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http://umj.imath.kiev.ua/authors/name/?lang=en&author_id=3997	2019 Том 71 № 11 # Kovalenko L. G. Articles: 1 Brief Communications (Russian) ### On the fractional integrodifferentiation of complex polynomials in $L_0$ Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 705-710 We establish Bernstein-type inequalities for the fractional integroderivatives of arbitrary algebraic polynomials in the space $L_0$.	2020-02-16 22:03:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.26649749279022217, "perplexity": 6994.612651804886}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875141430.58/warc/CC-MAIN-20200216211424-20200217001424-00000.warc.gz"}
https://newproxylists.com/algorithms-formula-to-find-the-number-of-colors-to-be-colored-on-a-map-so-that-two-adjacent-counties-do-not-have-the-same-color/	algorithms – Formula to find the number of colors to be colored on a map so that two adjacent counties do not have the same color Ironically, yes. Just use n colors. But I guess you mean the minimum number of colors. In this case, the problem is NP-complete. By the way, we know that: $$text {Let G text {be} X_G text {correspond to the number of colors of G and} D_G text {to the maximum number of adjacent cells in G,}$$we have that $$X_G the D_G + 1$$	2019-09-16 05:13:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8925380110740662, "perplexity": 162.39263794374625}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572484.20/warc/CC-MAIN-20190916035549-20190916061549-00168.warc.gz"}
https://www.physicsforums.com/threads/inductance-of-a-straight-wire.620356/	# Inductance of a straight wire 1. Jul 12, 2012 ### htg It is zero, contrary to published formulas. I assume that we are talking about a single wire, not about a twisted bunch of wires. Why is the misconception so popular? 2. Jul 12, 2012 ### cabraham Please elaborate. Why is it zero? Why would the published info be wrong if it is so obviously zero? Why do you get it & not the scientific community? Are you that much smarter than them, or do you have exclusive access to info we are unaware of? I'm just curious how a whole body of investigators, researchers, & practitioners can be so clueless, while the plain & simple truth is obvious to you but not the experts. You're making quite a claim, can you back it up? If there is merit to your claim, it may be merely semantics. How do you define inductance? What geometry do you use for the computations? What is wrong with published data? Please explain. Claude 3. Jul 12, 2012 ### yungman Say what?!!! 4. Jul 12, 2012 ### vk6kro Straight wires are used as inductors all the time at microwave frequencies. Usually as tracks on a printed circuit board. Even at 50 MHz a wire of 2 inch length can cause oscillation due to its inductance. Here is a calaculator for various wires: http://www.consultrsr.com/resources/eis/induct5.htm 5. Jul 12, 2012 ### truesearch "How do you define inductance?" Can someone give a definitive definition of inductance and we can take it from there. 6. Jul 12, 2012 ### htg In usual circuits there is a return path, so you practically always have a loop, which actually has inductance. To see why a straight wire has zero inductance, consider two charged balls, a switch - e.g. a MOSFET, and the straight wire. From the Biot-Savart-Laplace law you can calculate the magnetic field generated by the current in one part of the wire and see that it does not induce any voltage in another part of the wire. 7. Jul 12, 2012 ### jim hardy then why will a straight wire antenna resonate? 8. Jul 12, 2012 ### htg A dipole has capacitance, which, when connected to proper inductance gives you an LC circuit, which resonates at certain frequency. 9. Jul 12, 2012 ### yungman So you are saying a wire that is open does not form a circuit ( loop), thereby there is no inductance. So by the same analogy, you take a 10uH inductor and hang in the air, there is no connection and no circuit. Then there's no inductance? As it only has inductance if you have it in a circuit that form a loop where current flow. Same thing as your gold coin at home don't worth anything as it only worth a lot if you sell it. If you don't sell it, there is no value!!!! OK, I'll give you my address, please sent them to me, there is no value, it's only take up room and is dead weight!!!!:rofl: 10. Jul 12, 2012 ### jim hardy and i always figured that capacitance was the return path. The directors & reflectors in a yagi will resonate unconnected to anything. Inductance is flux linkages per ampere and the amperes definitely flow and i don't see why flux wouldn't link a straight conductor. it's just counterintuitive, that's all. 11. Jul 12, 2012 ### jim hardy http://www.ee.scu.edu/eefac/healy/indwire.html L = $\Phi$/I Being an old guy, that above is based on a 1908 work appeals to me. 12. Jul 12, 2012 ### cabraham But the definition of inductance, as correctly stated in a post above, is flux linkage per current. An inductance of 1 henry equals 1 weber-turn per amp. Induced voltage is computed per Faraday as v(t) = -Nd(phi)/dt. Induced voltage does not define inductance. Since Nphi = Li, then v(t) = -Nd(phi)/dt = -d(LI)/dt = -Ldi/dt - idL/dt. But dL/dt = 0 for an inductor of fixed value, leaving only the 1st term in the equation above. Nonetheless, L = Nphi/i, is how L is defined. You don't need a closed loop to have a current and/or a flux. It's been correctly stated above that capacitance can "close a loop" that is otherwise "open". Displacement current does not require a closed path, actually it is really an open path type of current. Inductance is well defined for an open circuit. Also, with wire tables, one can define the geometry as a pair of parallel wires, with a given spacing, and then tabulate an inductance per unit length. It is understood as conduction current in as closed path with a specific value of spacing. As the loop is lengthened the inductance increases linearly. Anyway, like I said, I would advise you & others against taking the viewpoint that for decades/centuries countless pros examined a topic but all missed something that is obvious to you. Even a very smart person is not in league of their own. If we held a science trivia contest with the world's best engrs, physicists, chemists, etc., I doubt that 1 person is heads & shoulder superior to all others. It isn't that way. I'll elaborate where necessary. BR. Claude 13. Jul 12, 2012 ### truesearch Any circuit in which a change of current is accompanied by a change of flux, and therefore by an induced emf, is said to be 'inductive' or to posess 'self inductance'. It is impossible to have a perfectly 'non-inductive' circuit. In cases where the inductance has to be reduced to the smallest possible value....for instance, in resistance boxes, (I have one by my side !!!) the wire is bent back on itself so that the magnetising effect of one conductor is neutralised by that of the adjacent conductor. 14. Jul 12, 2012 ### truesearch My text books define self inductance in terms of induced emf/rate of change of current. If the magnetic circuit has constant reluctance then this definition is equivalent to ???which post defined inductance???? Last edited: Jul 12, 2012 15. Jul 12, 2012 ### cabraham The above post by Jim Hardy gave the definition for inductance. By the way, flux linkage per unit current is the basis, & emf = rate of change of current times inductance is derived. L = N*∅/i is the basic definition of inductance. In units, 1 henry = 1 weber-turn/amp. Straight conductors can easily carry current & have an associated flux linkage. For a straight conductor, inductance is easily definable. No credible reference says otherwise. Again, if it were so simple that intuition alone can crack it, it would have been cracked in the latter 19th century. Claude 16. Jul 12, 2012 ### carlgrace You must be a theorist. Short wires called bond wires are used in integrated circuits to connect the bond pads on the surface of the integrated circuit to the pins on the package. These wires are not coils. They present an inductance of roughly 1 nH/mm. You ignore this at your peril. If you worked for me, and you tried to explain that there was no way your amplifier was oscillating because the wires have zero inductance, I'd fire you. Empiricism trumps half-cocked theory. 17. Jul 12, 2012 ### truesearch So it is confirmed that straight wires have an inductance of 1nH/mm. How many employees do you have carlgrace?? How many have been fired because of their half-cocked theoretical ideas ? Do you employ any 'theorists'; ?? 18. Jul 12, 2012 ### carlgrace Well it is confirmed that wires of a specific thickness have a specific inductance per unit length. I think it varies depending on material and geometry but it is always there. I was reacting to the OP's aggressiveness and certitude even while holding a notion that is widely known to be false. I say the OP is a theorist (I shouldn't have, it was rude) because this is a trait I have mostly seen in theorists (although of course not always). For example, there were strong theoretical arguments for why silicon transistors were not possible using the contemporary processing technology of the 1950s. That is, until Texas Instruments built them anyway. I work in academia now so I don't have any employees. At one time I lead a design team. I haven't seen anyone fired because of half-cocked theoretical ideas, but I have seen people eased out of their jobs due to inability to learn from empirical data and therefore change their views. Being inflexible is an express-lane to failure in the semiconductor business. There are many, many theorists where I work. I also worked with some system architects when I was in industry. Theory is extremely valuable, getting high off your own supply and "proving" things that are obviously false is actually destructive in practice. Look at the OP's post invoking Biot-Savart to "prove" a MOSFET doesn't have any inductance in the channel. For pete's sake! My post was too flippant, but my point was that while simple equations are useful, but they don't necessarily "mean" anything, especially when there is contradictory experimental evidence available. 19. Jul 12, 2012 ### truesearch You do not employ anyone but you feel that you are in a position to determine who should be employed !!! You are rude Dont quote irrelevant facts about silicon transistors !!! "theory is extremely valuable" .... many here will value this statement ....."they dont necessarily "mean" anything, especially when there is contradictory experimental evidence available". Can you back this up with hard evidence or is it a subjective, personal opinion? Have you read the guidelines to these forums.... agreemernt with text books ???? If you do not like what you hear there are complaint procedures. 