Maximum Order Research Articles

On elements of prescribed norm in maximal orders of a quaternion algebra

Abstract Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $ . We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p, $\mathcal {O}$ is spanned as a ${\mathbb {Z}}$ -module by elements with norm in S. The second says that $\mathcal {O}$ is determined up to isomorphism by its theta function.

NEUTROSOPHIC EXTENSION OF THE NEGATIVE BINOMIAL DISTRIBUTION: PROPERTIES AND APPLICATIONS TO INTERVAL-VALUED DATA

In real world while addressing the challenges of ambiguity and vagueness, the problems call for a scientific approach to quantify the uncertainties. The negative binomial distribution, a cornerstone in classical probability theory, traditionally examines the probability of achieving the kth success in n trials, following failures, with precise observations. However, the available literature does not address the occurrence of observations in the intervals and hence the handling of interval-valued data remained unattended. This study introduces the neutrosophic negative binomial distribution which is extended from the classical version by incorporating both indeterminacy and crisp intervals. We derive various properties of this generalized distribution which includes; characteristic function, probability function and moment generating function. This study also presents the reliability analysis metrics like cumulative hazard rate, Mills ratio, survival hazard rate, odds ratio and reversed hazard rate. This study explores the order statistics for neutrosophic negative binomial distribution, encompassing with joint, maximum, minimum and median order statistics. Special case studies are discussed which demonstrate the application of the proposed distribution, paving the way for addressing complex problems that combine conventional approaches with non-precisely determined information.

Algebraic algorithms for vector bundles over curves

In this paper, we represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and images of global homomorphisms, isomorphisms between vector bundles, cohomology groups, extensions, and splitting into a direct sum of indecomposables. Most algorithms are deterministic except for computing isomorphisms when the base field is infinite. Some algorithms are only polynomial time if we may compute Hermite forms of pseudo-matrices in polynomial time. All algorithms rely exclusively on algebraic operations in function fields. For applications, we give an algorithm enumerating isomorphism classes of vector bundles on an elliptic curve, and to construct algebraic geometry codes over vector bundles. We implement all our algorithms into a SageMath package.

On The Modeling of Biomedical Data Sets with a New Generalized Exponentiated Exponential Distribution

Introduction: Many experts in the field of distribution theory have focused on extending probability distributions utilizing extended families of continuous distributions to improve the modeling adaptability of the conventional probability distributions. Methods: This research employed a continuous family of probability distributions as introduced in the literature using the T-X methodology. Through this, some of statistical and mathematical properties of the model were derived and studied Results: This study had introduced a brand-new, five-parameter generalized exponentiated exponential distribution, which is a continuous probability distribution. With the aid of the quantile function, moments, moment generating function, survival function, hazard function, mean, and median, among other mathematical and statistical aspects, the new distribution's shape was deduced and researched. It was also possible to derive the probability density function for the minimum and maximum order statistics for this distribution. The method of maximum likelihood estimate was used to produce a conventional estimation of the unknown parameters. A simulation study was carried out to assess the efficiency and consistency of the estimation method used. To evaluate the fit and adaptability of the new model, it was applied to four real-world datasets in the field of medicine. Conclusion: The analysis's findings demonstrated that the new model performs better than its counterparts and offers a better fit than the Topp-Leone exponentiated exponential, Topp-Leone Kumaraswamy exponential, exponentiated exponential, and exponential distributions.

About One Multistep Multiderivative Method of Predictor-Corrector Type Constructed for Solving İnitial-Value Problem for ODE of Second Order

