induced dipolar interaction potential is axially anisotropic when the dipole moments have a nonzero projection orthogonal to the vortex line. This results in a corresponding elongation of the vortex core along this projection as well as an anisotropic superfluid phase and enhanced compressibility in the vicinity of the vortex core. Consequently, the trajectories of like-signed vortex pairs are described by a family of elliptical and oval-like curves rather than the familiar circular orbits. Similarly for opposite-signed vortex pairs their translation speeds along the binormal are found to be dipole-interaction dependent. We expect that these findings will shed light on the underlying mechanisms of many-vortex phenomena in dBECs such as quantum turbulence, vortex reconnections, and vortex lattices. Published by the American Physical Society 2024

Numerical simulation of optimizing the swing curve of a 3DOF biomimetic pectoral fin in drag mode

In response to the optimization problem of the motion law of the pectoral fins of biomimetic robotic fish, this paper uses computational fluid dynamics (CFD) methods to achieve the CFD optimization of two types of rigid biomimetic pectoral fins with three degrees of freedom (DOF): elliptical and 8-shaped swing curves. By analyzing the hydrodynamic characteristic curve of the pectoral fins, pressure distribution upstream and downstream of the fin surface, velocity vector of the surrounding flow field during pectoral fin swing, and three-dimensional vortex structure, it is found that when the pectoral fins are pushed forward in an elliptical swing curve, the resistance peak value decreases after two optimizations. However, at the same time, a certain amount of thrust is lost. When the pectoral fin advances in an 8-shaped swinging curve, after two optimizations, the pectoral fins resistance decreases while maintaining an enormous thrust, comprehensively improving the propulsive effect of the pectoral fin on the robotic fish. By substituting the optimized 8-shaped pattern into the robotic fish model, it is found that the flapping angle of 35° has better drag reduction performance, and the average swimming speed of the robotic fish is the fastest at this angle.

Classical trajectories from the zeros of the quantum potential: the 2D isotropic harmonic oscillator

In the first part of this work, using the quantum potential approach, we show that a solution to the time-independent Schrödinger equation determines a subset of classical solutions, only if the region corresponding to the zeroes of the quantum potential is tangent to the caustic region determined by the classical trajectories. Thus, the solutions of the time-independent Schrödinger equation, according to their caustic and the zeros of the quantum potential, can be classified in three different cases given by the following conditions: the two regions coincide, they are tangent at certain subset of points, and the two regions are not tangent at any point. In the second part, as examples of the first type of wave functions, we present the solutions of the Schrödinger equation for the 2D isotropic harmonic oscillator, which are eigenfunctions of both the Hamiltonian and the angular momentum operators. That is, we show that for this family of solutions, the zeroes of the quantum potential coincide with the caustic region. Furthermore, we find that the classical trajectories, determined from the quantum ones and the zeroes of the quantum potential, conform to a family of elliptical curves for a particle with energy, (2n + l + 1)ℏ ω, and orbital angular momentum l ℏ.

A Numerical Analysis on the Performance and Optimization of the Savonius Wind Turbine for Agricultural Use

Given the state of the world nowadays, renewable energy is becoming more and more essential rendering wind turbine electricity quite important. Its shape and the fact that the Savonius vertical axis wind turbine runs at relatively low wind speeds with high torque values makes it suitable for practical uses such as that of an irrigation system in agriculture industry. This paper utilizes numerical research with Computational Fluid Dynamics (CFD) to investigate the performance of a vertical-axis wind turbine. The ANSYS CFD program was engaged to construct the simulations during the pre- and post-processing stages. Wind speed remained constant while the angular velocity was altered to enable analysis of the flow through the wind turbine. Because of its mechanical simplicity, the primary profile of a semicircle has remained a typical option for turbines that generate high torque based on drag force. The effects of using elliptical curves and the fluctuation in thickness along the profile chord were both examined in this study. Equivalently, an attempt to optimize the rotor's design was made. After the performance of a numerical simulation, a geometry consisting of simple circle arcs was developed, with a 10.9% improvement in the power coefficient, analogous to prior optimizations with more complicated geometries. The numerical results derived include the torque coefficient evolution throughout a full rotation as well as the distribution of vorticity magnitude at different rotor points.

