Odd Number Research Articles

Investigation of a stress-strain state of a rubber-cable rope with cables of different tensile rigidity

Purpose. Establishment of dependences for stress-strain state parameters in a rubber-cable rope with cables of different tensile rigidity and cable breakage. Research methodology. Construction and development of an algorithm for solving a model of a stress-strain state of a rubber-cable rope with ropes of different tensile rigidity and breakage of one cable by using the methods of mechanics of layered composite materials with soft and hard layers. Findings. Analytical expressions are constructed, which allow determining the main stress-strain state indicators in a rope with an odd number of cables, while the middle cable is damaged, and tensile rigidity of the middle cable is half the tensile rigidity of the other cables. Analytical dependencies are established for determining the extreme angles of rubber shear between the cables, what allows determining the most dangerous tension states of a rope with a damaged cable, which has a different rigidity from the other cables. The level of reliability of the developed algorithm for solving the model of a stress-strain state of a rubber-cable rope with cables of different rigidity is confirmed by determining the maximum tensile forces of a rope made of three cables, in which the middle cable has a tensile rigidity half that of the other two. The force distribution coefficients between cables adjacent to the damaged one are equal to 1.25 per unit load on each cable, which corresponds to the only possible case of force distribution between parallel elements under the presented conditions. Scientific novelty. The character of influence of different rigidity and damage of the reinforcing cables on the main parameters of stress-strain state in a rubber-cable rope are established. Practical significance. An algorithm for determining the stress-strain state indicators of a rubber-cable rope, which has an odd number of reinforcing cables, its middle cable is damaged, and the tensile rigidity of the middle cable is different (lower) than the rigidity of the other cables, is developed. This makes it possible to increase the operational safety of rubber-cable ropes in hoisting and transporting machines, in particular when operating at significant lifting heights, and also contributes to justifying the use of a rope design as a stay rope in capital structures.

Guided waves in layered media: Twin ZGV and ambiguity of dispersion spectrum

The twin zero group velocity (ZGV) points are observed, which belong to the fundamental symmetric branch (S 0) of Lamb waves propagating in a composite traction-free layered plate containing odd number of layers with alternating physical properties. It is shown that in a vicinity of these ZGV points an ambiguity of dispersion spectrum takes place, resulting in the appearance of several waves, which (i) belong to the same fundamental mode; (ii) satisfy the same boundary conditions; and, (iii) propagate at the same frequency, however (iv) with different phase velocities. The analysis is based on a combination of Cauchy formalism and the exponential fundamental matrix method.

PATIENT'S DISCOMFORT ASSOCIATED WITH FIXED ORTHODONTIC APPLIANCES TREATMENT

To determine the prevalence and factors causing discomfort during fixed orthodontic treatment. Two hundred patients of fixed orthodontic patients in UiTM clinics who were willing to participate and could consent to the research were selected. The participants were randomly selected for this research in the odd number sequence from the population size of 740 individuals. Data was collected employing an Oral Impact on Daily Performance questionnaire to assess discomfort intensity and bio-psychosocial variables. Statistical analysis used was descriptive statistics and chi-squared statistics. The prevalence of discomfort in patients with fixed orthodontic appliances was 92% (mean: 0.92, standard deviation: 0.27), which covers 184 of the samples. Eating food (68%), speaking clearly (37%), cleaning teeth (64%), sleeping (21%), smiling and laughing, and showing teeth without embarrassment (21%) remained independently associated with a greater prevalence of discomfort (P ≤ 0.05). The patient wearing fixed orthodontics appliances has a high prevalence of discomfort. The factors associated with the discomfort included eating food, speaking, cleaning teeth, sleeping, and smiling and laughing, showing teeth without embarrassment.

Self‐dual association schemes, fusions of Hamming schemes, and partial geometric designs

AbstractPartial geometric designs can be constructed from basic relations of association schemes. An infinite family of partial geometric designs were constructed from the fusion schemes of certain Hamming schemes in work by Nowak et al. (2016). A general method to create partial geometric designs from association schemes is given by Xu (2023). In this paper, we continue the research by Xu (2023). We will first study the properties and characterizations of self‐dual association schemes. Then using the characterizations of self‐dual association schemes and the representation theory (character tables) of commutative association schemes, we obtain characterizations and classifications of self‐dual (symmetric or nonsymmetric) association schemes of rank 4 that produce as many as possible nontrivial partial geometric designs or 2‐designs. In particular, for a primitive self‐dual symmetric association scheme of rank 4, if is a power of 3 and each of , , and induces a partial geometric design, then we will prove that is algebraically isomorphic to a fusion scheme of the Hamming scheme for some odd number .