Considering the wide application of the initial-value problem for Ordinary Differential Equations second-order with a special structure, here for solving this problem constructed the special Multistep Multiderivative Methods. Many scientists studied this problem , but the most distinguishing is the Ştörmer. To solve this problem here is proposed to use the Multistep Second derivative Method with a special structure. This method has been generalized by many authors, which is called as the linear Multistep Multiderivative Methods with the constant coefficients. Many authors shave shown that the Multistep Second derivative Method can be applied to solve the initial-value problem for ODEs of the first order. Euler himself using his famous method discovered that, in his method when moving from one point to another local truncation errors add up, the results of which reach a very large value. To solve this problem, he suggested using more accurate methods. For this aim, Euler proposed calculating the next term in the Taylor series of the solutions of the investigated problem. Developing this idea and papulation of the Multistep Multiderivative Methods here to solve the named problem it is suggested to use MultistepThriedderivative Methods, taking into account that methods of this type are more accurate. For the demonstration above, receiving results here have constructed some concrete methods. Also by using some of Dahlquist’s and Ibrahimov’s results for Multistep Methods with the maximum order of accuracy were compared. Proven that the MultistepThriedderivative Methods are more accurate than the others. By using model problems have illustrated some results received here.

Challenges in reliable preoperative blood ordering: A qualitative interview study.

Presurgical blood orders are important for patient safety during surgery, but excess orders can be costly to patients and the healthcare system. We aimed to assess clinician perceptions on the presurgical blood ordering process and perceived barriers to reliable decision-making. This descriptive qualitative study was conducted at a single large academic medical center. Semi-structured interviews were conducted with surgeons, anesthesiologists, nurse anesthetists, nurse practitioners working in preoperative assessment clinics, and transfusion medicine physicians to assess perceptions of current blood ordering processes. Interview responses were analyzed using an inductive open coding approach followed by thematic analysis. Twenty-three clinicians were interviewed. Clinicians felt that the current blood ordering process was frequently inconsistent. One contributor was a lack of information on surgical transfusion risk, related to lack of experience in ordering clinicians, insufficient communication between stakeholders, high turnover in academic settings, and lack of awareness of the maximum surgical blood ordering schedule. Other contributors included differing opinions about the benefits and harms of over- and under-preparing blood products, leading to variation in transfusion risk thresholds between clinicians, and disagreement about the safety of emergency-release blood. Several barriers to reliable decision-making for presurgical blood orders exist. Future efforts to improve ordering consistency may benefit from improved information sharing between stakeholders and education on safe transfusion practices.

SBFEM with perturbation method for solving the Reynolds equation

AbstractRotordynamic simulations with nonlinear hydrodynamic bearing forces require a solution of the Reynolds equation at every time step. As a computationally efficient alternative to the standard numerical methods, a semi‐analytical solution based on the scaled boundary finite element method (SBFEM) was developed recently. Through a discretization of the hydrodynamic pressure (dependent variable) along the circumferential but not the axial coordinate, the partial differential equation is transformed into a system of ordinary differential equations. This system of differential equations is referred to as SBFEM equation and can be solved exactly if the influence of shaft tilting is neglected. In common numerical models, this influence can be taken into account without difficulties, but as far as this semi‐analytical approach is concerned, shaft tilting complicates the equations substantially. Therefore, previous studies on the SBFEM solution of the Reynolds equation were conducted without consideration of this effect. The formulation presented in the work at hand no longer requires this simplification. The terms representing the influence of shaft tilting in the SBFEM equation are handled by the perturbation method. The pressure field is expressed by a series expansion, where the solution of order correlates to the power of a perturbation parameter chosen proportional to the tilting angle. The differential equation governing the solution contains lower‐order solutions on its right‐hand side, implying a recursive computation of the series from lowest to highest order. A universal expression for the general solution is formulated, where only the coefficients and the maximum power of the axial coordinate differ for every . This allows the implementation of a general algorithm with no inherent limitation regarding the maximum order of perturbation. For verification, the pressure fields computed by the proposed method are compared to a numerical reference solution, showing that the series converges to the correct result for the investigated set of parameters.

On the positivity of B-spline Wronskians

A proof that Wronskians of non-zero B-splines are positive is given, using only recursive formulas for B-splines and their derivatives. This could be used to generalize the de Boor–DeVore geometric proof of the Schoenberg–Whitney conditions and total positivity of B-splines to Hermite interpolation. For Wronskians of maximal order with respect to a given degree, positivity follows from a simple formula.

