Multi-step policy evaluation for adaptive-critic-based tracking control towards nonlinear systems
Correspondence to: Prof. Ding Wang, Faculty of Information Technology, Beijing University of Technology, No.100, Pingleyuan, Chaoyang District, Beijing 100124, China. E-mail:
Currently, there are a large number of tracking problems in the industry concerning nonlinear systems with unknown dynamics. In order to obtain the optimal control policy, a multi-step adaptive critic tracking control (MsACTC) algorithm is developed in this paper. By constructing a steady control law, the tracking problem is transformed into a regulation problem. The MsACTC algorithm has an adjustable convergence rate during the iterative process by incorporating a multi-step policy evaluation mechanism. The convergence proof of the algorithm is provided. In order to implement the algorithm, three neural networks are built, including the model network, the critic network, and the action network. Finally, two numerical simulation examples are given to verify the effectiveness of the algorithm. Simulation results show that the MsACTC algorithm has satisfactory performance in terms of the applicability, tracking accuracy, and convergence speed.
In practical engineering applications, the controller design should not only meet the basic performance requirements but also need to further improve the control effect and reduce costs [1–4]. However, real industrial systems are often complex nonlinear systems [5, 6]. Solving the Hamilton-Jacobi-Bellman (HJB) equations is an unavoidable obstacle when designing optimal control policies for nonlinear systems . Considering that the analytical solution of the HJB equation is difficult to obtain, adaptive dynamic programming (ADP) based on the actor-critic framework approximates the solution of the HJB equation by the iteration [8, 9]. The ADP algorithm exhibits strong adaptive and optimization capabilities by incorporating the advantages of reinforcement learning, neural networks, and dynamic programming [10, 11]. Therefore, the ADP algorithm has a satisfactory performance in solving the HJB equation. Through continuous development, the ADP algorithm has become one of the key methods for solving optimal control problems of nonlinear systems [12–14]. In terms of the iterative form, the ADP algorithms can be categorized into value iteration (Ⅵ) and policy iteration (PI) [15, 16]. PI has faster convergence but requires an initial admissible control policy . However, it is difficult to obtain an initial admissible control policy for nonlinear systems. VI does not require an initial admissible control policy but has a slower convergence rate .
In the industry, we often need to solve more complex tracking problems for nonlinear systems [19–24], including the consensus tracking problem for multi-agent systems  and the output tracking control for single-agent systems . In fact, the regulation problem can be seen as a simplified special case of the tracking problem. To overcome these challenges, some scholars have started to develop optimal learning control algorithms based on the ADP framework for solving tracking problems . Currently, tracking control algorithms based on the ADP framework can be categorized into three groups. The first class of tracking control algorithms constructs the augmented system with respect to the original system and the reference trajectory and then builds the cost function based on the state vector and the control input of the augmented system [27, 28]. Although this type of tracking control algorithm has a well-established theory, it cannot completely eliminate tracking errors. The second class of tracking control algorithms utilizes the square of the tracking error at the next moment to build a novel utility function . With the novel utility function, the tracking error can be completely eliminated. However, the second class of tracking control algorithms is currently only applicable to affine nonlinear systems with known system models, which greatly reduces the application value of the algorithm. The third type of tracking control algorithm transforms the tracking problem into the regulation problem by constructing a virtual steady control . Thanks to the steady control, the third type of tracking control algorithm also completely eliminates tracking errors and is suitable for nonlinear systems. It should be emphasized that solving the steady control imposes a large computational burden on the algorithm.
Therefore, we expected that a speed-up mechanism could be designed for the tracking control algorithm with steady control to improve the convergence speed. Inspired by the idea of eligibility traces, Al-Dabooni
In this context, a multi-step adaptive critic tracking control (MsACTC) algorithm is established by introducing a multi-step policy evaluation mechanism into the steady tracking control algorithm. Compared to the general tracking control algorithm with steady control, the MsACTC algorithm exhibits higher learning efficiency as the evaluation step increases. Furthermore, the convergence proof of the MsACTC algorithm is given. The MsACTC algorithm is successfully implemented through neural networks and the least square method. Finally, we verify the optimization ability and learning efficiency of the proposed algorithm through simulations of two nonlinear systems. Table 1 explains the meaning of the mathematical symbols used in this article.
Summary of mathematical symbols
|The set of all
||The norm of vector
|The set of all
||The absolute value of
||The transpose symbol|
|The set of all non-negative integers||The set of all positive integers||A compact subset of
2. PROBLEM DESCRIPTION
Consider such a class of discrete-time nonlinear systems
The reference trajectory can be denoted as
We assume that there exists a steady control
The tracking control policy
According to equations (1)-(5), we can obtain the following error system:
The cost function is defined as
So far, the tracking problem of the system (1) has been successfully transformed into the regulation problem of the system (6).
3. MULTI-STEP ADAPTIVE CRITIC TRACKING CONTROL ALGORITHM
It should be noted that equation (8) is the HJB equation, and its analytical solution is difficult to obtain directly. In addition, since the system function
3.1. Algorithm design
According to equation (6), we have
Construct the sequence of iterative cost functions
The policy evaluation is expressed as follows:
By iterating continuously between equations (11) and (12), the cost function and the control policy converge to their optimal values as
3.2. Theoretical analysis
To ensure that the cost function
Theorem 1If the sequences
then the following conclusions hold:
Proof. First, we prove the conclusion 1). When (14) holds, we have
So far, we have shown that the conclusion 1) holds for
By further derivation, we get
Based on inequations (16) and (17), we have
According to mathematical induction, the proof of the conclusion 1) is completed.
Next, we prove the conclusion 2). Since the sequence of cost functions
By observation, equations (20) and (8) are equivalent. This means that
Theorem 2Based on equations (1)-(6),
Proof. According to the error system (6), we have
Besides, we have
The proof of Theorem 2 is completed. In this section, Theorem 1 is successfully proved by utilizing mathematical induction, and it provides a theoretical justification for the convergence of the MsACTC algorithm. Theorem 2 gives an equivalence between
Remark 1 To ensure that the condition (14) holds,
4. IMPLEMENTATION OF ALGORITHM
To implement the algorithm, three neural networks are constructed, including a model network (MN), a critic network (CN), and an action network (AN). Figure 1 illustrates the overall structure of the MsACTC algorithm. It should be noted that the training of the MN is carried out independently. The training of the CN and the AN need to cooperate with each other iteratively. It should be noted that the implementation approach used in this paper is similar to that in .
Figure 1. The structure diagram of the MsACTC algorithm. MsACTC: Multi-step adaptive critic tracking control.
The MN is used to estimate the dynamics of the system (1) and to solve the steady control
In order to make the MN with higher recognition accuracy, we utilize the neural network toolbox in Matlab to train the MN. The training parameter settings are given in Section 5.
After the MN is trained, according to equation (4), we can obtain the steady control
The CN is used to approximate the cost function, and its output can be expressed as
A sample set
The AN is used to approximate the tracking control policy, and its output can be expressed as
Considering the sample set
5. SIMULATION RESULT
In this section, we complete two simulations. Example 1 is to illustrate the effectiveness of the MsACTC algorithm and the correctness of Theorem 1. Example 2 is to show that the condition (14) is not sufficiently necessary for the convergence of the cost function.
5.1. Example 1
Consider the modified Van der Pol's oscillator system
The reference trajectory is defined as
Before the neural networks are trained, we make a sample set of
We construct the activation functions for the CN and the AN as
In order to satisfy the condition (14), the weight vector
In order to compare the convergence speeds under different values of
Figure 3. Convergence processes of network weights with different values of
We apply the optimal control policy obtained by the MsACTC algorithm for
5.2. Example 2
Consider the modified torsional pendulum system
Due to the excellent adaptive ability, the MsACTC algorithm does not have strict requirements for the settings of the parameters. Therefore, the parameter values and the setting of activation functions are kept the same as those in Example 1. In Example 2, we set
Figure 8. Convergence processes of network weights with different values of
Even if the condition (14) does not hold,
6. CONCLUSIONS AND OUTLOOK
In this paper, the MsACTC algorithm is developed by introducing the multi-step policy evaluation mechanism into the adaptive critic tracking control algorithm. For nonlinear systems with unknown models, the MN is built based on data to estimate the state vector in the future time. By solving the steady control
Formal analysis, validation, writing - original draft: Li X
Methodology, Supervision, Writing - review and editing: Ren J
Investigation, Supervision, Writing - review and editing: Wang D
Availability of data and materials
Financial support and sponsorship
This work was supported in part by the National Key Research and Development Program of China under Grant 2021ZD0112302 and in part by the National Natural Science Foundation of China under Grant 62222301, Grant 61890930-5, and Grant 62021003.
Conflicts of interest
All authors declared that there are no conflicts of interest.
Ethical approval and consent to participate
Consent for publication
© The Author(s) 2023.
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Cite This Article
Li X, Ren J, Wang D. Multi-step policy evaluation for adaptive-critic-based tracking control towards nonlinear systems. Complex Eng Syst 2023;3:20. http://dx.doi.org/10.20517/ces.2023.28
Li X, Ren J, Wang D. Multi-step policy evaluation for adaptive-critic-based tracking control towards nonlinear systems. Complex Engineering Systems. 2023; 3(4): 20. http://dx.doi.org/10.20517/ces.2023.28
Li, Xin, Jin Ren, Ding Wang. 2023. "Multi-step policy evaluation for adaptive-critic-based tracking control towards nonlinear systems" Complex Engineering Systems. 3, no.4: 20. http://dx.doi.org/10.20517/ces.2023.28
Li, X.; Ren J.; Wang D. Multi-step policy evaluation for adaptive-critic-based tracking control towards nonlinear systems. Complex. Eng. Syst. 2023, 3, 20. http://dx.doi.org/10.20517/ces.2023.28
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It is basically used to get the. The angle of depression is the angle between the horizontal line and the observation of the object from the horizontal line. We know that, ∠DAC=∠ADO (by using alternate interior . Solution: In the figure given above, there are two angles of depression formed from point A, which are ∠BAC=30° and ∠DAC=60°. President Franklin Delano Roosevelt initiated several acts that fixed the bank problems and helped the American people obtain jobs and relief during the Great Depression, according to PBS’s The Americ. · 2. From the top of a cliff m. From the top of a tower, a man finds that the angle of depression of a car on the ground is 30°. Angle of Depression · 1. Now, in right triangle PQR, QR = m ∠ R = 60° PQ = h (in m) Let ∠ R = θ = 60° We know, tan θ = (Opposite Side/Adjacent Side) Tan 60 = PQ/RQ = h/ √3 = h/ Or h = √3 x h = m Hence, the height of the tower is meters. The angle of depression, ∠ SPR = 60° So, The angle of elevation, ∠ QRP = Angle of depression, ∠ SPR = 60°. And. ∠O = Angle of depression. Or. θ = tan-1 (Opposite Side/Adjacent Side) See the below diagram, where θ is the angle of inclination, such as, ∠ ABO = Angle of elevation. Hence, ∠ABO = ∠O = θ. With angles of elevation, if two of the sides of the right triangle are known, then the formula for the angle of depression is given as below: Tan θ = Opposite Side/Adjacent Side. Angle of Depression Formula. RQ is the distance between the tower and the stone. PQ is the height of the tower. PS is the line of sight. Solution: In the above figure, R is a stone. The angle of depression of a stone from the top of a tower on the ground is If the stone is away from the building at a distance of meters, find the height of the tower. To begin solving the problem, select the appropriate trigonometric ratio. Aug 24, · Because the horizontal line is not directly the ground, add to the solution to the equation. Angle of Depression and Angle of Elevation are equal become they. In this section, we try to solve problems when Angles of depression are given. A variety of solutions for environmental problems exist including recycling, reduction of carbon emissions from fossil fuels, finding alternative energy solutions and the conservation of marine life.


