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# IEEE Transactions on Instrumentation and Measurement

## Filter Results

Displaying Results 1 - 25 of 32

Publication Year: 2014, Page(s):C1 - 249
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• ### IEEE Transactions on Instrumentation and Measurement publication information

Publication Year: 2014, Page(s): C2
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• ### Weighing Fusion Method for Truck Scales Based on Prior Knowledge and Neural Network Ensembles

Publication Year: 2014, Page(s):250 - 259
Cited by:  Papers (2)
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This paper presents a new approach of compensating truck scale's weighing errors based on prior knowledge and neural network ensembles (PKNNEs). Truck scale is a typical nonlinear system and it is fussy and labor intensive to compensate the weighing errors with the conventional method, leading to low accuracy of the weighing results. The general idea of this proposed approach consists of building ... View full abstract»

• ### Outdoor Insulators Testing Using Artificial Neural Network-Based Near-Field Microwave Technique

Publication Year: 2014, Page(s):260 - 266
Cited by:  Papers (1)
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This paper presents a novel artificial neural network (ANN)-based near-field microwave nondestructive testing technique for defect detection and classification in nonceramic insulators (NCI). In this paper, distribution class 33-kV NCI samples with no defects, air voids in silicone rubber and fiber glass core, cracks in the fiberglass core, and small metallic inclusion between the fiber core and s... View full abstract»

• ### A Practical Fundamental Frequency Extraction Algorithm for Motion Parameters Estimation of Moving Targets

Publication Year: 2014, Page(s):267 - 276
Cited by:  Papers (7)
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In this paper, a practical method is proposed for a moving target's fundamental frequency (MTFF) extraction from its acoustic signal. This method is developed for the application of motion parameters estimation. Starting from the analysis of the target frequency model and the acoustic Doppler model, the characteristics of moving target's signal are discussed. Based on the signatures of target's ac... View full abstract»

• ### Adaptive Calibration of Channel Mismatches in Time-Interleaved ADCs Based on Equivalent Signal Recombination

Publication Year: 2014, Page(s):277 - 286
Cited by:  Papers (12)
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In this paper, we present an adaptive calibration method for correcting channel mismatches in time-interleaved analog-to-digital converters (TIADCs). An equivalent signal recombination structure is proposed to eliminate aliasing components when the input signal bandwidth is greater than the Nyquist bandwidth of the sub-ADCs in a TIADC. A band-limited pseudorandom noise sequence is employed as the ... View full abstract»

• ### Design and Implementation of a Preprocessing Circuit for Bandpass Signals Acquisition

Publication Year: 2014, Page(s):287 - 294
Cited by:  Papers (9)
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The processing capabilities that are included into the acquisition block of the real-time digital oscilloscopes largely contribute to determine the overall performance of the instrument. Their remarkable improvement has made it possible to enhance the performance in terms of increased measurement rate, automation, and reduced measurement uncertainty related to quantization and noise. This paper pr... View full abstract»

• ### A 7.65-mW 5-bit 90-nm 1-Gs/s Folded Interpolated ADC Without Calibration

Publication Year: 2014, Page(s):295 - 303
Cited by:  Papers (7)
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Power consumption of high-speed low-resolution analog-to-digital converters (ADCs) can be reduced by means of calibration. However, this solution has some drawbacks such as time slot allocation for calibration and die area increase. This paper presents a 5-bit 1-Gs/s ADC without calibration, fabricated in 90-nm CMOS. Low power consumption has been ensured by operating at both architecture and comp... View full abstract»

• ### One-Class Classification of Mammograms Using Trace Transform Functionals

Publication Year: 2014, Page(s):304 - 311
Cited by:  Papers (10)
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Mammography is one of the first diagnostic tests to prescreen breast cancer. Early detection of breast cancer has been known to improve recovery rates to a great extent. In most medical centers, experienced radiologists are given the responsibility of analyzing mammograms. But, there is always a possibility of human error. Errors can frequently occur as a result of fatigue of the observer, resulti... View full abstract»

• ### A Novel Iterative Structure for Online Calibration of $M$-channel Time-Interleaved ADCs

Publication Year: 2014, Page(s):312 - 325
Cited by:  Papers (18)
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This paper proposes a computationally efficient calibration structure for online estimation and compensation of offset, gain and frequency response mismatches in M-channel time-interleaved (TI) analog-to-digital converters (ADCs). The basic idea of the proposed approach is to reserve some sampling instants for estimating and tracking the mismatch parameters of sub-ADCs with reference to a known in... View full abstract»

• ### Application of Cross Wavelet Transform for ECG Pattern Analysis and Classification

Publication Year: 2014, Page(s):326 - 333
Cited by:  Papers (37)
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In this paper, we use cross wavelet transform (XWT) for the analysis and classification of electrocardiogram (ECG) signals. The cross-correlation between two time-domain signals gives a measure of similarity between two waveforms. The application of the continuous wavelet transform to two time series and the cross examination of the two decompositions reveal localized similarities in time and freq... View full abstract»

• ### Temperature Distribution Reconstruction by Eigenfunction Interpolation of Boundary Measurement Data

Publication Year: 2014, Page(s):334 - 342
Cited by:  Papers (3)
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This paper deals with the inverse problem of evaluating the temperature distribution over time in a 3-D composite solid material having an arbitrary geometry. This approach is capable of evaluating the temperature distribution within the domain of the nonhomogeneous object under observation at each time instance. In this paper, we propose to use the eigenfunctions of the heat equation model, repre... View full abstract»

• ### Broadband 60 GHz Sounder for Propagation Channel Measurements Over Short/Medium Distances

Publication Year: 2014, Page(s):343 - 351
Cited by:  Papers (3)
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This paper presents a millimeter-wave propagation-channel sounder that uses an optical-fiber cable to achieve phase coherence between transmitting and receiving antennas using two modulated light waves in the vicinity of 1550 nm. In the described sounder, the optical-fiber cable approach permits a separation of up to almost 1000 m between the transmitter and receiver because of the low level of at... View full abstract»

• ### Power System Dynamic State Estimation With Synchronized Phasor Measurements

Publication Year: 2014, Page(s):352 - 363
Cited by:  Papers (24)
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The dynamic state estimation (DSE) applied to power systems with synchrophasor measurements would estimate the system's true state based on measurements and predictions. In this application, as phasor measurement units (PMUs) are not deployed at all power system buses, state predictions would enhance the redundancy of DSE input data. The significance of predicted and measured data in DSE is affect... View full abstract»

• ### Electrical Signal Source Separation Via Nonnegative Tensor Factorization Using On Site Measurements in a Smart Home

Publication Year: 2014, Page(s):364 - 373
Cited by:  Papers (9)
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Measuring the electrical consumption of individual appliances in a household has recently received renewed interest in the area of energy efficiency research and sustainable development. The unambiguous acquisition of information by a single monitoring point of the whole house's electrical signal is known as energy disaggregation or nonintrusive load monitoring. A novel way to look into the issue ... View full abstract»

• ### Survey on Voltage Dip Measurements in Standard Framework

Publication Year: 2014, Page(s):374 - 387
Cited by:  Papers (10)
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This paper reports the results of the analyses on the accuracy of algorithms commonly adopted in instruments devoted to the detection and characterization of voltage dips (also called sags). This analysis is particularly interesting because the results of dip measurements are used for calculation of severity levels and site indexes that are important parameters in the assessment of quality level o... View full abstract»

• ### Fast Synchrophasor Estimation by Means of Frequency-Domain and Time-Domain Algorithms

Publication Year: 2014, Page(s):388 - 401
Cited by:  Papers (22)
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This paper presents a performance comparison of two classes of synchrophasor estimators recently proposed in the literature, i.e., the frequency-domain algorithms known as interpolated discrete Fourier transform (IpDFT), and the time-domain algorithms based on the weighted least squares (WLSs) approach. The analysis is focused on fast phasor estimation only and it is performed under steady-state, ... View full abstract»

• ### Power System Frequency Estimation by Reduction of Noise Using Three Digital Filters