20. Jul 12, 2012 ### carlgrace I did employ people until a few years ago. Do you think that as soon as I took a position in academia I had to stop drawing conclusions based on my own previous experience? Yeah I apologized about that. I shouldn't have said that about theory. I don't believe it is irrelevant. My point for saying that was showing there are a lot of pitfalls in trusting in "theory" too much. It has direct, specific consequences. TI became the world leader in transistor manufacturing because they chose not to buy into the contemporary state of semiconductor theory. Yes, I can back it up. For example, • the OP used correct, but simplified reasoning to show there was no inductance in a straight wire. The OP is wrong. That is hard evidence. Do you think the OP is correct? • Standard pn-junction theory (at the level the OP was using to "prove" the nonexistance of straight wire inductance) doesn't predict avalanche breakdown. So, based on the level of theory the OP was using, the OP would categorically state "Avalanche Breakdown is a misconception". The OP would be wrong. This is not a strawman, since you asked for a concrete example. Here's another concrete example: • In the mid-1990s a lot of researchers believed that it would be impossible to build high-precision analog circuits in deep submicron CMOS technology because of excess thermal noise due to hot electron effects. This effect is called Drain Induced Barrier Lowering. Rather than give up, people built precision circuits anyway. We had a situation in the early 2000s where theory "proved" it was impossible, yet we had devices in the lab operating. Today, the theory has caught up and the equations modified to show that there is less excess thermal noise in very deep submicron circuits because of something now called the Reverse Short Channel Effect. This is all in the literature. My point here is that learning a little theory and then making strident statements about "misconceptions" is not only stupid, it's dangerous. 21. Jul 12, 2012 ### truesearch The op is not correct, a wire does have inductance.. "being rude"... that is a common weakness here... there is never any need to make personal comments as far as I am concerned (purely personal opinon), the physics should provide the answer. I apologise for raising it. I do not think that you need to be criticised for this. Distractive truisms (eg deep submicronCmos technologyransistors (I just made that up !!!!) should not be needed to disguise, back up, justify physics facts. Physics is difficult but there is an awful amount of information available in standard text books. I wish that responses here were backed upwith text book references. 22. Jul 12, 2012 ### carlgrace This was not a distracting truism. I said that simple theories "dont necessarily "mean" anything, especially when there is contradictory experimental evidence available". You then asked if I could "back this up with hard evidence or is it a subjective, personal opinion?". Essentially you were accusing me of arguing based on opinion, rather than facts. I produce the fact and then you claim it is a "distractive truism". Come on. The "now known to be not entirely correct" theory of excess noise in MOSFETS can be seen in the following papers: • A. A. Abidi, High-frequency noise measurements on FET's with small dimensions, IEEE Transactions on Electron Devices, vol. 3, no. 11, pp. 1801-1805, Nov. 1986. • R.P. Jindal, Hot-electron eff ects on channel thermal noise in fi ne-line NMOS transistors, IEEE Transactions on Electron Devices, vol. 3, no. 9, pp. 1395-1397, Sept. 1986. • Suharli Tedja, Jan Van der Spiegel, and Hugh H. Williams, Analytical and experimental studies of thermal noise in MOS-FETs, IEEE Transactions on Electron Devices, vol. 41, no.11, pp. 2069-2075, Nov. 1994. • Bing Wang, James R. Hellums, and Charles G. Sodini, MOS-FET thermal noise modeling for analog integrated circuits, IEEE Journal of Solid-State Circuits, vol. 29, no. 7, pp. 833-835, July 1994. These papers all support the theory that precision circuits would be impossible once devices got too small. Several explicitly state this. They were wrong. Lucky that the people who built these circuits didn't listen to the theory. If you want textbooks, these papers use as a basis the quantum theory of semiconductors and a method where you slice the transistor channel into small slices and associate each slice with an equilibrium noise source. This method can be traced back to Shockley himself, and here are two important books in the field that use it: • A. van der Ziel, Noise in solid state devices and circuits. New York: Wiley, 1986. • W. Shockley et al., in Quantum Theory of Atoms and Molecules and the Solid-State. New York: Academic, 1966, p.537. It turns out this theory is misleading. To deal with very small channels, you need to include the effect of non-equilibrium noise. Only very modern books would touch on this as it is an advanced correction. The old Shockley method is used because it gives good intuition. But there are circumstances under which it is wrong. I wonder if the OP would read a textbook and declare that precision analog circuits were impossible? You asked for an example in which established theory was wrong, but this wasn't known until people went ahead and did the "impossible" anyway and then theory caught up. Now you have one. With textbook references and everything. 23. Jul 12, 2012 ### The Electrician 24. Jul 13, 2012 ### Staff: Mentor A troll OP can lead to a difficult discussion. Thread closed for now.	2018-06-24 22:43:40	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5338273644447327, "perplexity": 1896.6189842957356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867095.70/warc/CC-MAIN-20180624215228-20180624235228-00350.warc.gz"}
