Odd Number Research Articles

Modal dynamics of self-excited thermoacoustic instabilities in even and odd numbered networks of lean-premixed combustors

Can-to-can acoustic interactions in a circumferential network of lean-premixed combustors have a prominent role in the development of complicated modal dynamics and large-scale pattern formations in can-annular gas turbine combustion systems. Despite recent progress in understanding these instabilities, however, previous relevant studies have been limited to an even number of constituent combustors; a fundamental characteristic of the modal dynamics in a can-annular arrangement consisting of an odd number of combustors remains unknown. To address this question, we compare key features of thermoacoustic instabilities in four-coupled (N = 4) and five-coupled (N = 5) can-annular combustor configurations obtained over a broad range of operating conditions. In conjunction with FEM-based Helmholtz simulations, our experimental data show that the dynamic behavior of the even-N case is characterized by the well-known interaction pattern, out-of-phase alternating modulations among the networked combustors. For the odd-N case, however, the can-annular system features complex phase dynamics originating from simultaneous excitation of the degenerate mode pair. We highlight that the development of self-excited azimuthal instabilities in the annular cross-talk section is associated with the modal interaction of the degenerate pair in the odd-N configuration. The initially degenerate pair can be split into two closely-spaced non-degenerate modes, and in consequence their modal interaction gives rise to a periodic transition between spinning and standing azimuthal modes in the annulus. Despite the formation of remarkably different inter-combustor interactions in the two can-annular configurations, however, the entire set of instability data collapses into very similar frequency regimes, suggesting the importance of flame-acoustic interactions occurring within each combustor, namely intra-combustor interactions.

Hermitian splines with exponential power links with an odd number of parameters

Hermitian splines with exponential power links with an odd number of parameters

Dynamics of coexisting rotating waves in unidirectional rings of bistable Duffing oscillators

We study the dynamics of multistable coexisting rotating waves that propagate along a unidirectional ring consisting of coupled double-well Duffing oscillators with different numbers of oscillators. By employing time series analysis, phase portraits, bifurcation diagrams, and basins of attraction, we provide evidence of multistability on the route from coexisting stable equilibria to hyperchaos via a sequence of bifurcations, including the Hopf bifurcation, torus bifurcations, and crisis bifurcations, as the coupling strength is increased. The specific bifurcation route depends on whether the ring comprises an even or odd number of oscillators. In the case of an even number of oscillators, we observe the existence of up to 32 coexisting stable fixed points at relatively weak coupling strengths, while a ring with an odd number of oscillators exhibits 20 coexisting stable equilibria. As the coupling strength increases, a hidden amplitude death attractor is born in an inverse supercritical pitchfork bifurcation in the ring with an even number of oscillators, coexisting with various homoclinic and heteroclinic orbits. Additionally, for stronger coupling, amplitude death coexists with chaos. Notably, the rotating wave speed of all coexisting limit cycles remains approximately constant and undergoes an exponential decrease as the coupling strength is increased. At the same time, the wave frequency varies among different coexisting orbits, exhibiting an almost linear growth with the coupling strength. It is worth mentioning that orbits originating from stronger coupling strengths possess higher frequencies.

Partitioning an even number of the new formulation into all pairs of odd numbers

Partitioning an even number of the new formulation into all pairs of odd numbers

On primitivity and regularity of wreath product groups of degree 5p that are not p- groups using numerical approach

Let p be an odd prime number. This work applies some group concepts to construct the Wreath Product of two permutation groups of prime degrees. We used numerical approach to investigate and determine the primitive and regular nature of the constructed Wreath Product Group of degree 5p. We apply Computational Group Theory (GAP) to facilitate as well as validate our results.

The debut and course of the neonatal form of propionic aciduria: a clinical case

Propionic aciduria (PA) is an autosomal recessive hereditary disease from the group of organic aciduria, caused by a deficiency of propionyl-CoA carboxylase, leading to impaired metabolism of methionine, threonine, valine, isoleucine, and fatty acids with an odd number of carbon atoms and cholesterol. The neonatal form of PA manifests itself during the first week of life, is characterized by an acute onset and a crisis course, which is accompanied by severe metabolic acidosis, hypoglycemia, hyperketonemia, hyperammonemia. Clinical symptoms are dominated by neurological disorders up to stupor or coma, which can lead to death. Since 2023, expanded neonatal screening has been introduced throughout the Russian Federation, which includes 36 groups of nosologies, as well as a number of hereditary metabolic diseases.
 Despite the inclusion of this pathology in expanded neonatal screening, doctors’ awareness of clinical manifestations and necessary therapy remains insufficient. Often such patients are diagnosed with, for example: hypoxic-ischemic damage to the central nervous system, acute meningoencephalitis, and others, which leads to inadequate therapy with the development of fatal neurological consequences.
 Therefore, the totality of knowledge and alertness of doctors regarding diseases from the group of hereditary metabolic diseases will help not only to suspect this pathology in a timely manner, but also to prescribe adequate therapy in time, which in the future will make it possible to prevent serious consequences and neurological disorders, as well as disability of patients.

