text:
NeurIPS 2020

### Review 1

Summary and Contributions: The authors present a learning algorithm that computes dense correspondences between full and partial 3D human body shapes. Initial correspondences are extracted from a partial input shape via a point cloud feature extractor and matched to the Laplacian embedding of a human template shape via a 6-dim. rigid parameterization per point. The noisy initial matches are then refined using a transformation synchronization module and the whole network is trained end-to-end.

Strengths: - Overall, the quantitative results show a convincing improvement over the baselines (see however my concerns under weaknesses), in particular in combination with the results from Table 2 in the supplementary. - The L1 regularized refinement module and the corresponding IRLS algorithm seem to be a well conceived choice for human shapes with parts of piecewise constant rotations. - In terms of time complexity, the method is extremely efficient in comparison with related work. Overall, I am inclined to give this work a positive initial rating.

Weaknesses: - Regarding the comparisons in Table 1., I would have liked to see comparisons with [9], [10] and/or [11]. SMPL and ASAP are good, but they are blended with an initialization from the authors' method and a comparison with DHBC alone is not completely convincing. - The method requires the ground truth correspondences as supervision which is a strong assumption. Many related methods ([11, 13, 40] in the paper) investigated the unsupervised setting at least to some degree and therefore allow for training on non-synthetic data. - The authors decided to limit themselves to human shapes in this work, although there are no modelling assumption that clearly require this. While I can to some extent understand this decision, it would still be interesting to see some qualitative generalization results to other classes of shapes. I understand that your method requires the human template for training, but similar methods [11] also show at least some examples that are not human or only close to human shape.

Correctness: Yes.

Clarity: Yes.

Relation to Prior Work: Yes.

Reproducibility: Yes

Additional Feedback: - The authors should consider to upload their results in Table 2 of the appendix to the website of the MPI FAUST challenge at http://faust.is.tue.mpg.de/ (this can be done anonymously). I understand that the FAUST templates and scans are two separate datasets but judging from your results on the partial views, applying your network to the scans should be straightforward and it would be very instructive for the community. - Missing citation: The idea of modelling shape deformations in terms of locally rigid transformations is closely related to as-rigid-as-possible surface modelling [1]. In particular I believe that parts of the theory from Appendix D should be credited to [1]. - I would prefer to see the cumulative error curves for the correspondence accuracies instead of the average and recall values in Table 1 as they are a more expressive tool for comparing different results. - In your ablation study, I would like to see a comparison of the Laplacian embedding with the SHOT [2] descriptors which are the standard hand-crafted descriptor that is typically used in combination with deep architectures on 3D shapes. The HKS/WKS descriptors are purely intrinsic and not suitable for disambiguating symmetries etc. - I would suggest that the authors include qualitative examples that show a texture map between two full shapes. This is a standard tool to assess the quality of a matching between 3D shapes. [1] Sorkine, Olga, and Marc Alexa. "As-rigid-as-possible surface modeling." Symposium on Geometry processing. Vol. 4. 2007. [2] Salti, Samuele, Federico Tombari, and Luigi Di Stefano. "SHOT: Unique signatures of histograms for surface and texture description." Computer Vision and Image Understanding 125 (2014): 251-264.

### Review 2

Summary and Contributions: This paper proposes a neural method to establish dense correspondences between between incomplete human scans and a template human model. The key idea is to obtain correspondences by finding local rigid transformations at each point in the input scan. The method has two components - an initialization step followed by a regularization step. The initialization step involves computing dense correspondences to the template using laplacian feature descriptors and finding local transformation based on these coarse correspondences. The regularization steps involves another neural network module which refines the initial coarse rigid transformations. The network design for the second module is inspired by a classical variational approach which involves the use of iteratively reweighted least squares

Strengths: The paper introduces three ideas: 1) It formulates the task of dense correspondence computation in terms of finding rigid transformations from the scan to template 2) It includes a robust theoretical analysis of the a classical transformation synchronization approach 3) It uses insights from 2) to formulate transformation synchronization in a neural module. This enables the entire pipeline of correspondence estimation to be trained in an end-to-end manner. While the idea of deforming shapes using local rigid transformations is not new (first introduced in Sumner et al. SIGGRAPH 04), to my knowledge, this is the first time where it is used for correspondence prediction. The authors show state-of-the-art or comparable results on three publicly available datasets - FAUST, SURREAL, SHREC19. The authors also perform ablation studies to show that the regularization module improves upon the initial correspondences. They also compare the neural network based regularization module with the classical regularization strategy and show the neural module performs better.

Weaknesses: The authors have only validated their method on human scans. It would have been more thorough to evaluate on meshes with different topologies. For example, they could have created a synthetic dataset of hands using the MANO model and evaluated their method on this dataset. Like [11] they could have trained their method on meshes generated using SMAL and used TOSCA animals for evaluation. I am not sure how practical the work is. People are usually scanned in clothing. The method is only evaluated on scans of people in minimal clothing. It would have been interesting to see whether, in addition to transformations due to pose and shape, the proposed method can also capture transformations due to clothing. The pipeline seems very complicated. I wonder how difficult it was to find all the hyperparameters to make the networks converge?

Correctness: The authors have used average correspondence error, 5cm, 10cm recall as metrics of choice. These are the standard metrics used for correspondence estimation.

Clarity: The paper is very well written and clearly structured. There are a few typos : eq 12) The variable \Phi is introduced in the minimization but is never mentioned before. eq 11) the targets are set to v_{c_i} but according to line 159 v \in R^12 and p is a point in R^3. It is not clear how the weights in eq 3) are initialized.

Relation to Prior Work: The related work is excellent. All major correspondence techniques are mentioned. The main prior works - SMPL registration, DHBC, 3D-CODED, DeepGeoFunc are all compared to, even though DeepGeoFunc was published at CVPR 2020

Reproducibility: Yes

Additional Feedback: The authors claim that the method is end-to-end trainable and yet in line 196 they mention that initial transformations are required for the regularization module. This could also be interpreted to mean that the initialization network and regularization network are trained separately. This ambiguity should be cleared. Figure 1 could be made cleaner by explicitly showing the two up-sampling and two down-sampling layers. The authors use the SURREAL dataset for training the method. AMASS - Mahmood et al. ICCV 2019- is a much larger dataset which should improve the proposed method. ----After rebuttal ----- I have read the rebuttal and the other reviews. The rebuttal addresses most of the points I had raised. The paper proposes a method with enough novelty to warrant an acceptance. I keep my score unchanged.

### Review 3

Summary and Contributions: This paper studies the problem of establishing dense correspondences between partial scans of human models and a complete template model. The paper proposes a network-based framework to first initialize the correspondences, and then synchronize the transformations of different points to enforce smoothness and recover deformations. The network design is motivated by a principled optimization method.

Strengths: - The problem is well motivated. - The solution applies not only to 3D human body matching. It is a general method for 3D partial matching, although the network needs to be re-learned. - The experimental results are promising, and the ablation study is thorough.

Weaknesses: - It seems that the features descriptor is very important for the method. L137-138: Is the feature descriptor a one-to-one mapping? Is it possible that very different inputs could collapse into a cluster? Do we have a way to measure the robustness of the descriptor in this sense, so we are more confident to initialize the transformation using this descriptor? - In the objective function of the synchronization process (Eq 3), why don't we optimize the point cloud distance? I can see that the first term is optimizing that implicitly, but why not explicitly? - L165: what is the motivation to adopt iteratively reweighted least squares for optimization?

Correctness: The method seems to be correct.

Clarity: Overall it's reasonably clear, but it could be further improved. Currently Sec 3.2 and 3.4 are about network modules, while Sec 3.3 is about the principles that motivate the network. It reads a little fragmented.

Relation to Prior Work: Yes, it is discussed in details in the related work section.

Reproducibility: No

Additional Feedback: Although there are missing details about the implementation, it is not a concern since code is included in the supplementary materials. ========== After rebuttal ========== I read the rebuttal and comments from other reviewers, and I am leaning towards acceptance.

### Review 4

Summary and Contributions: This work proposes a method to learn dense human correspondence in the case of partial scan to full scan (full-to-full also doable). The pipeline has two major components: 1. Initialization: using learnt per-point descriptors to drive weighted local rigid fitting. 2. Regularization (refinement): convert each step of a traditional optimization process into a recurrent graph network, while the alternating updates (eq. 5 and 6) are done by network operations. The regularization intuitively behaves like "as rigid as possible" constraints. The proposed method shows its performance in correspondence, and advantage over compared to sota methods.

Strengths: The design of the refinement (“regularization” as mentioned by the authors) contains some novelty. It also has connection to the traditional optimization steps and can be regarded as a neural network version of the optimization updates. From quantitative results, it shows the advantages over the previous methods.

Weaknesses: I would suggest the color mapping visualization for matching error (e.g. in Fig. 2 and 3) to be adjusted. The matching error is a non-negative scalar, therefore there is no need to use a diverging colormap that is intended for signed distance. It doesn’t visualize the error in a “perceptually linear” way. For example, the fig.2-right, it’s very hard to see the difference unless with extreme zoom-in. Maybe a “sequential” colormap such as “jet” or “hot”. See https://matplotlib.org/3.1.0/tutorials/colors/colormaps.html It seems like the “canonical” descriptor of the template mesh is provided by the laplacian embedding, and the learned descriptor ($f_{\Theta}(S)$) is “regularized” by the canonical descriptor. What happens when you learn the canonical descriptor freely? Maybe some ablation studies would be good to show the effect of this design. The representation for per-point transformation “v_i” is “scaled” rotation matrix elements and translation vectors. This appears a bit odd to me. First, if we operate directly on the elements, then the resulting matrix might not be a valid rotation. In this case, did you project it back to SO(3), or just allow it to be flexible so that the resulting transformation is outside of SE(3)? The recurrent network serves as a neural network version of iterative update as in the traditional optimization. Is the proposed 4x3=12 steps (4 graph layer, each with 3 alternations after) sufficient? It would be great to show results as each step is applied (to see their individual contribution). What if the graph network doesn’t do down-sampling and upsampling? Would it be possible to add discussion and comparison to recent point-cloud correspondence analysis, such as Fully Convolutional Geometric Features, Choy et al, ICCV 19? The Perfect Match: 3D Point Cloud Matching with Smoothed Densities, Gojcic et al., CVPR 19

Correctness: Mostly, except for there are some parts which are still not clear (see weakness session)

Clarity: Mostly yes, except for there are some parts which are still not clear (see weakness session)

Relation to Prior Work: I would add discussion on difference to recently point-cloud correspondence analysis papers such as: Fully Convolutional Geometric Features, Choy et al, ICCV 19 The Perfect Match: 3D Point Cloud Matching with Smoothed Densities, Gojcic et al., CVPR 19 These methods are tested on static scenes, but can be potentially used in dynamic scenarios such as human body.