O Endomorfismo de Frobenius para Curvas Elípticas e algumas propriedades

In this work we present the Frobenius endomorphism for Elliptical Curves defined over finite fields and some of their properties relative to their kernel and the condition of separability.

Wide-angle high-performance polarizers based on all-elliptical-metamaterial one-dimensional photonic crystals

As a class of all-dielectric metamaterials, elliptical metamaterials (EMMs) with elliptical iso-frequency curves provide a lossless platform to manipulate light with unity efficiency. Herein, we select two kinds of EMMs as the elements to constitute a special class of one-dimensional (1D) photonic crystals (PCs) called all-EMM 1D PCs and realize polarization-sensitive photonic bandgaps (PBGs). Enabled by the polarization-sensitive property of the PBG, we achieve a wide-angle high-performance polarizer. The maximum polarization extinction ratio reaches 1.82 × 104 and the width of the operating angle range reaches 36.07°. Besides, the performance of the polarizer demonstrates robustness against the layer thickness. The all-EMM 1D PCs are 1D nanostructures completely consisting of dielectric thin films, which can be much easily fabricated compared with 2D and 3D nanostructures. Our work provides a lithography-free recipe to design wide-angle high-performance polarizers.

Derivation of a normalised failure surface for square CECFST columns embedded with circular steel tube: A unified approach

Concrete-Encased Concrete-Filled Steel Tube (CECFST) columns are a new type of composite columns that possess great strength, stiffness, ductility, corrosion and fire resistance. To fully realise their benefits, an accurate design approach of calculating the strength of these members is necessary. In this paper, the plastic stress distribution (PSD) and load contour (LC) methods in the design code of predicting the cross-section strength of CECFST columns is first evaluated through numerical and published experimental data. Considering the shortcomings of conventional design methods, a unified design approach is then developed. Compared to the conventional PSD method, the developed method can yield conservative resistance predictions for uniaxial eccentric loaded (UEL) CECFST columns. In addition, instead of using linear moment interaction curves to predict the strength of biaxial eccentric loaded (BEL) CECFST columns, elliptical moment interaction curves are adopted in the proposed design approach. Most importantly, a normalised 3D axial-load–biaxial-moment (n–my–mz) interaction surface can be generated through the proposed method, which only requires determining 6 critical points on the interaction surface. Unlike the conventional design approach, in which the UEL and BEL composite columns need to be respectively designed through PSD and LC methods, a unified design procedure can be achieved through the proposed approach. Besides, the confinement effect from both the steel tube and the stirrups can also be considered in the proposed approach and much higher accuracy and consistency can be achieved in predicting the cross-section strength of CECFST columns.

Design of Metasurface with Nanoslits on Elliptical Curves for Generation of Dual-Channel Vector Beams.

The manipulations of nanoscale multi-channel vector beams (VBs) by metasurfaces hold potential applications in various important fields. In this paper, the metasurface with two sets of nanoslits arranged on elliptic curves was proposed to generate the dual-channel focused vector beams (FVBs). Each set of nanoslits was composed of the in-phase and the out-of-phase groups of nanoslits to introduce the constructive interference and destructive interference of the output light field of the nanoslits, focusing the converted spin component and eliminating the incident spin component at the focal point. The two sets of nanoslits for the channels at the two focal points were interleaved on the same ellipses, and by setting their parameters independently, the FVBs in the two channels are generated under illumination of linearly polarized light, while their orders and polarization states of FVBs were controlled independently. The generation of the FVBs with the designed metasurfaces was demonstrated by the finite-difference time domain (FDTD) simulations and by the experimental verifications. The work in this paper is of great significance for the generation of miniaturized multi-channel VBs and for broadening the applications of metasurfaces.

