Odd Number Research Articles

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 & 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 & sum of all odd natural number & 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.

Collatz Conjecture (3N+1) Solution

Collatz Conjecture (3x+1) or in some literature as 3N+1 is a problem because it works in the way that if you take any positive number, if it is an odd number you multiply it by three (3) then add one (1). On the other hand, if it is an even number, you divide it by two (2). Eventually, all positive numbers decrease to one (1). One (1) is odd, so multiply it by three (3) is three (3) and add one (1) is four (4). Four (4) is even, so divide it by two (2) is two (2). Two (2) is also even, so divide it by two (2) is one (1) again. All positive numbers end up in the loop (4-2-1). This loop is like a numerical lock. Therefore, the solution of this problem will have to be a numerical key results to all positive numbers.

A class of constacyclic codes of length 2pˢ over Fp^m[u,v] / ⟨u², v², uv − vu⟩

Let p be an odd prime number. This paper investigates λ-constacyclic codes of length 2ps over the ring Rᵤ,ᵥ = Fpm[u,v] / ⟨u², v², uv − vu⟩, where λ = ψ + ρu + ηv + τuv, with ψ, ρ, η, τ ∈ Fpm, ψ ∈ F*pm, and ρ, η not both zero. When λ = μ² for some μ ∈ Rᵤ,ᵥ, the structures of all λ-constacyclic codes of length 2ps over Rᵤ,ᵥ are determined and can be expressed as the sum of a (−μ)-constacyclic code and a μ-constacyclic code of length ps. When λ is not a square, we classify these codes into four distinct types, provide the number of codewords for each type, and give the details of their dual codes.

Comparison of the Effectiveness of Platelet-Rich Plasma Injection Versus Plantar-Specific Calf Stretching Exercises in Patients With Plantar Fasciitis: A Randomized Controlled Trial.

Background Plantar fasciitis is a common foot condition with multifactorial etiology. It is the most frequent cause of heel pain and has been categorized as an overuse syndrome. A clinical examination and history are crucial for diagnosis. There are several different forms of treatment available, two of which are frequently used: physical therapy and steroid injections. Recent research on platelet-rich plasma (PRP) has demonstrated encouraging outcomes and fewer side effects when compared to steroid injections. Methods A randomized controlled trial was conducted and randomization was done of indoor patients into two groups. Group 1, ending with odd numbers, was given PRP injections, and Group 2, ending with even numbers, was advised plantar-specific calf stretching exercises. Visual analog scale (VAS) scores were evaluated before and after the intervention and follow-up was done on the second, sixth, and 12th weeks. Results Comparing the VAS scores between the two groups, we found that in the pre-intervention phase, the VAS score of Group 1 was 5.4±0.56 and that of Group 2 was 5.4±0.59. In the post-intervention phase, the VAS score in Group 1 was 4.6±0.89, while in Group 2 it was 5.2±0.62. In the second week after intervention, the VAS score was observed to be 3.3±0.97 in Group 1, while in Group 2, it was 3.3±0.80. After the sixth week of intervention, the observed VAS score was 2.7±0.78, while in Group 2 it was 2.9±0.82. The mean VAS score after 12weeks of intervention was observed to be 2.3±0.91 in Group 1, while in Group 2, it was 2.2±0.80. Conclusion PRP injections and plantar-specific calf stretching exercises are equally effective in providing pain relief in plantar fasciitis. PRP injections have complications and problems which have been discussed. Exercises are devoid of such complications. No recurrences occurred in the exercise group and four cases had recurrence in the PRP group.

On exact products of a cyclic group and a dihedral group

A finite group G is an exact product of two subgroups A and B if G = AB and A ∩ B = { 1 G } . In this paper, for all odd numbers k ≥ 3 , using the theory of skew morphisms and associated extended power functions, we present a classification of exact products of a cyclic group and a dihedral group D 2 k . As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified.

