Odd Number Research Articles

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.

Differentiating Odd Dominating Sets in Graphs

Let $G=(V(G),E(G))$ be a simple and undirected graph. A dominating set \linebreak $S\subseteq V(G)$ is called a \textit{differentiating odd dominating set} if for every vertex $v\in V(G)$, \linebreak $|N[v]\cap S|\equiv 1(mod\ 2)$ and $N_G[u]\cap S\neq N_G[v]\cap S$ for every two distinct vertices $u$ and $v$ of $V(G)$. The minimum cardinality of a differentiating odd dominating set of $G$, denoted by $\gamma_D^o(G)$, is called the \textit{differentiating odd domination number}. In this paper, we discuss \linebreak differentiating odd dominating set in some graphs and give relationships between the \linebreak differentiating odd domination, odd domination, and differentiating-domination numbers. Moreover, we characterize the differentiating odd dominating sets in graphs resulting from join, corona, and lexicographic product of graphs and determine the differentiating odd domination numbers of these graphs.

Influence of the odd‐even effect of dicarboxylic acids as crosslinker on the physicochemical properties of polyvinyl alcohol

AbstractPolyvinyl alcohol (PVA) was crosslinked with a homologous series of four different dicarboxylic acids, HOOC(CH2) n‐2COOH (n = 2 to 5) viz. oxalic acid (OA; C2), malonic acid (MA; C3), succinic acid (SA; C4) and glutaric acid (GA; C5) containing odd and even number of carbon atoms. The characterization and assessment of synthesized crosslinked films were done using ATR‐FTIR, XRD, SEM, AFM, DSC, rheology, percent swelling and sessile drop contact angle. The odd‐even effect of dicarboxylic acids was observed to significantly influence the physicochemical properties of PVA when employed as a crosslinker. PVA films crosslinked using dicarboxylic acids containing an even number of carbon atoms, displayed translucency, roughness, and an irregular matrix. In contrast, films incorporating an odd number of carbon atoms were transparent, homogenous, and densely packed. Furthermore, films containing even acids exhibited relatively lower crystallinity, lower melting temperature (Tm) and displayed an unexpected irregular trend in swelling as crosslinker concentration increased compared to films containing dicarboxylic acids with an odd number of carbon atoms which exhibits a consistent swelling with increasing crosslinker concentration. These results are attributed to the difference in the solubility and crystal structures of the odd‐even series of dicarboxylic acids.

Bifurcation Mechanism at a Sustain Point of a Long Narrow Economy

In this paper, we investigate population agglomeration in a long narrow economy, in which an odd number of places are evenly distributed over a line segment. The bifurcation analysis of this economy elucidates the mechanism of the emergence of twin cities around the central city. The validity and usefulness of this analysis are confirmed using several well-known economic geography models that display various kinds of bifurcation behaviors. By this analysis, we investigate the historical change in the population distribution in a chain of cities on Japan’s Main Island.

Solid–Liquid Phase Equilibrium of the n-Nonane + n-Undecane System for Low-Temperature Thermal Energy Storage

The current article presents an exploration of the solid–liquid phase diagram for a binary system comprising n-alkanes with an odd number of carbon atoms, specifically n-nonane (n-C9) and n-undecane (n-C11). This binary system exhibits promising characteristics for application as a phase change material (PCM) in low-temperature thermal energy storage (TES), due to the fusion temperatures of the individual components, thereby motivating an in-depth investigation of the solid–liquid phase diagram of their mixtures. The n-nonane (n-C9) + n-undecane (n-C11) solid–liquid phase equilibrium study herein reported includes the construction of the phase diagram using Differential Scanning Calorimetry (DSC) data, complemented with Hot–Stage Microscopy (HSM) and low-temperature Raman Spectroscopy results. From the DSC analysis, both the temperature and the enthalpy of solid–solid and solid–liquid transitions were obtained. The binary system n-C9 + n-C11 has evidenced a congruent melting solid solution at low temperatures. In particular, the blend with a molar composition xundecane = 0.10 shows to be a congruent melting solid solution with a melting point at 215.84 K and an enthalpy of fusion of 13.6 kJ·mol–1. For this reason, this system has confirmed the initial signs to be a candidate with good potential to be applied as a PCM in low-temperature TES applications. This work aims not only to contribute to gather information on the solid–liquid phase equilibrium on the system n-C9 + n-C11, which presently are not available in the literature, but especially to obtain essential and practical information on the possibility to use this system as PCM at low temperatures. The solid–liquid phase diagram of the system n-C9 + n-C11 is being published for the first time, as far as the authors are aware.

Frustrating Quantum Batteries

We propose to use a quantum spin chain as a device to store and release energy coherently and we investigate the interplay between its internal correlations and outside decoherence. We employ the quantum Ising chain in a transverse field and our charging protocol consists of a sudden global quantum quench in the external field to take the system out of equilibrium. Interactions with the environment and decoherence phenomena can dissipate part of the work that the chain can supply after being charged, measured by the ergotropy. We find that overall, the system shows remarkably better performance, in terms of resilience, charging time, and energy storage, when topological frustration is introduced by setting antiferromagnetic interactions with an odd number of sites and periodic boundary conditions. Moreover, we show that in a simple discharging protocol to an external spin, only the frustrated chain can transfer work and not just heat. Published by the American Physical Society 2024

Analysis of Collatz Conjecture Rules

Abstract A proof of the Collatz Conjecture is presented. Changing the perspective of the problem from looking at the pattern of the positive integers to looking at the conjecture rules made the proof possible. The conjecture rule for even numbers organizes all positive integers into unique sets and the rule for odd numbers interconnects the unique sets into dendritic pathways to “1.” Infinite loops, other than the minor 4-2-1 loop after reaching “1,” and values continually increasing to infinity are shown to be mathematically impossible. The proof predicted a general equation that shows all positive integers reach a final value of “1," and calculates the values and locations of the odd positive integers during the iterations for each tested positive integer. JEL classification numbers: CO2; C65. Keywords: Collatz Conjecture, Proof, Rules, Dendritic pathway.

Near-field DOA estimation oriented sparse array design via symmetric augmentation of three ULAs

A symmetric sparse array, via symmetric augmentation of three (SAT) uniform linear subarrays, has been designed for direction of arrival (DOA) estimation of uncorrelated near-field or coexisted near-field and far-field narrowband source signals. With an explicit design of the subarray structures and positions, this SAT array 1) involves only two closely spaced antenna pairs (with quarter-wavelength spacing); 2) has a large uniform degree of freedom (uDOF) and difference co-array; 3) can have either even or odd number of antennas; and 4) has closed form expressions for antenna positions, uDOF and weight functions. Simulation results show that the SAT array is superior to the current symmetric sparse arrays in both antenna mutual coupling and consecutive virtual aperture.