Quadrature rules of Gaussian type for trigonometric polynomials with preassigned nodes

In this paper we consider Gaussian type quadrature rules for trigonometric polynomials where an even number of nodes is fixed in advance. For an integrable and nonnegative weight function w on the interval E=[a,a+2π), a∈R, these quadrature rules have the following form∫Et(x)w(x)dx=∑i=12kait(yi)+∑i=12(n+γ)Ait(xi),t∈T2(n+γ)+k−1, where the nodes yi∈E, i=1,2,…,2k, are fixed and prescribed in advance, γ∈{0,1/2} and Tn={cos⁡kx,sin⁡kx|k=0,1,…,n}, n∈N.Also, for γ=1/2, i.e., for the case of quadrature rules for trigonometric polynomials with odd number of nodes, we consider the optimal sets of quadrature rules in the sense of Borges (see [1,13]) for trigonometric polynomials with even number of fixed nodes. Let n=(n1,n2,…,nr), r∈N, be a multi-index and let W=(w1,w2,…,wr) be a system of weight functions on the interval E=[a,a+2π), a∈R. The optimal set of quadrature rules with respect to (W,n), with even number of fixed nodes, have the form∫Ef(x)wm(x)dx≈∑i=12kam,if(yi)+∑i=12|n|+1Am,if(xi),m=1,2,…r, where |n|=n1+n2+⋯+nr and the nodes yi∈E, i=1,2,…,2k, are fixed and prescribed in advance. For r=1 the optimal set of quadrature rules reduces to Gaussian quadrature rule for trigonometric polynomials with odd number of nodes.For all mentioned quadrature rules, in addition to the theoretical results, we will present the method for construction and give appropriate numerical examples.

Performance of production of polyhydroxyalkanoates from food waste fermentation with Rhodopseudomonas palustris

The use of waste as a carbon source can significantly reduce the cost of production of Polyhydroxyalkanoates (PHAs). In this study, an acidified hydrolysate solution derived from food waste (FW) was used as a carbon source for the synthesis of PHAs by Rhodopseudomonas palustris (R. palustris) and optimized the process parameters. The results showed that the PHAs yield reached 48.62% under optimal conditions (an incubation time of 30 days, volatile fatty acids (VFAs) in substrate concentration of 2202.21 mg⋅L−1, an initial pH of 8.0, and inoculum concentration of 15%). The fraction of VFAs affects the composition of PHAs, R. palustris first uses VFAs with an even number of carbons to synthesize poly(3-hydroxybutyrate)(3HB), and later uses VFAs with an odd number of carbons to synthesize poly-3-hydroxyvalerate (3HV). Pathways for the synthesis of PHAs by R. palustris were inferred. R. palustris is a strain with the potential to synthesize PHAs.

Odd covers of graphs

AbstractGiven a finite simple graph , an odd cover of is a collection of complete bipartite graphs, or bicliques, in which each edge of appears in an odd number of bicliques, and each nonedge of appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of by and prove that is bounded below by half of the rank over of the adjacency matrix of . We show that this lower bound is tight in the case when is a bipartite graph and almost tight when is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from . Babai and Frankl proposed the “odd cover problem,” which in our language is equivalent to determining . In this paper, we determine that is when and is when is equivalent to 1 or modulo 8.