Portrait of Damar Tradition in Welcoming the Arrival of Lailatul Qadar Night in the Community of Patipi Island Village

The purpose of the study was to explain the implementation of the Damar tradition in the community of Patipi, Fak-Fak Island Regency, West Papua Province. The Damar Tradition welcomes the night of Lailatul Qadar by lighting torches using resin sap. The research method used a qualitative model with a social phenomenological paradigm and a flow model analysis technique. This research produced several findings regarding implementing the Damar tradition as follows. The Damar tradition in Patipi Island developed orally from generation to generation. Islam, as a religion with plenty of followers in Patipi Island, adapts to its culture to form a tradition in enlivening the night of Lailatul Qadar. Damar tradition begins in the last ten days of Ramadhan month, exactly in the odd number of fasting days: the 21st day, the 23rd, the 25th and the 27th day of fasting, respectively. The Damar tradition can unite the community and form of joy to welcome the night of Lailatul Qadar. The absence of fixed and given goods is a moral idea and shows the citizen's generosity. The gift belongs to the mosque management, so they carry out their duties and responsibilities properly. There is reciprocity in the form of prayers read by the mosque management, a form of dependence between the community and the mosque management. The symbols in the tradition have meanings that can be interpreted by the Patipi island community, in the form of incense burned while reading prayers, Raun or a gift holder made of woven coconut leaves, torches, gifts, and cleaning and giving sand to the grave.

The Routh Hurwitz Criteria for Studying The Stability and Bifurcation in Multispiral Chua Chaotic Attractor

This article discusses the multispiral Chua Chaotic attractor’s hidden bifurcations that are generated by the sine function. The number of spirals (also known as a multiscroll attractor) that are controlled by the integer parameter c can be used to describe the basic shape of chaotic attractors. Since this parameter is an integer, increasing it by one does not allow the observation of bifurcations from n to n + 2 spirals. The method of hidden bifurcations, however, enables the observation of such bifurcations by adding a real parameter ε. Chaotic attractors with either an even or an odd number of spirals are visible along the marked paths of bifurcation. Moreover, this additional hidden parameter allows finding the bifurcation of the multispiral Chua attractor from a stable state to a chaotic state. Furthermore, the Routh-Hurwitz criteria are used to study the stability of the original equilibrium point of the Chua attractor.

Large-deviation theory of increasing returns.

An influential theory of increasing returns was proposed by the economist W. B. Arthur in the 1980s to explain the lock-in phenomenon between two competing commercial products. In the most simplified situation there are two competing products that gain customers according to a majority mechanism: each new customer arrives and asks which product they bought to a certain odd number of previous customers, and then buys the most shared product within this sample. It is known that one of these two companies becomes a monopoly almost surely in the limit of infinite customers. Here we consider a generalization [Dosi, Ermoliev, and Kaniovsky, J. Math. Econom. 23, 1 (1994)10.1016/0304-4068(94)90032-9] in which the new customer follows the indication of the sample with some probability, and buys the other product otherwise. Other than economy, this model can be reduced to the urn of Hill, Lane, and Sudderth, and it includes several models of physical interest as special cases, such as the Elephant Random Walk, Friedman's urn, and other generalized urn models. We provide a large-deviation analysis of this model at the sample-path level, and we provide a formula that allows us to find the most likely trajectories followed by the market share variable. Interestingly, in the parameter range where the lock-in phase is expected, we observe a whole region of convergence where the entropy cost is sublinear. We also find a nonlinear differential equationfor the cumulant-generating function of the market share variable, which can be studied with a suitable perturbation theory.

