Single Direction Research Articles

Abstract 128: Multi-Direction Diffusion Weighted Imaging on Portable, Low-Field Magnetic Resonance Imaging

Background and aims: Portable, low-field (LF) MRI has the potential to improve access to expeditious, definitive brain imaging and facilitate diagnosis of acute stroke. Currently available diffusion-weighted imaging (DWI) protocols at LF are limited to a single diffusion direction due to acquisition duration. However, single-direction diffusion has reduced sensitivity for detecting acute ischemic infarcts, particularly small lesions residing in white matter tracts. The purpose of this study was to establish the feasibility of acquiring multi-direction DWI compared with single-direction counterparts on LF-MRI. Methods: Patients presenting with a diagnosis of acute ischemic stroke between July and September 2023 were eligible. Consented patients underwent DWI acquisition on a 0.064T LF-MRI (Mk1.9; Hyperfine Research Inc). Three diffusion directions (x, y, and z) were acquired with a b weighting of 900 s/mm 2 and a single acquisition with a b 0 s/mm 2 . The b 900 images were co-registered to the b 0, trace and apparent diffusion coefficient (ADC) maps calculated, and the final images interpolated at 1 mm 3 . Results: Ten patients presenting to the Massachusetts General Hospital with acute ischemic stroke were consented and imaged within 72 hours of last known well. The total acquisition time was 14 minutes, with all subjects able to tolerate the scan duration. Ischemic lesions as small as 0.1 mL were detectable on the LF-MRI (17.5 +/- 18.2 mL). An example of each diffusion direction individually, the combined trace, and corresponding ADC maps are shown in Figure 1, compared with conventional high-field (HF) diffusion images acquired within 30 minutes of the LF acquisition. Conclusion: Multi-direction DWI imaging is feasible on a 0.064T LF-MRI scanner. Our experience suggests further modifications to the pulse sequence and scanner configuration may facilitate a reduction in acquisition time, improve resolution, or both.

The people think what I think: False consensus and unelected elite misperception of public opinion

AbstractPolitical elites must know and rely faithfully on the public will to be democratically responsive. Recent work on elite perceptions of public opinion shows that reelection‐motivated politicians systematically misperceive the opinions of their constituents to be more conservative than they are. We extend this work to a larger and broader set of unelected political elites such as lobbyists, civil servants, journalists, and the like, and report alternative empirical findings. These unelected elites hold similarly inaccurate perceptions about public opinion, though not in a single ideological direction. We find this elite population exhibits egocentrism bias, rather than partisan confirmation bias, as their perceptions about others' opinions systematically correspond to their own policy preferences. Thus, we document a remarkably consistent false consensus effect among unelected political elites, which holds across subsamples by party, occupation, professional relevance of party affiliation, and trust in party‐aligned information sources.

Design and analysis of magnetostrictive two-dimensional kinetic energy harvester

Conventional energy harvesters often require high ambient vibration frequencies and can only capture vibration energy in a single direction. To address these issues, this paper designs a magnetostrictive two-dimensional kinetic energy harvester placed under the floor and capable of capturing energy in both vertical and horizontal directions. In order to achieve higher electrical power output at low-frequency input forces, a two-stage force amplification mechanism is designed to amplify the walking kinetic energy of pedestrians and the main parameters of this structure are analyzed and optimized. On the other hand, by constructing different forms of bias magnetic field, the influence of bias magnetic field on the deflection and motion of the internal magnetic domain of Terfenol-D is systematically studied, and the best bias form that can make the material shows the strongest magnetization characteristics is determined. Next, a prototype harvester was built, and an experimental vibration system was set up to test and analyze the output characteristics of the harvester comprehensively. The experimental results show that the harvester produces 21.2 mW of peak output power under sinusoidal excitation at an operating frequency of 4 Hz. Under random excitation, a peak output voltage of 2.64 V and 170 mW peak power was obtained. Under actual pedestrian walking tests, 17.62 mW peak output power is obtained to power low-power devices. The study’s results provide preliminary evidence that the designed magnetostrictive energy harvester can stably harvest kinetic energy from pedestrian walking.