Publication Year: 2014, Page(s):402 - 409
Cited by:  Papers (12)
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Accurate estimation of power system frequency is essential for monitoring and operation of the smart grid. Traditionally, this has been done using discrete Fourier transform (DFT) coefficients of the positive fundamental frequency. Such DFT-based frequency estimation has been used successfully in phasor measurement units and frequency disturbance recorders in North America. Frequency errors in DFT... View full abstract»

• ### Measuring System for Microelectric Power

Publication Year: 2014, Page(s):410 - 421
Cited by:  Papers (7)
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In this paper, we are mostly concerned with the measurement of electric micropower and energy. This parameter is essential to evaluate the energy efficiency of low power devices, such as wireless operated monitoring and control systems. It is also important to measure the standby power consumed by appliances and equipment while switched off. The measurement of power generated by the harvesting sys... View full abstract»

• ### Load Characterization and Revenue Metering Under Non-Sinusoidal and Asymmetrical Operation

Publication Year: 2014, Page(s):422 - 431
Cited by:  Papers (4)
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This paper proposes an approach to load characterization and revenue metering, which accounts for the influence of supply deterioration and line impedance. It makes use of the Conservative Power Theory and aims at characterizing the load from the measurements done at the point of common coupling. Despite the inherent limitations of a single-point measurement, the proposed methodology enables evalu... View full abstract»

• ### Four-Terminal Imaging Using a Two-Terminal Electrical Impedance Tomography System

Publication Year: 2014, Page(s):432 - 440
Cited by:  Papers (5)
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Electrical impedance tomography (EIT) is a promising visualization measurement technique to reconstruct media distribution in a region of interest (ROI) through impedance measurements on its boundary. In this paper, a two-to-four-terminal mode is proposed for a two-terminal EIT system to take advantage of the four-terminal imaging mechanism. Using the two-to-four-terminal mode, data are acquired u... View full abstract»

• ### Design and Development of a Low-Cost Multisensor Inertial Data Acquisition System for Sailing

Publication Year: 2014, Page(s):441 - 449
Cited by:  Papers (5)
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This paper presents the development of an inertial measurement data acquisition system intended for use in sailboats. The variables of interest are three-axis acceleration, three-axis rotation, Global Positioning System (GPS) position/velocity, magnetic compass bearing, and wind speed/direction. The design focus is on low-cost microelectromechanical systems-based technology and demonstrating the v... View full abstract»

• ### Maximum Sample Volume for Permittivity Measurements by Cavity Perturbation Technique

Publication Year: 2014, Page(s):450 - 455
Cited by:  Papers (8)
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The maximum sample volume for accurate permittivity measurements of dielectric materials having various geometries (rod/bar, strip/disk, and sphere) by cavity perturbation technique has been investigated by determining the maximum volume ratio of sample to cavity (Vs/Vc)max based on analysis of the measurement theory. It is demonstrated that (Vs/Vc)max o... View full abstract»

• ### Two-Phase Air–Water Slug Flow Measurement in Horizontal Pipe Using Conductance Probes and Neural Network

Publication Year: 2014, Page(s):456 - 466
Cited by:  Papers (5)
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This paper presents a method to obtain gas and liquid flow rates of two-phase air-water slug flow in a horizontal pipe through conductance probes and neural network. Contrary to statistical features commonly used in other works, five characteristic parameters of the mechanistic slug flow model are extracted from conductance signals, i.e., translational velocity, slug holdup, film holdup, slug leng... View full abstract»

• ### Evaluation of a Calorimetric Thermal Voltage Converter for RF–DC Difference up to 1 GHz

Publication Year: 2014, Page(s):467 - 472
Cited by:  Papers (3)
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In this paper, we present a performance evaluation of the RF-DC transfer difference for a calorimetric thermal voltage converter (CTVC) designed by the National Research Council of Canada (NRC) at frequencies up to 1 GHz. In the first part of this paper, we describe a bilateral comparison of the RF-DC difference standards between the NRC and the National Metrology Centre of the Agency for Science,... View full abstract»

## Aims & Scope

Papers are sought that address innovative solutions to the development and use of electrical and electronic instruments and equipment to measure, monitor and/or record physical phenomena for the purpose of advancing measurement science, methods, functionality and applications.

Full Aims & Scope

Editor-in-Chief


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# What is exactly rotating in a rotating black hole?

https://arxiv.org/abs/gr-qc/9404041

In General Relativity the black hole solutions which have so far been found form a four parameter family called the generalized Kerr-Newman family of black holes. The four parameters are mass M, angular momentum J, charge Q, and the cosmological constant Λ [6].

A black hole is a region of spacetime exhibiting gravitational acceleration so strong, that nothing, not even light can escape from it.

https://en.wikipedia.org/wiki/Black_hole

There is no consensus whether the mass of the black hole lies in the singularity or the event horizon. For an outside observer, the mass of the black hole may lie on the event horizon.

How can a singularity in a black hole rotate if it's just a point?

The singultarity of a rotating Kerr black hole is not a point, but a ring in the plane of rotation, though still zero volume.

Some physicists even believe that singularities are not real.

Though, none of the answers on this site explain what is exactly rotating in the black hole. Is it the singularity, like a point particle (with the mass of a black hole), then it is like the spin of an elementary particle.

Or is it the event horizon, that is rotating, since to an outside observer, all the infalling mass is frozen on the horizon.

Some answers say that the gravitational field (potential) of the black hole (its stress energy) is determined by the (though the stress energy is determined by mass, angular momentum, charge, cosmological constant) event horizon (its size).

Some answers say that a singularity of a black hole is not made of any matter, since it is not a place in space, rather a place in time.

Like Luboš Motl's comment to this question:

What happens when a black hole and an "anti-black-hole" collide?

So there is a contradiction, because none of the answers specifically state what is actually rotating in a rotating Kerr black hole.

Question:

1. What is actually rotating in a rotating Kerr black hole, is it the singularity or the event horizon?
• Who says that there needs to be a physical "thing" that "rotates" in order for a system to have angular momentum? It isn't the case in electromagnetism - why would it need to happen in general relativity? – Emilio Pisanty Sep 21 at 11:08
• @EmilioPisanty Say, I take a loop wire with a current producing a magnetic field. Do electrons rotate there in a circle? Now I take the same electrons and make them rotate around a positive charge on the same radius producing the same magnetic field. Do they still rotate? But now I can use the Schrodinger equation to solve the electrons rotation around the charge and then... they no longer rotate? Rotation is a classical concept. Its quantum equivalent is spin. So spin is the quantum representation of rotation. Besides, this question is purely classical, so your comment is moot either way :) – safesphere Sep 27 at 7:41

So let us first ask ourselves: What makes us believe that anything is rotating in the Kerr space-time?

The answer is that we go very far away from the black hole and look at the asymptotics of the gravitational field as it becomes weaker and weaker. From there we can read off in a well defined way a set of mass and current multipoles "contained in the space-time". The lowest-order multipoles can be defined almost unambiguously and they would be interpreted as "total linear momentum in the space-time" and "total angular momentum in the space-time". Then what you read of for the Kerr metric is an angular momentum of the magnitude $$J =Ma$$.

What makes us assign such meaning to these quantities is matching in the limit when there are only small amounts of dilute matter (which can be treated by linearized gravity). In that limit these asymptotic quantities exactly match the meaning of total linear and angular momentum of the dilute matter as defined in your classical-mechanics course. So a similar asymptotic field as for the Kerr metric would be generated by a cloud of dilute matter with a total mass $$M$$ and rotating with a total angular momentum $$Ma$$. So that is what makes you believe that there is something rotating in Kerr space-time.

But as larger amounts of dense matter are added, you cannot just sum all the momenta of the matter particles in a space-time to get the total linear and/or angular momentum. The problem is that the space-time is curved and you stop being able to easily do things such as sum two vectors at different points of the space-time. But another issue is that no matter how hard you try to sum the contributions of momentum in the matter, you keep missing something, it just does not add up to the asymptotic momenta. It turns out that somehow the gravitational field itself carries momentum. But pinning down how and where this energy and momentum of the gravitational field is notoriously difficult. There simply is not a universally valid formula for that, the energy and angular momentum of the gravitational field is stored non-locally.