Generalization of the Synthetic Aperture Radar Azimuth Multi-Aperture Processing Scheme—MAPS

This paper analyzes the advantages and the drawbacks of using the Synthetic Aperture Radar (SAR) azimuth multichannel technique known as Multi-Aperture Processing Scheme (MAPS), in a set of relevant application cases that are far from the canonical ones. In the scientific literature on this topic, equally distributed azimuth channels with the quasi-monostatic deployment are assumed. With this research, we aim at extending the models from the current literature to (i) a generic bistatic acquisition geometry, (ii) a set of cases where the number of receiving tiles is not the same for each channel, or (iii) the tiles are shared between adjacent channels thus creating an overlapping configuration. The paper introduces the mathematical models for the listed non-conventional MAPS cases. Dealing with the bistatic MAPS, we first solve the problem by interpreting multichannel acquisition as a bank of Linear Time Invariant (LTI) filters. Then, a more physical approach, based on discrimination of the direction of arrivals (DoAs) is pursued. The effectiveness of the two methods and the advantages of the second approach on the first are proved by using a simplified 1D end-to-end simulation. Even limiting to the monostatic configuration, the azimuth antenna tiles have always been supposed equally partitioned among the RX channels. Overcoming this limit has two advantages: (i) more MAPS possible solutions in case few azimuth tiles are available, as in the ROSE-L mission; (ii) the number of channels can be designed independently of the number of tiles, also allowing asymmetric solutions, useful for a phase array antenna with an odd number of tiles such as in the SAOCOM-1 mission. Conversely, sharing one or more receiving tiles in different receiving channels makes the input noise partially correlated. The drawback is an increase in the noise level. A trade-off is determined for the different solutions obtained using simulations with real mission parameters. The theoretical performance and the end-to-end simulations are compared.

Molecular dynamics study on phase change properties and their nano-mechanism of sugar alcohols: Melting and latent heat

Sugar alcohol phase change materials have gained considerable attention for heat storage in recent years owing to their higher thermal conductivity and larger latent heat compared to traditional paraffin materials. However, knowledge about the reason of large latent heat and latent heat difference among diverse sugar alcohols as well as the mechanism of melting process remains elusive. In this study, molecular dynamics simulation was employed to investigate the melting process and latent heat of four sugar alcohols (glycerol, erythritol, arabinitol, and mannitol). Melting points and latent heat of fusion were determined first, different melting temperatures were observed under different one-dimensional heat conduction directions in several sugar alcohols. It was found that the strength of intermolecular interactions within molecular crystal layers has a positive correlation with the anisotropic melting tendencies, which can be more intuitively reflected by the difference in the number of hydrogen bonds within the molecular layers. In addition, the decomposition of the latent heat of fusion revealed that the latent heat is mainly composed of van der Waals and Coulombic contributions, where the interaction between hydroxyl groups accounts for a major part. Hydrogen bond analysis demonstrated that the latent heat of fusion is partly due to the breakage of hydrogen bonds during the phase transition of melting. It was also noted that the differences in the hydrogen bonding lifetimes between different sugar alcohols may be related to the magnitude of latent heat of fusion. Erythritol and mannitol which coincidentally happened to have an even number of carbon atoms can fulfill the properties of better PCMs compared to glycerol and arabinitol, which have an odd number of carbons. Those results will pave the way for better design optimization and performance evaluation of sugar alcohols as phase change materials.