Enumeration and Representation Theory of Spin Space Groups

Fundamental physical properties, such as phase transitions, electronic structures, and spin excitations, in all magnetic ordered materials, were ultimately believed to rely on the symmetry theory of magnetic space groups. Recently, it has come to light that a more comprehensive group, known as the spin space group (SSG), which combines separate spin and spatial operations, is necessary to fully characterize the geometry and underlying properties of magnetic ordered materials. However, the basic theory of SSG has seldom been developed. In this work, we present a systematic study of the enumeration and the representation theory of the SSG. Starting from the 230 crystallographic space groups and finite translation groups with a maximum order of eight, we establish an extensive collection of over 100 000 SSGs under a four-index nomenclature as well as international notation. We then identify inequivalent SSGs specifically applicable to collinear, coplanar, and noncoplanar magnetic configurations. To facilitate the identification of the SSG, we develop an online program that can determine the SSG symmetries of any magnetic ordered crystal. Moreover, we derive the irreducible corepresentations of the little group in momentum space within the SSG framework. Finally, we illustrate the SSG symmetries and physical effects beyond the framework of magnetic space groups through several representative material examples, including a candidate altermagnet RuO2, spiral spin polarization in the coplanar antiferromagnet CeAuAl3, and geometric Hall effect in the noncoplanar antiferromagnet CoNb3S6. Our work advances the field of group theory in describing magnetic ordered materials, opening up avenues for deeper comprehension and further exploration of emergent phenomena in magnetic materials. Published by the American Physical Society 2024

Maximum surgical blood order schedule: A key component of perioperative blood management in neurosurgery

Abstract: CONTEXT: Institute specific protocols are needed to reduce unnecessary cross matches. Type and screen policies for surgical procedures which does not require blood and maximum surgical blood order schedule (MSBOS) for procedures which require blood units are the two protocols in use currently. AIM: To compare the effect of maximum surgical blood order schedule in neurosurgery between pre- and post-implementation. SETTINGS AND DESIGN: It is a study of 5 years’ duration (4 years retrospective and 1 year prospective) conducted at the departments of transfusion medicine, neurosurgery and neuroanesthesia. SUBJECTS AND METHODS: The patients above 18 years for whom blood requisition were sent were included, whereas those who underwent emergency procedures, massive transfusion, or had underlying bleeding disorders were excluded. The MSBOS was developed based on 4 years’ retrospective data and the new protocol was followed for 1 year. The transfusion indices were compared between preimplementation and postimplementation. RESULTS: A total of 2462 patients were included in the retrospective period, whereas it was 746 in the prospective group. Overall, reduction of crossmatch to transfused ratio from 3.6 to 2.4 and increase of blood utilization from 27.40% to 41.55% was observed. The patients for whom type and screen were performed increased from 18% in the retrospective period to 75% in the prospective period. Eighteen out of 30 common procedures were managed with T and S and seven of them did not require even a single unit of red blood cells intraoperatively. CONCLUSIONS: Our study highlights the importance of including both diagnosis and procedure to formulate MSBOS and the use of modified formulas “transfusion index for type and screen and transfusion probability for type and screen” for revision of MSBOS.

Suppression of blowup by slightly superlinear degradation in a parabolic–elliptic Keller–Segel system with signal-dependent motility

In this paper, we consider an initial–Neumann boundary value problem for a parabolic–elliptic Keller–Segel system with signal-dependent motility and a source term. Previous research has rigorously shown that the source-free version of this system exhibits an infinite-time blowup phenomenon when dimension N≥2. In the current work, when N≤3, we establish uniform boundedness of global classical solutions with an additional source term that involves slightly super-linear degradation effect on the density, of a maximum growth order slogs, unveiling a sufficient blowup suppression mechanism. The motility function considered in our work takes a rather general form compared with recent works (Fujie and Jiang, 2020; Lyu and Wang, 2023) which were restricted to the monotone non-increasing case. The cornerstone of our proof lies in deriving an upper bound for the second component of the system and an entropy-like estimate, which are achieved through tricky comparison skills and energy methods, respectively.