This is one of the clearest definitions of a chaotic system that I’ve seen. I really like the timescale comparisons: seconds for pendulums, days for weather, Myrs for the solar system.
I’d love to see an XKCD table of timescales of various chaotic systems. 🙂 This is close but not quite: https://xkcd.com/1399/
Originally shared by Larry Phillips
A double pendulum is a pendulum with another pendulum attached to its end. It is one of the simplest systems that can have chaotic behavior.
This animation shows the calculated movement of two double pendulums that start from slightly different initial conditions. After a short time, the motion predicted for each of them is completely different. This is the hallmark of a chaotic system – differences in initial conditions, however small, eventually result in large differences in the system behavior.
Note that the mathematical model for the double pendulum is exact, and there is no inherent randomness in the motion. It’s just that we can never know the initial conditions precisely, so that the predicted motion will become less and less certain as we go further out in time.
Two examples of chaotic systems are
1) the Solar System, on a time scale of about 100 million years, and
2) the weather, on a time scale of around 12 days
Animation source: goo.gl/vspShs
‘Why chaotic. It’s motion is deterministic according to construction and initial conditions.
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Supervisor(s): Dr. Omer Bobrowski, Dr. Emil Saucan
The graph Laplacian and its spectrum have been proven to be powerful mathematical tools that reveal fundamental properties about the structure and functionality of networks. Spectral analysis has conduced a variety of applications in Computer Science and related fields, such as clustering algorithms, randomized approximation methods, error correcting codes, and studying computational complexity. The most significant and abundant applications are in Communication Networks and related problems. In particular, in the analysis of routing algorithms, for minimizing congestion and devising efficient parallel architectures.
One of the main limitations of the graph Laplacian is that it only uses information about pairwise interactions in networks (see Figure (a)). However, in modern networks one can often find connections of higher degrees, that cannot be encoded by a graph. Consider, for example, the case of a social network. Using a graph, we can represent the information that Alice and Bob are friends, Bob and Carol are friends, and Carol and Alice are friends. We cannot say, however, whether Alice, Bob, and Carol, all get together as a group.
Simplicial complexes are high-dimensional generalizations of graphs, that in addition to vertices and edges, may also contain triangles, tetrahedra, and higher dimensional simplexes. These objects provide a natural way to represent such high-degree interactions in network (see Figure (b)). Simplicial complexes have their own notion of Laplacian, and the goal of this project is to study its spectrum. The main question we are interested is: “What can we learn about the structure of such high-dimensional networks, by analyzing their Laplacians”? We will consider both simulated as well as real-world networks, and examine both the spectrum and its corresponding harmonics. Our goal is to look for insights about the network behavior, as well as study the potential of signal processing on such high-dimensional objects (in a similar spirit to signal processing on graphs).