So what is, really, rotating in Kerr space-time? The curvature singularity in the center of the space-time is a ring, and it would be very easy to say it is a rotating ring carrying the entire angular momentum $$Ma$$. But consider a very compact object very close to a black hole such as a neutron star. There you cannot assign the angular momentum only to the matter, an increasingly larger portion is stored nonlocally in the gravitational field, the space-time is rotating as well. On the other hand, the gravitational field (the space-time) would never rotate on its own, it will only rotate if the neutron star is rotating as well. This is true for any situation with matter - the field and its source rotate in tandem, each having a significant contribution to the angular momentum.

So what about black holes? There is really no rigorous argument to extend the observations of the last paragraph to the black hole, though. But consider this: in a rather precise sense, the nonsingular part of the black hole field "causes" the existence of the curvature singularity inside. You can slice space-time in a particular way, put one of the slices not including the black hole singularity in the computer and let it evolve the slice into the future according to GR. To your surprise, it will spontaneously evolve to the singularity (in the case of the Kerr space-time it would actually evolve a crumpled singularity called the Cauchy horizon). So in this sense it is the field that causes the singularity to exist rather than the other way around. So I am inclined to say that black holes can be seen as the limit where the fraction of mass and angular momentum in the gravitational field has actually converged to the entire mass and angular content of the space-time.

• If I understand it correctly, your conclusion is that what rotates is the spacetime around the horizon. This is also evident by the frame dragging effect +1. – safesphere Sep 27 at 7:26

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# Machine learning assisted design of γ′-strengthened Co-base superalloys with multi-performance optimization

## Abstract

Designing a material with multiple desired properties is a great challenge, especially in a complex material system. Here, we propose a material design strategy to simultaneously optimize multiple targeted properties of multi-component Co-base superalloys via machine learning. The microstructural stability, γ′ solvus temperature, γ′ volume fraction, density, processing window, freezing range, and oxidation resistance were simultaneously optimized. A series of novel Co-base superalloys were successfully selected and experimentally synthesized from >210,000 candidates. The best performer, Co-36Ni-12Al-2Ti-4Ta-1W-2Cr, possesses the highest γ′ solvus temperature of 1266.5 °C without the precipitation of any deleterious phases, a γ′ volume fraction of 74.5% after aging for 1000 h at 1000 °C, a density of 8.68 g cm−3 and good high-temperature oxidation resistance at 1000 °C due to the formation of a protective alumina layer. Our approach paves a new way to rapidly design multi-component materials with desired multi-performance functionality.

## Introduction

As the key materials of the twenty-first century, aerospace alloys are of great economic value and developmental potential. Among them, superalloys are considered the most state-of-the-art, and are used to manufacture aeroengine and industrial gas turbine hot-end parts1. To safely operate in the harsh service environment, various properties, such as the temperature capability, density, high-temperature strength, as well as satisfactory resistance to hot-corrosion and oxidation under severe service conditions are highly desired2,3. Great efforts have been put forth to balance different alloying elements such as Al, Ti, Fe, Cr, Nb, Mo, W, and Ta in order to promote precipitation of stable γ′-Ni3(Al, Ti), or a combination of γ′ and γ″-Ni3Nb, meanwhile optimizing different aspects of the material performance4. However, these alloys are reaching their temperature capability limits, and novel design ideas are required to further improve engine performance and efficiency.

In 2006, Sato et al.5 reported coherent, ordered L12 precipitate (γ′) strengthening in the Co-Al-W system, which has sparked interest in searching for novel γ′-strengthened Co-base superalloys with good performance to satisfy the aircraft and turbine application requirements and replace Ni-base superalloys. In the pursuit of higher temperature capability, many researchers have attempted to design γ′-strengthened Co-base superalloys with a higher γ′ solvus temperature. For example, Xue et al.6 reported a quinary γ′-strengthened Co-base superalloy, Co-7Al-8W-4Ti-1Ta, which has a γ′ solvus temperature of 1131 °C and maintains stable γ′ precipitates after aging at 1050 °C for 1000 h. Furthermore, Lass et al.7 reported the Co-30Ni-9Al-7W-4Ti-2Ta-0.1B alloy, which possesses an even higher γ′ solvus temperature, 1218 °C, and is devoid of any detrimental phase precipitation in the 900 °C to 1200 °C temperature range. However, lower density and improved oxidation resistance still remain as a major hurdle for these alloys, especially for long-term and higher temperature applications. As these various properties often compete, improving one is often at the expense of the other, and further improvement and optimization is needed8.

One of the largest obstacles to the design of superalloys is the complexity from its multiple components, each of which is added with aim of improving a particular property. The design task is even more difficult when the goal is to optimize multiple properties simultaneously2. The traditional experimental alloy design approach to search for novel advanced superalloys through “trial and error” or intuition is thus inefficient1. Although novel design ideas such as diffusion multiples9 and high-throughput thermodynamic calculations7 have been introduced, they are still not efficient enough to solve such a complicated problem. What is needed is a high-efficiency method that enables discovery of superior multi-component superalloys with satisfactory multi-performance in an efficient and accelerated manner.

Statistical inference and machine-learning (ML) algorithms have been recently applied to discover advanced materials10,11,12,13,14,15, and can adaptively guide the synthesis and characterization to select and optimize materials with desired properties. In the present study, utilizing data from both thermodynamic calculations and experiments, we combined ML with a global optimization algorithm to search for an optimized composition of a γ′-strengthened Co-base superalloy with desired multi-property performance. Considering the efficient balance of the ambiguous interaction among various elements and mutual inhibition between different properties, a design strategy of sequential filters composed of domain knowledge-driven empirical determination and ML models for different properties is employed to screen candidates, and an efficient global optimization (EGO) algorithm is then utilized to recommend potential target materials. The best alloy obtained from only three rounds of new experiments, Co-36Ni-12Al-2Ti-4Ta-1W-2Cr, outperforms many reported Co-base alloys in terms of the overall performance of this study. Our approach is general and applicable to various materials design problems aimed at optimizing multiple properties.

## Results

### Material design

Among all of the properties of the γ′-strengthened Co-base superalloys, the γ′ solvus temperature is of the greatest importance, as it determines the upper temperature capability limit. Therefore, in order to develop advanced single crystal materials, a high γ′ solvus temperature (>1250 °C) is needed. Additionally, the processing window (ΔTp = Tsolidus – Tγ′-solvus ≥ 40 °C), freezing range (ΔTf = Tliquidus – Tsolidus ≤ 60 °C), density (ρ ≤ 8.7 g cm−3), and microstructural stability were also considered simultaneously as targeted properties to be optimized by ML. Owing to insufficient experimental-data in thermodynamic databases, the oxidation resistance (giving priority to obtain protective alumina layer) and γ′ volume fraction (Vγ′ ≥ 60%) are considered empirically based on results of previous studies16,17,18,19,20, i.e., by setting thresholds for associated elements.

The CoaAlbWcNidTieTafCrg family was chosen as the system to be optimized. The potential composition space was defined as follows (the concentration of elements are in atomic percent): 30% ≤ a ≤ 100%, 10% ≤ b ≤ 15%, 1% ≤ c ≤ 8%, 20% ≤ d ≤ 50%, 1% ≤ e ≤ 8%, 1% ≤ f ≤ 4%, 2% ≤ g ≤ 10%, and each alloy in this space is constrained by a > d and a + b + c + d + e + f + g = 100%. The beneficial effect of Al and Cr on the environmental resistance of superalloys is most potent among the selected elements16. In order to meet the requirements for adequate oxidation resistance, and based on the study of Stewart et al.17, the lower and upper limits of the Al content in the composition space were set as 10 at.% and 15 at.%, respectively. Despite the negative effect of Cr on the γ′ solvus temperature, Cr still plays an irreplaceable role in the oxidation resistance of the alloy at elevated temperatures by stabilizing the Al2O3 oxide18. Therefore, we set the Cr content to be no <2 at.% and no >10 at.%. To ensure an appropriate γ′ volume fraction19,20, while ensuring lower density, the lower limits of the other γ′-forming elements—W, Ta, and Ti—were all set as 1%. In this composition space, we calculated the number of candidates that satisfy these two constraints exhaustively with a computer program. A composition variation step of 1 at.% for each element gives a total of 210792 unknown compositions to be evaluated.