Control of chaos with time-delayed feedback based on deep reinforcement learning

The time-delayed feedback control method, as one of popular methods for chaos control, is noninvasive and flexible for various dynamical systems from different fields of science and technology. In the method, however, an appropriate choice of the feedback gain is challenging, which requires the explicit mathematical model of the controlled system for stability analysis. Additionally, another limitation of the method is the so-called odd number limitation. The two problems restrict its application to practical situations. Fortunately, a technique called deep reinforcement learning is capable of learning the controlled environment (the dynamical system), by continually interacting with the controlled system, which makes it possible to obtain the right feedback gain without the requirement of the accurate mathematical model of the system. Hence, in this work, a time-delayed feedback control method based on deep reinforcement learning is put forward to solve the two problems. Compared with the traditional time-delayed feedback control, the proposed method, as a data-driven method due to the combination with deep reinforcement learning, offers a time-varying feedback gain according to the well-trained policy learned by the deep reinforcement learning algorithm. It maintains the non-invasive property of the time-delayed feedback control, but expands the operating range due to the time-varying feedback gain overcoming the odd number limitation. With numerical simulations, the proposed method is successfully applied to three different kinds of systems, the discrete logistic map, the non-autonomous Duffing oscillator and the autonomous Lorenz system.

Amorphous and Polycrystalline Routes toward a Chiral Spin Liquid.

We show that a chiral spin liquid spontaneously emerges in partially amorphous, polycrystalline, or ion-irradiated Kitaev materials. In these systems, time-reversal symmetry is broken spontaneously due to a nonzero density of plaquettes with an odd number of edges n_{odd}. This mechanism opens a sizable gap, at small n_{odd} compatible with that of typical amorphous materials and polycrystals, and which can alternatively be induced by ion irradiation. We find that the gap is proportional to n_{odd}, saturating at n_{odd}∼40%. Using exact diagonalization, we find that the chiral spin liquid is approximately as stable to Heisenberg interactions as Kitaev's honeycomb spin-liquid model. Our results open up a significant number of noncrystalline systems where chiral spin liquids can emerge without external magnetic fields.

Approximate analytical expressions for the Carr–Purcell–Meiboom–Gill sequences: Decay rates and modulation zeros of the echo train and the relation between the [formula omitted] and [formula omitted] relaxation times

Methods of multi-pulse sequences are widely used in magnetic resonance to study the local properties of magnetic particles and to increase the lifetime of spin coherence. Imperfect refocusing pulses lead to non-exponential signal decay due to the contribution of the coherence pathways in which T1 and T2 relaxation segments are mixed. Here, we present analytical approximations for echoes generated in the Carr-Purcell-Meiboom-Gill (CPMG) sequence. They provide simple expressions for the leading terms of the echo train decay and allow the relaxation times to be estimated for sequences with a relatively small number of pulses. For a given refocusing angle α, the decay times for the fixed phase and alternating phase CPMG sequences can be approximated as (T2-1+T1-1)/2 and T2O, respectively. The ability to estimate relaxation times from short pulse sequences can reduce the acquisition time, which is essential for the methods used in magnetic resonance imaging. In the case of a CPMG sequence with the fixed phase, the relaxation times can also be obtained from the points in the sequence at which the echo changes sign. Numerical comparison of the exact and approximate expressions shows the practical limits of the analytical formulas obtained. It is also shown that a double echo sequence in which the interval between the first two pulses is not equal to half the interval of the subsequent refocusing pulses provides the same information as two separate CPMG (or CP) sequences with fixed and alternating phases of the refocusing pulses. In addition, the two double-echo sequences differ in the parity of the number of intervals with longitudinal magnetization evolution (relaxation), i.e. the echo in one sequence is formed only from those coherence pathways that have an even number of intervals with longitudinal magnetization evolution, while the other sequence has an odd number of such intervals.

Validation of two preoxygenation techniques, 3 min tidal volume breath and eight vital capacity breath techniques in tribal and non-tribal population of Eastern India