http://www.ieee-jas.org/article/2019/5?pageType=en	A journal of IEEE and CAA , publishes high-quality papers in English on original theoretical/experimental research and development in all areas of automation ## Vol.6, No.5, 2019 Display Type： \| 2019, 6(5): 1081-1094. doi: 10.1109/JAS.2019.1911672 Abstract: Evacuation leaders and/or equipment provide route and exit information for people and guide them to the expected destinations, which could make crowd evacuation more efficient in case of emergency. The purpose of this paper is to provide an overview of recent advances in guided crowd evacuation. Different guided crowd evacuation approaches are classified according to guidance approaches and technologies. A comprehensive analysis and comparison of crowd evacuation with static signage, dynamic signage, trained leader, mobile devices, mobile robot and wireless sensor networks are presented based on a single guidance mode perspective. In addition, the different evacuation guidance systems that use high-tech means such as advanced intelligent monitoring techniques, AI techniques, computer technology and intelligent inducing algorithms are reviewed from a system’s perspective. Future researches in the area of crowd evacuation are also discussed. 2019, 6(5): 1095-1107. doi: 10.1109/JAS.2019.1911675 Abstract: The introduction of automated driving systems raised questions about how the human driver interacts with the automated system. Non-cooperative game theory is increasingly used for modelling and understanding such interaction, while its counterpart, cooperative game theory is rarely discussed for similar applications despite it may be potentially more suitable. This paper describes the modelling of a human driver’s steering interaction with an automated steering system using cooperative game theory. The distributed Model Predictive Control approach is adopted to derive the driver’s and the automated steering system’s strategies in a Pareto equilibrium sense, namely their cooperative Pareto steering strategies. Two separate numerical studies are carried out to study the influence of strategy parameters, and the influence of strategy types on the driver’s and the automated system’s steering performance. It is found that when a driver interacts with an automated steering system using a cooperative Pareto steering strategy, the driver can improve his/her performance in following a target path through increasing his/her effort in pursuing his/her own interest under the driver-automation cooperative control goal. It is also found that a driver’s adoption of cooperative Pareto steering strategy leads to a reinforcement in the driver’s steering angle control, compared to the driver’s adoption of non-cooperative Nash strategy. This in turn enables the vehicle to return from a lane-change maneuver to straight-line driving swifter. 2019, 6(5): 1108-1115. doi: 10.1109/JAS.2019.1911678 Abstract: In this paper, a data-driven control approach is developed by reinforcement learning (RL) to solve the global robust optimal output regulation problem (GROORP) of partially linear systems with both static uncertainties and nonlinear dynamic uncertainties. By developing a proper feedforward controller, the GROORP is converted into a global robust optimal stabilization problem. A robust optimal feedback controller is designed which is able to stabilize the system in the presence of dynamic uncertainties. The closed-loop system is ensured to be input-to-output stable regarding the static uncertainty as the external input. This robust optimal controller is numerically approximated via RL. Nonlinear small-gain theory is applied to show the input-to-output stability for the closed-loop system and thus solves the original GROORP. Simulation results validates the efficacy of the proposed methodology. 2019, 6(5): 1116-1126. doi: 10.1109/JAS.2019.1911681 Abstract: We consider quadratic stabilization for a class of switched systems which are composed of a finite set of continuoustime linear subsystems with norm bounded uncertainties. Under the assumption that there is no single quadratically stable subsystem, if a convex combination of subsystems is quadratically stable, then we propose a state-dependent switching law, based on the convex combination of subsystems, such that the entire switched linear system is quadratically stable. When the state information is not available, we extend the discussion to designing an output-dependent switching law by constructing a robust Luenberger observer for each subsystem. 