The Application of Frieze Groups and Crystallographic Groups in Generating Batak Toba Ornament Motifs Using a Matlab Graphical User Interface

Gorga is a carving or sculpture typically found on the exterior of a Toba Batak traditional house. The Batak people use fractal (geometric) dimensions in Batak gorga carvings. In mathematics, repetitive and symmetrical patterns in planes that result from transformations are included in the plane symmetry groups. Ethnomathematics is a cultural approach to the concept of mathematics. A frieze group can be defined as a symmetrical group which arises from a unidirectional translation and subsequently generates a linear pattern that recurs exclusively in a single direction. There are seven different pattern types in the frieze groups. Meanwhile, crystallographic patterns are flat two-dimensional patterns that form a lattice. There are 17 crystallographic types of patterns with five different types of unit lattices. The purpose of this study is to generate motifs for Batak ornaments based on frieze groups and crystallographic groups using a Matlab Graphical User Interface (GUI). A total of 119 new motifs were generated based on seven types of patterns in the frieze groups, namely, F1,F2,F3,F4,F5,F6, and F7. Meanwhile, in the crystallographic groups, 153 new motifs were generated based on nine types of patterns, namely, p1,p2,pm,pg,cm,pmg,pmm,cmm, and pgg. To keep with trends, the new motifs generated can be used in everyday life as decorations or business symbols that are characteristic of the Toba Batak region.

Application of Tesla Valve to Bidirectional Open-Channel Flows

Dams hinder water circulation and consequently disrupt the river continuum which adversely affects aquatic organisms and degrades water quality. An alternative new concept for a dam has been proposed with an optimal design that prevents seawater from flowing upstream while maintaining the continuity of the river–estuary ecosystem. A Tesla valve was used because its asymmetric geometry increases the flow resistance only in a single direction. The numerical model setup aimed to simulate wave-induced open-channel flows in OpenFOAM using the volume of fluid method and investigate the capability of the channel design to enhance the asymmetric flow resistance, and the model was validated by laboratory experiments. Various combinations, scales, shapes, and arrangements of the structures were considered for deployment. Subsequently, these designs were evaluated based on the difference in flow discharges in each flow direction. The optimal choice included a 30° angle between the sidewalls and the structures, and streamwise spacing between the structures at 5/3 times the channel width. Further, longer main structures with substructures and a slightly offset design can help increase the flow asymmetry in bidirectional open-channel flows. Although the present study has limitations in terms of methodology and complex structural factors, the proposed design suggests the basic design of a new concept for Tesla valve-like geometry.

Quaternion empirical Ramanujan Fourier decomposition and its application in gear fault diagnosis

In the field of gear health monitoring, the measured signals are traditionally derived from a single sensor or a single direction. However, with the increasing complexity and size of equipment structures nowadays, a single-channel signal often falls short in providing comprehensive status information. Furthermore, fault-induced vibrations are randomly indeterminate in both location and direction, which indicates the necessity of using multi-channel signals for effective gear fault diagnosis. As a single-channel method, Ramanujan Fourier transform (RFT) has been recognized for its effectiveness in extracting periodic components from signals. However, the RFT is limited to handling single-channel signals and lacks the ability to analyze nonstationary signals. To overcome these challenges, this paper proposes a novel multi-channel signal processing method called quaternion empirical Ramanujan Fourier decomposition (QERFD), which extends the RFT to the multi-channel field using the quaternion method. By fusing multi-channel signals while preserving the advantages of RFT, the proposed QERFD method enhances fault information extraction. In addition, QERFD shows high computational efficiency. Therefore, QERFD is of great practical value for the health monitoring of complex equipment in the present industrial context of big data. To substantiate the superiority of QERFD, simulations and experiments are carried out and compared against both single-channel and multi-channel methods.

Lorentz Transformation and time dilatation

We consider two inertial frames S and and suppose that frame moves, for simplicity, in a single direction: the X -direction of frame S with a constant velocity v as measured in frame S. Using homogeneity of space and time we derive a modified Lorentz Transformation (LT) between two inertial reference frames without using the second postulate of Einstein, i.e., we do not assume the invariant speed of light (in vacuum) under LT. Roughly speaking we suppose: (H) Any clock which is at rest in its frame measures a small increment of time by some factor s=s(v). As a corollary of relativity theory (H) holds with Lorentz factor 1/γ. For s=1 we get the Galilean transformation of Newtonian physics, which assumes an absolute space and time. We also consider the relation between absolute space and Special Relativity Theory, thereafter STR. It seems here that we need a physical explanation for (H). We introduce Postulate 3. The two-way speed of light in and -directions are c and outline derivation of (LT) in this setting. Note that Postulate 3 is a weaker assumption than Einstein's second postulate.