Background: Preoxygenation during anesthesia can be done by 3 min tidal volume breath and eight vital capacity breath in 1 min, conventionally. Population of our country is not homogenous. Aims and Objectives: The present study was conducted to validate preoxygenation processes in people of eastern India. Materials and Methods: Total 140 patients of ASA grade I and II, age group 20–50 posted for surgeries under GA were allotted in TV and VC group. Every odd number and even number patient was tested for TV method and VC method, respectively, with a Magill circuit and airtight face mask with 100% oxygen flow at a rate of 10 L/min. Time and number of breaths were noted when end tidal oxygen concentration ≥90% in TV and VC group, respectively. Results: In TV method, the mean time (in minute) was 3.10±0.27 and 3.13±0.24 for tribal and non-tribal sub-group (P=0.80); 3.02±0.37 and 3.10±0.27 for tribal and non-tribal male (P=0.57); and 3.18±0.05 and 3.16±0.21 for tribal and non-tribal female (P=0.57). In VC method, mean number of VC breaths was 8.32±0.60 and 8.43±0.74 for tribal and non-tribal sub-group (P=0.76); 8.75±0.49 and 8.23±0.70 for tribal and non-tribal male (P=0.42); and 8.20±0.60 and 8.60±0.72 for tribal and non-tribal female (P=0.16). Patients completing preoxygenation in TV and VC method – tribal: 22% and 53%, non-tribal: 4% and 62%, tribal male: 22% and 20%, non-tribal male: 8% and 68%, tribal female: 0% and 60%, and non-tribal female: 0 and 57%, respectively. Conclusion: No significant difference found between tribal and non-tribal groups of both sexes in each method. Completion of preoxygenation was higher in VC group, though not tested statistically.

Frequency Estimation with Co-prime DFT Filter Bank and Half Nyquist Frequency Shift

When processing wide sense stationary (WSS) signals, the sampled data rate of conventional MN bands DFT filter bank is at least MN, while the co-prime DFT filter bank (co-prime DFTFB) only needs M+N, that sharply lowers the sampling rate. Here, M and N are co-prime pair of integers. However, practical FIR filter imperfections and correlation detection mode arouse spurious peaks in signal power spectrogram, that greatly limits the application of co-prime DFTFB. In this paper, we propose a modified co-prime DFTFB by adding a real valued multiplier to the original one, where M and N are both odd numbers. Together with original co-prime DFTFB, we get a 2MN bands filter bank. In the simulation section, two BPSK signals with random carrier frequencies are employed. The results of detection probability versus SNR curves through 500 Monte Carlo experiments verify the effectiveness of our method.

Lacing topological orders in two dimensions: Exactly solvable models for Kitaev's sixteen-fold way

A family of two-dimensional (2D) spin-1/2 models have been constructed to realize Kitaev’s sixteen-fold way of anyon theories. Defining a one-dimensional (1D) path through all the lattice sites, and performing the Jordan-Wigner transformation with the help of the 1D path, we find that such a spin-1/2 model is equivalent to a model with \nuν species of Majorana fermions coupled to a static \mathbb{Z}_2ℤ2 gauge field. Here each species of Majorana fermions gives rise to an energy band that carries a Chern number \mathcal{C}=1𝒞=1, yielding a total Chern number \mathcal{C}=\nu𝒞=ν. It has been shown that the ground states are three (four)-fold topologically degenerate on a torus, when \nuν is an odd (even) number. These exactly solvable models can be achieved by quantum simulations.

Explicit Relation Between Two Resolvent Matrices of the Truncated Hausdorff Matrix Moment Problem

We consider the explicit relation between two resolvent matrices related to the truncated Hausdorff matrix moment problem (THMM) in the case of an even and odd number of moments. This relation is described with the help of four families of orthogonal matrix polynomials on the finite interval [a, b] and their associated second kind polynomials.

Nonexistence of Positive Integer Solutions of the Diophantine Equation p^x + (p + 2q)^ y = z^2 , where p, q and p + 2q are Prime Numbers

The Diophantine equation p^x + (p + 2q)^y = z^2 , where p, q and p + 2q are prime numbers, is studied widely. Many authors give q as an explicit prime number and investigate the positive integer solutions and some conditions for non-existence of positive integer solutions. In this work, we gather some conditions for odd prime numbers p and q for showing that the Diophantine equation p^x + (p + 2q)^y = z^2 has no positive integer solution. Moreover, many examples of Diophantine equations with no positive integer solution are illustrated.