P-147 Comparison of Fertilization rate, Blastocyst rate and number of Top Quality Embryo with Short (2 hours) and conventional co-incubation (16-20 hours)of gametes

Abstract Study question Does short Co-incubation of gametes increase quality and quantity of embryos by eliminating suboptimal culture conditions like increased reactive oxygen species seen with overnight incubation? Summary answer Although short co-incubation group had lower fertilization rate it yielded more blastocysts and top quality embryos on day 3 and 5. What is known already Successful fertilization can occur 20 minutes after co-incubation of gametes. Incubation time of 18 hours has been used as a routine combining it with fertilization check. However it is known to produce reactive oxygen species (ROS) released by cellular metabolism of inseminated sperms which affect embryo development and cause zona pellucida hardening. Studies have shown reduced oocyte exposure time caused improved embryo development. However, other studies have reported lower fertilization rate with no such advantage. Not many studies have been done with short co-incubation without denudation which might allow optimum fertilization and at the same time prevent increase in ROS. Study design, size, duration It is prospective controlled study. 62 patients (921 oocytes) were recruited. The sibling oocytes from each patient were divided into 2 groups. Group 1( Control) conventional co-incubation of gametes (16-20 hours) and group 2 short co-incubation (2 hours) after which they were transferred to a fresh media. Study period was from January 2022 to September 2022. Institutional scientific and ethical committee approval and written consent of couple taken. Participants/materials, setting, methods Couples undergoing IVF in tertiary infertility centre were recruited. Women under 40 years who yielded minimum 6 oocytes and males with normal semen parameters were included. Those excluded were couples requiring ICSI, poor responders with <6 oocytes retrieved and couples requesting day 3 transfer. Primary outcome was number and percentage of top quality embryos (TQE) and blastocyst rate. Secondary outcome was fertilization rate. SPSS used for data analysis. Results expressed as mean +/-SD or percentage Main results and the role of chance Total oocytes studied 921 in 62 patients. Since control and study group were from sibling oocytes of same patient they were matched for age BMI, ovarian reserve, stimulation protocol, semen parameters and laboratory handling. Extra oocytes in odd number oocytes were randomized to 2 groups. Mean age was 31 years. Group 1 had 425 out of 455 (93.4%) oocytes fertilized and group 2 had 412 out of 466 (88.4%) oocytes fertilized (p value<0.05). TQE on day 3 were 251 of 425 (59.05%) fertilized in group 1 and 277 of 412 (67.23%) fertilized in group 2 (p = 0.013). Blasts formed in group 1 were 41 of 425 (9.64%) and in group 2 were135 of 412 (32.76%) (p = 0.001). TQE on day 5 were 28/425 (6.67%) in group 1 (0.45 +/−0.9 blasts per patient) and 95/412 (23.05%) in group 2 (1.53+/-1.86 blasts per patient). (P = 0.001) Percentage of grade 2 embryos on day 3 and 5 was significantly higher in short co-incubation group. Although fertilization was higher in group 1 many embryos did not progress beyond day 2 as both grade 1 and 2 embryos on day 3 and 5 were less in percentage and absolute per patient value in this group. Limitations, reasons for caution Pregnancy rate could not be calculated as some patients wanted 2 embryos transferred and one group had only a single blast. For their transfer blasts were taken from both groups. Very poor responders (<6 oocytes) not studied. Further studies needed to establish best time duration for short co-incubation (1-8 hours) Wider implications of the findings Practice for co-incubation after 16-20 hours needs to be re-evaluated. The study shows more top quality blasts with short insemination period. This may become the practice if it is established by further studies assuming that TQE are known to yield better pregnancy rates and IVF results Trial registration number not applicable

Clifford group is not a semidirect product in dimensions N divisible by four

The paper is devoted to projective Clifford groups of quantum N-dimensional systems (with configuration space ). Clearly, Clifford gates allow only the simplest quantum computations which can be simulated on a classical computer (Gottesmann–Knill theorem). However, it may serve as a cornerstone of full quantum computation. As to its group structure it is well-known that—in N-dimensional quantum mechanics—the Clifford group is a natural semidirect product provided the dimension N is an odd number. For even N special results on the Clifford groups are scattered in the mathematical literature, but they mostly do not concern the semidirect structure. Using appropriate group presentation of it is proved that for even N the projective Clifford groups are not natural semidirect products if and only if N is divisible by four.

Connections between Linear Complementary Dual Codes, Permanents and Geometry

Linear codes with complementary duals, or LCD codes, have recently been applied to side-channel and fault injection attack-resistant cryptographic countermeasures. We explain that over characteristic two fields, they exist whenever the permanent of any generator matrix is non-zero. Alternatively, in the binary case, the matroid represented by the columns of the matrix has an odd number of bases. We explain how Grassmannian varieties as well as linear and quadratic complexes are connected with LCD codes. Accessing the classification of polarities, we relate the binary LCD codes of dimension k to the two kinds of symmetric non-singular binary matrices, to certain truncated Reed–Muller codes, and to the geometric codes of planes in finite projective space via the self-orthogonal codes of dimension k.