2019, 6(5): 1127-1138. doi: 10.1109/JAS.2019.1911684 Abstract: Eye center localization is one of the most crucial and basic requirements for some human-computer interaction applications such as eye gaze estimation and eye tracking. There is a large body of works on this topic in recent years, but the accuracy still needs to be improved due to challenges in appearance such as the high variability of shapes, lighting conditions, viewing angles and possible occlusions. To address these problems and limitations, we propose a novel approach in this paper for the eye center localization with a fully convolutional network (FCN), which is an end-to-end and pixels-to-pixels network and can locate the eye center accurately. The key idea is to apply the FCN from the object semantic segmentation task to the eye center localization task since the problem of eye center localization can be regarded as a special semantic segmentation problem. We adapt contemporary FCN into a shallow structure with a large kernel convolutional block and transfer their performance from semantic segmentation to the eye center localization task by fine-tuning. Extensive experiments show that the proposed method outperforms the state-of-the-art methods in both accuracy and reliability of eye center localization. The proposed method has achieved a large performance improvement on the most challenging database and it thus provides a promising solution to some challenging applications. 2019, 6(5): 1139-1151. doi: 10.1109/JAS.2019.1911687 Abstract: The paper proposes a novel \begin{document}$H_\infty$\end{document} load frequency control (LFC) design method for multi-area power systems based on an integral-based non-fragile distributed fixed-order dynamic output feedback (DOF) tracking-regulator control scheme. To this end, we consider a nonlinear interconnected model for multi-area power systems which also include uncertainties and time-varying communication delays. The design procedure is formulated using semi-definite programming and linear matrix inequality (LMI) method. The solution of the proposed LMIs returns necessary parameters for the tracking controllers such that the impact of model uncertainty and load disturbances are minimized. The proposed controllers are capable of receiving all or part of subsystems information, whereas the outputs of each controller are local. These controllers are designed such that the resilient stability of the overall closed-loop system is guaranteed. Simulation results are provided to verify the effectiveness of the proposed scheme. Simulation results quantify that the distributed (and decentralized) controlled system behaves well in presence of large parameter perturbations and random disturbances on the power system. 2019, 6(5): 1152-1165. doi: 10.1109/JAS.2017.7510724 Abstract: Structural controllability is critical for operating and controlling large-scale complex networks. In real applications, for a given network, it is always desirable to have more selections for driver nodes which make the network structurally controllable. Different from the works in complex network field where structural controllability is often used to explore the emergence properties of complex networks at a macro level, in this paper, we investigate it for control design purpose at the application level and focus on describing and obtaining the solution space for all selections of driver nodes to guarantee structural controllability. In accord with practical applications, we define the complete selection rule set as the solution space which is composed of a series of selection rules expressed by intuitive algebraic forms. It explicitly indicates which nodes must be controlled and how many nodes need to be controlled in a node set and thus is particularly helpful for freely selecting driver nodes. Based on two algebraic criteria of structural controllability, we separately develop an input-connectivity algorithm and a relevancy algorithm to deduce selection rules for driver nodes. In order to reduce the computational complexity, we propose a pretreatment algorithm to reduce the scale of network's structural matrix efficiently, and a rearrangement algorithm to partition the matrix into several smaller ones. A general procedure is proposed to get the complete selection rule set for driver nodes which guarantee network's structural controllability. Simulation tests with efficiency analysis of the proposed algorithms are given and the result of applying the proposed procedure to some real networks is also shown, and these all indicate the validity of the proposed procedure. 2019, 6(5): 1166-1178. doi: 10.1109/JAS.2019.1911690 Abstract: State convergence is a novel control algorithm for bilateral teleoperation of robotic systems. First, it models the teleoperation system on state space and considers all the possible interactions between the master and slave systems. Second, it presents an elegant design procedure which requires a set of equations to be solved in order to compute the control gains of the bilateral loop. These design conditions are obtained by turning the master-slave error into an autonomous system and imposing the desired dynamic behavior of the teleoperation system. Resultantly, the convergence of master and slave states is achieved in a well-defined manner. The present study aims at achieving a similar convergence behavior offered by state convergence controller while reducing the number of variables sent across the communication channel. The proposal suggests transmitting composite master and slave variables instead of full master and slave states while keeping the operator’s force channel intact. We show that, with these composite and force variables; it is indeed possible to achieve the convergence of states in a desired way by strictly following the method of state convergence. The proposal leads to a reduced complexity state convergence algorithm which is termed as composite state convergence controller. In order to validate the proposed scheme in the absence and presence of communication time delays, MATLAB simulations and semi-real time experiments are performed on a single degree-of-freedom teleoperation system. 