### Machine-learning strategy

The material design strategy for multi-performance optimization in Co-base superalloys is shown in Fig. 1, and includes three parts: sequential filters, EGO, experimental verification and feedback. The sequential filters comprise of domain knowledge-driven empirical determination (we define the rational composition space by the summarized experiences from predecessors to control the properties of compositions without available data), computational-data driven phase classification and experimental-data driven performance regression. Prior to any property optimization, the first consideration is the high-temperature and long-term microstructural stability of the alloys, as the precipitation of β, μ, and other related topologically close-packed (TCP) phases poses negative effects on the mechanical properties, and should be avoided as much as possible21. Thus, a ML classification model can be built and applied to obtain a potential composition space, which contains only γ and γ′ two-phase microstructures. However, as there is insufficient experimental-data, and the thermodynamic database could be a better choice. Li et al.22 used the thermodynamic database-Pandat™ to design multi-component Co-base superalloys without deleterious phases after aging at 1100 °C for 1000 h. Zhuang et al.23 also designed a novel γ′-strengthened Co-base wrought superalloy with desired microstructure by CALPHAD method. They mean good reliability of this thermodynamic database in controlling phase equilibrium. Therefore, we utilized thermodynamic computations to generate a training dataset for the ML classification model. Specifically, we used Pandat™ with the Co-base thermodynamic library PanCo2018 to produce 1000 phase fraction plots to be used as data. Within the desired temperature window between 900 °C and 1300 °C, there were 500 compositions corresponding to a γ/γ′ two-phase microstructures, which are labeled as “0”, and another 500 compositions exhibiting the precipitation of detrimental phases, which are labeled as “1”. In order to improve the adaptability and robustness of the phase classification model, the chemical compositions together with the composition-weighted mean value of three physical parameters (valence electron number, first energy ionization and effective charge nuclear (Clementi)) were used as the material descriptors. The four different classification algorithms were compared to obtain the optimal phase classification model. Receiver operating characteristic (ROC) curves revealing the classification capability are employed to select the optimal model. A large area under the ROC curve (AUC) ensures a better classifier. The mean value of 100 AUC for the gradient tree boosting (GTB) classification model is 0.96, as shown in Fig. 2a. The value is larger than the other models, indicating that a high accuracy and good classification performance (details of AUC for the four classification models are shown in the Supplementary Table 1). Therefore, the GTB classification model was chosen as the classifier for each alloy in the potential composition space to predict whether it possesses only a γ and γ′ two-phase microstructure.

Regression models were built to predict the γ′ solvus temperature, solidus, liquidus and density based on an experimental dataset assembled from the literature6,7,16,18,19,20,21,22,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54 and our own experiments. The number of data points in the original dataset for the γ′ solvus temperature, solidus, liquidus, and density was 134, 94, 68 and 31, respectively. We built six different regression models, and used the chemical compositions as the descriptors. The mean-squared error (MSE) was used to evaluate the quality of the regression models, and the model with the minimal MSE was selected. Among the six typical models, as shown in Fig. 2(b–e), the best model corresponding to the γ′ solvus temperature, solidus, liquidus and density were the GTB, random forest (RF), GTB and GTB, respectively (details of MSE values for each regression model are shown in the Supplementary Table 2). Then, the sequential filters were fully assembled by domain knowledge-driven empirical determination, computational-data driven phase classification and experimental-data driven performance regression for the optimization of multi-property performance. Owing to this rigorous screening by sequential filters, the composition space originally consisting of 210,792 unknown alloys was reduced to <6000 compositions.

In order to avoid the prediction result of the regression model trained by small data into local extremum, we adopt a global optimization algorithm EGO55 to search for desired alloys with high γ′ solvus temperature (the main targeted property). We utilized bootstrap sampling to attain the prediction value and the associated uncertainty. In detail, we generate a bootstrap training set by resampling the data from the original training data with replacement. We trained 1000 models based on 1000 bootstrapped datasets to obtain 1000 predictions of γ′ solvus temperature for each alloy in the unexplored composition space. The mean predicted value and standard deviation can be estimated from the 1000 predictions to obtain the expected improvement (EI) for each alloy (details are presented in “Methods”). In order to select a batch of alloys as the experimental group, the alloy with the largest EI value is selected to put into the training data with its predicted value, then the model is refined and another alloy with the largest EI value is chosen in the next iteration. We repeat this procedure until we obtain a recommendation of four candidate alloys as the experimental group. These alloys are synthesized and characterized, and the results are fed back to augment the training dataset. The loop is executed iteratively until the alloys with good targeted performance are discovered. As shown in Fig. 2f, we iterate our experimental group for a total of three times until the newly made alloy’s γ′ solvus temperature reaches >1250 °C, and the γ′ solvus temperature of the experimental alloys no longer increases significantly.

### Multi-performance optimization

The synthesized alloys from the iterative loop are given in Table 1 (more experimental results are collected in the Supplementary information). We conducted a total of three rounds of new experiments, fabricating four alloys each round (as for the abbreviation of each alloy, the first number gives the iteration round, and the second represents alloy with the highest EI value in the respective computational loop, e.g., Alloy 1–2 is the alloy with the highest EI value in the second computational loop of the first round). Based on the aim of optimizing multiple properties in this study, Alloy 1–2 is the preferred composition with the highest γ′ solvus temperature that simultaneously meets the demands of all targeted properties. The microstructure of Alloy 1–2 is shown in Fig. 3a. The γ′ volume fraction of the alloy is 74.5%, and it doesn’t exhibit any detrimental phase after aging at 1000 °C for 1000 h. The differential scanning calorimetry (DSC) heating curve of Alloy 1–2 is shown in Fig. 3b. Alloy 1–2 has a γ′ solvus temperature of 1266.5 °C, and the processing window and freezing range are 49.1 °C and 46.8 °C, respectively, which are suitable for practical manufacturing. The density of Alloy 1–2 is 8.68 g cm−3, which is less than the threshold set by the ML model and also less than the density of some advanced Ni-base single crystal superalloys (e.g., René N6 and CMSX-10)56. This is beneficial for reducing the weight of aero-engines and improving the overall aircraft efficiency. Owing to the lack of high-quality experimental-data with a unified standard, a prior knowledge approach was used to ensure good high-temperature oxidation resistance of the superalloy, by controlling the content of Al and Cr in the composition space based on the valuable experiences from previous studies16,17,18. The representative microstructure of the oxides and the oxidation mass gain curves of three selected alloys (Alloy 1–2, Alloy 3-3, and Alloy 3–4) after oxidation at 1000 °C are shown in Fig. 3(c–d). The oxidation mass gain of Alloy 1–2 is around 1.99 mg cm−2 which is the lowest in the three selected alloys. A protective layer of alumina formed after isothermal oxidation, representatively shown in Fig. 3d for Alloy 1–2.

Only three alloys did not precipitate detrimental phases after aging at 1000 °C for 1000 h, whereas for the other recommended alloys, all of the properties except the microstructural stability are acceptable. Based on the experimental results, the higher the γ′ solvus temperature, the higher the needed alloying content to achieve it, and hence the greater the propensity for precipitation of detrimental phases. Therefore, the predicted alloy compositions are likely located at the phase classification boundary. As it is not trivial to make an accurate phase classification model in a multi-component system using only thermodynamic calculation data where phase boundaries are often not known to a high degree of accuracy, most synthesized alloys precipitated deleterious phases. This can be alleviated with the use of more accurate thermodynamic data, or through improved fabrication techniques utilizing vacuum induction melting and electro-slag remelting for tight compositional control and reduction of tramp elements.