A Method for Micro Powder Dispensing by Using DC Motor with a 2-Axis Vibration

A comprehensive understanding of powder flow is essential, particularly in the synthesis of additives in the solid free-forming process and dosage dispensing in pharmaceutical industries. The vibration method is the most widely used for inducing powder flow. In this method, a hopper containing powder is vibrated by a piezoelectric system to allow powder flow. However, there are several drawbacks in this method, particularly in the degree of consistency of the resulting flow. This occurs because the vibration characteristics created have a single axis direction, resulting in which can lead to powder compaction. To resolve this problem, this research is conducted to determine the flow properties of titanium powder dispensed by 2-axis vibration. The titanium (Ti) powder having a size of less than 74 μm in the powder hopper was vibrated for 15 min by a DC motor and the weight of the dispensed powder was measured to evaluate the consistency of the resulting powder flow. The result shows that the powder flow generated by the DC motor was consistent during the period of dispensing. However, the powder flow rate dropped up to 8.6% during 5-10 min of dispensing at a speed of 1800 rpm. In conclusion, the 2-axis vibration by using a DC motor could prevent the powder compacting phenomenon and ensure a consistent flow of micro-sized powders during the dispensing process.

Deformation twins as a probe for tribologically induced stress states

Friction and wear of metals are critically influenced by the microstructures of the bodies constituting the tribological contact. Understanding the microstructural evolution taking place over the lifetime of a tribological system therefore is crucial for strategically designing tribological systems with tailored friction and wear properties. Here, we focus on the single-crystalline High-Entropy Alloy CoCrFeMnNi that is prone to form twins at room temperature. Deformation twins feature a pronounced orientation dependence with a tension-compression anisotropy, a distinct strain release in an extended volume and robust onset stresses. This makes deformation twinning an ideal probe to experimentally investigate the complex stress fields occurring in a tribological contact. Our results unambiguously show a grain orientation dependence of twinning under tribological load. It is clearly shown, that twinning cannot be attributed to a single crystal direction parallel to a sample coordinate axes. With deformation twins in the microstructure, stress field models can be validated to make them useable for all different tribological systems.

Research of direct and inverse problems of sensitivity using methods of practical stability by direction

The general statements of the problems of construction of extreme ranges of tolerances on parameters for a linear system of differential equations in the presence of dynamic restrictions on the spread of the state vector are considered. It was found that such sensitivity problems are closely related to the problems of estimating the maximum area of stability in the corresponding function space. In order to obtain upper estimates of the sought values, the problem of estimating tolerances from the positions of stability in a single direction is formalized. The cases of linear and nonlinear restrictions on the spread of the phase coordinate vector with initial linear initial conditions relative to the vector of the system parameters are studied. Necessary and sufficient conditions are formulated in the form of criteria for estimating tolerances on parameters covering the case of fixed initial conditions. The tasks of estimating the maximum volume of the stability region with respect to the deviations of the state vectors and system parameters are considered. For the case of compatible admissible constraints, the paper gives the corresponding numerical calculations for the upper estimates. In particular, for fixed initial conditions, the maximum set of tolerances for system parameters with compatible dynamic restrictions on fluctuations of state vectors and parameters are determined according to these evaluation criteria. From the standpoint of practical stability in direction, direct sensitivity problems are investigated, numerical estimates of the spread of the phase coordinate vector of the system in the presence of linear and fixed initial conditions relative to the parameter vector are given. It has been demonstrated that in the given specification of the problem of maximum estimation of tolerances it is completely covered by problem statements of practical stability by direction. Thus, to calculate areas of guaranteed sensitivity, the concept of practical stability in a single direction is extended to the space of sensitivity functions at the initial moment of time. The above algorithms are used to estimate the range of tolerances for the adjustment parameters of the discrete model of the induction acceleration system with the given limitations on the spread of the quality criterion; the variation of the quality indicator within the tolerance field is presented in the form of a linear form relative to the spread of the parameter vector.

Acousto-Optic Deflector-Principles, Working and Applications

Acoustic optic deflectors (AOD) have gained a lot of popularity as a fast way to scan the laser beam in optical coherence systems. It produces an acoustic wave that deflects the laser beam in a single direction. There are many factors to consider when choosing different solutions for acousto-optic spectral systems, but no comprehensive analysis or summary of these factors exists. In addition to summarizing the most popular optical schemes for acousto-optic spectral systems, this paper explains the basic principles of acousto-optic devices. A simulation study has also been performed to represent the characteristics of AOD in a practical environment. Additionally, applications of acousto-optic spectral systems in a few common fields are presented.