Anugerah Jaringan Industri dan Komuniti Berimpak Tinggi UPM 2014
INSTITUT PENYELIDIKAN MATEMATIK INSTITUT PENYELIDIKAN MATEMATIK
Pemenang Hadiah Karya Ilmiah Sains dan Teknologi 2014: Prof. Madya Dr. Mat Rofa Ismail
GALERI INSPEM GALERI INSPEM
Pakar matematik INSPEM dapat pengiktirafan Eropah Para saintis matematik kebanggaan UPM ini baru sahaja mendapat pengiktirafan daripada European Mathematical Society (EMS) melalui anugerah “Emerging Regional Centre of Excellence.”
Conference on Agricultural Statistics 2015 In recent years, Agricultural Statistics has emerged as an important research field and there has been a rapid growth in this field over the last few decades. This is evidenced by the number of international conferences that have been organised and one such prominent conference is the International Conference of Agricultural Sciences (ICAS), which is a triennial event since 1998.
Seams School Algebra and their Applications (Quantum Physics, Cryptography and Statistics) The South East Asian Mathematical Society initiates the SEAMS School of Mathematics as a series of intensive 7-day workshop. The purpose of this school is to provide opportunities for advanced undergraduate and postgraduate students to have an advanced learning experience in mathematics, and to introduce a research-based learning.
This school will introduce fundamental notions of Algebras, Quantum Physics, Cryptography and Statistics. It is addressed to advanced undergraduate and graduate students as well as young researchers from South East Asian countries. It will provide them with some of the knowledge necessary to further study and research.


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Ising Machines based on networks of mutually coupled oscillators
Note, this is a work in progress, like a diary to formulate our thoughts and collect ideas, questions, hypothesis and results as we work towards gaining insight to the topic.
[collaborators: Dimitris Prousalis, Lucas Wetzel]
This post is about our exploration into the realm of so called Ising Machines. My curiosity into the topic was sparked by discussions with genius Bernd Ulmann, Mathematician, self-appointed Analog Evangelist, Inventor, Founder, museum director and most importantly a nice person.
An Ising machine is a physical implementation of a spin-system and its interactions to compute minima of the Ising Hamiltonian. As reviewed and shown, e.g., in “Ising formulations of many NP problems” by Andrew Lucas, many NP hard problems can be mapped onto the Ising Hamiltonian. Hence, if an Ising Machine can be build and scaled, it can act as an analog computer to solve these NP hard problems by finding/relaxing to ground states of the system. (Which is not guaranteed since the system may get trapped in a local minima. In this case some “shaking may help”.)
First of all I would like to link the first publications that I read to start with the topic. To be honest, they gave me a bit of a headache since by no means I was able to confirm the results that they presented when I stared looking at them in early 2021 — but more on that later. Here is the list to the open access manuscripts:
- OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems, Tianshi Wang, Jaijeet Roychowdhury
Hypotheses, issues and questions
- When scaling Ising Machines to large numbers of spins (represented by oscillators whose phases are binarized) and to operate at high frequencies, the systems (spatial) dimensions may lead to non-negligible transmission time delays in the connections between the oscillators. We know, that such time delays can lead to multistability of in- and anti-phase synchronized states and therein. Furthermore, as the problem is usually encoded via the adjacency matrix of the network, these time delays will most likely be heterogeneous. Also, it has been shown that inert responses in, e.g., electronic oscillators, that stem from signal processing affect the self-organized dynamics in such systems.
The question is then, whether the mapping to the Ising Hamiltonian can still be justified and if not, which measures can be taken to achieve a sufficient (to solve a problem) approximation.
TODO: compile a list of open questions and problems related to the presence of time delayed coupling and inertia, quantify their effects in simulations and look for ways to address these.
- When simulating delay-coupled phase oscillators using the Kuramoto model we made the following observations:
a) The results in Oscillator-based Ising Machine, chapter IV: Examples can be reproduced for sufficiently strong dynamical noise. If the noise was too small, we did not observe the convergence to the correct MAX-CUT solutions in all cases (actually we observed that only rarely). This could be a sign for rugged energy landscapes that require some ‘shaking’ to find the absolute minima and hence the correct solutions to the MAX-CUT problem.
b) When simulating using a second order Kuramoto model that takes into account inertia like effects introduced by, e.g., a loop filter in a phase-locked loop or the mass of a pendulum, the phases did not converge to the correct solutions for arbitrary loop filter cut-off frequencies. Increasing the cut-off frequency, or in other words, reducing the oscillators inert behavior, led again to the correct solutions. My hypothesis is, that loop filters with cut-off frequencies far below the second harmonic injection frequency will damped this too strongly and prevent he binarization of the oscillators phases. Hence, it may be beneficial to mix the second harmonic injection locking signal directly into the control signal of the PLL. In principle it may be possible to use a ‘copy’ of the feedback signal which is mixed with itself. The resulting output can then be filtered by a band-pass that allows only the second harmonic. These are however details the implementation of such an idea, in e.g., a PLL type of setup.
c) Adding small transmission delays seemed not to hinder convergence to the correct solutions, independently of whether the time delay would have led to in- or anti-phase synchronized states.
Notes for publication:
- issues to be addressed: subharmonic injection locking in the presence of transmission delays that affect the frequencies of the oscillators – in that case the injection locking has to happen at roughly twice the instantaneous frequency of the oscillators
- approaches discussed so far in the literature with electronic oscillators:
a) T. P. Xiao, Optoelectronics for refrigeration and analog circuits for combinatorial optimization, Ph.D. thesis, UC Berkeley (2019).
b) relaxation oscillators with capacitive coupling: A. Parihar, N. Shukla, M. Jerry, S. Datta, and A. Raychowdhury, Scientific reports 7, 1 (2017).
c) CMOS LC oscillators on chimera graph: N=240 in T. Wang, L. Wu, and J. Roychowdhury, in Proceedings of the 56th Annual Design Automation Conference 2019 (2019) pp. 1–2. and N=5 in J. Chou, S. Bramhavar, S. Ghosh, and W. Herzog, Scientific reports 9, 1 (2019).
d) RLC oscillators: LQ English, A. V. Zampetaki, K. P. Kalinin, N. G. Berloff, P. G. Kevrekidis, arXiv preprint arXiv:2204.00153
Playground, simulations & observations
Here I collect some results obtained from simulating different types of Kuramoto models. The code used here can be found on Github. The documentation process is still underway and I am being supported in cleaning up and organizing the code by Deborah Schmidt.
Along the results presented in Oscillator-based Ising Machine obtained by Matlab simulations of the Kuramoto model we start here and ask whether these results can be reproduced. Our model reads in the most general form [1,2,3]:
where denotes the instantaneous frequency, and the intrinsic frequency of oscillator , with . denotes the coupling strength or sensitivity towards input signals to oscillator from other oscillators in the network, as given by , the components of the adjacency matrix that are either one or zero depending on whether there is a connection from oscillator to or not, respectively. The integral represents the signal processing or filtering within the feed-forward path of the oscillators, e.g., a low pass filter in a phase-locked loop oscillator with impulse response function . denotes the phase of oscillator , the transmission time delay for signals propagating from oscillator to , and the feedback time delay within oscillator . denotes the strength of the harmonic injection locking. Finally, denotes additive Gaussian white noise with variance .
In terms of the model, we aim to address how the following parameters affect finding the ground states and relate to the classical Ising model:
- how to appropriately increase or , which one preferably and at what rate, does it depend on the network size?
- the role of the loop filter via the , the filter transfer function – if the cut-off frequency is below the harmonic frequency, then the injection locking signal will be damped/filtered and binarization may be impaired (here we should consider a modified PLL architecture where the harmonic injected signal does not pass trough the loop filter, e.g., multiplication by itself and band-pass to extract harmonic)