RADIOGRAPHIC ASSESSMENT OF THE COURSE AND VISIBILITY OF THE MANDIBULAR CANAL BY PANORAMIC RADIOGRAPHY

Background and Objectives: The mandibular canal is a canal inside the mandible that contains the inferior alveolar artery, the inferior alveolar nerve and the inferior alveolar vein. The anatomy and variations of the human mandible are very important for planning various surgical measures such as extraction of the third molar, dental implants and mandibular reconstruction. This study was conducted to determine the frequency of the different courses of the mandibular canal and to investigate the visibility of the mandibular canal by panoramic radiography, among a sample of Yemeni adults.
 Subjects and Methods: A retrospective cross-sectional study was conducted in Sana'a city on a sample of Yemeni adults. The study consisted of panoramic radiographs of 928 patients; the number of males was 340 (36.6%) and 588 females (63.4%). The mandibular canal course, as shown in the panoramic radiographs, was classified into four types: elliptical, linear, turning curves and spoon-shaped. The visibility of this canal from the first to the third molar region was assessed through visual determination of whether the lower canal was clearly visible, possibly visible, or invisible.
 Results: Elliptical curves were the most observed along the mandibular canal course at 62.2% of the total, followed by the linear curve (24.8%) while the lower rate curve spoon (7.9%) and the turning curve (5.2%). There were no statistically considerable differences between the distribution of the mandibular canal courses between males and females except in the spoon curve where the ratio was 10.8% in males versus 6.1% in females (p <0.05). The percentage of clearly visible mandibular canals was the highest among the spoon-shaped curves (49.9% in the first molar, 63% in the second, 78.1% in the third molar) and the lowest among the linear curves (10.9% in the first, 18.7% in the second and 33.5% in the third molar). On panoramic radiographs, the invisible mandibular canals are found in 198 (21.3%) of the examined sites in the first molar region, in 85 (11%) in the second molar region, and in only 6 (0.64%) in the third molar region.
 Conclusion: It was observed that elliptical curves along the path of the mandibular canal were the most present, while a decrease in the ratio of the two types of spoon curve and turning curve was observed. It was found that spoon curve was significantly more common among males than females. It was also found that the visibility of madibular canal is fewer in the first molar region than in the third molar region.
 Peer Review History: 
 Received: 21 September 2020; Revised: 8 October; Accepted: 29 October, Available online: 15 November 2020
 UJPR follows the most transparent and toughest ‘Advanced OPEN peer review’ system. The identity of the authors and, reviewers will be known to each other. This transparent process will help to eradicate any possible malicious/purposeful interference by any person (publishing staff, reviewer, editor, author, etc) during peer review. As a result of this unique system, all reviewers will get their due recognition and respect, once their names are published in the papers. We expect that, by publishing peer review reports with published papers, will be helpful to many authors for drafting their article according to the specifications. Auhors will remove any error of their article and they will improve their article(s) according to the previous reports displayed with published article(s). The main purpose of it is ‘to improve the quality of a candidate manuscript’. Our reviewers check the ‘strength and weakness of a manuscript honestly’. There will increase in the perfection, and transparency.
 Received file 
 
 Average Peer review marks at initial stage: 6.5/10
 Average Peer review marks at publication stage: 8.0/10
 Reviewer(s) detail:
 Dr. Sunita Singh, Baylor College of Medicine, Houston, Texas, USA, sunita.nccs@gmail.com
 Dr. Ali Abdullah A. Al-Mehdar, Thamar University, Yemen, alialmehdar2006@yahoo.com
 
 Comments of reviewer(s): 
 
 Similar Articles:
 BIOFILM FORMATION AND ANTIBIOTIC SUSCEPTIBILITY OF UROPATHOGENS IN PATIENTS WITH CATHETER ASSOCIATED URINARY TRACT INFECTIONS IN IBB CITY -YEMEN

Design and power loss evaluation of a noncircular gear pair for an infinitely variable transmission

A novel noncircular gear design method is proposed in this work based on a modified high-order elliptical pitch curve for an infinitely variable transmission (IVT), and power loss evaluation of a noncircular gear pair is investigated. Modified high-order elliptical pitch curves of noncircular gears are developed by using kinematic equations of the IVT to eliminate cusps between every two neighboring lobes of pitch curves, and improve smoothness of pitch curves and manufacturability of the noncircular gear pair. A power loss evaluation method is developed for the noncircular gear pair by calculating sliding friction power loss of meshing tooth surfaces. The time-varying friction force of every contact position on meshing tooth surfaces of the noncircular gear pair is calculated by using their variable coefficients of friction and sliding velocities. The instantaneous sliding friction power loss of the noncircular gear pair is calculated based on time-varying friction forces and sliding velocities along contact paths of tooth profiles of the noncircular gear pair.