Synergistic control of microstructures and properties in eutectic high-entropy alloys via directional solidification and strong magnetic field

The AlCoCrFeNi2.1 eutectic high-entropy alloy (Ni2.1 EHEA), as an exemplary representative of the high-entropy alloy family, has garnered significant research attention owing to its exceptional comprehensive properties. In this study, we investigated the influence of various growth velocities on the microstructure, lamellar spacing, and mechanical properties of the Ni2.1 EHEA. We observed that at lower growth velocities, the structure consisted of an alternating face-centered-cubic (FCC) phase and B2 phase lamellae aligned in a single direction, with the lamellae orientation parallel to the direction of the heat flow. The yield strength increased with the growth rate, while the ultimate tensile strength decreased with increasing the growth velocity. Ductility remained relatively consistent, and a double yield phenomenon was observed in the elastic-plastic deformation region. Under conditions of high growth velocities, the microstructure transitioned from a single-directional full lamellar structure to a multi-stage lamellar arrangement. The most favorable comprehensive mechanical properties were achieved at a growth rate of 200 μm/s, resulting in a yield strength of 450 MPa, an ultimate tensile strength of 1092 MPa, and a remarkable ductility of ∼32% in the directionally solidified samples—double that of the arc-melted sample. The evolution law of directional solidification structures under the coupling effect of different magnetic fields and different growth rates was studied. The interaction of the thermoelectric-magnetic force and thermoelectric-magnetic convection and the potential mechanism of microstructure evolution under the effect of magnetic field were deeply analyzed. The results reveal that at a growth rate of 2 μm/s, the spacing between eutectic layers decreases as the magnetic induction intensity increases, leading to the transformation of some regular layers into spherical layers. Similarly, at a growth velocity of 10 μm/s, the eutectic structure exhibited a Columnar-to-equiaxed transition (CET). However, as the growth rate further increases, the limited exposure time to the magnetic field prevented significant structural changes.

«ЗЕЛЕНЫЕ» ЦИФРОВЫЕ ФИНАНСЫ, КАК ФАКТОР ОБЕСПЕЧЕНИЯ УСТОЙЧИВОГО РАЗВИТИЯ ЭКОНОМИКИ РЕСПУБЛИКИ ТАДЖИКИСТАН

Limited economic resources, global warming, a rapidly growing population, and significant changes in the global economy lead to the need to reorient countries' economic policies towards achieving the Sustainable Development Goals set by the United Nations. Undoubtedly, financial sector reform plays a key role in this process. Recently, the influence of digital technologies on all spheres of economic activity has been increasing, which determines the need for their careful research, including in the field of finance. The article reveals the essence of the relatively new concept of "green digital finance" and their role in the sustainable development of the country's economy. "Green finance" and "digital finance", having previously been separate concepts, have recently been implemented as a single direction for achieving sustainable development of the financial sector and the economy as a whole. The author identifies and substantiates the advantages of introducing "green" digital finance for banking institutions, enterprises and the state as a whole. The impact of digital technologies on the development of "green" financing, possible risks and potential areas of interaction, such as the creation of investment platforms for sustainable projects, the use of improved analytics tools to search for new types of financial products and services, solving problems related to global warming and limited resources, etc., are considered. The author also examines the tools of "green" digital finance: "green" bonds, "green" mutual funds, "green" leasing/leasing, "green" loans. The main problems of the Republic of Tajikistan hindering the spread of sustainable digital finance in the country have been identified. Based on the conducted research, the author provides recommendations for government agencies, financial sector participants, research institutes, educational institutions and innovation centers to accelerate the process of introducing "green digital finance" into the country's economy. Thus, it was proposed to strengthen state support in this area, strengthen cooperation with international organizations, provide consulting support to businesses, continue to develop digitalization of all sectors of the economy, etc.