2019, 6(5): 1179-1186. doi: 10.1109/JAS.2017.7510667 Abstract: This paper deals with analyzing a newly introduced method for tuning of fractional order [proportional derivative] (FO[PD]) controllers to be used in motion control. By using this tuning method, not only the phase margin and gain crossover frequency are adjustable, but also robustness to variations in the plant time-constant is guaranteed. Conditions on the values of control specifications (desired phase margin and gain crossover frequency) for solution existence in this tuning method are found. Also, the number of solutions is analytically determined in this study. Moreover, experimental verifications are presented to indicate the applicability of the obtained results. 2019, 6(5): 1187-1195. doi: 10.1109/JAS.2019.1911693 Abstract: It is well known that automatic speech recognition (ASR) is a resource consuming task. It takes sufficient amount of data to train a state-of-the-art deep neural network acoustic model. As for some low-resource languages where scripted speech is difficult to obtain, data sparsity is the main problem that limits the performance of speech recognition system. In this paper, several knowledge transfer methods are investigated to overcome the data sparsity problem with the help of high-resource languages. The first one is a pre-training and fine-tuning (PT/FT) method, in which the parameters of hidden layers are initialized with a well-trained neural network. Secondly, the progressive neural networks (Prognets) are investigated. With the help of lateral connections in the network architecture, Prognets are immune to forgetting effect and superior in knowledge transferring. Finally, bottleneck features (BNF) are extracted using cross-lingual deep neural networks and serves as an enhanced feature to improve the performance of ASR system. Experiments are conducted in a low-resource Vietnamese dataset. The results show that all three methods yield significant gains over the baseline system, and the Prognets acoustic model performs the best. Further improvements can be obtained by combining the Prognets model and bottleneck features. 2019, 6(5): 1196-1208. doi: 10.1109/JAS.2019.1911696 Abstract: Stochastic iterative learning control (ILC) is designed for solving the tracking problem of stochastic linear systems through fading channels. Consequently, the signals used in learning control algorithms are faded in the sense that a random variable is multiplied by the original signal. To achieve the tracking objective, a two-dimensional Kalman filtering method is used in this study to derive a learning gain matrix varying along both time and iteration axes. The learning gain matrix minimizes the trace of input error covariance. The asymptotic convergence of the generated input sequence to the desired input value is strictly proved in the mean-square sense. Both output and input fading are accounted for separately in turn, followed by a general formulation that both input and output fading coexists. Illustrative examples are provided to verify the effectiveness of the proposed schemes. 2019, 6(5): 1209-1219. doi: 10.1109/JAS.2019.1911699 Abstract: This paper presents a new Long-range generalized predictive controller in the synchronous reference frame for a wind energy system doubly-fed induction generator based. This controller uses the state space equations that consider the rotor current and voltage as state and control variables, to execute the predictive control action. Therefore, the model of the plant must be transformed into two discrete transference functions, by means of an auto-regressive moving average model, in order to attain a discrete and decoupled controller, which makes it possible to treat it as two independent single-input single-output systems instead of a magnetic coupled multiple-input multiple-output system. For achieving that, a direct power control strategy is used, based on the past and future rotor currents and voltages estimation. The algorithm evaluates the rotor current predictors for a defined prediction horizon and computes the new rotor voltages that must be injected to controlling the stator active and reactive powers. To evaluate the controller performance, some simulations were made using Matlab/Simulink. Experimental tests were carried out with a small-scale prototype assuming normal operating conditions with constant and variable wind speed profiles. Finally, some conclusions respect to the dynamic performance of this new contro-ller are summarized. 