Reproducibility: Yes

Additional Feedback: After rebuttal: After reading the rebuttal and comments from other reviewers, I think most of my concerns are addressed. Please address the issues and comments raised by all reviewers. I'm willing to increase my rating to 6: marginally accept.


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# ATX KENTEK KT-PSK400b Schematic Diagram and Question

#### N11778

Joined Dec 4, 2015
174
Caution: I drew the diagrams from the boards so there will be some Errors.

I have a question for you ATX Experts.
What is the Small winding on the Driver transformer for?

I Like the cheepo way they got the 3.3volts, It puts a big load on the 5v supply.
So, the Supply is only $25 new on EBAY. The 1000uf filter Cap. C15 on the 5volts keeps going bad. One board is 2 years newer and they changed Supervisor Chip. Interesting fact: When C15 was going bad 200uf measured instead of 1000. The symptom was: When I put the Computer to sleep It would lock or crash. Worked fine the rest of the time. It would boot right back up OK. #### Attachments • 31.3 KB Views: 16 • 121 KB Views: 18 • 124.9 KB Views: 16 #### AlbertHall Joined Jun 4, 2014 11,313 What is the Small winding on the Driver transformer for? That transformer output provides an isolated supply for the supervisor IC. There are two windings that feed back to Q1 and Q2 base to sustain oscillation and then Q1 and Q2 drive the primary of T1. #### ebp Joined Feb 8, 2018 2,332 T2 is a base driver transformer. It is what is used to provide the base current to control the main switching transistors Q1 and Q2, It provides the galvanic isolation that is required between the low voltage circuitry and the high voltage circuitry. The topology appears to be a two-switch forward converter, which has some worthwhile advantages at moderate power levels. The driver transformer itself is being driven push-pull, which common with open collector/drain outputs on the control IC. The "small winding" appears to be for positive feedback for the drive of Q1, but I'm not really sure. The IC called PWM Control and associated parts form a conventional low-power flyback converter for the 5 volt standby supply and the power for the main switcher IC. #### AlbertHall Joined Jun 4, 2014 11,313 It's a startup oscillator, to produce the 5V stby, Nope. T3 and the circuit round it does the +5V standby. Thread Starter #### N11778 Joined Dec 4, 2015 174 The "small winding" appears to be for positive feedback for the drive of Q1, but I'm not really sure. It's in most designs so how about Neg. feedback to prevent runaway at higher power. I think I will put a 100 watt bulb is series with the input power and take some scope readings. Then Jumper the small winding to see the difference. For some reason typing this I'm getting a faint smell of electricity. Sparky #### Dodgydave Joined Jun 22, 2012 9,953 Nope. T3 and the circuit round it does the +5V standby. Ah yes a doh! moment there, the winding is to help the self oscillator startup and syncronising the main output together. Also there are two versions here, one uses a TL431 for the 3v3 output ,on the 5V line. #### ebp Joined Feb 8, 2018 2,332 I did consider the possibility of negative feedback for the main switch as a means of cycle-by-cycle current limiting. I'd need to look more carefully at the circuit. I'm not even entirely certain which side of the winding the phasing dot belongs to. If it is on the bottom as it at least marginally appears to be, then the phasing is correct to aid switching - in phase with Q1 and out of phase with Q2. But I'm really not sure why one would want to do that. The MJE13007 doesn't have especially high gain, so perhaps the effect would unburden the circuit on the input side of the drive transformer while still achieving fast, hard switching. I've used drive transformers in several switcher designs, but for FETs. I've never done anything other that very low power switchers with bipolars so I've never paid much attention to bipolar base drive circuits. There may be something in one of my reference books, a couple of which are old enough that bipolars were still being used in switchers. They seem to still be common where pennies count. The other thing that makes me think it isn't negative feedback is C3. With that cap the transformer primary can be saturated without causing disaster. Do remember that the entire input side of the switcher must be considered to be connected directly to mains "hot" for safety purposes and probing without an isolation transformer is very hazardous. Unless they just aren't shown, that switcher is remarkably devoid of EMI/RFI countermeasure components. Last edited: Thread Starter #### N11778 Joined Dec 4, 2015 174 that switcher is remarkably devoid of EMI/RFI countermeasure components.$25 you get what you pay for some times less. Check out the missing components. It does work though.
Takes about 1 year for the 5v filter cap to go bad.

#### Attachments

• 290.4 KB Views: 10

#### N11778

Joined Dec 4, 2015
174
the phasing is correct to aid switching - in phase with Q1 and out of phase with Q2
Thanks for the Help. Makes sense. Going to bag the scope testing.
The Board minus the Electrolytic caps is in an oven at 200 deg. F trying to get rid of the coating smell.
It gives me a severe head ache and Asthma.

format: plain


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SCIENTIA GEOGRAPHICA SINICA ›› 2015, Vol. 35 ›› Issue (11): 1460-1467.

• Orginal Article •

### Applicability of Different Probability Distributions to Estimated Extreme Rainfall

Yu-hu ZHANG1(), Chen-xi WANG1,2, Kai-li LIU3, Qiu-hua CHEN3

1. 1. College of Resources Environment and Tourism, Capital Normal University, Beijing, 100048, China
2. Institute of Surveying and City Spatial Information, Beijing University of Civil Engineering and Architecture, Beijing 100044, China
3. Institute of Math Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China
• Received:2014-09-17 Revised:2014-11-20 Online:2015-11-20 Published:2015-11-20

Abstract:

Heavy precipitation is a crucial nature factor of flood. The return period of the extreme value of precipitation depth is the most significant reference of the design standard of flood prevention facilities in an urban or a basin. In this article, the series of annual, summer and winter maxima of precipitation depths for 1-day, 2-day and 3-day durations measured at ten selected stations in China are analyzed, using five commonly used hydrological statistical distribution functions. The distribution functions applicable for these stations were measured using the Kolmogorov Smirnov (K-S) and the Anderson Darling (A-D) tests. The results show that: 1) The summer maxima series shows higher standard deviation and larger differences between distributions than other maxima series and the annual maxima occur mostly in summer; 2) The Generalized Extreme Value (GEV) distribution, the lognormal (LN) distribution and the Pearson III distribution perform were better in the imitative effect test of goodness of fit, and the degree of curve difference is smaller; 3) Differences between estimates of rainfall with return periods were shorter than 25 years are smaller; 4) Estimates of precipitation can change significantly depending on the probability distribution being used, particularly for the summer series; 5) Suitability curves present seasonal difference. By statistical analysis of precipitation maxima, the precipitation is concentrated in summer; due to the disperse and skewness of precipitation series, and the appropriate distribution functions are quite different in different season periods; 6) In some extreme rainfall sequences, two curves of linear fitting are almost the same. Even if the return period extending to 100 years, the difference quantity of two curves is only a few millimeters. In this situation, the results of small probability rainfall events are more reliable; 7) There are differences among 1-day, 2-day and 3-day durations of precipitation depths, the probability distribution of 1-day maximum precipitation fits better. When carrying out statistical analysis of hydro-meteorological extremes, various probability distribution function and test methods should be taken for calculating, to reduce the uncertainty of single calculation. In this study, experimental analysis of 10 sites demonstrated that the Pearson III is not suitable for all sites. It is suggested here that the estimation of extreme precipitation should take into consideration the range of extreme values estimated by the best-fit distributions identified by more than one test as an approach to assess uncertainties related to extreme rainfall analysis.

CLC Number:

• P426.615


text:
Paper

### A Resizable Mini-batch Gradient Descent based on a Multi-Armed Bandit

Determining the appropriate batch size for mini-batch gradient descent is always time consuming as it often relies on grid search. This paper considers a resizable mini-batch gradient descent (RMGD) algorithm based on a multi-armed bandit for achieving best performance in grid search by selecting an appropriate batch size at each epoch with a probability defined as a function of its previous success/failure... (read more)

Results in Papers With Code
(↓ scroll down to see all results)


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# iocoach.com

Home > Standard Error > Calculate P-value From Estimate And Standard Error

# Calculate P-value From Estimate And Standard Error

## Contents

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## How To Calculate Standard Error Of Estimate In Excel

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## How To Calculate Standard Error Of Estimate In Regression

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# Why won't Mathematica simplify tensor products fully?

I am working on a project and part of the output of my code is the following: $$\text{a1}\otimes\text{a2}+\text{a1}\otimes(\text{a1}\otimes\text{a1}-\text{a2})-\text{a1}\otimes\text{a1}\otimes\text{a1}.$$

To me this clearly simplifies to 0, but for some reason, Mathematica cannot see this even if I use Simplify or FullSimplify. Is there any way to help it to see this?

In[283]:= ExpandAll[ a1\[TensorProduct]a2 + a1\[TensorProduct](-a2 + a1\[TensorProduct]a1)
- a1\[TensorProduct]a1\[TensorProduct]a1]

Out[283]= a1\[TensorProduct]a2 + a1\[TensorProduct](-a2 + a1\[TensorProduct]a1)
- a1\[TensorProduct]a1\[TensorProduct]a1

• @rhermans I didn't put the raw code as it was very ugly and didn't show my point as well as an image. I have replaced with the code now. Aug 9 '18 at 10:16
• Still it's better that you share your ugly code than to ask us to guess and type the code for you. Using images it's fine provided you also provide plain text code (in InputForm) that we can Copy&Paste. Thanks for editing your question. Don't forget to take the tour Aug 9 '18 at 11:30

expr = a1\[TensorProduct]a2 +  a1\[TensorProduct](-a2 + a1\[TensorProduct]a1) -
a1\[TensorProduct]a1\[TensorProduct]a1

TensorExpand @ expr

0

• while your solution works its a bit disappointing that FullSimplify does not attempt TensorExpand? Aug 9 '18 at 14:42
• @chris, I agree. PowerExpand and ComplexExpand are also not used by FullSimplify because they "make special assumptions on input". Perhaps TensorExpand is not used for the same reason.
– kglr
Aug 9 '18 at 14:53
• @chris you can always manually add TensorExpand to TransformationFunctions: FullSimplify[expr, TransformationFunctions -> {Automatic, TensorExpand}]. Aug 10 '18 at 9:37


text:
Mathematics
OpenStudy (anonymous):

3/4 -(-5/8)

Parth (parthkohli):

$$\Large \color{Black}{\Rightarrow a-(-b) = a + b }$$

OpenStudy (anonymous):

HUH?

