Odd Number Research Articles

Fuss’ Relation for Bicentric Tridecagon and Bicentric Pentadecagon

This article describes a general analytical derivation of the Fuss’ relation for bicentric polygons with an odd number of vertices. In particular, we derive the Fuss’ relations for the bicentric tridecagon and the bicentric pentadecagon.

Micromagnetic study of the dynamics of toron chains

This work analyzes the nucleation and stabilization of torons in cylindrical nanopillars with bulk DMI and easy-plane anisotropy. A micromagnetic study reveals the dependence of toron chains on the nanopillar length and the different behaviors when nucleating an even or odd number of torons in a magnetic nanopillar. Spin wave resonant modes in these systems are explored, evidencing differences according to the number of torons. The interplay between torons, chiral bobbers, and their associated spin wave modes is analyzed, which is relevant for applications demanding dynamical modes in the range of gigahertz.

Selection criteria for rotor number and rotational direction in arrays of closely spaced Darrieus turbines

Abstract The efficiency gains observed in a pair of closely spaced Darrieus turbines suggest the deployment of multiple turbines as an appealing solution for wind and, particularly, hydrokinetic applications, where the inflow direction is constant. The present study develops some design guidelines for closely spaced hydrokinetic Darrieus turbines by analyzing the trends of both power augmentation and wake development within arrays of multiple rotors, including not only an even number of rotors, which is the usual case in literature, but also an odd one. The analysis is carried out by means of two-dimensional Computational Fluid Dynamics simulations and includes not only the assessment of instantaneous blade forces but also locally sampled flow fields past each blade that allowed the reconstruction of dynamic polar data, contributing to a more comprehensive understanding of the physical mechanisms at play in such compact setups. The study demonstrates a consistent power augmentation mechanism across different layouts, even in the case of an odd number of rotors. This enhancement originates from flow blockage in the mutual interaction areas, favorably altering the inflow angle and subsequently increasing the angle of attack and lift generation. While this mechanism aligns with previous observations on arrays of counter-rotating pairs, its application to multiple turbines introduces complexities due to potential asymmetries in inflow, leading to an uneven power enhancement across turbines within the array. The identified efficiency improvement pattern suggests that maximizing leeward mutual interactions within an array of multiple Darrieus turbines would enhance the overall efficiency of the setup.

The compassionate care framework plus quality control group improves the quality of postoperative nursing management for patients in anorectal surgery: An observational study.

This study aimed to investigate the effectiveness of the joint application of the compassionate care framework (CCF) and a quality control (QC) team in improving the quality of postoperative nursing management for patients in anorectal surgery. Patients who underwent anorectal surgery at the China-Japan Friendship Hospital between May 2022 and December 2022 were selected by convenience sampling. A digital random generator was used for random grouping. The odd numbers entered the control group (n = 53) and the even numbers entered the study group (n = 53). The control group received postoperative traditional nursing, and the study group received postoperative nursing and QC team management. A questionnaire survey was used to analyse patients' satisfaction with nursing. The two groups were compared for nursing quality scores, patient satisfaction with nursing care and incidence of complications. After management, compared with the control group, the study group scored significantly higher for the nursing process, environmental management, emergency response and safety protection (p < 0.05 for all). The study group also exhibited higher satisfaction scores with the nursing process, nurse-patient relationship and service attitude compared with the control group (p < 0.05 for all). The complication incidence rate in the study group was significantly lower than in the control group (p < 0.05). The combined use of the CCF and a QC team can effectively improve the quality of postoperative nursing management for patients in anorectal surgery, reduce the incidence of complications and enhance patient satisfaction with nursing care, thus holding significant potential for wider application.

Differences in elemental abundances between CI chondrites and the solar photosphere

AbstractCI chondrites have been a proxy for the solar system since the mid‐20th century. The photospheric and CI chondrite abundances (P and CI, respectively) show a strong correlation. CI as a proxy is also justified by the (i) smoothness of their abundances plotted as a function of odd mass number and (ii) agreement within the error of P as determined spectroscopically. But our statistical assessment of spectroscopic studies and solar wind from the Genesis mission indicates that the small, ~10%–30%, differences (residuals) between CI and P depend on the 50% condensation temperature (Tc50). So, if CI is to be used as a proxy for P, Cosmochemists may want to add a correction to individual elements. Our work is consistent with two published hypotheses: that (i) residuals are linear with Tc50 and (ii) that elements having Tc50 &gt; 1343 K are depleted relative to those with 495 K &lt; Tc50 &lt; 1343 K in CI. We discuss other interpretations which are also feasible. Understanding these small differences of the CI and P for different elements and their variation with Tc50 can help constrain future models of solar system formation and the history of CI chondrites.