- the role of feedback and transmission time delays, as the size of a system grows it becomes increasingly difficult to arrange the oscillators such that they can be coupled to any other in the network with the same transmission time delay; question: if there are transmission time delays that are heterogeneous within one connection (asymmetric), i.e., between an oscillator and and oscillator , this leads to phase-difference between these oscillators for the case of pure mutual coupling without injection locking – how does that affect a networks ability to relax to a ground state of the Ising model? the same holds for the case and
- what about the coupling capacity/normalization of the inputs? does an oscillator couple more strongly the more inputs it receives?
- should the coupling between the oscillators be repelling, i.e., should ?
In the publication Oscillator-based Ising Machine, see chapter IV, they set radHz, radHz and simulate different network topologies, using simple MAX-CUT problems to benchmark. Note, that here the coupling strength is increased linearly / ramped up once the simulation has started. We chose for the noise strength (radHz).
Using the same parameters we find the following results for a system of oscillators, where oscillator receives input from all other oscillators while all other only receive from oscillator . Hence, we have a star-like topology with oscillator being the center. In this case, the MAX-CUT problem is solved if all oscillators evolve to a phase different by from that of oscillator . The results were promising and lead to the correct solution of the MAX-CUT problem in most cases, however not in all cases as we show in the figures below:
We then realized the importance of the process of ramping up the coupling strength for finding the correct solution successfully. Also, adding dynamical frequency noise seemed to help convergence. Moreover, we simulated some realizations with time delay, signal filtering and ramping up the harmonic injection coupling strength instead of the coupling strength of the first harmonic coupling terms. This had profound impact on whether the correct solution to the optimization problem was found. Therefore we decided to develop ideas on how to approach the impact of these parameters in a more structured way. Our preliminary results is what we will present in the following.
How we measure the success probability and time to solution as we change different system parameters?
We decided to run a number of realization for a fixed set of parameters and solve the MAX-CUT problem for a system of oscillators arranged in a star like topology, where an oscillator indexed by is located in the middle of the star and hence receives input from all others, while all others only receive input from , the center oscillator. After computing realizations we ask how many have led to a correct solution and in what average time that was achieved. A realization is categorized at being solved correctly when the magnitude of the order parameter settles at a value of in that particular case, i.e., all phases have separated correctly. The Kuramoto order parameter is defined as
where denotes the magnitude of the order parameter and the mean phase. measures the synchrony of the oscillators and approaches if all oscillators are in-phase. In our benchmark case, all outer oscillators of the star topology form a subset that is shifted by a phase difference of with respect to oscillator . One can then calculate
which yields the result for the particular MAX-CUT problem used for benchmarking here mentioned above and visible in Fig.~1.
A few more words on how we obtained the statistics on the realizations automatically from the simulations.
In order to determine whether the correct solution to MAX-CUT problem was found we checked the asymptotic values of the phase differences. Hence, if the phase differences of all oscillators with respect to the phase of are equal to asymptotically, the correct solution has been obtained. We quantify this by checking these phase-differences and the value of the order parameter. Since the phase-configuration should be constant once the system has settled to the ground state, we compute the average over one and a half period of the intrinsic oscillations of the oscillators.
Extracting the mean time until a solution is obtained is measured as follows. We consider only those realizations that led to a correct solution of the MAX-CUT problem. For these, we calculate the time-derivative of the order parameter which we then average over half a period of the intrinsic frequency of the oscillators. This yields a more smooth curve. We then track from the end of the time-series towards the beginning, i.e., backwards in time, when the order parameter changes for the first time. This represents the end of the transient dynamics that led to the solution. To calculate the time to solution, we choose the difference between the time when the transient dynamics have finished and the time where we started to ramp up the coupling strength.
Here is an example how the resulting plots look like:
Results for networks of oscillators that respond instantaneously and where no transmission time delays are present
In a network of mutually coupled oscillators with second harmonic injection locking (SHIL) we show that the Maximum cut problem can be reliably solved. The oscillators have an infinite modulation bandwidth and there are not time delays associated to the mutual coupling between the oscillator nodes of the network. We benchmark with the network topology shown in Fig. 7.
The correct MAX cut solution to this problem looks as shown in Fig. 8, where the different colors separate the two populations of nodes defined by the Maximum cut.
The result of the realizations shown in the plots were obtained for random initial conditions, choosing the initial phases of all oscillators as independent and identically distributed random variables from the interval . For some initial time we simulate the oscillators dynamics in the presence of noise and the SHIL perturbation with constant radHz. Then we ramp up the coupling strength of the mutual interaction terms from radHz to radHz, as shown in Fig. 9.
Given the ‘ground truth’ above, we hence expect that oscillators align to one group and the other to the group that is separated by from the first. The order parameter is then expected to be equal to zero. Here we show the plots of the phase differences with respect to the first oscillator in the system, see Fig. 10. Note that in the plots the oscillator numbering starts from and ends at .
The order parameter plots associated to the phase difference plots in Fig. 10 reveal that there are indeed two populations with oscillators in each. The populations are separated by a phase-difference and hence the order parameter , see Fig. 11.
An overview plot of all the order parameters for all realizations and the ‘statistics’ (it’s only 18 realizations) concludes that all cases were solved correctly by the relaxation dynamics, see Fig. 12. Note that the automatized measurement of the relaxation time, i.e., the time to solution failed here due to the fluctuations of the order parameter. This feature will be reworked in the next iteration.
Hence we have to rework the way the automatic estimation of the time to solution is performed. We will use smoothing functions in the future to obtain a moving average that accounts for boundary effects. These results show that with the right choice of parameters , , and , the slope of the linear ramp the network of mutually coupled oscillators solves the Maximum cut problem correctly with probability one.
The results shown so far correspond to the following architecture design schematic of an oscillator with SHIL, see next Fig. XY.
Adding inert oscillator responses
In that case we obtain the following set of dynamical equations
where the same notation holds as provided following Eq. (1). Note that in the case presented here, the SHIL signal is also filtered by the LF that filters the cross-coupling signals between the oscillators, see schematic Fig. XY1. This is an architecture design choice. We will later also discuss the case where the SHIL signal is added after the LF and is not filtered by the low pass.
Adding inert oscillator responses and transmission time delay
In that case we obtain the following set of dynamical equations
where the same notation holds as provided following Eq. (1). Note that also in the case presented here, the SHIL signal is also filtered by the LF that filters the cross-coupling signals between the oscillators, see schematic Fig. XY2. This is an architecture design choice. We will later also discuss the case where the SHIL signal is added after the LF and is not filtered by the low pass.
Inert oscillator responses, transmission time delay and no LF filtering of the SHIL signal
In this case the SHIL signal is not fed through the LF together with the cross-coupling signals, however instead there is a band-pass filter that ensures that only the second harmonic contributions from the self-mixing of the oscillators passes to the control signal, see Fig. XY3.


