Odd Number Research Articles

On Remoteness Functions of k-NIM with k + 1 Piles in Normal and in Misère Versions

Given integer n and k such that 0<k≤n and n piles of stones, two players alternate turns. On each move, a player is allowed to choose any k piles and remove exactly one stone from each. The player who has to move but cannot is the loser in the normal version of the game and (s)he is the winner in the misère version. Cases k=1 and k=n are trivial. For k=2, the game was solved for n≤6. For n≤4, the Sprague–Grundy function was efficiently computed (for both versions). For n=5,6, a polynomial algorithm computing P-positions was obtained for the normal version. Then, for the case k=n−1, a very simple explicit rule that determines the Smith remoteness function was found for the normal version of the game: the player who has to move keeps a pile with the minimum even number of stones; if all piles have an odd number of stones, then (s)he keeps a maximum one, while the n−1 remaining piles are reduced by one stone each in accordance with the rules of the game. Computations show that the same rule works efficiently for the misère version too. The exceptions are sparse. We list some. Denote a position by x=(x1,…,xn). Due to symmetry, we can assume wlog that x1≤…≤xn. Our computations partition all exceptions into the following three families: x1 is even, x1=1, and odd x1≥3. In all three cases, we suggest formulas covering all found exceptions, but it is not proven that there are no others.

Mechanistic Insights into the cis-Selective Catalytic Ring-Opening Metathesis Polymerization.

Cis-selective ring-opening metathesis polymerization (ROMP) with the commercial Grubbs "nitrato catalyst" has shown promise for synthesizing stereoregular materials, but it comes with the drawback of losing control over the molecular weight due to the poor initiation rate of the catalyst and the need for stoichiometric ruthenium complex loading. To address these issues, we developed a chain transfer polymerization method that allows for the catalytic synthesis of polymers while controlling the degree of polymerization. This allowed us to produce shorter polymers with exceptional chain-end control. Analysis of the polymers revealed a novel double monomer addition mechanism for this catalyst. MALDI-ToF mass spectrometric measurements showed that when using small monomers like norbornene, the polymer chains contained only odd numbers of monomers. In contrast, the polymerization of norbornene-imide-type monomers shows a major distribution with odd numbers of monomers along with a minor distribution of even numbers. This unique distribution of polymer chain types had not been previously observed in ROMP. We explain this phenomenon by the chiral nature of the catalyst that yields two isomeric catalytic species with dissimilar reactivities toward monomer and chain transfer agents.

Laser hemorrhoidoplasty versus hemorrhoidectomy in the treatment of surgically indicated hemorrhoids in inflammatory bowel patients: a randomized comparative clinical study.

This study aims to assess and compare the outcomes of traditional surgery and laser surgery for anal hemorrhoids in patients with inflammatory bowel disease. This is a single-center prospective randomized comparative clinical trial performed at Zagazig University Hospitals from September 2023 to September 2024. The study comprised 48 patients who were admitted during this period and had inflammatory bowel disease along with hemorrhoids. The patients were randomly allocated into two groups: Group I consisted of 24 patients who underwent laser hemorrhoidoplasty, selected based on odd numbers, while Group II consisted of 24 patients who underwent traditional surgery, selected based on even numbers. Group I, which underwent laser treatment for hemorrhoids in inflammatory bowel patients, demonstrated significantly better outcomes than traditional surgery (p < 0.001) regarding operation time, pain score and duration, postoperative anal discharge, return to work time, and patient satisfaction. Moreover, laser treatment was found to be more effective than traditional surgery for treating hemorrhoids in inflammatory bowel patients in terms of postoperative bleeding and anal stenosis. Additionally, there was a higher frequency of residual hemorrhoids in laser group. Laser treatment is mostly superior to traditional surgery for hemorrhoids in inflammatory bowel patients. Clinical Trial.gov (NCT06216223), registered on December 27, 2023, last updated in May 12, 2024. PROTOCOL REGISTRATION ID: #101080-5-9-2023.

Odd chromatic number of graph classes

Odd chromatic number of graph classes

Atrial Topology for a Unified Understanding of Typical and Atypical Flutter.

Macroreentry stands as the predominant mechanism of typical and atypical flutter. Despite advances in mapping, many aspects of macroreentrant atrial tachycardia remain unsolved. In this translational study, we applied principles of topology to understand the activation patterns, entrainment characteristics, and ablation responses in a large clinical macroreentrant atrial tachycardia database. Because the atrium can be topologically seen as a closed sphere with holes, we used a computational fixed spherical mesh model with a finite number of holes to induce and analyze macroreentrant atrial tachycardia. The ensuing insights were used to interpret high-density activation maps, postpacing interval-tachycardia cycle length values (difference between postpacing interval and tachycardia cycle length), and ablation response in 131 cases of typical and atypical flutter (n=106 left atrium, n=25 right atrium). Modeling of macroreentrant atrial tachycardia revealed that reentry on closed surfaces consistently manifests itself as paired rotation and that an odd number of critical boundaries is mathematically impossible. Together with mathematical confirmation by the index theorem, this led to a unifying construct that could explain the number of loops, difference between postpacing interval and tachycardia cycle length values, and ablation outcomes (termination, no change, or prolongation in tachycardia cycle length) in all 131 cases. Combining topology with the index theorem offers a novel and cohesive framework for understanding and managing typical and atypical flutter.

Sensitivity to Number Parity in Young Myrmica Sabuleti Ants

Young Myrmica sabuleti worker ants aged at most a few weeks and settled in small single-cohort colonies, were learned, by operant conditioning, to recognize either 2 or 3 dots displayed on a stand. When tested in front of two stands bearing different numbers of dots, one even and the other odd, their response was always higher towards the stand bearing the same parity as the one learned. They thus reacted to the parity corresponding to the number of dots they have learned to associate with the reward. These young ants were sensitive to the parity of a number of dots ranging from 1 to at least 6, but could not discriminate their parity when in front of 5 and 6. They were somewhat more responsive towards an even number than towards an odd number, having learned a number with this parity. A preceding study showed that older workers (experienced foragers) are sensitive to the parity of numbers of dots ranging from 1 to at least 7, and this equally whether the numbers are odd or even. Their limit of parity discrimination of close numbers is in front of 6 and 7. It therefore appears that, although present early on, the sensitivity of young workers to parity of numbers of elements and to number discrimination has to improve over the course of their lives, perhaps through an age-related maturation process, or through acquired experience.

Triangular Carrier Based Synchronous PWM for Operating High Power IM Drives at Odd Non-Triplen Pulse Numbers

Triangular Carrier Based Synchronous PWM for Operating High Power IM Drives at Odd Non-Triplen Pulse Numbers

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.