Numerical Analysis and Experimental Validation of Trimaran and Pentamaran Resistance at Various Separation Distances

Trimaran and pentamaran are multihull types with an odd number of hulls, namely three and five hulls, generally consisting of one center hull and two or four side hulls smaller than the center hull, which can reduce ship resistance. The trimaran and pentamaran have a more complex phenomenon than the monohull, because of the interaction between the main hull and the side hull, which causes interference caused by changes in flow velocity, pressure changes, and wave interactions generated by each hull. The objective of this study is to analyze the resistance of trimaran and pentamaran NPL-4b models with transom-symmetrical hull and separation distances, specifically S/L ratios of 0.2, 0.3, and 0.4. Since a more limited base of experience exists for multihull ships, experimental or numerical modeling techniques are essential for designers. Numeric investigations were conducted using Numeca software, and experiments were performed in a towing tank. Both methods follow ITTC procedures. Furthermore, the numerical analysis with CFD simulation modifies the Navier-Stokes equation using the turbulence model k-ꞷ SST to generate the RANS equation so that unsteady fluid flow problems can be calculated. It can be implemented in predicting resistance in Froude numbers (Fr) 0.2 to 0.6 at the systematic series of trimaran and pentamaran hull shapes at the model scale. The simulation results were compared to experimental data to validate the resistance of the ships. Overall, good resistance was produced by the trimaran and pentamaran in the C configuration at an S/L ratio of 0.4 with a total resistance coefficient CT of 6.60 x 10-3 and 7.37 x 10-3, respectively. The two results are in good agreement, both methods have a discrepancy of 3.70 %. Fluctuations influence the resistance in the wetted surface area (WSA), wave interactions between hulls, ship velocity, and wave propagation in the aft hull.

Crystal structure and compressibility of magnesium chloride heptahydrate found under high pressure.

The odd hydration number has so far been missing in the water-rich magnesium chloride hydrate series (MgCl2·nH2O). In this study, magnesium chloride heptahydrate, MgCl2·7H2O (or MgCl2·7D2O), which forms at high pressures above 2 GPa and high temperatures above 300 K, has been identified. Its structure has been determined by a combination of in-situ single-crystal X-ray diffraction at 2.5 GPa and 298 K and powder neutron diffraction at 3.1 GPa and 300 K. The single-crystal specimen was grown by mixing alcohols to prevent nucleation of undesired crystalline phases. The results show orientational disorder of water molecules, which was also examined using density functional theory calculations. The disorder involves the reconnection of hydrogen bonds, which differs from those in water ice phases and known disordered salt hydrates. Shrinkage by compression occurs mainly in one direction. In the plane perpendicular to this most compressible direction, oxygen and chlorine atoms are in a hexagonal-like arrangement.

Synthetic Approaches to and Structures of Diplatinum Polyynediyl Complexes trans,trans‐[(C6F5)(R3P)2Pt(C≡C)nPt(PR3)2(C6F5)] With Odd Numbers of Triple Bonds; Avoiding Complicating Ethynediyl Extrusions

AbstractReactions of trans‐(C6F5)(p‐tol3P)2Pt(C≡C)nSiEt3 (PtC2nSi; n=5, 7, 9) and excess PtCl in the presence of wet n‐Bu4N+ F− (to effect protodesilylation) under Sonogashira‐type conditions (CuCl, base, other additives) afford the title compounds PtC10Pt, PtC14Pt, and PtC18Pt in 42–32 % yields. A four‐fold substitution of the phosphine ligands in PtC10Pt by PEt3 affords Pt'C10Pt’ (78 %), and a Sonogashira reaction of Pt'C2H and Pt'Cl affords Pt'C2Pt’ (68 %). The analogous reaction with PtC2Si and PtCl is unsuccessful, presumably for steric reasons. The crystal structures of PtC10Pt, PtC14Pt, Pt'C10Pt′, and Pt'C2Pt’ exhibit a number of interesting trends and features. Certain sp chain extension reactions that lead to or employ the precursors PtC10Si, PtC12Si, PtC14Si, and PtC18Si sometimes give byproducts derived from C2 loss, and possible origins are discussed. Related phenomena have been reported by others in the course of synthesizing extended conjugated polyynes.