Quantization for Uniform Distributions on Hexagonal, Semicircular, and Elliptical Curves

In this paper, first we have defined a uniform distribution on the boundary of a regular hexagon and then investigate the optimal sets of n-means and the nth quantization errors for all positive integers n. We give an exact formula to determine them, if n is of the form $$n=6k$$ for some positive integer k. We further calculate the quantization dimension, the quantization coefficient, and show that the quantization dimension is equal to the dimension of the object, and the quantization coefficient exists as a finite positive number. Then, we define a mixture of two uniform distributions on the boundary of a semicircular disc and obtain a sequence and an algorithm, with the help of which we determine the optimal sets of n-means and the nth quantization errors for all positive integers n with respect to the mixed distribution. Finally, for a uniform distribution defined on an elliptical curve, we investigate the optimal sets of n-means and the nth quantization errors for all positive integers n.

Efficient trochoidal milling based on medial axis transformation and inscribed ellipse

Trochoidal milling is a promising strategy for machining pockets with difficult-to-cut materials. Conventional trochoidal toolpath is composed of a set of circular curves and transition curves. However, the actual radial cutting width has a large variation along processing path due to geometric features of inscribed circles, which results in large fluctuation of cutting width and low material removal rate. In this paper, a new trochoidal toolpath based on elliptical curve is proposed to overcome the mentioned drawbacks. Medial axis transformation is adopted to represent the 2D pocket geometry. An algorithm is developed to determine a sequence of inscribed ellipses according to pre-defined machining parameters. Hermit curve is utilized to generate the transition curves between two adjacent elliptical curves. The overall toolpath length is significantly reduced. Machining experiments are conducted to validate the feasibility and efficiency of the proposed method. The experimental results showed an improvement of 12.7% in machining efficiency without increasing the maximum cutting force, compared with conventional trochoidal toolpath.

A Proof for the Birch Swinnerton-Dyer Conjecture

The Birch Swinnerton-Dyer Conjecture states that the number of rational points on an elliptic curve are infinite. This paper develops the logical formalism for a reinforcement learning system embedded with a quantum search algorithm. The formalism proves the rational points on high dimensional elliptical curves are infinite.

Four grades of ignorance-involvement and how they nourish the cognitive economy

In the human cognitive economy there are four grades of epistemic involvement. Knowledge partitions into distinct sorts, each in turn subject to gradations. This gives a fourwise partition on ignorance, which exhibits somewhat different coinstantiation possibilities. The elements of these partitions interact with one another in complex and sometimes cognitively fruitful ways. The first grade of knowledge I call “anselmian” to echo the famous declaration credo ut intelligam, that is, “I believe in order that I may come to know”. As construed here, one knows in this anselmian way that E = mc2 just in case one knows that sentence expresses a true statement, but without having to understand the proposition it expresses. Most epistemologists ignore the significance of this grade of epistemic involvement. In a second grade of epistemic involvement, knowing that E = mc2 is knowing what that sentence means and understanding the proposition it express. This is knowledge in the propositional or semantic sense, and is the dominant target of epistemological investigation. Tacit and implicit (TI) knowledge occupies another tier. A typical example would be something that someone has “known all along” but, until now, hasn’t had occasion to put her mind to it or formulate in words. TI-knowledge remains a minority interest in today’s epistemology. Operating at a fourth grade of epistemic involvement is what I call “impact”-knowledge, which is the knowledge of a matter at its deepest and most widespread. An example, to be discussed below, is the knowledge that was generated by the Wiles proof of Fermat’s last theorem. Its true importance lies not only, or even mainly, in its verification of a commonly accepted fact about numbers, but rather in its enrichment of the mathematics of elliptical curves and the promise it holds for greater advancement into the mathematical unknown. Knowledge of this fourth grade has yet to find a seat in the parliaments of epistemology. Knowledge of the anselmian sort is independent of the other three. Tacit and implicit knowledge is incompatible with anselmian and semantic knowledge but coinstantiable with impact-knowledge. Semantic knowledge is incompatible with tacit and implicit knowledge but coinstantiable with the others. Impact-knowledge is pairwise coinstantiable with the others. Below I will bring the ignorance partitions into such alignment as they have with these ones. In doing so, I’ll propose a naturalized causal response epistemology designed to give these interactive distinctions the theoretical air they need to breathe.