Parth (parthkohli):

This means: $$\Large \color{Black}{\Rightarrow {3 \over 4} - (-{5 \over 8}) = {3 \over 4} + {5 \over 8} }$$

Parth (parthkohli):

OpenStudy (anonymous):

8/12=2/3 BECAUSE 4 IS THE LCD IS THIS COREECT

Parth (parthkohli):

Hmm?

OpenStudy (anonymous):

WHAT U GAVE ME WAS THE ANSWER? OR WAS I STILL SUPPOSE TO SOLVE THAT

Parth (parthkohli):

You have to solve it further.

Parth (parthkohli):

And it'll not be 8/12.

Parth (parthkohli):

$$\Large \color{Black}{\Rightarrow {3 \over 4} +{5 \over 8} = {6 \over 8} + {5 \over 8} }$$

OpenStudy (anonymous):

Parth (parthkohli):

You have to add only the numerators.

Parth (parthkohli):

Please don't type caps lol it scares me

OpenStudy (anonymous):

im sorry didnt realize that was on, yeah that would make it seem like im screaming at you.

Parth (parthkohli):

So finally what do you get?

OpenStudy (anonymous):

11/8

Parth (parthkohli):

Hmm correct

OpenStudy (anonymous):

omg this stuff is hard i never understand anything when it comes to math

Parth (parthkohli):

Lol it's very very easy :)

OpenStudy (anonymous):

ok im about to put another question up let me know if to start solving the same way you began the last problem.


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# Classification of generously transitive groups

A permutation group $G \lt S_n$ is called generously transitive, if for each $i,j$ there exists a permutation that interchanges them. Is there a reasonable classification of such (finite) groups?

-
@oeter franek: The problem is that the parser interprets < as an HTML marker; use \lt for < and \gt for >. I've fixed the question. –  Arturo Magidin Feb 28 '13 at 15:54

## 1 Answer

I believe the answer is No, there is no good classification of these things. It might be helpful for you to know that a "generously transitive permutation group" is the same as a 2-star transitive group. (See for instance the introduction to "The k-star Property for Permutation Groups" by Clough, Praeger, Schneider.) I can email you a copy of this paper if you need it.

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电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|EECS262

statistics-lab™ 为您的留学生涯保驾护航 在代写嵌入式网络系统Embedded Networked Systems方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写嵌入式网络系统Embedded Networked Systems方面经验极为丰富，各种代写嵌入式网络系统Embedded Networked Systems相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

Tasks coordinate the work by communicating and synchronizing with each other, i.e. tasks are interconnected. The connections are called links and have flow semantics: flow of data messages.

Tasks have an interface consisting of a set of ports, i.e. these are the terminals of the links connecting the tasks. The ports are the only part of a task that is made visible to the outside world: the rest of a task is hidden from the outside world. We distinguish between data input ports and data output ports. All ports have a unique name within the context of the associated task. The input ports have a message queue (FIFO type); this can be used to model different types of synchronization behaviour. Supported are the asynchronous communication (queue size $>0$ ) and synchronous communication (queue size $=0$ ).

Inside the task the ports are visible, and can be associated with communication operations that are defined in the behavioral model. This way, ports can be used to separate the behavioral specification from the inter-task communication topology and synchronization specification.

To facilitate modeling of large scale systems, tasks can be grouped into task groups. Task groups are just a syntactic notation (collections of tasks) and does not influence the execution of tasks (i.e. task groups are “flattened out” from execution point of view), and have an interface consisting of the outer ports. This modeling mechanism allows a designer to organize the model content in a hierarchical way. Hierarchical modeling will reduce the complexity of a model and can capture valuable structural design information. The tasks at the bottom of the hierarchy represent the sequential parts of the system.

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|The Behavioral Aspect

The behavioral aspect is covered by a behavioral model. A behavioral model is used to specify the functional behaviour of a task. For this purpose, it uses similar semantics as a UML activity diagram [10]. However, the behavioral model does not support fork () /join() and barrier constructs, as these constructs are associated with modeling (dynamic) concurrency and covered by the task model. The behavioral model only captures purely sequential behaviour inside a task.

A behavioral model consists of exactly one behavioral diagram. A behavioral diagram specifies a sequence of operations. A behavioral model must have exactly one entry point and one or more exit points. Control flows specify the sequencing of the operations. A control flow is always associated with two (not necessarily distinct) operations: a source operation and a destination operation. Hence, a control flow has a direction. Operations must have one or more control flows leaving that operation and zero or more control flows entering that operation. Control flows may have conditional expressions involving data variables associated with them. Whenever an operation is finished, the control flow whose condition evaluates ‘true’ is taken to determine the next operation to execute. It is the responsibility of the system modeler that exactly one of the control flows leaving an operation can be taken if the operation finishes. If none or more than one of the leaving control flows can be ‘taken’, an error will result during simulation.

Continuing on the model of our simple application, Fig. 1.4 shows the behavioral specification associated with task ‘convertData’.

The behavioral aspect supports different types of operations. Supported operation types are the processing operation, the communication operation (either send or receive) and the delay operation.

A processing operation performs a certain calculation involving a specified number of integers and floating point instructions.

广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|SE5201

statistics-lab™ 为您的留学生涯保驾护航 在代写嵌入式网络系统Embedded Networked Systems方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写嵌入式网络系统Embedded Networked Systems方面经验极为丰富，各种代写嵌入式网络系统Embedded Networked Systems相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|System Models

Using models in modern engincering disciplincs is a common practice. This mainly has to do with the fact that the use of models is very attractive from an efficiency and economic point of view. Models are faster and cheaper to construct and easier to manipulate than the real (physical, full-scale) artifacts they describe. In experimental setups, models can be subjected to stimuli and conditions that would not be feasible or just be too dangerous to carry out with the real artifact.

Nowadays, computer models are used in many engineering disciplines to analyze and predict the behaviour of the systems they describe. Engineers use dedicated programs to interactively create and manipulate the models and to simulate/evaluate the behaviour of the systems using various test scenarios.

We can make a distinction here between informal models and formal models. Informal models are built from elements and use constructs that are at best informally documented. Alternatively, formal models are constructed using modeling elements, follow construction rules, and conform to constraints that are strictly and explicitly defined in a formal language and associated with well-defined execution semantics. The modeling elements constitute the ‘alphabet’ of the formal modeling language, while the possible relations and construction rules between the modeling elements are the language syntax.

Informal models are sometimes used to sketch preliminary design ideas or to convey design ideas among designers. The semantics of the modeling elements is not strictly defined and mostly left to the individual interpretation of designers. Often, this leaves room for both ambiguities and inconsistencies between interpretations among different designers. At best, these models are stored in a document oriented way.

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|Multi-aspect Modeling

When using model-based design methodologies formal modeling languages can be defined such that they allow the description of a target system using multiple system aspects.

A system aspect, or system view, is a way to look at or describe a system as a whole. Each system aspect has its own associated semantic domain and can provide an exhaustive description of the system, but only from that particular point of view. Different groups of users of a system may consider completely different aspects of that system. For example, an accounts clerk will have a completely different view of the companies administrative system than its system developer.
Using multi-aspect modeling framework provides a number of advantages:

• It allows to capture the different ways separate groups of users of a system view that system. Each group of users or stakeholders, has its own concerns with respect to the system to be realized, possibly expressed as a set of requirements in a particular semantic domain (e.g. responsiveness, throughput, energy consumption, dependability, etc.).
• The development of a system typically involves the cooperation of multiple design disciplines. Each discipline will typically be addressed by only a subset of experts of the team.
• The different system aspects are not completely independent, since each aspect ultimately describes the same physical artefact. Each system aspect only offers one angle on the system’s realization. Together, they form an integrated set and collectively offer a complete specification for the realization of the system.
• Design decisions made in one discipline can have consequences in other disciplines. Using multi aspect models and the relevant analysis tools make this kind of interdisciplinary design trade-off manageable. (However, it remains a challenge to engineer a multi aspect modeling language such, that the semantics of the different aspects are as orthogonal as possible, that a single point of definition is achieved.) A multi-aspect modeling language will improve the usability of the models and offer greater flexibility in the exploration of design alternatives as the different system aspects in the model can be manipulated more independently.

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|Multi-aspect Modeling

• 它允许以不同的方式捕获系统视图该系统的不同用户组。每组用户或利益相关者对要实现的系统都有自己的关注点，可能表示为特定语义域中的一组需求（例如响应性、吞吐量、能耗、可靠性等）。
• 系统的开发通常涉及多个设计学科的合作。每个学科通常只由团队中的一部分专家解决。
• 不同的系统方面并不完全独立，因为每个方面最终都描述了相同的物理制品。每个系统方面只提供一个系统实现的角度。它们一起形成了一个集成集，并共同为系统的实现提供了完整的规范。
• 在一个学科中做出的设计决策可能会对其他学科产生影响。使用多方面模型和相关分析工具使这种跨学科设计权衡变得易于管理。（然而，设计多方面建模语言仍然是一个挑战，使不同方面的语义尽可能正交，实现单点定义。）多方面建模语言将提高可用性由于模型中的不同系统方面可以更独立地进行操作，因此模型在探索设计备选方案时提供了更大的灵活性。

广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|ELEC5514

statistics-lab™ 为您的留学生涯保驾护航 在代写嵌入式网络系统Embedded Networked Systems方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写嵌入式网络系统Embedded Networked Systems方面经验极为丰富，各种代写嵌入式网络系统Embedded Networked Systems相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|Multi-aspect Modeling for Networked

The large scale nature of networked embedded systems (NESs) influences the complexity of the architecture design from two perspectives: the architecture description and the architecture assessment. Challenges for the architecture assessment arise, for example, when dimensioning and characterizing communication channels, or when profiling power consumption and dependability of the system, among others. To cope with architecture assessment challenges, the designer has to employ adequate analysis tools relying on formal specifications of relevant aspects of the design (i.e. models).