Dynamic Arithmetic Optimization Algorithm for Truss Optimization Under Natural Frequency Constraints
Nima Khodadadi, Snasel Vaclav, Seyedali Mirjalili
27 January 2022
Metaheuristic algorithms have successfully been used to solve any type of optimization problem in the field of structural engineering. The newly proposed Arithmetic Optimization Algorithm (AOA) has recently been presented for mathematical problems. The AOA is a metaheuristic that uses the main arithmetic operators’ distribution behavior, such as multiplication, division, subtraction, and addition in mathematics. In this paper, a dynamic version of the arithmetic optimization algorithm (DAOA) is presented. During an optimization process, a new candidate solution change to regulate exploration and exploitation in a dynamic version in each iteration. The most remarkable attribute of DAOA is that it does not need to make any effort to preliminary fine-tuning parameters relative to the most present metaheuristic. Also, the new accelerator functions are added for a better search phase. To evaluate the performance of both the AOA and its dynamic version, minimizing the weight of several truss structures under frequency bound is tested. These algorithms ’ efficiency is obtained by five classical engineering problems and optimizing different truss structures under various loading conditions and limitations.


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Terahertz laser technologyGrade 3 » Operations & Algebraic Thinking » Understand properties of multiplication and the relationship between multiplication and division. » 5 Print this page. Apply properties of operations as strategies to multiply and divide. 2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication ...
- 1. Commutative Property The order in which you add or multiply numbers does not change their sum or product. a + b = b + a Example: 7+2=2+7 ab = ba Example: 3 x 2 = 2 x 3
- Grade 3 » Operations & Algebraic Thinking » Understand properties of multiplication and the relationship between multiplication and division. » 5 Print this page. Apply properties of operations as strategies to multiply and divide. 2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication ...
- Free printable Commutative and Associative properties of addition practice examples worksheet. with answers key. Associative property of addition worksheets
- Multiplication is commutative because 2 × 7 is the same as 7 × 2. In other words. 2 × 7 = 7 × 2. The associative property. An operation is associative when you can apply it, using parentheses, in different groupings of numbers and still expect the same result. The two Big Four operations that are associative are addition and multiplication.
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Studying for the GRE
This isn't necessarily a post about music, but as I am taking the Graduate Record Examine(GRE) next Monday in order to complete my application to Yale I thought I'd include some of my thoughts about the study process here.
The GRE is a test of analytical and verbal skills. As I have been writing term papers for university, posts on this blog and various other articles, I'm not particularly concerned about the verbal side of the exam. I have gone over some of the topics for the Analysis portion (analysing a short statement). It's the analytical portion of the test (maths) that has me concerned.
I understand all the math necessary for the test, but a good portion of it I've not used since I was in high school (nearly 30 years ago). The biggest concern for me is the quadradic equation. I remember working with the quadradic equation back in Algebra 2, but until I was reviewing the study guide for the GRE, I hadn't thought about it since then. If you'd asked me the formula for solving a quadratic equation I wouldn't have been able to tell you. I'm sure, at some point in class, we went through the process of solving quadratic equations, and proving the formula works, but going back over the information now, it is amazing to me that anyone ever thought of it in the first place.
Not limiting my research to the study guide, I took some time to look up quadratic equations on the internet and was stunned to find mathmeticians have been solving these problems for thousands of years. This is hardly new math. The Babylonians were working algebra back in 1800BC, with Sulba Sutras in India solving more difficult quadratic equations in 800BC.
I have no idea how well I'll do on the text next week, but I am enjoying the relearning of some of the stuff I learned in high school. I don't know how relevant it will be to my musical education, but perhaps taking a second look at it will mean I will retain it better this time.


As a baseball fan, I like the idea of this, but if we're going to have an "objective" ranking system that goes beyond simple win% I really do want to know what goes into these numbers, and I suspect others will too. If you could explain how you arrived at these numbers that would be great. I can sort of get the picture from your strikethrough section, but it's not completely clear. The Awesomest (talk) 17:05, May 18, 2013 (UTC)
Easily done! Talk page is probably the best place for this anyway.
So, it's a modified version of the Elo rating system that is used in Chess, with a base/starting ranking of 1000, a K-factor of 200, and an expectation scaling factor (I forgot the technical term for it) of 500.
When a match is played, it calculates the expected win probability of each wrestler, based on the formula:
Win% = 1 / (1 + 10 ^ ( (rating_opponent - rating_self) / 500 )
That 500 is the expectation scaling factor mentioned above. So, if someone who is at a rating of 1000 plays someone with a 1500 rating, their win % is
1 / (1 + 10 ^ ( (1500 - 1000) / 500) = 1 / (1 + 10) = 1/11, or 0.091. You can do the same thing with all pairs of ratings, if they're the same then both have a 0.5 chance of winning.
One person wins and one person doesn't. The one that wins, has an Actual Win% of 1, the one that doesn't, 0. Each wrestler's rating is updated with the formula
New Rating = Old Rating + 200 * (Actual Win% - Expected Win%)
In the example above, if the person who was 500 points worse going into the match won, his new rating would be
1000 + 200 * (1 - 0.091) = 1181.8
That's the core of the system. I give extra weight to #1 contender matches (150% as much of a swing, so effectively their K is 300), to title matches (200% as much, K~400) and to more-than-2 wrestler fights (also 150%). Matches with more than two wrestlers are a bit tetchy, but how I'm handling them here is by treating them as a match with the average opponent's ranking; that is, if a 1000, 1200, 1100, and 1300-rated wrestler are in a four way, it's treated as though the 1000-rated wrestler is fighting someone with an (average of 1100, 1200, 1300 = 1200) rated wrestler, the 1300-rated wrestler is fighting someone with a 1100 rating, and so on.
This is fantastic stuff! Perhaps we can covert these numbers to moneyline odds (for all those who are sportsbettors like me) and set up a vBookie system so we can bet our hard earned Bison dollars on some of these matches. Uknowwhatitis (talk) 19:59, May 18, 2013 (UTC)
Here's another question: you say you don't count handicap matches. I assume you mean handicap matches that end in losses? In the event that, say, Barret had won over The Practice, how would you have ruled that? Django Boigas (talk) 20:08, May 19, 2013 (UTC)
I don't have an account on the SA forums, but based on the % to win posted there, here would be the sportsbetting moneyline odds for both matchups in the "GurlGamer" #1 contenders tournament (if there was a sports bookie setting odds and stuff):
Terra with a 68% chance to win:
Terra -213/Jessie+203 (in sportsbetting talk, -213 means you are betting $2.13 to win $1; for Jessie, +203 means you are betting $1 to win $2.03).
Daisy with a 55% chance to win:
Daisy -122/Morrigan +112 (For Daisy, -122 means you are betting $1.22 to win $1; for Morrigan, +112 means you are betting $1 to win $1.12).


8:30 a.m. - 2004-10-31
Yesterday I spent a lot of the day trying to rectify the calculations on two sweaters, my T-Top with kimono sleeves, and Abby's short-sleeved raglan (Berroco Book 207, Sylvia). Finally, I found a solution that involved me having some computer skills (I know, scary). I loaded a program from the Sweater Wizard folks called Print-A-Grid and printed out graph paper for the original gauge and for the 7 st/inch gauge I wanted to change it to. I printed in half scale so I made 14/18 paper and 9/11 paper. I drew the original instructions out in the original gauge, cut out the pattern piece, put this on top of the smaller gauge graph paper, and drew around it. Then I neatened up the lines and figured out where the decreases etc were. Much faster and it worked! I really like that Print-A-Grid, which I bought last spring at Stitches but did not install until now.
I made rolls yesterday, but discovered that the knitting machine sound makes it impossible to hear the timer, so they are a little brown, but still edible. I made a loaf of bread, which came out fine. He asked me to make him some cookies so I am going to try out a recipe from Have Your Cake And Eat It Too for oatmeal cookies. Other that that, I did a mountain of laundry, most of which still needs to be put away.
There are some cold artichokes in the fridge so I think I will have some for breakfast.
previous - next


Maybe if I wasn’t so weird I’d be better at math, but numbers are the definition of insanity. I can’t think of a caveman sitting around just looking at a rock and deciding, “this is one thing, it could be a part of infinite things, and it is made up of a finite amount of infinite things.”
Oh. How do these things come to be? Is it a flaw or is it a twisted intention? Are we close to discovering the secrets of the universe or have we made up a diversion to keep ourselves busy? What do you we actually know?
Oh how I really don’t want to do math.
School is starting.