We used just three rounds of experiments to find new desired alloys that satisfy our design targets under the ML guidance, and the γ′ solvus temperatures of these alloys are higher when compared with previously reported alloys7,22, without exhibiting detrimental phases after high-temperature long-term aging, as shown in Fig. 3e. Compared with oxidation-resistant alloy SB-CoNi-10 that has a low density (8.65 g cm−3) and a high γ′ solvus temperature (1196 °C)57, the γ′ solvus temperature of Alloy 1–2 is about 70 °C higher than that of alloy SB-CoNi-10. The different properties optimized in this study (using Alloy 1–2 as an example) are comparable to advanced Ni-base single crystal superalloys56,58,59,60, such as CMSX-10, as shown in Fig. 3f. Such rapid optimization and ability to achieve well-balanced properties is a breakthrough for this class of superalloys, although a small amount of the γ′ solvus temperature, freezing range and oxidation resistance were sacrificed, compared to CMSX-10, in order to improve the other targeted properties. Therefore, the best performer, Co-36Ni-12Al-2Ti-4Ta-1W-2Cr alloy, warrants further research whether it meets other unstudied properties requirements to use as single crystal material, and to serve as a base for further optimization and development.

## Discussion

In order to further understand the elemental distribution to the γ and γ′ phases of the experimental alloys, elemental mapping was performed with high-angle annular dark field (HAADF) scanning transmission electron microscope (STEM) nanoprobe and accurate phase compositions were measured using atom probe tomography (APT) for Alloy 1–2. From this data, the partitioning coefficients $$K_{\gamma \prime /\gamma }^X = C_{\gamma \prime }^X/C_\gamma ^X$$ of each element, where $$C_{\gamma \prime }^X$$ and $$C_\gamma ^X$$ are equilibrium concentrations of element X in the γ′ and γ phases, were calculated. As shown in Fig. 4a, distinct cuboidal γ′ precipitates are distributed in the γ phase matrix, and elements such as Ta, Ti, W, Al, and Ni preferentially partition to the γ′ phase, while elements such as Co and Cr partition strongly to the γ phase. Based on the quantitative APT results (Fig. 4(b–d)), the partitioning coefficient of Ta is the largest ($$K_{\gamma \prime /\gamma }^{Ta} = 5.90$$). Hence, it has the greatest positive effect on the stabilization of the γ′ phase as a γ′-forming element, and is also advantageous to increase the Tγ′-solvus. This is further supported by the Ta content of the alloy, which has reached the upper limit of our potential composition space during the optimization by the ML model. As a γ phase forming element, Cr mainly partitions to the γ phase, resulting in the smallest partitioning coefficient ($$K_{\gamma \prime /\gamma }^{Cr} = 0.40$$) among the elements, hence it is not favorable for improving the γ′ phase stability and Tγ′-solvus of the alloy. Consequently, in agreement with the trends obtained by ML, the content of Cr in the alloy has reached the lower limit in the composition space. Compared to the partitioning coefficients reported by other studies in the existing literatures61,62,63, the partitioning trend and extent of each element is consistent even though these elements contained in different Co-base alloys. Under the same aging conditions, the partitioning of each element is mainly affected by the chemical compositions and ambiguous interaction among various elements. These results of partitioning coefficients demonstrate the black-box ML models could detect the inner partitioning mechanism of each element, and provide the active guidance in the optimization of multiple properties and compositions. Comparing the Co and Ni contents in the two phases, Co is the major element of the γ matrix (59.4 at.%), followed by Ni (25.6 at.%), while their contents are nearly equal in the γ′ phase. A higher Ni content in the alloy expands the γ-γ′ two-phase field and increases the stability of the alloy22,64, which is clearly captured by the ML approach. This also suggests that the future research direction is likely the design of Ni-rich Co-base superalloys (also known as CoNi-base superalloy) with desired multi-properties.

To summarize, a machine-learning framework was used to efficiently obtain optimized γ′-strengthened Co-base superalloys through three rounds of new experiments based on a multi-performance criterion. These alloys are comparable with some advanced Ni-base single crystal superalloys, and warrant further research for additional properties that were not involved in this study (e.g., creep and fatigue) to determine whether they meet the requirements for industrial application. Benefiting from the use of sequential filters and EGO algorithm, multiple optimized properties are balanced efficiently, and the time and cost of material discovery are greatly reduced. Specifically, the introduction of sequential filters driven by domain knowledge and machine-learning models greatly narrows the vast composition space to be optimized, resulting in the best performer, Co-36Ni-12Al-2Ti-4Ta-1W-2Cr, obtained by only three rounds of new experiments assisted with EGO. Our research provides a fast and efficient machine-learning method to optimize multiple properties of superalloys, which also showcases a typical case study for utilizing machine learning in extremely complex material design problems. This material design strategy driven by machine learning is general and is not limited to the multi-performance optimization of superalloys but also applicable to other multi-component materials such as high-entropy alloys, superconductors, ceramics and perovskite-type materials, etc. It provides a new material design method for the research and development of advanced materials.

## Methods

### Machine-learning models

The logistic regression, decision tree, AdaBoost and GTB classification algorithms were used to train these classification models, and the hyperparameters of each model were tuned by ten-fold cross-validation. The ROC curve is very effective in evaluating the performance of the binary classification model. The closer the curve is to the upper left corner, the higher the reliability of the classification model. The AUC value is the area under the ROC curve, and can be used to quantify the merit of the ROC curve. The closer the AUC value is to 1, the better the performance of the model. We utilized the method of hold-out to obtain a reliable AUC value, 90% of the original data were randomly chosen as training set to train the model and 10% of the data were used as a testing set to verify the predictive capability of the model. Thus, 100 models of four different algorithms were obtained. The mean value of 100 AUC values was taken as the metric to reflect the classification capability of the model. The method of hold-out was also used to train the performance regression models of solidus, liquidus and density. The 1000 models of the AdaBoost, decision tree, support vector machine with a kernel of radial basis function, k-nearest neighbors, RF and GTB regression algorithms were used to build the regression models based on the original dataset. The MSE = $$\mathop {\sum}\nolimits_{i = 1}^n {\left( {y_i - \hat y_i} \right)}$$, (where (yi) is the true value and ($$\hat y_i$$) is the predicted value) was used to evaluate the predictive capability of different regression models, and the models with the minimal MSE (the mean MSE value of 1000 models) was selected for use.

### Efficient global optimization

In the present study, we use a powerful optimization strategy called EGO, which can balance the exploitation and exploration by an adaptive iteration loop, shown in Fig. 1. Expected improvement (EI) is employed to select the next experimental candidate, such as an alloy with higher γ′ solvus temperature. Subsequent alloy synthesis and experiments allow an iterative improvement of the surrogate model by incorporating the measured results into the training dataset. The calculation of EI is given by EI = σ[φ(z)+(z)], where z = Tγ′-solvus – μ*)/σ and μ* is the maximum γ′ solvus temperature in the original dataset, φ(z) and Φ(z) are the standard normal density and cumulative distribution functions, respectively. It can be seen that the EI is not only related to the exploitation (model-predicted value u), but also to the exploration (uncertainty of prediction σ). Maximizing EI provides an optimization approach to balance these two extremes and help to navigate to the target alloy efficiently. This machine-learning procedure was implemented in Python using the publicly available library of Numpy/Scipy and scikit-learn (version 0.19.1).