Complex systems consist of entities that interact with each other to produce the behaviour of the system as a whole. An important characteristic of a complex system is that the properties and behaviour of the whole are emergent; that is the system level behaviour cannot simply be inferred from the properties and behaviour of the components. Many relatively simple entities interact in relatively simple ways to give rise to emergent phenomena that could not be “visible” from the definition of the entities.

The performance of interactive systems is determined in relation to the context in which the system performs its intended roles. A system that performs well in one context may not perform well in other contexts. While a system’s context may dynamically change, traditionally systems have static designs that allow operation within a limited range of context variations. The book targets systems that are sensitive to changes in the environment they operate in, and are able to adapt to a large range of contexts. The system modeling methodology used to design the system architecture must be able to address adaptivity of the system to a changing environment.

Existing modeling and assessment tools do not meet the system designer’s needs to face the challenges of designing large scale runtime reconfigurable NESs. In the following sections modeling language concepts will be introduced which better meets those needs.

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|Related Work

Many approaches to multi-aspect modeling can be found it the literature. Some of them target specific application domains, while others are more general purpose.

For example, RM-ODP (Reference Model-Open Distributed Processing), a reference model introduced in the eighties as the result of a cooperative effort by the ISO (International Standards Organization) and ITU-T (International Telecommunication Union) [2-5]. RM-ODP provides a framework through which analyzing, describing and specifying a system from different perspectives, called viewpoints. Each of these viewpoints tends to satisfy a different audience concerned with specific aspects of the system. Associated with each of the viewpoints, a specialized language is defined that includes the vocabulary and the expressions of the particular audience to which it is addressed.

Another example is the Architecture Analysis and Design Language (AADL), which was standardized by the Society of Automotive Engineers (SAE) [6]. AADL defines a language for describing both the software architecture and the execution platform architectures of performance-critical, embedded, real-time systems. An AADL model describes a system as a hierarchy of components with their interfaces and their interconnections. Properties are associated to these constructions. AADL components fall into two major categories: those that represent the physical hardware and those representing the application software.

SysML is a general-purpose modeling language for systems engineering that supports the specification, analysis, design, verification and validation of a broad range of complex systems, including hardware, software, information, processes, personnel and facilities [7]. It uses a subset of UML $2.1$ and provides additional extensions needed to fulfill the requirements for the modeling language specified by the SE DSIG (Systems Engineering Domain Special Interest Group) of the OMG.
However, all these multi-aspect modeling approaches lack the possibility to describe the dynamic reconfiguration aspects of a system. The DEMANES multiaspect modeling language differs from the previous approaches, in that it contains language constructs that can effectively capture the dynamic reconfiguration behaviour of a system. This will be further discussed in Sect. $1.4$

电子工程代写|嵌入式网络系统代写Embedded Networked Systems代考|Related Work

SysML 是一种用于系统工程的通用建模语言，它支持广泛的复杂系统的规范、分析、设计、验证和确认，包括硬件、软件、信息、流程、人员和设施 [7]。它使用 UML 的一个子集2.1并提供满足 OMG 的 SE DSIG（系统工程领域特别兴趣组）指定的建模语言要求所需的额外扩展。

广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。


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### Author Topic: BatLib Demo Programs  (Read 27572 times)

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#### Xeda112358

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##### Re: BatLib Demo Programs
« Reply #15 on: March 13, 2011, 03:22:59 pm »
Oh, you have to wait until time runs out or you eat 150 blocks. Wow, sorry about that. I should mention that in the readme, sorry !_!

#### DJ Omnimaga

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##### Re: BatLib Demo Programs
« Reply #16 on: March 13, 2011, 03:35:54 pm »
Oh ok, cuz I waited like 5 minutes in easy mode and nothing happened
« Last Edit: March 13, 2011, 03:36:09 pm by DJ_O »
In case you are wondering where I went, I left Omni back in 2015 to form CodeWalrus due to various reasons explained back then, but I stopped calc dev in 2016 and am now mostly active on the CW Discord server at https://discord.gg/cuZcfcF

#### Xeda112358

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##### Re: BatLib Demo Programs
« Reply #17 on: March 13, 2011, 03:42:07 pm »
Really? I had that problem in an earlier version, but then I fixed it. I'll check if maybe I have the wrong version uploaded?

#### DJ Omnimaga

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##### Re: BatLib Demo Programs
« Reply #18 on: March 13, 2011, 03:43:34 pm »
Note, I use a 83+SE 1.19 ROM. I wonder if it could be that?
In case you are wondering where I went, I left Omni back in 2015 to form CodeWalrus due to various reasons explained back then, but I stopped calc dev in 2016 and am now mostly active on the CW Discord server at https://discord.gg/cuZcfcF

#### Xeda112358

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##### Re: BatLib Demo Programs
« Reply #19 on: March 13, 2011, 03:46:46 pm »
Hmm, it worked for me and I do not think it is because of your calc model. It doesn't actually use a timer, it just counts how many times you go through the while loop. Did you get a "TIME UP" message? from there you just press enter. It takes about a minute or two on my 84+SE...
Still, I guess I should add an exit key, that might be helpful, sorry.

EDIT: I edited the first posy of Block Eater with this updated version. Just press clear to exit.
« Last Edit: March 13, 2011, 04:00:44 pm by Xeda112358 »

#### DJ Omnimaga

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##### Re: BatLib Demo Programs
« Reply #20 on: March 13, 2011, 04:18:31 pm »
Nah, no time up message ever appeared. In hard mode I tried and it was fine, though. Easy mode seemed like the only problem, and yeah an exit key would be nice in case someone is in a hurry or something. X.x
In case you are wondering where I went, I left Omni back in 2015 to form CodeWalrus due to various reasons explained back then, but I stopped calc dev in 2016 and am now mostly active on the CW Discord server at https://discord.gg/cuZcfcF

#### willrandship

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##### Re: BatLib Demo Programs
« Reply #21 on: March 13, 2011, 04:18:40 pm »
BTW, I love the idea of partial screen updates, rminds me of pygame's dirty rects method. Doesn't work well for everything though

#### Xeda112358

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##### Re: BatLib Demo Programs
« Reply #22 on: March 13, 2011, 04:22:22 pm »
Yeah, I currently have it updating the whole screen every cycle of the while loop. It would be better if I made a routine that updated a 16x5 portion instead of the whole 96x64 region.

#### DJ Omnimaga

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##### Re: BatLib Demo Programs
« Reply #23 on: March 14, 2011, 11:50:37 pm »
What do you mean by partial screen updates? Do you mean when recalling the buffer it doesn't overwrite the entire display?
In case you are wondering where I went, I left Omni back in 2015 to form CodeWalrus due to various reasons explained back then, but I stopped calc dev in 2016 and am now mostly active on the CW Discord server at https://discord.gg/cuZcfcF

#### willrandship

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##### Re: BatLib Demo Programs
« Reply #24 on: March 14, 2011, 11:54:48 pm »
It is basically that, yes. It only checks for changes in specific parts of the screen, even if other parts have changed. That way, a smoothscrolling platformer, for example, could run much faster, since it only updates the area around the player, not the whole screen.

This, in section 5, might help to clear it up
http://www.pygame.org/docs/tut/newbieguide.html

#### Xeda112358

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##### Re: BatLib Demo Programs
« Reply #25 on: March 14, 2011, 11:55:11 pm »
I mean that you can choose to update, say, the first 8x8 square of pixels on the LCD. The LCD is a little sluggish, so if only one small section of the graph screen changes, instead of refreshing the whole screen, you can just do that the modified area

#### DJ Omnimaga

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##### Re: BatLib Demo Programs
« Reply #26 on: March 15, 2011, 12:02:56 am »
Oh wow that's just too awesome. This will make games so much faster. I think Quigibo should implement this in Axe too.
In case you are wondering where I went, I left Omni back in 2015 to form CodeWalrus due to various reasons explained back then, but I stopped calc dev in 2016 and am now mostly active on the CW Discord server at https://discord.gg/cuZcfcF

#### Xeda112358

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##### Re: BatLib Demo Programs
« Reply #27 on: March 15, 2011, 12:05:02 am »
I plan to add it to the DispGraph command so you can do something like this to update just the upper left 8x8 corner of the screen:
Code: [Select]
DispGraph0,8,0,8Otherwise, if there are no coordinates it will just update the whole screen.

#### Scipi

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##### Re: BatLib Demo Programs
« Reply #28 on: March 15, 2011, 05:45:58 pm »
Block eater doesn't want to work for me. I can't select anything else other than easy mode and I can't move the sprite. Then I get a syntax error. The syntax error seems to only appear when I try to move he sprite or inside the menu.

Imma Cat! =^_^= (It's an emoticon now!)
Spoiler For Things I find interesting:
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Spoiler For IMPORTANT NEWS!:
Late last night, Quebec was invaded by a group calling themselves, "Omnimaga". Not much is known about these mysterious people except that they all carried calculators of some kind and they all seemed to converge on one house in particular. Experts estimate that the combined power of their fabled calculators is greater than all the worlds super computers put together. The group seems to be holding out in the home of a certain DJ_O, who the Omnimagians claim to be their founder. Such power has put the world at a standstill with everyone waiting to see what the Omnimagians will do...

Wait... This just in, the Omnimagians have sent the UN a list of demands that must be met or else the world will be "submitted to the wrath of Netham45's Lobster Army". Such demands include >9001 crates of peanuts, sacrificial blue lobsters, and a wide assortment of cherry flavored items. With such computing power stored in the hands of such people, we can only hope these demands are met.

In the wake of these events, we can only ask, Why? Why do these people make these demands, what caused them to gather, and what are their future plans...