They balance out their numbers by mixing them up. They don’t use all their numbers in the same number group and they don’t use triple numbers. The winners look at a pattern of what numbers hit in the past several weeks and they track the numbers by playing at least 80% of winning numbers.
Once you really understand about it, the things that have been holding you back fade away. Let’s have a look at the 3 main reasons that people decide to have a online gambling strategy.
If you follow the frequency theory, you need to study the game and choose the “hot” numbers instead of “cold” numbers. “Hot” numbers are numbers which have been drawn most frequently based on the past results. On the other hand, “cold” numbers are numbers which are least drawn in the past.
You are visiting casinos more and more frequently. You have tried to stop gambling by promising yourself that you would not go anymore, but it never works. You keep going back to the casino, regardless of the negative consequences.
They are patient and persistent and they don’t miss a drawing. They play on the days that are the least busiest. So if their lottery is drawn twice a week, they play on a weekday. That’s because the payouts are larger on those days.
The Hoosier lottery that uses Indiana State’s nickname, carries a 6/48. Michigan lottery is 6/47, Arizona Lottery and Missouri lottery are 6/44, Maryland Lottery is 6/43, and Colorado Lottery is 6/42. Compare this to the Florida Lottery.
online gambling Then review the folders you’ve set up in your e-mail system. In Outlook, they’re on the left side of the screen. What have you been keeping over there? Be sure to clear out e-mails and folders you no longer need.
Third, apply mathematical rules. Based on the past statistics, numbers that have won the lottery in a previous week are not precluded from winning again in the following week. However, it is also a recognized fact that in a lottery game, the numbers are drawn randomly. This means every number has an equal chance of winning the lottery. If a number that has won the lottery a week before is drawn again this week, this means the chances for other numbers to win the game are reduced. This is something that doesn’t sit well with the numbers and random theory. So, while it is still possible for the same winning number to appear as winner again, the chances are not as good as numbers that have not won the game before.
Are these tasks and dates locked in stone? Of course not! Will you move them around? Absolutely! That’s the beauty and flexibility and power of the system.
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|HOW MANY MOLECULES OF AIR ARE IN THE ATMOSPHERE?
METEOROLOGIST JEFF HABY
Molecules are extremely small. Much of the air is composed of Nitrogen and Oxygen but there is also Argon, Carbon Dioxide, Water Vapor
and many other trace gases. The number of molecules in Earth’s atmosphere must be incredibly large. This writing aims to estimate the
amount of air molecules.
First we start by finding the mass of the atmosphere
Pressure = Force / Area
Pressure = 101,325 Pascals
Area = Area of Earth
The earth is a near sphere. A sphere has an area of 4*PI*radius^2. Earth has a radius of 6,371 km = 6,371,000 meters.
Area of Earth = 4*PI*6,371,000^2 meters^2 = 5.1 * 10^14 m^2
Force = Pressure * Area
Force = (101,325 Pascals * 5.1 * 10^14 m^2) = 5.17 * 10^19 Newtons
Force = mass * gravity
mass = Force / gravity
mass = 5.17 * 10^19 Newtons / 9.8 ms^-2 = 5.27 * 10^18 kilograms
Now that we have a mass in kilograms, we need to convert this number into grams
(5.27 * 10^18 kilograms) * (1000 g / 1 kg) = 5.27 * 10^21 grams
The molar mass of air is around 29 grams / mole
(5.27 * 10^21 grams) * (1 mole / 29 grams) = 1.81 * 10^20 moles
Finally, multiply by Avogadro ’s number to convert moles to molecules
(1.81 * 10^20 moles) * (6.02214179*10^23 molecules/mole) = 1.09 * 10^44 molecules
Written out and rounded, there are about this many molecules of air in the atmosphere:
That is a lot of molecules!!


Tuesday, July 20, 2010
17 | Number Detective . 2
Hey there! I'm Chychy.
Welcome to Number Detective.
My beat is the city streets. The job is to bring you the best numbers we can find out there. Then share all sorts of cool stuff about them with you.
Wow. This is a really old looking number, isn't it? But it's only 17! I found it above a door of a church on Hanover Street in the North End. That's the Italian part of Boston.
17 has some mysteries about it, one of which is that it's the least random number. A random number is any number that just pops into your head.
Try this on your friends, the more the better - simply ask them to pick a number between 1 and 20. Write down all the answers then look at how many times each number was chosen. You might find out, as many people do, that 17 comes out way, way ahead.
That's surprising because each number should have an equal chance of being chosen, therefore making it appear about 5% (1/20th) of the time. Some tests, however, show 17 appearing almost 20% of the time - 4 times the expected rate! A huge difference.
No one really knows why 17 is the least-random random number. But the Number Detective will keep looking into it.


FF14 Mini Cactpot Calculator
Update: I made a web app for this program hosted on PythonAnywhere. Here’s the link: Mini Cactpot Calculator. It does the same thing as this program but it looks much better and is more user-friendly.
Final Fantasy 14 MMO has a minigame called Mini Cactpot The ticket plays out on a three by three matrix with numbers from one to nine. The interesting thing about the game is that you cannot reveal all the possible numbers on the ticket. You get a number for a new ticket and can scratch off three additional numbers. Therefore the ticket still has five hidden numbers that you have to guess.
Thoughts on the Game
The game piqued my curiosity. Since the ticket is just another form of a 3×3 matrix. There has to be a way to calculate the line with the best possible payout. So I thought to write the program myself. It was a challenge due to the fact that there is not a right solution to the problem. With many unknown variables, I can only use a brute force program to calculate the potential payout for each line. It is not perfect but it works.
In the long run, I believe you can get a better payout with the program than guessing randomly. In this post I’m only going to put the code. I’ll write about the algorithm of the game in separate posts. If you see how improvements can be made to the code, please let me know.


There is and has been a lot of debate about the magical number of five users for usability testing. Jeff explains the discussion from a statistical perspective.
The five user number comes from the number of users you would need to detect approximately 85% of the problems in an interface, given that the probability a user would encounter a problem is about 31%.
However, when you release a new product the proability that a user encounters a certain problem might be closer to 10%. In that case you would need up to 18 users to detect the problem.
The advice is to test with 5 users, solve problems and test again with 5 users.


A Slot is a machine that allows players to bet money on a random number generated by a random number generator (RNG). A standard slot machine pays out when a player gets the same number of spins as the machine’s jackpot. This means that if you are lucky enough to win the jackpot, you will get a payout that is one-fifth of the total bet. This means that a person can win big money while losing nothing, and there is no way to tell if a machine is cheating.
Unlike a roulette wheel, a slot machine’s pay table lists the credits that a player receives if several symbols line up on the screen. While some symbols represent many different other symbols, others are specific to one theme, and you must know what your machine pays for them all. Generally, the pay table is on the machine’s face, or above and below the wheels, while video slot machines show the pay table in the help menu.
The payout percentages of slot machines are based on long-term mathematical expectations, and the payout percentage is less than one hundred percent. Real slot machines tend to produce more winning combinations, but pay out less than a hundred percent of the money that a player spends. This percentage is known as the payback percentage. Modern machines use the same math, but use computer programs to determine the payouts. While this increases the player’s odds of winning, it also has its drawbacks.