Adsorption of Ag, Au, Cu, and Ni on MoS2: theory and experiment

Here, we present results of a computational and experimental study of adsorption of various metals on MoS2. In particular, we analyzed the binding mechanism of four metallic elements (Ag, Au, Cu, Ni) on MoS2. Among these elements, Ni exhibits the strongest binding and lowest mobility on the surface of MoS2. On the other hand, Au and Ag bond very weakly to the surface and have very high mobilities. Our calculations for Cu show that its bonding and surface mobility are between these two groups. Experimentally, Ni films exhibit a composition characterized by randomly oriented nanoscale clusters. This is consistent with the larger cohesive energy of Ni atoms as compared with their binding energy with MoS2, which is expected to result in 3D clusters. In contrast, Au and Ag tend to form atomically flat plateaued structures on MoS2, which is contrary to their larger cohesive energy as compared to their weak binding with MoS2. Cu displays a surface morphology somewhat similar to Ni, featuring larger nanoscale clusters. However, unlike Ni, in many cases Cu exhibits small plateaued surfaces on these clusters. This suggests that Cu likely has two competing mechanisms that cause it to span the behaviors seen in the Ni and Au/Ag film morphologies. These results indicate that calculations of the initial binding conditions could be useful for predicting film morphologies. In addition, out calculations show that the adsorption of adatoms with odd electron number like Ag, Au, and Cu results in 100% spin-polarization and integer magnetic moment of the system. Adsorption of Ni adatoms, with even electron number, does not induce a magnetic transition.

On Broadcast Schemes of Knödel Graphs

Knödel graphs on even number of vertices serve as crucial components in constructing various broadcast graphs, particularly those on an odd number of vertices. The efficacy of these constructions heavily relies on the chosen broadcast scheme. Dimensional broadcast schemes are widely employed for broadcasting Knödel graphs. This paper delves into the investigation of standard and reverse dimensional broadcast schemes identifying numerous valid schemes. Specifically, let [Formula: see text] be a Knödel graph of order [Formula: see text] and [Formula: see text]. We explore the impact of [Formula: see text] cyclic shifts on both standard and reverse dimensional broadcast schemes of Knödel graphs, where [Formula: see text], with repetition of the second dimension.

The homology of a Temperley–Lieb algebra on an odd number of strands

The homology of a Temperley–Lieb algebra on an odd number of strands

An Efficient Algorithm for Constrained CASSCF(1,2) and CASSCF(3,2) Simulations as Relevant to Electron and Hole Transfer Problems.

We propose an efficient algorithm for the recently published electron/hole-transfer Dynamical-weighted State-averaged Constrained CASSCF (eDSC/hDSC) method studying charge transfer states and D1-D0 crossings for systems with odd numbers of electrons. By separating the constrained minimization problem into an unconstrained self-consistent-field (SCF) problem and a constrained nonself-consistent-field (nSCF) problem, as well as accelerating the direct inversion in the iterative subspace (DIIS) technique to solve the SCF problem, the overall computational cost is reduced by a factor of 8-20 compared with directly using sequential quadratic programming (SQP). This approach should be applicable for other constrained minimization problems, and in the immediate future, once gradients are available, the present eDSC/hDSC algorithm should allow for speedy nonadiabatic dynamics simulations.

Equilibrium configurations of two-dimensional bubbles in a channel: N -bubble case

Two-dimensional liquid foam systems comprised of an odd number N of bubbles arranged in a staircase configuration are studied as they are packed within a confined straight channel. Energy minimization is employed to characterize the permitted topology and geometry of the system at equilibrium. It is known that in an infinite staircase, bubble films are flat. The configuration for a finite number of bubbles, however, deviates from that of an infinite staircase at the edges. It is known that films near the edges are not flat and structures with a few bubbles (up to N = 3 ) are well characterized. To study what happens as more bubbles are added, and how the behaviour approaches that of an infinite staircase, structures comprised of N = [ 5 , 7 , … ] bubbles are studied here. For these N values, minimum and maximum bubble areas are found that can pack in a staircase configuration within a straight channel. However, for an infinite staircase and also for N = 3 , there are maximum areas, but no minima. Staircases with very short films, which make the structure susceptible to break in various ways, are also identified.