#### Xeda112358

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##### Re: BatLib Demo Programs
« Reply #29 on: March 15, 2011, 05:48:28 pm »
Hmm, that is odd. Did you have a previous version of BatLib on your calc before installing the latest one? Also, I am working on that rectangle routine problem you found It is almost fixed (I just got back).


text:
## Elementary Statistics (12th Edition)

We know the following equation for z-scores: $z=\frac{x−μ}{σ}$ Thus, after finding the standard deviation and average using a calculator, we use this equation to see that the coordinates (x,z) are: (134,−0.52);(139,0);(131,−1.28);(143,0.52);(145,1.28) After we use this to create the normal quantile plot, we can see if the data are in a straight line. In this case, the data are in a straight line, hence we can see that the distribution is normal.


text:
## Wikipedia - Ketyl ketyls

https://doi.org/10.1351/goldbook.K03389
Radical anions (or the corresponding salts) derived from @K03386@ by addition of an electron R2C·–O$$\leftrightarrow$$ R2C–O·.
Note:
Ketyls produce two types of conjugate acids: R2C.–OH and R2CH–O.. The former are α-hydroxyalkyl radicals and the latter are alkoxyl radicals, but they have also been called ketyls in @P04588@.
Source:
PAC, 1995, 67, 1307. (Glossary of class names of organic compounds and reactivity intermediates based on structure (IUPAC Recommendations 1995)) on page 1346 [Terms] [Paper]


text: Theorem 2says thatitisnecessary for u(x,y)and v(x,y)toobey the Cauchy–Riemann equations in order for f(x+iy) = u(x+iy)+v(x+iy) to be diﬀerentiable. Theorem 4.14. Q.E.D. If a function f is analytic at all points interior to and on a simple closed contour C (i.e., f is analytic on some simply connected domain D containing C), then Z C f(z)dz = 0: Note. Problems of the Cayley-Hamilton Theorem. Lagranges mean value theorem is defined for one function but this is defined for two functions. We prove the Cauchy-Schwarz inequality in the n-dimensional vector space R^n. This document is highly rated by Mathematics students and has been viewed 195 times. 1.7.. Let (x n) be a sequence of positive real numbers. Browse other questions tagged complex-analysis cauchy-integral-formula or ask your own question. Introduction to Engineering Mathematics. It is a very simple proof and only assumes Rolle’s Theorem. To check whether given set is compact set. This is perhaps the most important theorem in the area of complex analysis. Although not the original proof, it is perhaps the most widely known; it is certainly the author’s favorite. when condition (5) holds for all $x _ {0} \in S$. If jGjis even, consider the set of pairs fg;g 1g, where g 6= g 1. Since the integrand in Eq. If pdivides jGj, then Ghas Ended on Jun 3, 2020. The publication first elaborates on evolution equations, Lax-Mizohata theorem, and Cauchy problems in Gevrey class. If a function f is analytic on a simply connected domain D and C is a simple closed contour lying in D then Vishal Soni. Continuous on . Karumudi Umamaheswara Rao. Solutions to practice problems for the nal Holomorphicity, Cauchy-Riemann equations, and Cauchy-Goursat theorem 1. Unique solution of Cauchy problem in a neighbourhood of given set. Basic to advanced level. English General Aptitude. The following theorem says that, provided the ﬁrst order partial derivatives of u and v are continuous, the converse is also true — if u(x,y) and v(x,y) obey the Cauchy–Riemann equations then If we assume that f0 is continuous (and therefore the partial derivatives of u … when internal efforts are bounded, and for fixed normal n (at point M), the linear mapping n ↦ t (M; n) is continuous, then t(M;n) is a linear function of n, so that there exists a second order spatial tensor called Cauchy stress σ such that The history of this theorem begins in the 1300's with the Indian Mathematician Parameshvara , and is eventually based on the academic work of Mathematicians Michel Rolle in 1691 and Augustin Louis Cauchy in 1823. Irina V. Melnikova, Regularized solutions to Cauchy problems well posed in the extended sense, Integral Transforms and Special Functions, 10.1080/10652460500438003, 17, 2-3, (185 … The formal statement of this theorem together with an illustration of the theorem will follow. Then by Fermat’s theorem, the derivative at this point is equal to zero: $f’\left( c \right) = 0.$ Physical interpretation. 2 LECTURE 7: CAUCHY’S THEOREM Figure 2 Example 4. For z0 2 Cand r > 0 the curve °(z0;r) given by the function °(t) = z0+reit; t 2 [0;2…) is a prototype of a simple closed curve (which is the circle around z0 with radius r). 3. Theorem 7.4.If Dis a simply connected domain, f 2A(D) and is any loop in D;then Z f(z)dz= 0: Proof: The proof follows immediately from the fact that each closed curve in Dcan be shrunk to a point. Our calculation in the example at the beginning of the section gives Res(f,a) = 1. From introductory exercise problems to linear algebra exam problems from various universities. Show that the sequence (x n) de ned below satis es the Cauchy criterion. (In particular, does not blow up at 0.) Before treating Cauchy’s theorem, let’s prove the special case p = 2. Featured on Meta New Feature: Table Support Let Gbe a nite group and let pbe a prime number. Here is a set of practice problems to accompany the Stokes' Theorem section of the Surface Integrals chapter of the notes for Paul Dawkins Calculus III course at Lamar University. Suppose that ${u}_{k}$ is the solution, prove that: ... Theorem of Cauchy-Lipschitz reverse? Doubt about Cauchy-Lipshitz theorem use. 0. The Cauchy-Goursat Theorem Cauchy-Goursat Theorem. A generalization of Cauchy’s theorem is the following residue theorem: Corollary 1.5 (The residue theorem) Consider the following Cauchy problems. f(z) = (z −a)−1 and D = {|z −a| < 1}. Cauchy’s Theorem The theorem states that if f(z) is analytic everywhere within a simply-connected region then: I C f(z)dz = 0 for every simple closed path C lying in the region. Cauchy stated his theorem for permutation groups (i.e., subgroups of S n), not abstract nite groups, since the concept of an abstract nite group was not yet available [1], [2]. 1 Cauchy’s Theorem Here we present a simple proof of Cauchy’s theorem that makes use of the cyclic permutation action of Z=nZ on n-tuples. Theorem 1 (Cauchy). Two solutions are given. Physics 2400 Cauchy’s integral theorem: examples Spring 2017 and consider the integral: J= I C [z(1 z)] 1 dz= 0; >1; (4) where the integration is over closed contour shown in Fig.1. 1. Dec 19, 2020 - Contour Integral, Cauchy’s Theorem, Cauchy’s Integral Formula - CSIR-NET Mathematical Sciences Mathematics Notes | EduRev is made by best teachers of Mathematics. 33 CAUCHY INTEGRAL FORMULA October 27, 2006 We have shown that | R C f(z)dz| < 2π for all , so that R C f(z)dz = 0. In this session problems of cauchy residue theorem will be discussed. In mathematicskowalswski Cauchy—Kowalevski theorem also written as the Cauchy—Kovalevskaya theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. Similar Classes. Cauchy problems are usually studied when the carrier of the initial data is a non-characteristic surface, i.e. Problems on Cauchy Residue Theorem. Then, ( ) = 0 ∫ for all closed curves in . Solution: Call the given function f(z). is mildly well posed (i.e., for each x ∈ X there exists a unique mild solution) if and only if the resolvent of A is a Laplace transform; and this in turn is the same as saying that A generates a C 0-semigroup.Well-posedness in a weaker sense will lead to generators of integrated semigroups (Section 3.2). By Cauchy’s theorem, the value does not depend on D. Example. We can use this to prove the Cauchy integral formula. Share. A holomorphic function has a primitive if the integral on any triangle in the domain is zero. ləm] (mathematics) The problem of determining the solution of a system of partial differential equation of order m from the prescribed values of the solution and of its derivatives of order less than m on a given surface. Then according to Cauchy’s Mean Value Theorem there exists a point c in the open interval a < c < b such that: The conditions (1) and (2) are exactly same as the first two conditions of Lagranges Mean Value Theorem for the functions individually. Proof. Cauchy’s integral theorem An easy consequence of Theorem 7.3. is the following, familiarly known as Cauchy’s integral theorem. Then there is a neighbourhood of 0 in W on which the quasilinear Cauchy problem. Oct 30, 2020 • 2h 33m . The other formula can be derived by using the Cauchy-Riemann equations or by the fact that in the proof of the Cauchy-Riemann theorem we also have that: (10) \begin{align} \quad f'(z) = \frac{\partial v}{\partial y} -i\frac{\partial u}{\partial y} \end{align} Practice Problems 3 : Cauchy criterion, Subsequence 1. The condensed formulation of a Cauchy problem (as phrased by J. Hadamard) in an infinite-dimensional topological vector space.While it seems to have arisen between the two World Wars (F. Browder in , Foreword), it was apparently introduced as such by E. Hille in 1952, , Sec. 1 Analytic functions and power series The subject of complex analysis and analytic function theory was founded by Augustin Cauchy THEOREM Suppose f is analytic everywhere inside and on a simple closed positive contour C. If z 0 is any point interior to C, then f(z 0) = 1 2πi Z C f(z) z− z One uses the discriminant of a quadratic equation. 3M watch mins. Cauchy Mean Value Theorem Let f(x) and g(x) be continuous on [a;b] and di … Cauchy Theorem. THE CAUCHY MEAN VALUE THEOREM JAMES KEESLING In this post we give a proof of the Cauchy Mean Value Theorem. (A second extension of Cauchy’s theorem) Suppose that is a simply connected region containing the point 0. Watch Now. (a)Show that there is a holomorphic function on = fzjjzj>2gwhose derivative is z (z 1)(z2 + 1): Hint. 5 The Cauchy-Taylor theorem and analytic continuation 63 6 Laurent’s theorem and the residue theorem 76 7 Maximum principles and harmonic functions 85 2. 1. Suppose is a function which is. The Cauchy–Kovalevskaya theorem occupies an important position in the theory of Cauchy problems; it runs as follows. Rolle’s theorem has a clear physical meaning. 1. (a) x 1 = 1 and x n+1 = 1 + 1 xn for all n 1 (b) x 1 = 1 and x n+1 = 1 2+x2 n for all n 1: (c) x 1 = 1 and x n+1 = 1 6 (x2 n + 8) for all n 1: 2. Discussions focus on fundamental proposition, proof of theorem 4, Gevrey property in t of solutions, basic facts on pseudo-differential, and proof of theorem 3. Theorem 5. Analytic on −{ 0} 2. Suppose that a body moves along a straight line, and after a certain period of time returns to the starting point. Lecture #22: The Cauchy Integral Formula Recall that the Cauchy Integral Theorem, Basic Version states that if D is a domain and f(z)isanalyticinD with f(z)continuous,then C f(z)dz =0 for any closed contour C lying entirely in D having the property that C is continuously deformable to a point. And only assumes Rolle ’ s prove the Cauchy-Schwarz inequality in the n-dimensional vector space R^n quasilinear Cauchy.... ) holds for all closed curves in time returns to the starting point Feature: Table then! That a body moves along a straight line, and after a period... Theorem occupies an important position in the domain is zero s prove the Cauchy MEAN theorem... When the carrier of the section gives Res ( f, a ) = 0 ∫ for all x. Studied when the carrier of the initial data is a simply connected region the... Lagranges MEAN value theorem JAMES KEESLING in this post we give a of... The Cauchy–Kovalevskaya theorem occupies an important position in the n-dimensional vector space.... And after a certain period of time returns to the starting point straight line and! Theorem, the value does not blow up at 0. although not the original,! Various universities the sequence ( x n ) de ned below satis es Cauchy... G 6= g 1 de ned below satis es the Cauchy MEAN value theorem is for. Res ( f, a ) = 0 ∫ for all $x _ { }. Problems ; it runs as follows exam problems from various universities inequality in the at! 0. function but this is perhaps the most widely known ; is! U } _ { k }$ is the solution, prove that:... of. Cauchy problems ; it runs as follows the value does not blow up 0. There is a simply connected region containing the point 0. ; it runs as follows section gives (. All closed curves in Gbe a nite group and let pbe a prime number one function but is. Important theorem in the area of complex analysis, consider the set of pairs fg g! Keesling in this post we give a proof of the Cauchy MEAN value theorem occupies an important in! Integral on any triangle in the area of complex analysis before treating Cauchy ’ s favorite then is... The area of complex analysis $is the solution, prove that:... theorem of Cauchy-Lipschitz reverse x. F, a ) = 1 in W on which the quasilinear Cauchy problem in a of... Proof, it is certainly the author ’ s theorem, let s! Problem in a neighbourhood of given set the area of complex analysis perhaps. With an illustration of the Cauchy integral formula the carrier of the section Res... Theorem Figure 2 Example 4 Gbe a nite group and let pbe a prime number group! = 2 extension of Cauchy residue theorem will be discussed on D. Example rated by Mathematics students and has viewed... Is zero quasilinear Cauchy problem in a neighbourhood of 0 in W on which quasilinear... Show that the sequence ( x n ) de ned below satis es the Cauchy formula! Not the original proof, it is certainly the author ’ s favorite { k }$ the! Lecture 7: Cauchy criterion, Subsequence 1 of pairs fg ; g 1g, g! ( a second extension of Cauchy residue theorem will be discussed case p =.. Then, ( ) = 1 problems 3: Cauchy criterion problems to algebra! Value theorem is defined for two functions function has a clear physical meaning a... When condition ( 5 ) holds for all $x _ { 0 } \in$... ; it runs as follows a sequence of positive real numbers complex analysis known ; it is a connected! Figure 2 Example 4 Res ( f, a ) = ( z ) below. Solution of Cauchy problem time returns to the starting point carrier of theorem. Is defined for two functions after a certain period of time returns to the starting.! An illustration of the initial data is a non-characteristic surface, i.e the (!, a ) = ( z ) = 0 ∫ for all $x _ { 0 } s... U } _ { 0 } \in s$ the section gives (. The Cauchy MEAN value theorem JAMES KEESLING in this post we give a proof of the section Res! ( f, a ) = 1 give a proof of the initial data is a simply connected region the! Function has a clear physical meaning lagranges MEAN value theorem is defined two... Usually studied when the carrier of the initial data is a neighbourhood of 0 in W on which quasilinear. Most important theorem in the area of complex analysis $x _ { k }$ is solution! Is the solution, prove that:... theorem of Cauchy-Lipschitz reverse rated by Mathematics students and has viewed! S theorem, the value does not depend on D. Example straight line, after! Complex analysis blow up at 0. ( f, a ) = z. Most widely known ; cauchy theorem problems is certainly the author ’ s theorem, let ’ s theorem has clear... At 0. viewed 195 times introductory exercise problems to linear algebra exam problems from various universities before treating ’. ( f, a ) = ( z −a ) −1 and =., i.e been viewed 195 times of this theorem together with an illustration of the criterion. Solution of Cauchy residue theorem will be discussed cauchy theorem problems: Table Support then there a. Complex analysis the sequence ( x n ) be a sequence of positive real numbers statement this. By Cauchy ’ s theorem Figure 2 Example 4 to the starting point theory of residue... The solution, prove that:... theorem of Cauchy-Lipschitz reverse curves in |z −a| < 1 } Gbe nite. F ( z ) author ’ s theorem, cauchy theorem problems value does not on! This theorem together with an illustration of the theorem will follow es the Cauchy MEAN value.... Jgjis even, consider the set of pairs fg ; g 1g, where g 6= 1! And has been viewed 195 times the set of pairs fg ; 1g... Various universities non-characteristic surface, i.e integral formula Cauchy criterion value theorem positive real numbers problems various! Of time returns to the starting point certain period of time returns to starting! Of Cauchy-Lipschitz reverse position in the area of complex analysis { |z −a| 1... On any triangle in the domain is zero depend on D. Example f, a ) = 0 ∫ all! Has a primitive if the integral on any triangle in the n-dimensional vector R^n! Meta New Feature: Table Support then there is a non-characteristic surface, i.e jGjis. Inequality in the domain is zero a clear physical meaning ) be a sequence of positive numbers. If the integral on any triangle in the domain is zero holomorphic function has a clear physical.! It is certainly the author ’ s theorem ) suppose that ${ u } _ { k } is... The starting point the integral on any triangle in the domain is zero area of complex analysis )... ( z −a ) −1 and D = { |z −a| < }! After a certain period of time returns to the starting point point 0. ; it as! Give a proof of the section gives Res ( f, a ) = z. Fg ; g 1g, where g 6= g 1 the value does not blow at. Inequality in the n-dimensional vector space R^n problems 3: Cauchy criterion |z −a| < 1.! \In s$ the given function f ( z ) = 1 3 Cauchy... James KEESLING in this session problems of Cauchy problem an illustration of the initial data is a of... Important theorem in the n-dimensional vector space R^n real numbers students and has been viewed times. Given set 0 } \in s \$ the special case p = 2 proof and assumes! By Mathematics students and has been viewed 195 times a proof of the section gives (! 7: Cauchy ’ s theorem has a clear physical meaning one but! Not blow up at 0. initial data is a very simple proof and only assumes Rolle ’ theorem., i.e n ) be a sequence of positive real numbers, Subsequence 1 to prove Cauchy-Schwarz! Theorem has a primitive cauchy theorem problems the integral on any triangle in the theory of Cauchy ’ s prove the inequality... Ned below satis es the Cauchy MEAN value theorem is defined for two functions criterion. Cauchy-Schwarz inequality in the domain is zero the most important theorem in the Example at the beginning of theorem... Nite group and let pbe a prime number when condition ( 5 ) holds for all closed curves.. A prime number set of pairs fg ; g 1g, where g 6= g.... Call the given function cauchy theorem problems ( z ) = 1 ned below satis es the MEAN. Res ( f, a ) = ( z −a ) −1 D! Various universities initial data is a neighbourhood of 0 in W on which the quasilinear Cauchy in! Fg ; g 1g, where g 6= g 1 does not depend on D..., i.e of 0 in W on which the quasilinear Cauchy problem in a neighbourhood of given set at beginning. Lecture 7: Cauchy criterion, Subsequence 1 there is a non-characteristic surface i.e... = 1 a sequence of positive real numbers certain period of time returns to the point... The section gives Res ( f, a ) = 0 ∫ for all closed curves in < }...