Analysis of Random Fields Using CompRandFld
Adler , Eliran Subag , and Jonathan E. Taylor More by Robert J. Adler Search this author in:.
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We provide a new approach, along with extensions, to results in two important papers of Worsley, Siegmund and coworkers closely tied to the statistical analysis of fMRI functional magnetic resonance imaging brain data. These papers studied approximations for the exceedence probabilities of scale and rotation space random fields, the latter playing an important role in the statistical analysis of fMRI data.
The techniques used there came either from the Euler characteristic heuristic or via tube formulae, and to a large extent were carefully attuned to the specific examples of the paper.
This paper treats the same problem, but via calculations based on the so-called Gaussian kinematic formula. This allows for extensions of the Worsley—Siegmund results to a wide class of non-Gaussian cases. In addition, it allows one to obtain results for rotation space random fields in any dimension via reasonably straightforward Riemannian geometric calculations.
Modeling and statistical analysis of non-Gaussian random fields with heavy-tailed distributions
Previously only the two-dimensional case could be covered, and then only via computer algebra. By adopting this more structured approach to this particular problem, a solution path for other, related problems becomes clearer. Source Ann. Zentralblatt MATH identifier Sign In. Access provided by: anon Sign Out.
Stochastic Geometry, Spatial Statistics and Random Fields
Restriction of a Markov random field on a graph and multiresolution statistical image modeling Abstract: The association of statistical models and multiresolution data analysis in a consistent and tractable mathematical framework remains an intricate theoretical and practical issue. Several consistent approaches have been proposed previously to combine Markov random field MRF models and multiresolution algorithms in image analysis: renormalization group, subsampling of stochastic processes, MRFs defined on trees or pyramids, etc.
- Extremes in Random Fields: A Theory and Its Applications.
- Black Bodies and Quantum Cats - Tales from the Annals of Physics!
- The Best Man (Blue Heron, Book 1).
- Random field - Wikipedia;
For the simulation or a practical use of these models in statistical estimation, an important issue is the preservation of the local Markovian property of the representation at the different resolution levels. It is shown that this key problem may be studied by considering the restriction of a Markov random field defined on some simple finite nondirected graph to a part of its original site set. Several general properties of the restricted field are derived.
The general form of the distribution of the restriction is given. Sufficient conditions for the new neighborhood structure to be "minimal" are derived. Several consequences of these general results related to various "multiresolution" MRF-based modeling approaches in image analysis are presented.


This low-prep activity is an engaging way for students to practice working with systems of equations! Students become suspects after a hotel staff member is murdered. The murderer left a cluesheet and students must work together, solving the systems of equations problems on the cluesheet to figure out who the murderer is! The clue sheet includes a variety of problems, including solving systems from word problems, solving by graphing, matrices, and substitution.
Students are given character cards when they enter the room and assume the role of a bus passenger. Their leisurely trip is brought to a sudden halt when one of the passengers is found dead. This product is aligned to CCSS 8.
Murder Mystery! Solving Equations Mystery! Teachers Pay Teachers is an online marketplace where teachers buy and sell original educational materials.
Solve Systems of Equations by Graphing (Lesson Plan with Homework)
Are you getting the free resources, updates, and special offers we send out every week in our teacher newsletter? All Categories. Grade Level. Resource Type. Log In Join Us. View Wish List View Cart. Systems of Equations Activity! Mini Mystery.
View Preview. MathAlgebra. Grade Levels. ActivitiesFun StuffGames. File Type. CCSS 8. Also included in:. CCSS Aligned! View Bundle. Algebra 1 Activities Bundle! This Algebra I activity bundle will keep your students engaged throughout the year with 7 Mystery Activities! Product Description Standards NEW This low-prep activity is an engaging way for students to practice working with systems of equations!
Log in to see state-specific standards only available in the US. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.
Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.By Mary Jane Sterling.
Solving Systems of Equations
A solution of a system of two linear equations consists of the values of x and y that make both of the equations true — at the same time. Graphically, the solution is the point where the two lines intersect. The two most frequently used methods for solving systems of linear equations are elimination and substitution:.
Elimination also called add-subtract : This method involves adding the two equations together — or multiples of the two equations — so that in the sum, the coefficient of one of the variables becomes 0. That variable drops out is eliminatedso you can solve for the other variable.
Then you plug the solution back into one of the original equations and solve for the variable you eliminated. Substitution: This method has you set one of the equations equal to x or y. You can then substitute the equivalent of the variable from one equation for that variable into the other equation. You end up with a single-variable equation, which you can solve. Then plug that answer into one of the original equations and solve for the other variable.
You can use either method to solve linear systems, and you choose one over the other if a method seems to work better in a particular system substitution works best if the coefficient on one of the variables is 1 or —1.
The following examples show the same system of equations solved using both methods. You choose the number —2 as a multiplier because it makes the coefficient of the x term in the first equation equal to —2, while the coefficient on x in the second equation is 2.Iis reverse proxy outbound rule
The numbers —2 and 2 are opposites, so adding the equations together eliminates the x term:. You can also solve for the x -value by putting the 3 into the second equation — you get the same result. To use substitution, select a variable in one of the equations with a coefficient of 1 or —1.
The only variable that qualifies in this system is x in the first equation. Solve for x in terms of y in that equation. Substitute that equivalent of x into the second equation. That answer should look familiar. Multiply the terms in the first equation by 2 and the terms in the second equation by 3.
As a result, you end up adding —6 y and 6 y together, which eliminates the y terms when you add the two equations. The second equation is already solved for y. Solving Two Linear Equations Algebraically.Solve systems of linear equations using substitution when one equation is already solved for a variable. Turn content from Match Fishtank lessons into custom handouts for students in just a few clicks.
Download Sample. The essential concepts students need to demonstrate or understand to achieve the lesson objective. Understand substitution as an algebraic approach to solving a system of equations. Understand that the method of substitution replaces one of the variables in the system with an expression in order to solve for the remaining variable. Look for and make use of the structure of equations in a system to identify which expression to substitute into which equation MP. Use substitution to solve systems of equations.
This is the first of three lessons on substitution. In Lesson 5, students focus on the concepts behind substitution and why it works. In Lesson 6, students will solve an equation first before substituting, and they will look at examples with no or infinite solutions. Finally in Lesson 7, students will apply this strategy of solving systems to real-world applications. Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding.
Before you substitute, how many variables do you have? Can you solve for the value of either variable when you have two unknowns? Can you solve for the value of the single unknown? This Anchor Problem introduces students to the idea of using substitution to create a single-variable equation that can be solved. Students are familiar with substituting numerical values in for variables; in this Anchor Problem, they extend on that idea to substitute in an algebraic expression.
Solve the system below using substitution. Write your final answer as a coordinate point.In some word problems, we may need to translate the sentences into more than one equation. If we have two unknown variables then we would need at least two equations to solve the variable.
In general, if we have n unknown variables then we would need at least n equations to solve the variable. In the Substitution Method, we isolate one of the variables in one of the equations and substitute the results in the other equation.Simultaneous Equations Math Lesson
We usually try to choose the equation where the coefficient of a variable is 1 and isolate that variable. This is to avoid dealing with fractions whenever possible. If none of the variables has a coefficient of 1 then you may want to consider the Addition Method or Elimination Method. Steps to solving Systems of Equations by Substitution: 1. Isolate a variable in one of the equations.
Substitute the isolated variable in the other equation. This will result in an equation with one variable. Solve the equation. Substitute the solution from step 3 into another equation to solve for the other variable.Cisco ucl basic licensing
Recommended: Check the solution. Step 1: Try to choose the equation where the coefficient of a variable is 1. Step 2: From equation 3we know that y is the same as 3 x - 8. Step 3: Remove brackets using distributive property. Step 4: Combine like terms. Step 5: Isolate variable x. Step 7: Check your answer with equation 1.
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations. We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. Related Topics: Worksheets to practice solving systems of equations More Algebra Lessons These algebra lessons introduce the technique of solving systems of equations by substitution.
The following example show the steps to solve a system of equations using the substitution method. Scroll down the page for more examples and solutions.
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.Learn Zillion Instructional Videos. Graphing to solve systems of equations 8.
Solve pairs of simultaneous linear equations; understand why solutions correspond to points of intersection 8. Analyze and solve pairs of simultaneous linear equations; solve systems in two equations algebraically 8. Guided Practice. IXL Lessons.
Make sure you sign-in at the IXL website prior to clicking the specific lessons. Link to sign-in. Ready Common Core. Search this site. Grade 5. Grade 6. Grade 7.
Systems of Linear Equations
Grade 8. Student Book pages - C Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Learn Zillion Instructional Videos Graphing to solve systems of equations 8.
Interactive Manipulatives. Click "Select Grade" to enter 8th Grade Click "Backgrounds" if you want to have a graphic organizer in the background. Optional Click "Manipulatives" to select the type of manipulatives. Solve a system of equations by graphing: word problems Eighth grade - Y.Today we are going to extend solving systems of linear equations to non-linear equations.
I begin the lesson by giving students a linear system solve.
Solving Systems of Equations by Substitution
As students work I identify the methods used by students. Most students either rearrange the equations to put the equations into a calculator or solve by elimination also called linear combination. After a few minutes I have students share their process in solving. I give students the names of the 3 methods learned in Algebra 1 and 2. Namely elimination, graphing and substitution.Oppo k1 fingerprint fast kaise kare
I ask students to identify which process is demonstrated on the board. Since I rarely have students who do substitution on this problem we go through the substitution method on the problem. I ask students why substitution is not the most efficient method for this system.
Now that we have reviewed solving systems I give students a problem with one equation being non linear. I ask students to decide how they can solve the problem. Students discuss ideas with each other and some start to work on the problem. After a couple of minutes I have students share ideas.
Some solve the second equation for y and then substitute. Other try elimination by subtracting the second equation from the first. Students that have started working the problem put their process on the board for discussion. To help students see how the graphs intersect, I graph page 2 the functions in Desmos. Desmos allows the equations to be typed without rearranging which helps in the class.
We identify the points of intersection and use the graphs to verify what happens in the algebra.Empty Layer.
Home Professional Learning. Professional Learning. Learn more about. Sign Up Log In. English Language Arts. Blended Learning. Measurement and Data. The Number System. Eighth Grade The Number System. The Complex Number System. HS Functions Interpreting Functions.
Building Functions. Trigonometric Functions. HS Geometry Congruence. Expressing Geometric Properties with Equations. Modeling with Geometry.
Using Probability to Make Decisions. Eighth Grade. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
Big Idea: Students will identify zero pair coefficients to solve systems of equations by adding. Standards 8.
Big Idea: Students move about in a scavenger hunt, solving systems using different algebraic methods. Big Idea: Students use data to create equations describing a concrete scenario.
Students analyze the equations as a system to solve a problem. Big Idea: Assess what students have learned about systems, and find where they may still need help conceptually. Elimination 1 - add or subtract only 8th Grade Math. Big Idea: Why does elimination work? Use sensitive nieces, peppermint bark, and gingerbread men to illustrate this mathematical concept.
Baseball Helmets Day 1 of 2 8th Grade Math. Big Idea: Connect solving systems of equations to real life baseball helmets and reading tables to find the equations for solving by substitution. Elimination 4 - multiply one equation only 8th Grade Math. Big Idea: More practice!Png images hd
Get kids fluent in solving systems using elimination
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CoinDCX Pro will support the Monthly NPXS Token Unlock Program, the calculations will be made on a daily basis.
We will take a snapshot of the number of NPXS tokens held by each user at 11:59:59 pm every day of each month. The tokens that the user is eligible to receive, will be credited on the 15th day of the following month.
The tokens will automatically reflect in your CoinDCX Pro NPXS balance.
Monthly NPXS distribution program
The calculation for NPXS generated by each user, ie. the reward is as follows :
Daily NPXS generated by each user = Daily NPXS generated by CoinDCX Pro * Daily User NPXS holdings ratio*
Daily user NPXS holdings ratio* = Daily User NPXS holdings at the time of daily screenshot / Total NPXS held on CoinDCX Pro
To simplify this let’s take an example.
User A has 100 NPXS tokens reflecting as balance on CoinDCX Pro when the snapshot is taken at 11:59:59 pm on 08/10/2018, User A’s Daily NPXS generated would be calculated by the above formula.
Daily NPXS generated by User A = Daily NPXS generated by CoinDCX Pro * Daily User NPXS holdings ratio*
Daily NPXS holdings ratio for User A* = 100 (Daily User NPXS holdings at the time of daily screenshot) / Total NPXS held on CoinDCX Pro
If total NPXS held on CoinDCX Pro = 10000000000
Then Daily NPXS holdings ratio for User A* = 100/ 10000000000 = 0.00000001
If Daily NPXS generated by CoinDCX Pro = 100000
Hence Daily NPXS generated by User A = 100000 * 0.00000001 = 0.001 NPXS
User A’s daily NPXS generated for 08/10/2018 is 0.001.
- User’s with balance < 1 NPXS at the time of the snapshot will not be eligible for distribution.
- Distribution will be solely based on the formulae mentioned above and the snapshots clicked.
Disclaimer: The distribution will be based solely on the calculation and discretion of the team.