Efficient grain size evaluation based on single direction measurement of ultrasonic backscattering coefficient

<sec>GH4742 nickel-based superalloy exhibits excellent mechanical properties, and grain size is a key factor affecting its performance. A physical model-based ultrasonic backscattering method makes grain size measurement accurate and efficient. Nevertheless, it is constrained by complex models or multiple measurements taken from various beam angles. As a result, a backscattering coefficient method that requires only a single measurement for grain size evaluation is proposed. In contrast to the existing methods, the proposed method solely focuses on the backscattering coefficient component of the backscattering signal. It effectively eliminates the influence of unrelated factors, such as the measurement system and the acoustic field, through the utilization of reference signals.</sec><sec>The independent scattering model is employed to derive the backscattering coefficient, which solely pertains to the material itself. The relationship between grain size and backscattering coefficient is described by using a spatial correlation function. To consider the irrelevant factors, an experimental measurement method is developed by using the reference signals. Through numerical calculation and analysis, it has been observed that the backscattering coefficient is closely related to the frequency. When the product of the wavenumber and the grain size is significantly greater than 1 (<inline-formula><tex-math id="M1">\begin{document}$ ka\gg 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M1.png"/></alternatives></inline-formula>), a Stochastic scattering limit is reached. Conversely, when <inline-formula><tex-math id="M2">\begin{document}$ ka\ll 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="7-20231959_M2.png"/></alternatives></inline-formula>, a Rayleigh scattering limit is observed. Furthermore, the backscattering coefficient is directly proportional to the grain size. As a general trend, larger grain sizes result in higher backscattering coefficient.</sec><sec>Three sets of GH4742 specimens with different grain sizes are prepared for phased array ultrasound experiments. It can be observed that the experimental backscattering coefficients, root mean square (RMS) values, and the amplitude trend of time domain signal are consistent. To perform grain size inversion, the backscattering coefficients in the effective bandwidth range of the probe are selected. By utilizing the least-square method, the theoretical backscattering coefficient is employed to fit the curves of the experimental backscattering coefficients. The evaluation results are compared with those obtained by metallographic analysis. The results show that the grain sizes obtained by the proposed method have a maximum relative error of –22.7% and a minimum relative error of –3.7%.</sec>

Effects of scanning methods on cracking, microstructures and microhardness of Inconel 625 parts formed by direct laser metal deposition

This study aims at investigating the effects of scanning methods on the stress distribution and microstructures of Inconel 625 thin-walled part fabricated by direct laser metal deposition. The results showed that with the single direction scanning (SDS) method, the residual stress at both the ends of the thin wall was relatively high while the stress at the middle was smaller with a stress difference of about 1800 MPa. In contrast, with the reverse direction scanning (RDS) method, the residual stress in the thin wall was distributed relatively evenly, with a stress difference of about 350 MPa between both the ends and the center. The experimental results showed that, in the SDS method, cracks occurred at both ends and in the middle of the thin wall, whereas in the RDS method, warping and cracks phenomena were eliminated. The microstructure of the Inconel 625 in the forming layer is characterized by a columnar crystal structure that has a small length and grows perpendicularly to the scanning direction. This growth is continuous between the forming layers. In both cases, the microhardness increases with the height of the formed layers. The microhardness values are relatively uniform with values ranging from 420 to 450 HV.

A high-output performance disc-shaped liquid-solid triboelectric nanogenerator for harvesting omnidirectional ultra-low-frequency natural vibration energy

Low-frequency vibrations, such as ocean waves, are widely distributed in nature and serve as a promising renewable energy source. Triboelectric nanogenerators (TENGs) have been shown to be an effective method for harvesting low-frequency vibration energy. However, most TENGs can only effectively harvest vibration energy in a single direction and a specific vibration mode. To solve the problem, we developed a disc-shaped liquid-solid triboelectric nano-generator (DLS-TENG) that can harvest vibration energy omnidirectionally. Benefiting from the high flexibility of the fluid and the ultra-high surface utilization of the disc structure, the DLS-TENG has the highest volumetric charge density and load driving capability in harvesting omnidirectional vibrational energy. Driven by seesaw vibrations with a frequency of 0.5 Hz and an amplitude of 30°, one DLS-TENG successfully lit up 544 LEDs simultaneously, with a transferred charge of 1263 nC. Considering the random, irregular, and variable characteristics of most ambient vibrations, our omnidirectional TENG is more practical. The proposed DLS-TENG can achieve long-term and efficient natural vibration energy harvesting to power an increasing number of microsensors widely distributed in the field.