The Galois group of X^{p^2}+aX+a

Let p be an odd prime number, and a be an integer divisible by p exactly once. We prove that the Galois group G of the trinomial X^{p^{2}}+aX+a over the field Q of rational number is either the full symmetric group S_{p^{2}} or G lies between AGL(1,p^{2}) and AGL(2,p)$. Furthermore, we establish conditions when G is S_{p^{2}}.

Robust formulation of Wick's theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions.

Numerical difficulties associated with computing matrix elements of operators between Hartree-Fock-Bogoliubov (HFB) wavefunctions have plagued the development of HFB-based many-body theories for decades. The problem arises from divisions by zero in the standard formulation of the nonorthogonal Wick's theorem in the limit of vanishing HFB overlap. In this Communication, we present a robust formulation of Wick's theorem that stays well-behaved regardless of whether the HFB states are orthogonal or not. This new formulation ensures cancellation between the zeros of the overlap and the poles of the Pfaffian, which appears naturally in fermionic systems. Our formula explicitly eliminates self-interaction, which otherwise causes additional numerical challenges. A computationally efficient version of our formalism enables robust symmetry-projected HFB calculations with the same computational cost as mean-field theories. Moreover, we avoid potentially diverging normalization factors by introducing a robust normalization procedure. The resulting formalism treats even and odd number of particles on equal footing and reduces to Hartree-Fock as a natural limit. As proof of concept, we present a numerically stable and accurate solution to a Jordan-Wigner-transformed Hamiltonian, whose singularities motivated the present work. Our robust formulation of Wick's theorem is a most promising development for methods using quasiparticle vacuum states.

Rashba Band Splitting and Bulk Photovoltaic Effect Induced by Halogen Bonds in Hybrid Layered Perovskites.

Non-covalent interactions play an essential role in directing the self-assembly of hybrid organic-inorganic crystals. In hybrid halide perovskites, hydrogen bonding has been the paramount non-covalent interaction. Here, we show another non-covalent interaction, namely, the halogen bond interaction, that directs a symmetry-breaking assembly in a new series of two-dimensional (2D) perovskites (ICH2 CH2 NH3 )2 (CH3 NH3 )n-1 Pbn I3n+1 (n is the layer thickness, n=1-4). Structural analysis shows that the halogen bond strength varies with the layer thickness. For the odd number (n=1, 3) layered perovskites, stronger halogen interaction leads to centrosymmetric structures, whereas for the n=2 layered perovskites, weaker halogen bonds result in non-centrosymmetric structures. Transient reflection spectroscopy shows a suppressed radiative recombination rate (k2 ≈0) and prolonged spin lifetime for n=2 structure, suggesting an enhanced Rashba band splitting effect. The structural asymmetry is further confirmed with a reversible bulk photovoltaic effect. Our work provides a new design strategy to enable hybrid perovskites with emerging properties and functionalities associated with structural asymmetry.

Rashba Band Splitting and Bulk Photovoltaic Effect Induced by Halogen Bonds in Hybrid Layered Perovskites

AbstractNon‐covalent interactions play an essential role in directing the self‐assembly of hybrid organic–inorganic crystals. In hybrid halide perovskites, hydrogen bonding has been the paramount non‐covalent interaction. Here, we show another non‐covalent interaction, namely, the halogen bond interaction, that directs a symmetry‐breaking assembly in a new series of two‐dimensional (2D) perovskites (ICH2CH2NH3)2(CH3NH3)n−1PbnI3n+1 (n is the layer thickness, n=1–4). Structural analysis shows that the halogen bond strength varies with the layer thickness. For the odd number (n=1, 3) layered perovskites, stronger halogen interaction leads to centrosymmetric structures, whereas for the n=2 layered perovskites, weaker halogen bonds result in non‐centrosymmetric structures. Transient reflection spectroscopy shows a suppressed radiative recombination rate (k2≈0) and prolonged spin lifetime for n=2 structure, suggesting an enhanced Rashba band splitting effect. The structural asymmetry is further confirmed with a reversible bulk photovoltaic effect. Our work provides a new design strategy to enable hybrid perovskites with emerging properties and functionalities associated with structural asymmetry.