Piti With Pmi Calculator PITI – Official Site – PITI calculator calculates your monthly mortgage payment with principal, interest, taxes, insurance and PMI if needed. Calculator provides mortgage amortization schedule. learn if you qualify for the full piti mortgage payment based on selected mortgage amount and interest rate.
Compound Interest Calculator | Tools and Calculators. – Calculate Compounded Amount (Hint: Enter the opening balance in the "Principal" box below to calculate what the compounded amount will be.): Calculate Opening Balance (Hint: Enter the compounded amount in the "Principal" box below to calculate the opening balance you need to obtain your goal.): Base Calculations on a 360-Day Year: Base Calculations on a 365-Day Year
Commercial Property Refinance Rates Property Investment Calculator – Home Loan Experts – Will your investment property be positively or negatively geared? Calculate your weekly profit or loss with our property investment calculator.
All of our products calculate interest daily. Ending Balance: The calculator will calculate the ending balance, including interest earned, after you enter the Initial Deposit and Interest Rate, then select Compounded Daily. Annual Percentage Yield (APY): The calculator will calculate the APY, which is the actual interest earned per year.
The Ally Bank Savings Interest Calculator | Ally – As your savings grows, you want to be sure it’s working as hard as it can for you by earning a great interest rate. A savings interest calculator like this one from Ally Bank can help you understand what you can expect to earn on your deposit. Using the savings interest calculator is simple.
30 Year Mortgage On 150 000 You have a loan for $150,000 at 5% on a 30-year mortgage. You. – Exactly five years ago, you obtained a $150,000 30-year mortgage with a fixed rate of 10%. Today, you can get a 30-year loan for the currently outstanding loan . asked by Bob on January 20, 2016; MATH. a borrower received a 30 year arm mortgage loan for 200,000.
If your bank calculates interest monthly and adds earnings to your account monthly, as many banks do, a simple interest calculation is not accurate. Calculate compound interest compounding happens when you earn interest, and then you earn even more interest on the interest earnings you previously received.
Savings Account Interest Calculator with Dynamic Growth Chart – Savings Account Interest Calculator. This calculator will calculate the compound interest earnings on saving accounts given the rate, length of time, initial deposit, periodic deposits, and compounding frequency (including daily compounding). The learn tab includes: How a savings account can save you money and worries.
Student loan interest rates to drop – The new fixed interest rates are dropping by more than half a percentage point from a year ago. And while about 90% of college borrowers rely on Uncle Sam, students who are applying to banks.
Loan Calculator – Free loan calculator to determine repayment plan, interest cost, and amortization schedule of conventional amortized loans, deferred payment loans, and bonds. Also, learn more about different types of loans, experiment with other loan calculators, or explore other calculators addressing finance, math, fitness, health, and many more.
Debt Service Payment Calculator DSCR Calculator – Calculate Debt Service Coverage Ratio – Debt service coverage ratio (DSCR) is the ratio of cash accessible for servicing a loan or an entity’s debt. It is used to measure an entity’s capability to pay off a loan. A higher ratio makes it easier to obtain a loan.
Compound Interest Calculator (Daily, Monthly, Yearly Compounding) – Calculate compound interest on your savings or investments and discover how. on this, so you may wish to check with your bank or financial institution to find.