Mechanical model and experimental investigation of a novel pneumatic foot

Almost all forms of soft crawling robots and animals rely on frictional anisotropy to achieve terrestrial locomotion. However, the electroadhesion force of actuators is limited to the materials, roughness of various terrain, and patterning high- and low- friction materials on the ventral side of the robot can only enable the robot to crawl along a single direction. This paper presents a novel pneumatic foot that can regulate the friction between itself and the crawling surface to achieve frictional anisotropy in two opposite directions. The design scheme, working mechanism and analytical model based on the energy method of the pneumatic foot are described. The accuracy of the analytical model is verified by numerical simulation and experimental study. A soft crawling robot, including its locomotion mode and control system, is designed. Crawling experiments on various terrain and payload experiments are conducted to prove the feasibility of the pneumatic foot. The pneumatic foot can enable the robot to crawl on various terrain at a maximum speed of 0.128 BL/s. When the payloads are 4.29 times of the robot’s weight, the crawling speed is still relatively high (0.108 BL/s). This pneumatic foot can be applied to biomimetic inchworm or earthworm robots on various terrain and in some limited space, such as the air ducts of central air-conditioning.

The Linkage Between Fiscal Policy and Financial Development: Exploring the Moderating Role of Institutional Quality in Emerging Economies

This paper investigates the role of fiscal policy on financial development in Sub-Saharan African economies, drawing on a sample of 23 countries from 2000 to 2021 using the panel ARDL method after evidencing stationarity and co-integration properties among the variables. Our results show that an increase in fiscal policy and institutional quality decreases financial development in the long run. An increase in taxation and expenditure by the government affects the development of finance in SSA countries. Our results also show that an increase in foreign capital and industrial growth increases financial development in the long term. The outcome evidence that the interaction between fiscal policy and institutional quality exhibits a positive effect on financial development. Causality results reveal no directional link between fiscal policy, foreign capital, industrialization, and financial development with institutional quality indicating a single direction. The study suggested that SSA countries should focus on developing policies to track the implementation of adequate fiscal policy systems and structures. Institutional coherence within and between SSA nations is required for efficient fiscal policy development.

Mixed method examination of alcohol and suicidality among actively suicidal adults who engage in heavy episodic drinking

Suicide is a serious public health problem in the United States. Alcohol use has been substantially documented as a risk factors for suicide, yet it is unclear how alcohol is associated with suicidal ideation (SI) and behavior (SIB) at the event level. We examined the association between alcohol use and SI using a mixed methods approach that included daily assessments from 13 adults who engage in heavy episodic drinking with current SI and qualitative interviews among 12 of those adults. Participants were recruited on social media. Separate mixed effects logistic regression models indicated that individuals’ alcohol use on a given day was associated with SI (OR = 1.37), and suicidal urges (OR = 1.41). Adjusting for repeated measures, the expected marginal mean for intensity of SI (EMM = 3.33) and urges (EMM = 2.94) were higher on days with reported drinking behavior than days without reported drinking (EMM = 2.68 and EMM = 2.62 respectively). Qualitative data indicated that the association between alcohol use and SIB is more complex than a single directionality. Instead, the association can be unidirectional, bidirectional, and/or dependent on factors including mental health and amount of alcohol consumed. Overall, these findings emphasize a need for integrated alcohol and SIB interventions while providing insight on possible daily, just-in-time adaptations.

Saturation of flames to multiple inputs at one frequency

Existing experimental results show that swirling flames in annular combustors respond with a different gain to acoustic azimuthal modes rotating in either the clockwise or anti-clockwise direction. The ratio $R$ of these two gains is introduced, with $R=1$ being the conventional case of flames responding the same to the two forcing directions. To allow a difference in response to the different directions ( $R\neq 1$ ), a multiple-input single-output azimuthal flame describing function is successfully implemented in a quaternion valued low-order model of an annular combustion chamber in the current work. Theoretical studies have explored this kind of symmetry breaking between the two acoustic wave directions in the past, but it has not been backed by experimental data. One of the main features of the new model proposed in this work is the potential difference in mode shapes between the acoustic and the heat release rate modes, which has recently been observed experimentally. This results in a gain-dependent equation for the nature of the mode, which has a significant influence on the fixed points of the system. For example, one of the spinning solutions and the standing solution can disappear through a saddle node bifurcation as the parameters are varied. The presence of only a single direction for the spinning solution matches experimental observations better than the conventional models, and the proposed model is shown to qualitatively describe experimental measurements well.