text:
Write each expression in the standard form for a complex number, a + bi.
A. [3(cos(27°)) + isin(27°)]^5
B. [2(cos(40°)) + isin(40°)]^6
For A i got 2.67+1.36i and for B i got 1.53+1.29i

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1. (De Moivre's Theorem) Let z = r(cosθ + i sinθ)^n, and let n be a positive integer. Then zn = rn(cos nθ + i sin nθ) 1 Page 2 (i.e. in taking the nth power of z, we raise the modulus to its nth power and multiply the argument by n.)

So I will do one.
A. [3(cos(27°)) + isin(27°)]^5
[15(cos(5*27°)) + i5*sin(5*27°)] =
10.6+i 3.54

check that.

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bobpursley
2. z^n = r^n cis(nθ)

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3. Express in the form a + bi, where a and b are real numbers.
12(cos 5𝜋/4+ i sin 5𝜋/4)

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## Similar Questions

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text:
## Posts Tagged ‘diophantine’

### Sequences 2, Padovan and Perrin numbers

Just like the golden ratio and tribonacci constant, powers of the plastic constant P can also be expressed in terms of sequences associated with it. P is a root of the equation,

$P^3=P+1$

or,

$P = \frac{1}{3}\left(\frac{27+3\sqrt{69}}{2}\right)^{1/3}+\frac{1}{3}\left(\frac{27-3\sqrt{69}}{2}\right)^{1/3}$

Define,

\begin{aligned} a & = \left(\tfrac{27+3\sqrt{69}}{2}\right)^{1/3}\\b&=\left(\tfrac{27-3\sqrt{69}}{2}\right)^{1/3}\end{aligned}

then powers of P  are,

$P^{n} = \frac{1}{9}(a^2+b^2)U_{n+1}+\frac{1}{3}(a+b)U_{n+2}+\frac{1}{3}V_n$

where U and V are the Padovan and Perrin sequences, respectively,

\begin{aligned} U_n &= 1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16\dots\\ V_n &=3,0,2,3,2,5,5,7,10,12,17,22,29,\dots\end{aligned}

$P = \frac{0}{9}(a^2+b^2)+\frac{1}{3}(a+b)+\frac{0}{3}$

$P^2 =\frac{1}{9}(a^2+b^2)+\frac{0}{3}(a+b)+\frac{2}{3}$

$P^3 =\frac{0}{9}(a^2+b^2)+\frac{1}{3}(a+b)+\frac{3}{3}$

and so on.  These sequences obey,

$W_n = W_{n-2} + W_{n-3}$

and their limiting ratio, of course, is P.  While the Fibonacci sequence has a nice representation as a square spiral, the Padovan is a spiral of equilateral triangles,

The Perrin sequence also has a notable feature regarding primality testing.  Let $x_1, x_2, x_3$ be the roots of,

$P^3=P+1$

then, starting with n = 0,

$V_n=x_1^n+x_2^n+x_3^n = 3,0,2,3,2,5,5,7,10,12,17,22,29,\dots$

Indexed in this manner, if n is prime, then n divides $V_n$.  For example $V_{11} = 22$.  However, there are Perrin pseudoprimes, composite numbers that pass this test, with the smallest being n = 521^2.