CRYPTOGRAPHIC AUTHENTICATION PROTOCOL ZERO-KNOWLEDGE SECRET ON ELLIPTIC CURVES USING PUBLIC KEYS AND RANDOM MESSAGES

We propose a cryptographic protocol with zero-knowledge proof (ZKP) on elliptic curves (EC) using public keys and random messages, allowing to establish the truth of a statement not conveying any additional information about the statement itself. The cryptographic protocols based on zero-knowledge proof allow identification, key exchange and other cryptographic operations to be performed without leakage of sensitive information during the information exchange. The implementation of the cryptographic protocol of the zero-knowledge proof on the basis of the mathematical apparatus of elliptic curves allows to significantly reduce the size of the protocol parameters and increase its cryptographic strength (computational complexity of the breaking). The security of cryptosystems involving elliptic curves is based on the difficulty of solving the elliptic curve discrete logarithm problem. We determine the completeness and correctness of the protocol and give an example of the calculation is given. The cryptographic protocol was modeled in the High-Level Protocol Specification Language, the model validation and verification of the protocol were also performed. The software verification of the cryptographic protocol was performed using the software modules On the Fly Model Checker and Constraint Logic based Attack Searcher. In order to validate the cryptographic protocol resistance to intruder attacks, we used the Security Protocol Animator package for Automated Validation of Internet Security Protocols and Applications. The security of the proposed cryptographic protocol ZKP EC is based on the difficulty of solving the elliptic curve discrete logarithm problem). The recommended elliptical curves according to DSTU 4145-2002 may be used to implement such cryptographic protocol.

Криптографический протокол доказательства с нулевым разглашением на эллиптических кривых с использованием односторонней хэш-функции

Proposed cryptographic protocol with zero-knowledge proof on elliptic curves using one-way hash function, allowing to establish the truth of allegation and does not convey any additional information about the approval. Cryptographic protocols based on zero-knowledge proof allow identification, key exchange and other cryptographic operations to be performed without leakage of sensitive information during the information exchange. The implementation of the cryptographic protocol of the zero-knowledge proof on the basis of the mathematical apparatus of elliptic curves allows to significantly reduce the size of the protocol parameters and increase the cryptographic stability (computational complexity of the hacking problem). The security of cryptosystems on elliptic curves is based on the difficulty of solving the elliptic curve discrete logarithm problem. The completeness and correctness of the protocol is determined in the work, an example of calculation is given, the cryptographic protocol is modeled on the High-Level Protocol Specification Language, the model validation and verification of the protocol are performed. Software verification of the cryptographic protocol was performed using the software modules On the Fly Model Checker and Constraint Logic based Attack Searcher. To validation the cryptographic protocol for resistance to intruder attacks was used the Security Protocol Animator package for Automated Validation of Internet Security Protocols and Applications. The security of the proposed cryptographic protocol is based on the difficulty of solving the elliptic curve discrete logarithm problem and the cryptographic stability of the hash function. To implement the cryptographic protocol, you can use the recommended elliptical curves according to DSTU 4145-2000 and the hash function GOST 34.311-95.

Modern ways of improving the procedure for the formation and verification of digital signature

In the article, a comparative analysis of existing algorithms for the formation and verification of the digital signature has been carried out, namely, widespread algorithms and the criteria for comparison was the mathematical basics on which the algorithm is based and the size of the public key. Based on the analysis, it is established that there are schemes with the addition of messages and schemes with the restoration of the information message. The conducted studies made it possible to determine that the schemes with message recovery differ from the schemes with the addition that they do not completely hash messages, but instead use masking functions and redundancy of the message. It was determined that the most effective and optimal for further use is the Nimberg-Ruppel scheme, which is based on elliptical curves (ECNR). The article describes the theoretical basis for creating a system for ensuring the integrity, reliability and confidentiality of information based on digital signatures based on elliptical curves using the standard DSTU 4145-2002 with the possibility of restoring an information message based on ECNR.