Ramanujan Sums of All Natural Numbers with Grandi Series

We are all know that the Ramanujan’s theory of calculating the sum of all the natural number. When we came to known that sum of all-natural number is ,that time everyone thinks about two things, one is how it possible that sum of positive number is negative &amp; other is how it is very close to zero. Another research paper gives value of all natural number is then I started the study of this sum, then I find it is as zero. Quit interestingly but this value has more important than Ramanujan’s sum because I did not consider the Grandi series for calculating this sum. And then it is very easy to calculate the sum of negative integer &amp; sum of all odd natural number &amp; sum of all even natural number.

ENSO modulates soil organic carbon retention and deposition in the East China Sea

Accurately quantifying the contribution of terrestrial soil organic carbon (SOC) in marine environment is still a challenge. Moreover, reassessing the effect of climate change on SOC variability is necessary, with the increasing variability of El Niño under greenhouse warming. Here, we present new insights from bacterial 3-hydroxy fatty acids (3-OH-FAs) by systematically analyzing the distributions of 3-OH-FA in soils, lake/river sediments and marine sediments, and propose a novel proxy (RIA, the ratio of odd carbon number iso to anteiso 3-OH-FAs) to quantificationally evaluate the contribution of OC from different components. The reliability of the RIA proxy is tested by using a three end-member model and Monte Carlo simulations along a transect from the Yangtze River estuary to the ECS shelf. Then the RIA proxy is applied to a core in the East China Sea covering the past 700 years, and the results reveal that El Niño (La Niña) induced more (less) precipitation and increased (decreased) SOC deposition. This study provides a novel approach for quantifying terrigenous OC inputs and burial in marginal seas and shows the impact of El Niño-Southern Oscillation (ENSO) modes on SOC loss and burial in marginal seas under global warming.

Görtler Vortices in the Shock Wave/Boundary-Layer Interaction Induced by Curved Swept Compression Ramp

This study builds on previous research into the basic flow structure of a separated curved swept compression ramp shock wave/turbulence boundary layer interaction (CSCR-SWBLI) at the leading edge of an inward-turning inlet. We employ the ice-cluster-based planar laser scattering (IC-PLS) technique, which integrates multiple observation directions and positions, to experimentally investigate a physical model with typical parameter states at a freestream Mach number of 2.85. This study captures the fine structure of some sections of the flow field and identifies the presence of Görtler vortices (GVs) in the CSCR-SWBLI. It is observed that due to the characteristics of variable sweep angle, variable intensity interaction, and centrifugal force, GVs exhibit strong three-dimensional characteristics in the curved section. Additionally, their position is not fixed in the spanwise direction, demonstrating strong intermittence. As the vortices develop downstream, their size gradually increases while the number decreases, always corresponding to the local boundary layer thickness. When considering the effects of coupling of bilateral walls, it is noted that the main difference between double-sided coupling and single-sided uncoupling conditions is the presence of a large-scale vortex in the central plane and an odd number of GVs in the double-sided model. Finally, the existence of GVs in CSCR-SWBLI is verified through the classical determine criteria Görtler number (GT) and Floryan number (F) decision basis.

Hidden magnetism and split off flat bands in the insulator metal transition in VO2

Transition metal d-electron oxides with an odd number of electrons per unit cell are expected to form metals with partially occupied energy bands, but exhibit in fact a range of behaviors, being either insulators, or metals, or having insulator-metal transitions. Traditional explanations involved predominantly electron-electron interactions in fixed structural symmetry. The present work focuses instead on the role of symmetry breaking local structural motifs. Viewing the previously observed V-V dimerization in VO2 as a continuous knob, reveals in density functional calculations the splitting of an isolated flat band from the broad conduction band. This leads past a critical percent dimerization to the formation of the insulating phase while lowering the total energy. In VO2 this transition is found to have a rather low energy barrier approaching the thermal energy at room temperature, suggesting energy-efficient switching in neuromorphic computing. Interestingly, sufficient V-V dimerization suppresses magnetism, leading to the nonmagnetic insulating state, whereas magnetism appears when dimerization is reduced, forming a metallic state. This study opens the way to design novel functional quantum materials with symmetry breaking-induced flat bands.