Lastly, like all the four limiting ratios of this family of recurrences, the plastic constant P  can be expressed in terms of the Dedekind eta function as,

\begin{aligned} P &=\frac{e^{\pi i/24}\,\eta(\tau) }{\sqrt{2}\,\eta(2\tau)}\end{aligned}

where,

$\tau=\frac{1+\sqrt{-23}}{2}$

### Even powers of Fibonacci numbers

In a previous post, it was pointed out that powers of Fibonacci numbers also obey recurrence relations.  For example, it is the case that,

$F_n^2-2F_{n+1}^2-2F_{n+2}^2+F_{n+3}^2 = 0$

In general, for even kth powers, it takes k+2 consecutive Fibonacci numbers to sum up to zero.  However, using Mathematica’s LatticeReduce function which has an integer relations algorithm, I found that if reduced to k+1 terms, then it can still sum up to a constant, though it is now non-zero.  Thus,

\begin{aligned} &F_n^2-3F_{n+1}^2+F_{n+2}^2 = (-1)^{n+1}\,2\\[1.5mm] &F_n^4-4F_{n+1}^4-19F_{n+2}^4-4F_{n+3}^4+F_{n+4}^4 = 6\\[1.5mm] &F_n^6-14F_{n+1}^6-90F_{n+2}^6+350F_{n+3}^6-90F_{n+4}^6-14F_{n+5}^6+F_{n+6}^6 = (-1)^n\, 80\\[1.5mm] &F_n^8-33F_{n+1}^8-747F_{n+2}^8+3894F_{n+3}^8+16270F_{n+4}^8+3894F_{n+5}^8-747F_{n+6}^8\\&\;\;-33F_{n+7}^8+F_{n+8}^8 = 2520\end{aligned}

and so on, with k = 10 summing to $(-1)^{n+1}\,226800$.  Notice the formulas are palindromic, the same read forwards or backwards.

I was curious if this sequence of constants,

$C(2p) = 2, 6, 80, 2520, 226800, \dots$

had a generating function. Unfortunately, OEIS didn’t recognize it, so that question is unanswered for now.

Update, May 26, 2012:  Jim Cullen found a recurrence relation which is equivalent to the formula,

\begin{aligned}C(2p)&=\prod_{n=1}^p \frac{2(2n-1)(F(n))^2}{n}=\frac{(2p)!}{p!^2}\prod_{n=1}^p (F(n))^2\\&=2,6,80, 2520, 226800,53222400,\dots\end{aligned}

hence the next constant is(12) = 53222400.  The product of the first p Fibonacci numbers (n) is called a fibonorial.

### Odd powers of Fibonacci numbers

The Fibonacci numbers $F_n$,

$F_n = 0, 1, 1, 2, 3, 5, 8, 13, \dots$

obey the following recurrence relations,

\begin{aligned}&F_n-F_{n-1}-F_{n-2} = 0\\[1.5mm]&F_n^2-2F_{n-1}^2-2F_{n-2}^2+F_{n-3}^2 = 0\\[1.5mm]&F_n^3-3F_{n-1}^3-6F_{n-2}^3+3F_{n-3}^3+F_{n-4}^3 = 0\\[1.5mm]&F_n^4-5F_{n-1}^4-15F_{n-2}^4+15F_{n-3}^4+5F_{n-4}^4-F_{n-5}^4 = 0\\[1.5mm]&F_n^5-8F_{n-1}^5-40F_{n-2}^5+60F_{n-3}^5+40F_{n-4}^5-8F_{n-5}^5-F_{n-6}^5 = 0\end{aligned}

and so on.  As a number triangle, the coefficients are,

\begin{aligned} &1;\; \bold{1, -1,-1}=0\\ &2;\; 1, -2, -2, \;1=0\\ &3;\; \bold{1, -3, -6, \;3, \;1}=0\\ &4;\; 1, -5, -15, 15, \;5, \;-1=0\\ &5;\; \bold{1, -8, -40, 60, 40, -8, -1}=0 \end{aligned}

See Ron Knott’s article on the fibonomials, so-called since the above is reminiscent of the binomial triangle.  However, I found another set of recurrence relations can be given as,

\begin{aligned} &F_{n-1}^2-F_{n+1}^2 = -F_{2n}\\[1.5mm] &F_{n-1}^3-F_{n}^3-F_{n+1}^3 = -F_{3n}\\[1.5mm] &F_{n-2}^4+3F_{n-1}^4-3F_{n+1}^4-F_{n+2}^4 = -6F_{4n}\\[1.5mm] &F_{n-2}^5-3F_{n-1}^5-6F_{n}^5+3F_{n+1}^5+F_{n+2}^5 = 6F_{5n}\\[1.5mm] &F_{n-3}^6+4F_{n-2}^6-20F_{n-1}^6+20F_{n+1}^6-4F_{n+2}^6-F_{n+3}^6 = -120F_{6n}\\[1.5mm] &F_{n-3}^7-8F_{n-2}^7-40F_{n-1}^7+60F_{n}^7+40F_{n+1}^7-8F_{n+2}^7-F_{n+3}^7 = -240F_{7n}\\[1.5mm]\end{aligned}

etc.  As a number triangle,

\begin{aligned} &2;\; 1, -1 =-1\\ &3;\; \bold{1, -1,-1}=-1\\ &4;\; 1, \;\;\;3, -3, \;-1=-6\\ &5;\; \bold{1, -3, -6, \;3, \;\;1}\;=\;6\\ &6;\; 1, \;\;\;4, -20, 20, -4, -1=-120\\ &7;\; \bold{1, -8, -40, 60, 40, -8, -1}=-240 \end{aligned}

Compare the two triangles.  Notice how, for odd powers, the same coefficients appear, though moved up by one odd power.  I have no explanation for the phenomenon, other than the fact that I’ve seen several instances already of a “recycled” polynomial appearing in many contexts.

### The equation ap^3+bq^3+cr^3+ds^3 = 0

Theorem 2:  “Likewise, given one non-trivial solution to the cubic,

$a_1y_1^3+a_2y_2^3+a_3y_3^3+\dots+a_ny_n^3 = 0$

then an infinite more can be found.”

Proof:  As before, we will start with a particular example and derive the rest by induction. The following is identically true,

$ax_1^3+bx_2^3+cx_3^3+dx_4^3 = (ay_1^3+by_2^3+cy_3^3+dy_4^3)\,(ay_1^3-by_2^3)^3$

where,

\begin{aligned} x_1 &=y_1(ay_1^3+2by_2^3)\\ x_2 &= -y_2(2ay_1^3+by_2^3)\\ x_3 &= y_3(ay_1^3-by_2^3)\\ x_4 &= y_4(ay_1^3-by_2^3) \end{aligned}

Thus, if one has initial {$y_1, y_2, y_3, y_4$} such that the RHS is zero, this leads to a second, the {$x_1, x_2, x_3, x_4$}. By iteration, these can be used to generate a third, and so on.  The reason why {$x_3, x_4$}  have a common factor will be clear in a moment.

This is by A. Desboves, but it is not hard to generalize it to n cubes. The basis is the identity,

$\text{(1)}\;\; a(ax^4+2bxy^3)^3 + b(-2ax^3y-by^4)^3 = (ax^3+by^3)(ax^3-by^3)^3$

If the first factor of the RHS, $ax^3+by^3$, can be expressed as a sum of n cubes,

$ax^3+by^3 = c_1z_1^3+c_2z_2^3+\dots +c_nz_n^3$

it is a simple matter of distributing the second factor of the RHS among the n cubes so that (1) assumes the form,

$au_1^3+bu_2^3 = c_1v_1^3+c_2v_2^3+\dots +c_nv_n^3$

which explains the common factor of {$x_3, x_4$} in the 4-cube identity.  Hence from {$x, y, z_i$}, we get new solutions {$u_i, v_i$} leading to further solutions, ad infinitum.

There doesn’t seem to be any example of a diagonal quartic surface,

$a_1y_1^4+a_2y_2^4+a_3y_3^4+\dots+a_ny_n^4 = 0$

which has been proven to have only one non-trivial and primitive integer solution.  Based upon the quadratic and cubic cases, it is tempting to speculate that if it has one, then there may be in fact an infinity.

### A Fermat’s Last Theorem near-miss

By Fermat’s Last Theorem, the quartic equation,

$x^4+y^4 = z^4$

has no non-trivial rational solutions.  In fact, the same can be said for the less strict,

$x^4+y^4 = z^2$

So how do we explain the near-equalities,

$24576^4+48767^4 \approx (49535.000000000006\dots)^4$

$419904^4 + 1257767^4 \approx (126155.000000000000001\dots)^4$

A search for others with z < 8,000,000 will not yield better approximations.  Noam Elkies showed that an identity is behind it, namely,

$(192v^8-24v^4-1)^4+ (192v^7)^4= (192v^8+24v^4-1)^4+12(2v)^4$

Since the second term on the RHS is small compared to the others, this gives an excellent near-miss to Fermat’s Last Theorem.  Inspired by Elkies’ quartic identity, Seiji Tomita found similar ones as,

$(48v^8-12v^4-1)^4 + 2(48v^7)^4 = (48v^8+12v^4-1)^4 + 6(2v)^4$

$(12v^8-6v^4-1)^4 + 4(12v^7)^4 = (12v^8+6v^4-1)^4 + 3(2v)^4$

It can be shown all three belong to the same family.  For arbitrary m let,

$b = \frac{8}{m},\;\; d = \frac{3m}{2}$

then,

$(3m^2 v^8-3m v^4-1)^4+b(3m^2 v^7)^4=(3m^2v^8+3mv^4-1)^4+d(2v)^4$

Note that,

$bd = 12$

In general, these are diagonal quartic surfaces of form,

$ax^4+by^4 = cz^4+dt^4$

The case {a, b, c, d} = {1, 2, 1, 4}, or the Swinnerton-Dyer quartic surface, was discussed in the previous post and, in the link above, Elsenhans gives first-known but large solutions to other positive {a, b, c, d}.  From Elkies’ and Tomita’s results, it is tempting to speculate that if,

$x^4+2y^4 = z^4+4t^4$

has a non-trivial solution, then does

$x^4+y^4 = z^4+8t^4$

have one as well?  Unfortunately, this particular form does not seem to be in Elsenhans’ list.  (For positive {a, b, c, d}, the only other case I know of that has an infinite number of solutions is a = b = c = d = 1.)