Evaluation of a multiphase cascaded H-bridge inverter for induction motor operation

Multi-phase systems are becoming more popular for applications requiring high power and precise motor control, even if single-phase AC power is still frequently utilized in households and some enterprises. While both systems have benefits over single-phase, there are trade-offs associated with each. Because of its balanced operation and effective power transfer, the three-phase (3-Φ) system is the most widely used multi-phase system. Nevertheless, different phase values can be investigated for particular applications where reducing torque ripple and harmonic content is essential. Using odd numbers of phases (such as 5-Φ) that are not multiples of three is one method. This design has the ability to reduce torque ripple by producing a more balanced magnetic field as compared with even-numbered phases. But adding more phases also makes the system design and control circuitry more complex. Systems with five phases (5-Φ) provide a compromise between performance and complexity. Applications such as electric ship propulsion, rocket satellites, and traction systems may benefit from their use. Nevertheless, choosing a multi-phase system necessitates carefully weighing the requirements unique to each application, taking into account elements like cost, power transmission, control complexity, and efficiency. The increasing popularity of electric vehicles and renewable energy technologies has led to the need for inverters in current electric applications. Conventional inverters provide square wave outputs, which cause the drive system to become noisy and cause harmonics. Multi-phase multilevel inverters can be used to enhance inverter functioning and produce an improved sinusoidal output. This study focuses on an induction motor drive powered by a five-phase multilevel cascaded H-Bridge inverter. With less torque and current ripples in the motor rotor, the power conversion harmonics are reduced and the switching components of the inverter are under less stress. However, in comparison to traditional inverters, it does require a greater number of legs. Because the switches needed for the cascaded H-Bridge inverter are less expensive in five-phase systems, they are favoured over higher phase orders. Furthermore, the suggested inverter removes 5th order harmonics, something that is not possible with traditional inverters. A five-phase induction motor appropriate for variable speed driving applications is also suggested by this research. Lastly, utilizing pulse width modulation (PWM) converters and an FPGA controller, an experimental study is carried out to assess the dynamic performance of the suggested induction motor drive. Particular attention is paid to the In-Phase Opposition Disposition (IPD) PWM technique.

Some computational results on a conjecture of de Polignac about numbers of the form p + 2k

Let U be the set of positive odd numbers that can not be written in the form p+2k. Recently, by analyzing possible prime divisors of b, Chen proved b≥11184810 and ω(b)≥7 if an arithmetic progression a(modb) is in U, with ω(b)=7 if and only if b=11184810, where ω(n) is the number of distinct prime divisors of n. In this paper, we take a computational approach to prove b≥11184810 and provide all possible values of a if a(mod11184810) is in U. Moreover, we explicitly construct nontrivial arithmetic progressions a(modb) in U with ω(b)=8, 9, 10, or 11, and provide potential nontrivial arithmetic progressions a(modb) in U such that ω(b)=s for any fixed s≥12. Furthermore, we improve the upper bound estimate of numbers of the form p+2k by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.

Combinatorial Identities with Multiple Harmonic-like Numbers

Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are immediate consequences of the main result. Finally, combinatorial identities involving harmonic-like numbers and other prominent sequences like hyperharmonic numbers and odd harmonic numbers are offered.

Thermal resilience of buildings: The role of partition walls and slabs in the optimization of the building external envelope

Nowadays it is advisable for buildings to be thermally resilient, so as to passively resist to the daily external air temperature fluctuations, reducing the interventions of air conditioning systems to a minimum. The decrement factor and time lag are parameters notoriously used to evaluate the responses of the building envelope to periodic oscillations in external air temperature. However, in their traditional definition, these parameters do not consider the effects of internal elements, such as partition walls and slabs, which instead influence the passive thermal behaviour of the building. For this reason, a choice of the external wall stratigraphy, made exclusively on the basis of the traditional decrement factor and time lag, seems to be simplistic. In this paper, problems concerning the passive behaviour of buildings, in case of sinusoidal outdoor air temperature variations, are investigated. Within a lumped-capacitance scheme and using the heat transfer matrix method, the problem of the determination of a wall stratigraphy minimizing the decrement factor for given values of the overall thermal resistance and capacity, usable for the realization of both external and internal walls of a given room, is solved. After developing an analysis on the influence of the internal walls and slabs on the room thermal balance, the resistance-capacity distribution, within the external wall, minimizing the decrement factor is determined, for assigned thermal resistances and capacities of internal elements and the external wall. Results indicate that there is no resistance-capacity distribution that is absolutely better than the others, but it is dependent on the thermo-physical characteristics of the internal elements. The optimized external wall stratigraphy can be composed of an even number of resistive and capacitive layers, alternating with each other, or of an odd number of layers (with one more resistive layer than the capacitive ones). Even the number of layers that makes optimized the passive behaviour of the external wall varies with the characteristics of the internal elements, the results show that already with a number of layers varying from 2 to 5 high performances can be obtained.