### Experimental process

Raw metals with purity higher than 99.95% were used, and the oxides and impurities on the surface of the raw metals were removed before processing the alloy. In order to ensure the homogeneity of the alloy composition and facilitate comparison, the alloy button ingots were prepared by vacuum arc melting, where each 30 g alloy was melted at least six times. The Archimedes displacement principle was applied to measure the density of each alloy five times and the mean value was taken. After ultrasonic cleaning, the as-cast ingot was sealed in quartz tube filled with high purity argon, and then subjected to solution heat treatment at 1200–1280 °C for 24 h followed by air cooling. All samples were cut and subsequently aged at 1000 °C for 50 h followed by water cooling. The γ′ solvus, solidus and liquidus temperatures were determined by DSC (NETZSCH STA 449C) with high purity Ar flow. The samples for DSC of size φ 3 mm × 1 mm were tested in a temperature range of 800–1400 °C at a heating rate of 5 °C min−1. The line intercept method was used to measure the transformation temperatures based on the DSC heating curves. All samples were metallographically prepared using standard techniques, and etched for a few seconds using a solution of HNO3: HCl: H2O = 1: 1: 1. The microstructure was characterized with a Zeiss GeminiSEM 300 field-emission scanning electron microscope (FE-SEM) in secondary electron imaging mode, and equipped with energy-dispersive X-ray spectroscopy (EDS) detector, used to measure the alloy composition. The crucible used for isothermal oxidation was pre-fired at 1050 °C until the weight change of the crucible was <2 × 10−4 g, and then considered to be constant weight. Isothermal oxidation experiments were carried out at 1000 °C for 100 h. The weight of the sample and crucible was measured by an electronic analytical balance with sensitivity down to 10−5 g before and after oxidation. The oxidized samples Ni-plated to preserve the oxide layers. The oxides microstructure was characterized with the FE-SEM in backscattered electron imaging mode, and EDS was used to detect the element distribution of the oxides. Thin foil specimens for the TEM observation were prepared by ion-thinning. STEM observations and energy-dispersive X-ray spectrometry (EDS) analysis were performed in a Cs-corrected FEI Titan G2 60–300 ChemiSTEM, equipped with Super-X EDS detectors and operated at 300 kV. Specimens for APT analysis were then extracted and fabricated from select sites using a dual beam SEM/focused-ion-beam instrument via an in-situ lift-out protocol. A final cleaning procedure was carried out at 5 kV and 15 pA beam current to remove regions severely damaged by the high-energy (30 kV) Ga ion beam. The APT measurements were conducted using a LEAP™ 5000XR instrument (Cameca Instruments). Laser pulsing mode operation was applied at a pulse repetition rate of 125 kHz and a pulse energy of 40 pJ. The specimen’s base temperature was kept at 30 K, and the target detection rate was set to five ions detected every 1000 pulses. Data analysis was performed using the software package IVAS™ 3.8.2. To further study the microstructural stability of the alloy, the experimental samples were subjected to long-term aging of 1000 h at 1000 °C, respectively. The γ′ volume fraction was obtained from the mean value of five different microstructure images by Image-Pro plus 6.0 software.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

The codes that support the findings of this study are available from the corresponding author upon reasonable request.

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## Acknowledgements

We gratefully acknowledge the financial support of National Key Research and Development Program of China (2016YFB0700505 and 2017YFB0702902), Guangdong Province Key Area R&D Program (2019B010940001) and Scientific and Technological Innovation Foundation of Shunde Graduate School, USTB (BK19BE030). We would like to thank Prof. Yuan Wu and Huihui Zhu for their help on APT characterization. We are grateful to Prof. Ryosuke Kainuma for helpful discussions.

## Author information

Authors

### Contributions

The study was planned and designed by Y.J. and Q.F., machine-learning programs and experiments were performed by P.L., the manuscript prepared by P.L., Y.J., H.Y., and S.A. The STEM characterization was implemented by H.W. All authors discussed the results and commented on the manuscript.

### Corresponding authors

Correspondence to Haiyou Huang or Yanjing Su.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Liu, P., Huang, H., Antonov, S. et al. Machine learning assisted design of γ′-strengthened Co-base superalloys with multi-performance optimization. npj Comput Mater 6, 62 (2020). https://doi.org/10.1038/s41524-020-0334-5

• Accepted:

• Published:

• ### Accelerated design of L12-strengthened Co-base superalloys based on machine learning of experimental data

• Jinxin Yu
• , Chenglei Wang
• , Yuechao Chen
• , Cuiping Wang
•  & Xingjun Liu

Materials & Design (2020)