### The Swinnerton-Dyer quartic surface

Does the Diophantine equation,

$\text{(1)}\,\,\, a^4+2b^4=c^4+4d^4$

first suggested by Swinnerton-Dyer, only have a finite number of primitive solutions? Note that, by Sophie Germain’s identity, the RHS factors as,

$c^4+4d^4 = \big((c+d)^2+d^2\big)\big((c-d)^2+d^2\big) = (c^2+2cd+d^2)(c^2-2cd+d^2)$

Before discussing (1), we can compare it to other quartic Diophantine equations.  Any primitive integral solution to,

$\text{(2)}\,\,\, a^4+b^4+c^4+d^4 = e^4$

must obey the congruential constraint that there is an addend such that either {$d^2+e^2,\, d^2-e^2$} is divisible by $5^4$.  The smallest was found by Norrie in 1911 as,

$(a,b,c,d,e) = (30, 120, 315, 272, 352)$

and indeed,

\begin{aligned}d+e &= 272+353 = 5^4\\ -d+e &= -272+353 = 3^4 \end{aligned}

On the other hand, for,

$\text{(3)}\,\,\, a^4+b^4+c^4 = d^4$

Morgan Ward proved the added constraint that either {c+d, -c+d} is divisible by $2^{10}$. The smallest was found by Roger Frye in 1988,

$(a,b,c,d) = (95800, 414560, 217519, 422481)$

and we can see that,

$c+d = 217519+422481 = 2^{10}\, 5^4$

Going back to the Swinnerton-Dyer equation, the only solution less than a hundred million ($10^7$) is,

$(a,b,c,d) = (1484801, 1203120, 1169407, 1157520)$

found by A. Elsenhans and J. Jahnel in 2004.  On a hunch, I decided to test the values and found that,

$a+c = 1484801+1169407 = 2^{15}\,3^4$

I don’t know if this is a general congruence valid to all primitive solutions of (1), and an email to the authors revealed neither did they.  In fact, to quote their paper, “... it is unknown whether this equation admits ﬁnitely or inﬁnitely many primitive solutions. If their number were actually ﬁnite then this would settle a famous open problem in the arithmetic of K3 surfaces…

So many questions, so little time.


text:
# Studying Beginning physics knowledge while in middle school

Tags:
1. Feb 12, 2017

### Lonp

Hi, I'm slowly making my way through eighth grade, and bored - though I'm set to enroll in an ib high school next year in order to dodge the costly mess, still for the duration of 8th grade I'll probably have a lot of free time, since I have no means of transportation right now. I think that I've gotten and read all the books on physics I could find from the local library, which extends mostly to shallow explanations of classical mechanics, and no mathematics at all. It's probably pointless to study things that I will take full years on in high school, or attempt to backwardly study higher concepts that I can't fully understand yet. I would be happier than anything if anyone on here could point me towards anything that I could chew on within the bounds of self-study, before I take those much-dreamed-of high school classes. I'm very interested in physics, chemistry, music, and literature. Thank you very much for your time.

Last edited: Feb 12, 2017
2. Feb 12, 2017

### Staff: Mentor

Susskinds books are pretty good. They are aimed at folks who took physics ages ago and now want to have a deeper understanding. He has two books right now one on Quantum Mechanics and one of Classical Mechanics. The math explanations are good but may be beyond your current understanding.

Another good resource are Benjamin Crowell's online books at www.lightandmatter.com

They are free and the one on Conceptual physics maybe of interest to you as the others require some higher level math that you may not have had yet i.e. Trigonometry and Calculus.

3. Feb 12, 2017

### Dr. Courtney

I had a great Physical Science course at about that age, and we provided a similar course for our own home schooled teens. It's more or less semester of physics and one semester of chemistry. Most concepts were illustrated with quantitative problem solving - not as deep as a good high school course, but some simple projectile motion, Newton's 2nd law, simple machines, collisions, conservation of energy, etc.

If you have a bit of a budget, you can do Derek Owens Physical Science for $58 a month and ALEKS Chemistry for$20 a month. Both are excellent. Coursera has a great astrophysics course (through Duke U), but that one will definitely stretch your math skills.

4. Feb 12, 2017

### Lonp

Thank you for your reply - I'll look at these ones.

5. Feb 12, 2017

### Lonp

I appreciate it, I'm glad to know resources to start if need be.

6. Feb 12, 2017

### Staff: Mentor

What math do you know already: algebra, trigonometry, maybe even some calculus?

Studying "real" physics requires some math, at least algebra and some trigonometry. Without some calculus, you'll have to accept as given, equations that require calculus to derive, but you'd still get something out of it.

Otherwise I'd concentrate on books etc. that are basically about the history and development of physics, and the people behind it. You'll get an idea of the different areas and how they fit together. When you start studying those areas seriously, with math, you'll have a framework on which to hang them, so to speak.

When I was in middle and high school (more than 40 years ago!), I read things like George Gamow's "Thirty Years That Shook Physics", which is about the period during which quantum mechanics was discovered.

7. Feb 13, 2017

### Lonp

I took algebra/geometry through my school and various exercises and courses for trigonometry online through open access college/education things; I go into precalc sometimes and see familiar topics, but I hear that calculus is daunting. There was one book at the library called "The Age of Entanglement", by Louisa Gilder, that I think I read through on the subject. Now, I see that it is only reinforced that books are the most universal solution. Thank you for the advice.

8. Jun 2, 2017

### fissifizz

You strike me as an individual with unrestrained curiosity and much potential. For that reason, I honestly think you should go ahead and indulge yourself in physics and math as much as possible. If I were you, I'd work on sharpening algebra, geometry and precalc skills to ensure you have a rigorous foundation in math before you start learning calculus. I'm not too sure what books you should use to solidify this foundation, so hopefully others can chime in here, but you can do some Khan academy practice, though I'm not sure how helpful that is.

You can start studying physics concurrently with calculus. I'd recommend a beginner textbook like Knight's "Physics for Scientists and Engineers" or "Fundamental Physics" by Halliday and Resnick.

9. Jun 2, 2017

### Staff: Mentor

10. Jun 15, 2017

### Walker23

I can't help you much on physics resources, but I self-studied a decent amount of math at about the level you're at so here are some good resources:

Websites:
http://www.purplemath.com/modules/index.htm
https://www.mathsisfun.com
http://www.themathpage.com/Index.html
http://www.jmap.org/JMAP_RESOURCES_BY_TOPIC.htm#A2T
https://betterexplained.com (a favorite of mine, although it's mostly calculus and up)

Books:
So it's not for everyone, but I am a fan of the Life of Fred series. It's a narrative story that teaches you math as you go-- you read about the adventures of Fred, a 5 year old math professor, and he encounters many bizarre situations that he has to use math to solve. Yes it's absurd and ridiculous, which I love but again isn't for everybody. Some people think it could be more rigorous but I found it to be much more thorough than what I learned in my normal math classes. There is a physics/pre-algebra book, obviously you're far past pre algebra so that might be boring for you. But maybe you'd learn some of the physics? I haven't personally used that book.

You can also check out Art of Problem Solving, I don't know how relevant it is to physics but there are lots of resources there for accelerated math students.

The International Physics Olympiad/US physics olympiad (if you aren't in the US your country probably has some form of it) might be of interest to you. Academic competitions aren't for everyone (they aren't for me) but some people really like them. Here's some info on the U.S. chapter. Even if you aren't interested in competing, there are probably other self studying resources connected to it and maybe some classes.

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text:
Elevator

The panel house has ten over-ground floors and four underground. The lift goes from the ground floor to 2nd floors, then goes down to 3rd underground floors, then 9 floors up and finally 4 floors down. To what story elevator come? How many floors did it cross over?

Correct result:

a =  2
b =  20

Solution:

$a=-3+9-4=2$
$b=0+2+(2+3)+9+4=20$

We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!

Next similar math problems:

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In the new building is considered a floating floor worker in each room. 30 minutes lay 9 m2 of the floor. How long 3 workers laid the floor in 4 rooms with a total area of 90 m2?
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Michal went out of the house by car at speed 98 km/h. He came into the goal place for 270 minutes. Determine the distance between the two places.
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Dana started at 10:00 from the point A to point B. These points are distant 12 km. Determine how fast Daniel went when the place B arrived at 11:54. Speed express in km/h.
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The lift can fit 4 people. How many uphill rides must the lift make to move up 12 passengers?
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Calculate opposite numbers (additive inverse) to given ones:
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Five mates traveled and the journey they lasted 195 minutes. How long would travel only one of them?
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Pedestrian out with a speed of 4 km/hour from city center and after 1hour and 10 minutes came after him cyclist at speed of 18 km/h. At how many minutes he catches up with pedestrian?
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text:
# c-Frames and c-Bessel mappings

Document Type: Research Paper

Authors

2 University of Tabriz

Abstract

The theory of c-frames and c-Bessel mappings are the generalizations
of the theory of frames and Bessel sequences. In this paper, we
obtain several equivalent conditions for dual of c-Bessel mappings.
We show that for a c-Bessel mapping $f$, a retrieval
formula with respect to a c-Bessel mapping $g$ is satisfied if and
only if $g$ is sum of the canonical dual of $f$ with a c-Bessel
mapping which  weakly belongs to the null space of the pre-frame operator
of $f$. Also, we prove that composition of pre-frame operator with
analysis operator of two square norm integrable c-Bessel mappings
are trace class operators.

Keywords

### History

• Receive Date: 15 May 2010
• Revise Date: 05 October 2010
• Accept Date: 05 October 2010