Modeling the Determination of Wavelength Using a Diffraction Grating in Python

To model a laboratory experiment for determining the wavelength of light using a diffraction grating in Python, it is proposed to use the libraries numpy, matplotlib, and ipywidgets. A simulation is created to model the positions of maxima on the screen. The developed code can be run in an environment that supports ipywidgets, such as Jupyter Notebook, to obtain an interactive simulation for determining the wavelength of light using a diffraction grating. The code includes the following steps: 1. Import the necessary libraries: numpy for numerical calculations, matplotlib for plotting graphs, ipywidgets for creating interactive widgets. 2. Define constants: d - the distance between the slits of the diffraction grating, L - the distance from the grating to the screen. 3. Create the function calculate_maxima to calculate the positions of the maxima. 4. Create the function plot_diffraction_pattern to plot the diffraction pattern. 5. Create interactive widgets: wavelength_slider to change the wavelength of light, order_max_slider to change the maximum order of the interference maxima. 6. Run the interactive simulation, which allows parameters to be adjusted and results to be observed in real-time.

Maximum odd induced subgraph of a graph concerning its chromatic number

AbstractLet be the maximum order of an odd induced subgraph of . In 1992, Scott proposed a conjecture that for a graph of order without isolated vertices, where is the chromatic number of . In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that for a connected graph of order . Scott's conjecture is open for graphs with chromatic number at least 3.

Understanding the surface tension reduction of the water/air interface due to the synergistic effect of the SDS-CTAB mixture: A molecular dynamics simulation study

Using molecular dynamics simulation, the surface tension is calculated and compared with experimental data to provide a microscopic interpretation of the synergistic effect on water surface tension reduction caused by a binary mixture composed of anionic and cationic surfactants. For the anionic and cationic surfactants, the sodium dodecyl sulphate (SDS) and the cetyl trimethylammonium bromide (CTAB) were used, respectively. Besides the electrostatic attraction between the anionic and cationic surfactants, resulting in a synergistic effect, other properties were analysed to study the reduction of the surface tension from a molecular level. For different SDS:CTAB composition of distinct properties of the pseudobinary mixture, such as the hydrogen bonds, order of the tails, and the arrangement of the surfactants at the interface were calculated. Those properties present different behaviours at the maximum and minimum values of the surface tension. The CTAB tails present a higher order than the SDS tails at all SDS:CTAB compositions however, the first ones show a maximum order at the minimum in the surface tension. The excess configurational energy was also calculated and shows a minimum value in the same composition as the minimum value of the surface tension.

Improvement for modeling the damping of the wake oscillator based on the Van der Pol scheme

Given the importance of risers and umbilical cables in the exploitation of deep-sea resources, the vortex induced vibration (VIV) of long flexible cylinders has been systematically studied, and it is acknowledged that the wake oscillator is a satisfactory tool in practically predicting the VIV for offshore engineering applications. Based on the conventional wake oscillator with different damping term formulations, the present study systematically explores the influences of the coefficients and the maximum order of a polynomial damping term within the Van der Pol type wake oscillator. More specifically, the coefficients of the second-order polynomial are adjusted to vary inside a reasonable range, and the polynomial order is increased from the conventional specification of 2–4, 6, and 8. The vibrations of the flexible cylinder predicted by the revised wake oscillator are compared to the measurements taken from an experiment reported in the literature. The comparison indicates that increasing polynomial coefficients generally reduce VIV dominant mode numbers. In addition, increasing the polynomial order aligns the dominant mode more closely with experimental data, although this effect diminishes when the polynomial order exceeds 4. It is argued that the gradual change in phase differences along the cylinder induced by increasing either polynomial coefficient or maximum order could be the reason. The present study sheds light into the mechanism for the damping effect observed in hydrodynamic forces observed in VIVs and lays the foundation for suggesting an optimal formulation of the damping terms as 0.45q2+0.6q−0.3 compared to the conventional formulation of 0.3q2−0.3.

Finding orientations of supersingular elliptic curves and quaternion orders

AbstractAn oriented supersingular elliptic curve is a curve which is enhanced with the information of an endomorphism. Computing the full endomorphism ring of a supersingular elliptic curve is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $$\mathfrak {O}$$ O -orientable for a fixed imaginary quadratic order $$\mathfrak {O}$$ O provides non-trivial information towards computing an endomorphism corresponding to the $$\mathfrak {O}$$ O -orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at p and $$\infty $$ ∞ . We provide code implementations in Sagemath (in Stein et al. Sage Mathematics Software (Version 10.0), The Sage Development Team, http://www.sagemath.org, 2023) which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to O(p), even for cryptographically sized p.