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The noise matrix M M? any R object; a matrix for the matrix method. If A has any missing values, \code{is.symmetric.matrix} returns NA. Applicable to: m-by-n matrix A with linearly independent columns Decomposition: = where Q is a unitary matrix of size m-by-m, and R is an upper triangular matrix of size m-by-n Uniqueness: In general it is not unique, but if is of full rank, then there exists a single that has all positive diagonal elements. 1. a. Nov 27, 2009 at 2:45 am: Hi, I'd like to store large covariance matrices using Matrix classes. For more information, see vignette(ggasym-stats) Usage [R] Symmetric Matrix classes; Gad Abraham. Berrie Zielman . logical indicating if object is symmetric or not. A quick short post on making symmetric matrices in R, as it could potentially be a nasty gotcha. R: convert asymmetric list to matrix - number of elements in each sub-list differ. For all a and b in X, if a is related to b, then b is not related to a.; This can be written in the notation of first-order logic as ∀, ∈: → ¬ (). #' \item \code{is.positive.definite} returns TRUE if a square symmetric real matrix A is positive-definite. complex matrix Z must be “Hermitian” for The matrix method is used inside eigen by The reason For example has only the parts of y where attract attention. {\displaystyle R=\exp(A)=\sum _{n=0}^{\infty }{\frac {A^{n}}{n!}}.} considered, see all.equal.numeric. Setting it to length zero will skip the pre-tests. numeric scalar >= 0. isSymmetric.matrix() R > mat = as.matrix(cbind(x, y,z)) R > mat x y z [1,] 1 2 3 [2,] 2 4 7 [3,] 3 6 9 R > t(mat)%*%diag(j)%*%mat x y z x 180 360 560 y 360 720 1120 z 560 1120 1750 share | … In mathematics, a relation is a set of ordered pairs, (x, y), such that x is from a set X, and y is from a set Y, where x is related to yby some property or rule. ‘ggasym’ (pronounced “gg-awesome”) plots a symmetric matrix with three different fill aesthetics for the top-left and bottom-right triangles and along the diagonal. Inequalities on real-valued functions also give us inequalities on matrices. colnames are identical. Smaller differences are not considered, see all.equal.numeric.. tol1: numeric scalar >= 0. isSymmetric.matrix() ‘pre-tests’ the first and last few rows for fast detection of ‘obviously’ asymmetric cases with this tolerance. Asymmetric Matrix Plotting in ‘ggplot’. Note that a matrix m is only symmetric if its rownames and Currently only a matrix method is implemented, where a either a $$p\times p$$ matrix whose columns contain the eigenvectors of x, or NULL if only.values is TRUE. symmetric argument is missing. In mathematics, an asymmetric relation is a binary relation on a set X where . At least some elements in the upper-triangle are different from the corresponding elements in the lower triangle. Another quite interesting opportunity is based on the following mathematical fact: if A is some matrix, then A multiplied by its transpose is always symmetric. For real asymmetric matrices the vector will be complex only if complex conjugate pairs of eigenvalues are detected. 2.4 OpenFOAM code example. At least some elements in the upper-triangle are different from the corresponding elements in the lower triangle. A matrix is Skew Symmetric Matrix if transpose of a matrix is negative of itself. R Davo January 22, 2014 3. Transpose of A = – A. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be quite bene cial. symmetric: if TRUE, the matrix is assumed to be symmetric (or Hermitian if complex) and only its lower triangle is used.If symmetric is not specified, the matrix is inspected for symmetry. commutation.matrix: Commutation matrix for r by c numeric matrices creation.matrix: Creation Matrix direct.prod: Direct prod of two arrays direct.sum: Direct sum of two arrays D.matrix: Duplication matrix duplication.matrix: Duplication matrix for n by n matrices elimination.matrix: Elimination matrix for lower triangular matrices E.matrices: List of E Matrices Note. It might not be appropriate in all To modify the off-diagonal of a symmetric matrix, first test which pointer is active using hasUpper() and hasLower(). An antisymmetric matrix is a square matrix that satisfies the identity A=-A^(T) (1) where A^(T) is the matrix transpose. Claim 15 Let f : R !R and g : R !R satisfy f(x) g(x) for all x2[l;u] ˆR. This list of fathers and sons and how they are related on the guest list is actually mathematical! Consider using unname(m). for all indices and .. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. So we see that the inverse of a non-singular symmetric matrix is obtained by inverting its eigenvalues. 2Fakulta¨t fu¨r Physik, Universita¨t Duisburg-Essen, 47048 Duisburg, Germany Abstract. default to test symmetry of matrices up to rounding error, using In that case, actually is t is not so important. This function prepares the results of a statistical test for plotting using 'geom_asymmat' from the ggasym package. is.symmetric.matrix(x) Arguments x an R object . An instance of the asymmetric matrix partition problem consists of a matrix $$A \in \mathbb{R}_+^{n \times m}$$ and a probability distribution p over its columns. to link to this page. Keywords: asymmetric matrices, eigenaluev perturbation, entrywise eigenvector perturbation, linear forms of eigenvectors, heteroscedasticit.y Contents 1 Introduction 2 The default is "U" unless x already has a uplo slot (i.e., when it is symmetricMatrix, or triangularMatrix), where the default will be x@uplo. So in R, there are two functions for accessing the lower and upper triangular part of a matrix, called lower.tri () and upper.tri () respectively. isSymmetric(Z) to be true. Sample Problem Question : Show that the product A T A is always a symmetric matrix. Usage. If is square, also is unique. It operates within the Grammar of Graphics paradigm implemented in ‘ggplot2’. How can I convert the list to a matrix? situations. In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. Having no balance or symmetry: an asymmetric design. Test for symmetric numeric matrix . The vectors are normalized to unit length. An asymmetric matrix has the same number of rows and columns, and these rows and columns refer to the same set of objects. Let Abe a symmetric matrix for which all eigenvalues lie in [l;u] (i.e., lI A uI). Consider a matrix A, then. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.. Generic function to test if object is symmetric or not. dsyMatrix seems like the right one, but I want to specify just the upper/lower triangle and diagonal and not have to instantiate a huge Formally, a binary relation R over a set X is symmetric if: ∀, ∈ (⇔). numeric scalar >= 0. Keywords math. Smaller differences are not vectors. If the row and column The matrix exponential of a skew-symmetric matrix is then an orthogonal matrix: R = exp  ( A ) = ∑ n = 0 ∞ A n n ! more general rank- r case. For example, in calculating the diffusion of a quantity , the governing equation is: . is composed of independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This decomposition provides a justification for separate analyses of S and A. Then for that distance measure it we look at r + s divide by q + r + s. That means all the cases with t cases removed, okay. x: a matrix whose spectral decomposition is to be computed. A matrix is positive-definite if its smallest eigenvalue is greater than zero. further arguments passed to methods; the matrix method We leverage our observations to propose an asymmetric relative motion space, which enables the asymmetric resolution of the relative motion without resorting to a redefinition of the absolute motion space, Sec. View source: R/asymmetrise_stats.R. Please use the canonical form : only.values: if TRUE, only the eigenvalues are computed and returned, otherwise both eigenvalues and eigenvectors are returned. all.equal. Multidimensional scaling models and methods for the visualization for asymmetric data . I have an asymmetric list, i.e., the number of elements in each sub-list differ. eigen which calls isSymmetric when its Our model can be … names of object are allowed to differ for the symmetry check Read More on Symmetric Matrix And Skew Symmetric Matrix. . any square matrix (of numbers), either “"traditional"” or inheriting from Matrix. It operates within the Grammar of Graphics paradigm implemented in 'ggplot2'. The goal is to find a partition scheme that maximizes the resulting partition value. cal adj. Viewed 3k times 5. In ggasym: Asymmetric Matrix Plotting in 'ggplot2' Description Usage Arguments Value. ggasym: Asymmetric Matrix Plotting in 'ggplot2' Plots a symmetric matrix with three different fill aesthetics for the top-left and bottom-right triangles and along the diagonal. A logically equivalent definition is ∀, ∈: ¬ (∧). An example is a student migration table, where the rows correspond to the countries of origin of the students and the columns to the destination countries. A symmetric relation is a type of binary relation.An example is the relation "is equal to", because if a = b is true then b = a is also true. Furthermore, a heat map for skew-symmetric data, and the decomposition of asymmetry are provided for the analysis of asymmetric tables. This package provides the slide-vector model , a scaling model with unique dimensions and the asymscal model for asymmetric multidimensional scaling. Setting it to length zero will skip the pre-tests. This might arise if, for example, we have two independent samples for each entry of M? Checkout the documentation and vignettes at the pkgdown website https://jhrcook.github.io/ggasym/. So for these asymmetric variables, they are different as r + s, but they both not appearing. Active 3 years, 1 month ago. Ask Question Asked 8 years, 5 months ago. ‘pre-tests’ the first and last few rows for fast detection of We solve a family of Gaussian two-matrix models with rectangular N × (N + ν) matrices, having real asymmetric matrix elements and depending on a non-Hermiticity parameter µ. Here's something interesting! uplo: optional string, "U" or "L" indicating which “triangle” half of x should determine the result. 3. An asymmetric matrix has the same number of rows and columns, and these rows and columns refer to the same set of objects. 4.The new space leads to the proposal of a novel relative Jacobian formulation and a corresponding differential IK algorithm, Sec. A matrix is positive semi-definite if its smallest eigenvalue is greater than or equal to zero. Abstract. do use check.attributes = FALSE! For example, A=[0 -1; 1 0] (2) is antisymmetric. This decomposition is a useful tool for data analysis, and a second application is to the study of an asymmetric matrix of … object: any R object; a matrix for the matrix method.. tol: numeric scalar >= 0. The image of the exponential map of a Lie algebra always lies in the connected component of the Lie group that contains the identity element. perturbed version M is observed. Suppose that Riverview Elementary is having a father son picnic, where the fathers and sons sign a guest book when they arrive. Description. Caution: Accessing the opposite pointer without a const modifier will convert the matrix to an asymmetric matrix. ‘obviously’ asymmetric cases with this tolerance. and arrange them in an asymmetric … The decomposition into a skew-symmetric and a symmetric component is written as: Q=S+A, where Q is an asymmetric matrix, S is a symmetric matrix, and A is a skew-symmetric matrix. Value. TRUE or FALSE. This function returns TRUE if the argument is a numeric symmetric square matrix and FALSE otherwise. Let me illustrate: https://CRAN.R-project.org/package=asymmetry If we let F be the set of … Antisymmetric matrices are commonly called "skew symmetric matrices" by mathematicians. passes these to all.equal. Symmetry check do use check.attributes = FALSE in ‘ ggplot2 ’ test symmetry of matrices to. Eigenvalues and eigenvectors are returned uplo: optional string,  U '' or  L '' indicating “! 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## [Pol. p target] Modeling Microwave Unit Signal

a period is 23ms. with in this period, the modulation signal voltage is

$V_{ms}(b,s) = b + ( t - 12.5) s [V]$

$Max(V_{ms}) = 29.2 [V]$   $Min(V_{ms} = 0 [V]$

on the Gunn Oscillator, there is a mechanical switch, which can adjust the base frequency but changing the length.

$f_{base} (l) = 0.61 + 6.433 [GHz]$

this data is provided by 3 data point in the manual. the output frequency of the microwave is

$f_{out} = f_{base} + F_m ( V_{ms} )$

where $F_m$ is the modulation function, that we have to find out. linear?quadratic? at least get a good approximation for it.

the resonance frequency and its FWHM should depend only on the microwave cavity. an a absorption signal can be formulated by a Lorentzian distribution. and this signal will be converted to voltage by a linear conversion factor. ( the green words is an assumption )

$L( f_{res} , f_{out} , FWHM_{res} ) = 1/ ( 1 + (\frac { f_{res} - f_{out} } {FWHM_{res}} )^2 )$

From the relation between the length and voltage at peak. we can find out the modulation function. since the output frequency is equal to the peak frequency. thus, the output frequency is fixed

$f_{out} = f_{res} = 0.6 l + 6.433 + F_m (V_{ms})$

if we measure l and V_{ms} we can find out F.

by further measurement,  the modulation is non-linear. That’s also explained the FWHM on the CRO change with frequency. since the FWHM of the microwave cavity should be same and the change of the FWHM in CRO reflected that the gradient of the frequency output. for a linear frequency output, the FWHM should be the same. but if the gradient change with due to the modulation signal, the FWHM will change.

## Mircowave Unit

the Gunn Oscillator and the modulation signal were studied.

the problem is the IV converter. it should be a high-pass.

but there are 2 problems

1. the reflected microwave signal is negative voltage.
2. there are 2 things to determine the microwave frequency. – the modulation signal and the knob is Gunn Oscillator.

## [Pol. p target] Finished Circuit Connection and Test on MW system

The Above diagram is the Setup. The upper one is laser unit, the 2 devices between Sample Chamber and CRO is field Sweep Unit. and the lower part is the Microwave unit. and of course the NMR unit and the Power and Cooling unit for the Static B-field.