Emergent (2+1)D topological orders from iterative (1+1)D gauging

Gauging introduces gauge fields in order to localize an existing global symmetry, resulting in a dual global symmetry on the gauge fields that can be gauged again. By iterating the gauging process on spin chains with Abelian group symmetries and arranging the gauge fields in a 2D lattice, the local symmetries become the stabilizer of the XZZX-code for any Abelian group. By twisting the gauging map, we obtain codes that explicitly confine anyons, whose local creating operators violate an odd number of plaquettes. Their fusion results in either mobile dipole excitations twisting only half of the plaquette terms, or complete immobile Sierpiński-like excitations if we twist all the terms. Our construction naturally realizes any gapped boundary by taking different quantum phases of the initial (1+1)D globally symmetric system. In addition, our method also establishes a promising route to obtain higherdimensional topological codes from lower ones and to identify their gapped boundaries and their tensor network representations.

Topological boundary states in engineered quantum-dot molecules on the InAs(111)A surface: Odd numbers of quantum dots

Atom manipulation by scanning tunneling microscopy was used to construct quantum dots on the InAs(111)A surface. Each dot comprised six ionized indium adatoms. The positively charged adatoms create a confining potential acting on surface-state electrons, leading to the emergence of a bound state associated with the dot. By lining up the dots into N-dot chains with alternating tunnel coupling between them, quantum-dot molecules were constructed that revealed electronic boundary states as predicted by the Su-Schrieffer-Heeger (SSH) model of one-dimensional topological phases. Dot chains with odd N were constructed such that they host a single end or domain-wall state, allowing one to probe the localization of the boundary state on a given sublattice by scanning tunneling spectroscopy. We found probability density also on the forbidden sublattice together with an asymmetric energy spectrum of the chain-confined states. This deviation from the SSH model arises because the dots are charged and create a variation in on-site potential along the chain—which does not remove the boundary states but shifts their energy away from the midgap position. Our results demonstrate that topological boundary states can be created in quantum-dot arrays engineered with atomic-scale precision. Published by the American Physical Society 2024

Analytical and Numerical Investigation of Star Polymers in Confined Geometries.

The analysis of the impact of the star polymer topology on depletion interaction potentials, depletion forces, and monomer density profiles is carried out analytically using field theory methods and techniques as well as molecular dynamic simulations. The dimensionless depletion interaction potentials and the dimensionless depletion forces for a dilute solution of ideal star polymers with three and five legs (arms) in a Θ-solvent confined in a slit between two parallel walls with repulsive surfaces and for the case where one of the surfaces is repulsive and the other inert are obtained. Furthermore, the dimensionless layer monomer density profiles for ideal star polymers with an odd number (f˜ = 3, 5) of arms immersed in a dilute solution of big colloidal particles with different adsorbing or repelling properties in respect of polymers are calculated, bearing in mind the Derjaguin approximation. Molecular dynamic simulations of a dilute solution of star-shaped polymers in a good solvent with N = 901 (3 × 300 + 1 -star polymer with three arms) and 1501 (5 × 300 + 1 -star polymer with five arms) beads accordingly confined in a slit with different boundary conditions are performed, and the results of the monomer density profiles for the above-mentioned cases are obtained. The numerical calculation of the radius of gyration for star polymers with f˜ = 3, 5 arms and the ratio of the perpendicular to parallel components of the radius of gyration with respect to the wall orientation for the above-mentioned cases is performed. The obtained analytical and numerical results for star polymers with an odd number (f˜ = 3, 5) of arms are compared with our previous results for linear polymers in confined geometries. The acquired results show that a dilute solution of star polymer chains can be applied in the production of new functional materials, because the behavior of these solutions is strictly correlated with the topology of polymers and also with the nature and geometry of confined surfaces. The above-mentioned properties can find extensive practical application in materials engineering, as well as in biotechnology and medicine for drug and gene transmission.