On the maximum local mean order of sub‐k $k$‐trees of a k $k$‐tree

AbstractFor a ‐tree , a generalization of a tree, the local mean order of sub‐‐trees of is the average order of sub‐‐trees of containing a given ‐clique. The problem whether the maximum local mean order of a tree (i.e., a 1‐tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any ‐tree , does the maximum local mean order of sub‐‐trees containing a given ‐clique occur at a ‐clique that is not a major ‐clique of ? In this paper, we give it an affirmative answer.

Maximum surgical blood ordering schedule for elective surgical procedures in Omdurman teaching hospital, Sudan

BackgroundThe need for blood during a surgical procedure is greater than what blood banks are able to provide. There is an excessive amount of blood being ordered for elective surgeries, surpassing the actual requirements. Only 30% of the cross matched blood is actually used in these surgeries. The accuracy of estimating the transfusion needs before a surgical procedure can be determined by looking at the cross match to transfusion ratio and the transfusion index. “These indicators play a crucial role in developing the maximum surgical blood ordering schedule; in this study, these indicators were tested.”Aim of studyIs to determine the efficiency of blood ordering and transfusion practices for patients undergoing elective surgeries.MethodsThis study is a prospective cross-sectional hospital-based study done at Omdurman Teaching Hospital-Sudan. Conducted for the duration of 6 months period from July to December 2019.The study participants were patients who underwent elective surgical procedures in general surgery and Urology departments as total coverage sample over a period of study duration. Ethical clearance obtained from ethical committee of Sudan Medical Specialization Board.ResultsTwo hundreds seven patients included in this study, the amount of blood units requested were 443-unit, cross matching for 98.6% (n 437) of units were done. Only 100 unit were Transfused (22,8%). The calculated CT ratio was 4.4, transfusion index was 1.6 and transfusion probability was 29.9%.ConclusionTransfusion probability and transfusion index of the present study were optimal but comparatively higher than the standard guidelines as most of the cross matched blood was not utilized.

On the order of recursive differentiability of finite binary quasigroups

The recursive derivatives of an algebraic operation are defined in [1], where they appear as control mappings of complete recursive codes. It is proved in [1], in particular, that the recursive derivatives of order up to r of a finite binary quasigroup (Q, ·) are quasigroup operations if and only if (Q, ·) defines a recursive MDS-code of length r + 3. The author of the present note gives an algebraic proof of an equivalent statement: a finite binary quasigroup (Q, ·) is recursively r-differentiable (r ≥ 0) if and only if the system consisting of its recursive derivatives of order up to r and of the binary selectors, is orthogonal. This involves the fact that the maximum order of recursive differentiability of a finite binary quasigroup of order q does not exceed q − 2.

Thickness of Nanoscale Poly(Dimethylsiloxane) Layers Determines the Motion of Sliding Water Drops.

Layers of nanometer thick polydimethylsiloxane (PDMS) are applied as hydrophobic coatings because of their environmentally friendly and chemically inert properties. In applications such as heat exchangers or fog harvesting, low water drop friction on surfaces is required. While the onset of motion (static friction) has been studied, the knowledge of dynamic friction needs to be improved. To minimize drop friction, it is essential to understand which processes lead to energy dissipation and cause dynamic friction? Here, the dynamic friction of drops on PDMS brushes of different thicknesses is measured, covering the whole available velocity regime. The brush thickness L turns out to be a predictor for drop friction. 4-5nm thick PDMS brush shows the lowest dynamic friction. A certain minimal thickness is necessary to form homogeneous surfaces and reduce the attractive van der Waals interaction between water and the substrate. The increase in dynamic friction above L = 5nm is also attributed to the increasing viscoelastic dissipation of the capillary ridge formed at the contact line. The height of the ridge is related to the brush thickness. Fluorescence correlation spectroscopy and atomic force measurements support this interpretation. Sum-frequency generation further indicates a maximum order at the PDMS-water interface at intermediate thickness.