Analytical Approach to Parameter Determination in Kaiser Function for Power-weighted Antenna Array Design

Window methods that are frequently used in the design of finite impulse response filters are also applicable to antenna array designs. This paper explores the application of a Kaiser function in a power-weighted antenna array design, focusing on the determination of the Kaiser function’s β parameter. The determination, which includes the calculation, optimization, and validation of the β parameter, was carried out based on a specific configuration of a linear antenna array design. The observation of this exploration emphasized the suppression of the sidelobe level (SLL) and the width of main lobe (WML) performance. By changing the β parameter, the Kaiser function is capable of approximating different window methods, since it plays an important role in defining the set of weighting coefficients for a specifically targeted SLL. Kaiser function application in power-weighted antenna array designs with a linear arrangement indicates the need of β parameter optimization because of the disagreement between the obtained SLL and the targeted SLL. The optimized β parameter produced a smaller SLL error for even and odd numbers of elements. From the validation, the average SLL error percentage for a targeted SLL of 25 dB, 35 dB, and 45 dB was 6%, 4.31%, 6.10%, respectively.

ФОРМУЛА ДЛЯ НИЖНЕЙ ОЦЕНКИ ПЕРВОЙ ЧАСТОТЫ СОБСТВЕННЫХ КОЛЕБАНИЙ ФЕРМЫ ТРЕХГРАННОЙ БАШНИ

A scheme of a statically determinate tower-type spatial truss is proposed. Three lateral faces of the structure have double diagonal lattices, from below the truss rests on six racks and three additional horizontal lateral braces. The problem of the first natural frequency of free oscillations of the structure is solved. The inertial properties of the truss are modeled by the masses in its nodes. Each mass has two horizontal degrees of freedom. To obtain an analytical expression for the dependence of the lower limit of the oscillation frequency on the number of panels, the approximate Dunkerley method is used. The forces in the truss rods are found from the solution of the system of equilibrium equations for the nodes. The matrix of the system of equations is compiled in the Maple computer mathematics system. The rigidity of the structure is determined using the Mohr integral. A series of solutions obtained for a different number of panels is generalized to an arbitrary case by induction. For sequences of coefficients in the solution, linear recurrent equations are compiled and solved. The analytical solution is compared with the numerical solution of the problem of the vibration spectrum of the structure. It is shown that the first frequency of the spectrum is higher than the analytical estimate by no more than 30%. The accuracy of the estimate depends to a greater extent on the number of panels. As the number of panels increases, the accuracy of the resulting estimate increases. A trusses with only an even number of panels are considered. It is noticed that with an odd number of panels, the determinant of the system of equilibrium equations of nodes vanishes, which indicates the kinematic degeneration of the structure.

Tunable electron-flexural phonon interaction in graphene heterostructures.

Peculiar electron-phonon interaction characteristics underpin the ultrahigh mobility1, electron hydrodynamics2-4, superconductivity5 and superfluidity6,7 observed in graphene heterostructures. The Lorenz ratio between the electronic thermal conductivity and the product of the electrical conductivity and temperature provides insight into electron-phonon interactions that is inaccessible to past graphene measurements. Here we show an unusual Lorenz ratio peak in degenerate graphene near 60 kelvin and decreased peak magnitude with increased mobility. When combined with ab initio calculations of the many-body electron-phonon self-energy and analytical models, this experimental observation reveals that broken reflection symmetry in graphene heterostructures can relax a restrictive selection rule8,9 to allow quasielastic electron coupling with an odd number of flexural phonons, contributing to the increase of the Lorenz ratio towards the Sommerfeld limit at an intermediate temperature sandwiched between the low-temperature hydrodynamic regime and the inelastic electron-phonon scattering regime above 120 kelvin. In contrast to past practices of neglecting the contributions of flexural phonons to transport in two-dimensional materials, this work suggests that tunable electron-flexural phonon couping can provide a handle to control quantum matter at the atomic scale, such as in magic-angle twisted bilayer graphene10 where low-energy excitations may mediate Cooper pairing of flat-band electrons11,12.

Sums of Consecutive Odd Numbers

Summary We show visually that certain pairs of consecutive subsets of consecutive odd numbers have equal sums.

Asymptotic Fermat for signature (4,2,p) over number fields

Let K be a number field. Using the modular method, we prove asymptotic results on solutions of the Diophantine equation x4−y2=zp over K, assuming some deep but standard conjectures of the Langlands programme when K has at least one complex embedding. On the other hand, we give unconditional results in the case of totally real extensions having odd narrow class number and a unique prime above 2. When modularity of elliptic curves over K is known, for example when K is real quadratic or the r-layer of the cyclotomic Z2-extension of Q, effective asymptotic results hold.