Research and application of the flatness target curve discrete dynamic programming based on two-dimensional decision making

The flatness target curve (FTC) is a fundamental process model in cold rolling flatness control, which directly affects the quality control effect of the strip. The accurate and efficient setting of the flatness target curve is the key to ensuring continuous rolling and is a common problem faced by various production companies. This paper establishes a discrete dynamic planning (DDP) setting method for FTC with two-dimensional decision-making. To quantitatively analyze the roles of the coefficients in the FTC, the preprocessing process is established using standardized processing methods and statistical analysis methods. The flatness target curve discrete dynamic programming (FTC-DDP) is based on Bellman optimization theory with a dual structure of local and system decision-making. The decision-making functions are established based on the effect of each coefficient on FTC geometric characteristics. The concepts of “curvature factor” and “wedge factor” are innovatively proposed to obtain the decision-making value functions. The regression analysis establishes the solution equations of even number coefficients, odd number coefficients, and gain coefficients, respectively. Roll bending and tilting roll analyze the FTC’s ability to correct flatness defects. Meanwhile, utilizing a 1450 mm five-stand six-roll cold rolling mill, it is verified that FTC-DDP can quickly and accurately set the FTC, which provides an effective solution for obtaining high-precision and high-quality cold-rolled strips. Application results show that FTC-DDP can reduce the difficulty of setting up FTC by more than 70 %, reduce the setup time to milliseconds, and improve the setup accuracy by more than 50 %.

Sophie Germain prime p and the permutation of product of first p cycles

For a natural number n, the permutation (n!) is defined as the left-to-right product of the first n cycles, namely, . In this article, we prove that for any natural number n, 2 is a primitive root of 2n + 1 if and only if 2n + 1 = pk for some odd prime number p and for some natural number k such that the permutation (n!) has exactly k orbits. We also prove that a prime number p is a Sophie Germain prime if and only if the permutation (p!) has at most two orbits.

--------Comparison of Hybrid-Simulation-Based Teaching with Traditional Teaching of Pharmacology.

To compare the outcome of traditional teaching with hybrid simulation-based teaching for undergraduate medical students. Place and Duration of the Study: Department of Pharmacology, Jinnah Medical and Dental College, Karachi, Pakistan, from June to August 2023. Quasi-experimental study. One hundred students from MBBS 3rd year were included in the study after taking the informed consent. Participants were divided into two cross-over groups and sampling was done randomly. Group A: (even roll numbers, n = 50) was the control group, taught by traditional lecture on positive inotropic medicines. Group B: (with odd roll numbers, n = 50) was the intervention group, taught the same topic by simulation-based teaching through 5 case scenarios. The teaching of this group was reinforced by role plays. Scores of post-test and retention test were compared by applying the Student's t-test. Students taught by traditional lectures i.e., Group A, their post-test mean scores were 30.7 ± 5.6, whereas Group B scored 45.7 ± 3.3, taught by hybrid stimulation (p <0.001). Retention test (MCQs based) was conducted after one month in which Group A obtained a mean score of 18.8 ± 9 with a passing percentage of approximately 30, whereas Group B obtained a score of 41.3 ± 5.6 (p <0.001). Hybrid-simulation-based teaching improved the immediate test scores as well as retention. Traditional teaching, Hybrid-simulation, MCQs, Retention test, Intervention group.