Kolintang Symbolic Construction of The Union Odd Numbers (Three) in Socio-cultural Meaning of Minahasa Community

This research examines the construction of Kolintang through the symbol of the number Three in the socio-cultural life system of the Minahasa people, whose embodiment in Kolintang music. A qualitative method through a historical approach in social and anthropological writing is used in this research. The data sourcewas obtained from field observations for approximately three years in Lembean Village, namely Nine taranak or Minamerot.Interviews and recordings do well; researchers are also active in direct observation. The research found that the number one is a transcendent and intangible symbol, the God of the Minahasa people Empung, WailanWangko, EmpungRenga-Rengan, God is Glorious, God is Great. The number one does not appear in the form of kolintang but is unifying. The number three is seen in the three forms of the Minahasa grandmothers Karema, Lumimuut, and Toar; also in the three Minahasa leaders, Tunduan/Tonaas, Wadian/Walian, and Teterusan. It is seen in the three first descendants of Minahasa, namely the Tombulu, Toutewoh, and Tounsawang sub-ethnicity. This symbolisation can be seen in the construction of Kolintang instruments using three wooden sticks (Teken) singing Tatembaken, the use of three-wood Kolintang in Waleposan, and traditional modes/scales such as (1 - 2 - 3) as well as the use of three wood in modern Kolintang Music Accompaniment called Wooden Kolintang Music Ensemble (AMKK). In conclusion, the construction of Kolintang music is a form of symbolisation of the Unification of the number three in the social and cultural life of the Minahasa people.

Physico-chemical characterization of Synperonic™ 91/5 self-assembly behaviour in water

We studied the self-assembled aggregates of Synperonic™ 91/5 (SYN) in water solution. This surfactant presents the peculiarity of having an odd number of aliphatic carbons, with a tail composed of C9H19 and C11H23, and the headgroup having an average of 5 ethoxy units. While the general family of CiEOj nonionic surfactants has been widely studied, there is little knowledge for the case of surfactants with odd-number carbons. For this reason, we carried out an in depth physico-chemical characterization of water solutions containing SYN in the wide range of concentration from 2 % to 50 % (i.e. 0.05–1.4 M), considering that high concentrated systems are relevant for industrial applications. This class of surfactants is also interesting as remediation agent since they can incorporate different polluting molecules, both polar and nonpolar, such as volatile Organic Compounds (VOCs) and Dense Non Aqueous Phase liquids (DNAPLs). The rheological and transport properties of SYN solutions were investigated through diffusion coefficients and viscosity, while shape and size of self-assemblies were determined by Dynamic Light Scattering and by Small Angle Scattering of Neutrons (SANS) and X-rays (SAXS). Results show that SYN generally behaves as other nonionic surfactants in dilute solutions, but shows some peculiarity of aggregate packing at high concentration. We recently demonstrated the efficiency of the Synperonic™ family of surfactants in solubilising trichloroethylene (TCE), in particular above a critical concentration of 0.1 M, thus being a good candidate for remediation purposes. Here, the solubility of TCE was correlated with the number and the size of micelles at different surfactant concentrations. Our work helps filling a gap in the literature about technical grade surfactants with odd-number carbons, and the results can help in assessing the performances of Synperonic™ products for technical tasks.

Non-Imaginary unit Circle and Distribution Odd Natural Numbers

This paper introduces a non-Imaginary unit circle partitioning as proof for the distribution of odd natural numbers in relation to an imaginary unit circle in a complex plane. First, we will introduce the concept of a non-imaginary unit circle and its relation to an imaginary unit circle in a complex plane. Then we will go through some examples to prove that for any N odd natural number at N/2, we only have the imaginary part for any complex number on the complex plane if we use our technique of portioning for the non-imaginary unit circle.

In-Ear SpO₂ for Classification of Cognitive Workload

Classification of cognitive workload promises immense benefit in diverse areas ranging from driver safety to augmenting human capability through closed loop brain computer interface. The brain is the most metabolically active organ in the body and increases its metabolic activity and thus oxygen consumption with increasing cognitive demand. In this study, we explore the feasibility of in-ear SpO2 cognitive workload tracking. To this end, we preform cognitive workload assessment in 8 subjects, based on an N-back task, whereby the subjects are asked to count and remember the number of odd numbers displayed on a screen in 5 second windows. The 2 and 3-back tasks lead to either the lowest median absolute SpO2 or largest median decrease in SpO2 in all of the subjects, indicating a robust and measurable decrease in blood oxygen in response to increased cognitive workload. Using features derived from in-ear pulse oximetry, including SpO2, pulse rate and respiration rate, we were able to classify the 4 N-back task categories, over 5 second epochs, with a mean accuracy of 94.2%. Moreover, out of 21 total features, the 9 most important features for classification accuracy were all SpO2 related features. The findings suggest that in-ear SpO2 measurements provide valuable information for classification of cognitive workload over short time windows, which together with the small form factor promises a new avenue for real time cognitive workload tracking.