(some incorrect in here, the modulator is Gunn Oscillator, and the Gunn Oscillator is the power supply and Modulation signal. )

## Laser Unit

The Laser we use is a Class 4 Ar ion Laser ( Coherent TSM25 ), wavelength is 454.5 nm to 528.7 nm. Maximum power is 25W.

the laser beam was reflected by a mirror to fit the space of the lab. and it pass through a lens with 100mm focal length so that the beam gets converged on the position of the beam chopper. after the chopper, the beam diverged and it passes another 100mm focal length lens to be come parallel.

before it goes to the sample, a 1:99 beam splitter splits the beam such that 1% of it go to a power meter. the signal of the power meter is used to control the microwave and field sweep.

this part is the optical pumping for polarizing the electron.

although the chopper is moving at high frequency, when the chopper cut the laser, the power meter signal will be decreased. and this event will trigger the microwave and field sweep to transfer polarization to protons. the triggering is by the Controller.

## Field Sweep Unit

The field sweep unit is all-in-one. the field will be monitored by the CRO and controlled by the Controller.

## Microwave Unit

This part contains most devices and trouble me. since i just follow the menu and not fully understand the function of each devices. i try to explain.

Power Supply of the Gunn Oscillator and Modulation signal

this is the starting point. the voltage of the power supply can be adjusted by a knob. the Modulation signal can be adjusted by a “Bias ADJ” and “Output Cont”.

The “Bias ADJ” change the level. and “Output Cont” change the slope.

the frequency of the Modulation Signal is 23ms.

and the signal will chop out negative value.

the signal also monitored by a cable from the back to the CRO. the monitored signal is a fixed signal with fixed “Bias ADJ” and “Output Cont”.

the maximum power output is 150W to the Gunn Oscillator.

Gunn Oscillator

Gunn Oscillator is used for general microwave according to the modulation signal. when the voltage of the modulation signal is low, the frequency of the microwave is low.

Attenuator

This is for reduce the microwave power. the reason for using this, is that the RF amplifier has maximum power input 100W for pulse and 20W for CW signal. or to say, the working range is that.

RF amplifier

This is for amplifier the MW signal. and the RF amplifer can be controlled by a remote device, so that the amplify ratio can be change to on or off.

this is important, since after the laser polarized the electron, the chopper will cut the laser, then microwave and field sweep come in to transfer the polarization to the proton. therefore a remote controllable device is needed.

i don’t know the amplify ratio and maximum output.

Circulator

the RF signal will go to the circulator. there are 3 ports on the circulator. the function for the circulator is circulate the signal. the signal from port 1 will only go to port 2, port 2 will only go to port 3, and port 3 will only go the port 1.

we have to detect the reflected microwave to find out the resonance frequency of the microwave cavity in side the sample chamber. and we don’t want to mixed up the signals.

Directional Coupler

there are 4 ports in Directional Coupler, 1, 2, 3, 4. most of the signal from port 1 will go to part 2 and little bit of it will got to port 3. but most of the signal of form port 2 will go to port 4 and only less to the port 1.

$1\rightarrow 2 : 1\rightarrow 3 >>1$

$2\rightarrow 1 : 2\rightarrow 4 << 1$

i don’t understand this part.

Microwave cavity inside the Sample Chamber

the microwave then enter the cavity and get reflected when the input frequency is not the natural frequency of the cavity. the reflection signal will go to the port 2 of the directional coupler and then the port 2 of the circulator. the signal will be diverted to port 3 and enter an other attenuator, for decrease the power.

This should be corrected to “Crystal Converter”. it convert the microwave into current signal.

the maximum input is 20W. therefore, it need an attenuator before the microwave come in.

I-V converter

it simple convert a current signal into a voltage signal, so that we can wee in CRO.

## NMR Unit

an easy part is the NMR system. since we have 1 single machine for generate NMR pulse and analyzing the NMR signal.

the only thing we have to do is adjust the tuner. the tuner is a variable capacitor, so that it can matching the impedance of the NMR frequency and the NMR coil ( which inside the chamber) to give Maximum power delivery.

## Power supply & Cooling Unit

this is for the static B-field.

____________________________________

when we run the microwave system, the absorption of the reflected wave is too small and not symmetric. we tried every possible setting and check the cables. not thing wrong.

may be the microwave coil inside the chamber has problem. that is what we are going to do.

# Motivation

In order to convert the NMR single to the absolute magnetic field strength of the polarization. the thermal polarization ( polarization at thermal equilibrium ) should be measure and used to calibrate the NMR signal.

the thermal polarization is given by the Boltzmann statistic. The excited and the ground state population is:

$\frac{ N_{\uparrow}} { N_{\downarrow} } = Exp \left( - 2 \frac { \mu_p B } {k_B T } \right)$

the thermal polarization is the ratio of the different of spin-up and spin-down to the total spin.

$P_{thermal} (B , T ) = tanh \left( \frac{ \mu_p B }{ k_B T } \right)$

where $\mu=p$ is the proton magnetic moment.

$\mu_p = g_p \mu_N = g_p \frac{e \hbar }{2 m_p} = 1.410606662 \times 10^{-26} J T^{-1}$

$k_B$ is the Boltzmann constant.

$k_B = 1.3806504 \times 10^{-23} J K^{-1}$

The proton magnetic moment is small, the thermal polarization can be approximated as a linear function:

$P_{thermal} ( B, T) = \frac{\mu_p B}{ k_B T}$

since our polarization is on $T = -5 ^oC$ and $B = 0.05 T$, thus, the thermal polarization is:

$P_{thermal} (0.05, 268.15) = 1.90509 \times 10^{-7}$

which is very small to be detected, or to say, the signal is smaller then the noise level.

the small system has a better sensitivity, down to $10^{-7}$.

# TO-do

• Connect the Static field power and water cooling system
• connect the Controler Unit
• connect the magnetic field sweeping
• Test Run

## First experiment of 6He with a polarized proton target

this paper reported a first spin polarized proton solid target under low magnetic field ( 0.08 T ) and hight temperature ( 100K )

the introduction overview the motivation of a solid target.

• a polarized gas target is ready on many nuclear experiment.
• on the radioactive beam ( IR beam ), the flux of a typical IR beam is small, since it is produced by 2nd scattering.
• a solid target has highest density of solid.
• most solid target can only be polarized on low temperature ( to avoid environmental interaction to reduced the polarization )
• increase the experimental difficult, since a low temperature should be applied by a cold buffer gas.
• high field ( the low gyromagnetic  ratio ).
• high magnetic field make low energy scattered proton cannot get out from the magnetic field and not able to detect.
• a solid target can be polarized at high temperature and low magnetic field is very useful

the material on use is a crystal of naphthalene doped with pentacene.

the procedure of polarizing the proton is :

1. use optical pumping the polarize the electron of pentacene
• the population of the energy states are independent of temperature and magnetic field.
2. by Dynamic Nuclear Polarization (DNP) method  to transfer  the polarization of the electron to the proton.
• if the polarization transfer is 100% and the relaxation time is very long. the expected polarization of proton will be 72.8%

The DNP method is archived under a constant microwave frequency with a sweeping magnetic field. when the magnetic field and  microwave frequency is coupled. the polarization transfer will take place.

the next paragraph talks about the apparatus’s size and dimension, in order to fit the scattering experiment requirements.

the polarization measurement is on a scattering experiment with 6He at 71 MeV per nucleons. By measuring the polarization asymmetry $\epsilon$, which is related to the yield. and it also equal to the polarization of the target $P_t$  times the analyzing power $A_y$.

$\epsilon = P_t \times A_y$

with a reasonable guess of the target polarization. the analyzing power of  6He was found.

the reason why the polarization-asymmetry is not equal to the analyzing power is that, the target is not 100% polarized, where the analyzing power is defined. when the polarization of the target is 100%, both are the same.

in the analysis part. it used optical model and Wood-Saxon central potential to simulate the result. And compare the result from 6He to 6Li at same energy. the root mean square of 6Li is larger then 6He. it suggest the d-α core of 6Li may responsible for that.

they cannot go further discussion due to the uncertainly on the polarization of the target.