Концепція впровадження автоматизованої системи розпізнавання мовця у процес автентифікації для доступу до критичної системи

In the article, ISO/IEC 27001: 2013, an effective and user-friendly concept of reliable authentication for accessing a multi-server information critical system, which includes, in particular, a set of users, a set of servers and a dedicated server —a registration center for a system objects registration without the maintenance of verification tables, is proposed. Information exchange between the objects of the system is organized using secure sessions based on an secret keys with a matching mechanism based on one-way hash functions and cryptography of elliptic curves, which now provides the best correlation between encryption reliability and a size of a secret key among existing cryptosystems. Work with users is personalized using individual access cards, which are protected based on the provisions of the cryptographic theory of elliptical curves. Access cards, in addition to identification information and password, store biometric information about the features of the user's voice in the form of a information features vector. The application for authentication, which is installed on the user's computing device, is initiated by a two-step user recognition procedure (with an identification card) — according to the individual features of his voice and the entered password, which together with convenience ensures the reliability of the authentication process. The basic concept of introduction of the automated speaker recognition system in the authentication process for access to the critical system was proposed in the article, and according to the results of the tests it was concluded that it conforms to the requirements of the ISO/IEC 27001: 2013 standard. However, experience in the practical use of the authentication system for accessing information critical system found a number of vulnerabilities in the basic concept, for the purpose of eliminating them, an improved concept was created that also fully meets the requirements of ISO/IEC 27001: 2013. The article evaluates the computational efficiency of the proposed concepts on the basis of calculating the number of the most computationally complex operations — scalable multiplication Tm and calculating the hash function Th for authentication of objects as part of the information critical system (user, server, registration center and all these objects together). The final computational complexity of the basic concept was 4Tm + 15Th, and of the advanced one — 6Tm + 17Th, which in absolute terms is an average result in comparison with analogues. However, the use of cryptography of elliptical curves makes hashing operations about 6 times faster than hashing with the use of a public key, as in analogues.

Calibration of Binocular Vision Sensors Based on Unknown-Sized Elliptical Stripe Images.

Most of the existing calibration methods for binocular stereo vision sensor (BSVS) depend on a high-accuracy target with feature points that are difficult and costly to manufacture. In complex light conditions, optical filters are used for BSVS, but they affect imaging quality. Hence, the use of a high-accuracy target with certain-sized feature points for calibration is not feasible under such complex conditions. To solve these problems, a calibration method based on unknown-sized elliptical stripe images is proposed. With known intrinsic parameters, the proposed method adopts the elliptical stripes located on the parallel planes as a medium to calibrate BSVS online. In comparison with the common calibration methods, the proposed method avoids utilizing high-accuracy target with certain-sized feature points. Therefore, the proposed method is not only easy to implement but is a realistic method for the calibration of BSVS with optical filter. Changing the size of elliptical curves projected on the target solves the difficulty of applying the proposed method in different fields of view and distances. Simulative and physical experiments are conducted to validate the efficiency of the proposed method. When the field of view is approximately 400 mm × 300 mm, the proposed method can reach a calibration accuracy of 0.03 mm, which is comparable with that of Zhang’s method.

Experimental investigation of evolving anisotropy in unsaturated soils

This paper investigates the ‘initial’ and ‘evolving’ mechanical anisotropy of a compacted unsaturated soil. A wide campaign of triaxial compression and extension tests, involving different stress and suction paths, has been performed on both isotropically and anisotropically compacted samples of unsaturated Speswhite kaolin. The first objective is the definition of the initial yield surface of the compacted soil after suction equalisation and before any plastic loading/wetting path takes place. This is followed by the investigation of the evolution of the yield surface induced by plastic straining along different loading/wetting paths. Experimental results are interpreted by using two alternative stress variables, namely net stresses [Formula: see text] and Bishop's stress [Formula: see text] (where [Formula: see text] is Kronecker delta, Sr is the degree of saturation and s is suction). Constant suction cross-sections of the yield surface are represented as distorted ellipses not passing through the origin in the q:p plane of deviator stress plotted against mean net stress, and by distorted ellipses passing through the origin in the q:p* plane of deviator stress plotted against mean Bishop's stress. The inclination of these distorted elliptical yield curves evolves with plastic straining but remains the same at all suction levels for a given level of plastic deformation. The critical state lines in the planes q:p and q:p*, or in the semi-logarithmic v:lnp and v:lnp* planes (v is the specific volume), are generally independent of initial anisotropy or stress history, suggesting that fabric memory tends to be erased at critical state.