2019, 6(5): 1220-1229. doi: 10.1109/JAS.2018.7511273 Abstract: This paper presents a new composite nonlinear bilateral control method based on the nonlinear disturbance observer (NDOB) for teleoperation systems with external disturbances. By introducing the estimations of NDOB and systems' nominal nonlinear dynamics into controller design, a NDOB based composite nonlinear bilateral controller is constructed to attenuate the influence of disturbance and uncertain nonlinearities. As compared with the existing bilateral control methods which usually achieve force haptic (i.e., contact force tracking) through a passive way, the newly proposed method has two major merits: 1) asymptotical convergence of both position and force tracking errors is guaranteed; 2) disturbance influence on force tracking error dynamics is rejected through the direct feedforward compensation of disturbance estimation. Simulations on a nonlinear teleoperation system are carried out and the results validate the effectiveness of the proposed controller. 2019, 6(5): 1230-1239. doi: 10.1109/JAS.2019.1911702 Abstract: This paper discusses the design and software-in-the-loop implementation of adaptive formation controllers for fixed-wing unmanned aerial vehicles (UAVs) with parametric uncertainty in their structure, namely uncertain mass and inertia. In fact, when aiming at autonomous flight, such parameters cannot assumed to be known as they might vary during the mission (e.g. depending on the payload). Modeling and autopilot design for such autonomous fixed-wing UAVs are presented. The modeling is implemented in Matlab, while the autopilot is based on ArduPilot, a popular open-source autopilot suite. Specifically, the ArduPilot functionalities are emulated in Matlab according to the Ardupilot documentation and code, which allows us to perform software-in-the-loop simulations of teams of UAVs embedded with actual autopilot protocols. An overview of realtime path planning, trajectory tracking and formation control resulting from the proposed platform is given. The software-inthe-loop simulations show the capability of achieving different UAV formations while handling uncertain mass and inertia. 2019, 6(5): 1240-1250. doi: 10.1109/JAS.2017.7510454 Abstract: In this study, we present a Pareto-based chemical-reaction optimization (PCRO) algorithm for solving the multi-area environmental/economic dispatch optimization problems. Two objectives are minimized simultaneously, i.e., total fuel cost and emission. In the proposed algorithm, each solution is represented by a chemical molecule. A novel encoding mechanism for solving the multi-area environmental/economic dispatch optimization problems is designed to dynamically enhance the performance of the proposed algorithm. Then, an ensemble of effective neighborhood approaches is developed, and a self-adaptive neighborhood structure selection mechanism is also embedded in PCRO to increase the search ability while maintaining population diversity. In addition, a grid-based crowding distance strategy is introduced, which can obviously enable the algorithm to easily converge near the Pareto front. Furthermore, a kinetic-energy-based search procedure is developed to enhance the global search ability. Finally, the proposed algorithm is tested on sets of the instances that are generated based on realistic production. Through the analysis of experimental results, the highly effective performance of the proposed PCRO algorithm is favorably compared with several algorithms, with regards to both solution quality and diversity. 2019, 6(5): 1251-1260. doi: 10.1109/JAS.2019.1911705 Abstract: In this paper, we use the cycle basis from graph theory to reduce the size of the decision variable space of optimal network flow problems by eliminating the aggregated flow conservation constraint. We use a minimum cost flow problem and an optimal power flow problem with generation and storage at the nodes to demonstrate our decision variable reduction method. The main advantage of the proposed technique is that it retains the natural sparse/decomposable structure of network flow problems. As such, the reformulated problems are still amenable to distributed solutions. We demonstrate this by proposing a distributed alternating direction method of multipliers (ADMM) solution for a minimum cost flow problem. We also show that the communication cost of the distributed ADMM algorithm for our proposed cycle-based formulation of the minimum cost flow problem is lower than that of a distributed ADMM algorithm for the original arc-based formulation. 2019, 6(5): 1261-1267. doi: 10.1109/JAS.2018.7511093 Abstract: Tracking control is a very challenging problem in the networked control system (NCS), especially for the process with blurred mechanism and where only input-output data are available. This paper has proposed a data-based design approach for the networked tracking control system (NTCS). The method utilizes the input-output data of the controlled process to establish a predictive model with the help of fuzzy cluster modelling (FCM) technology. Then, the deduced error and error change in the future are treated as inputs of a fuzzy sliding mode controller (FSMC) to obtain a string of future control actions. These candidate control actions in the controller side are delivered to the plant side. Thus, the network induced time delays are compensated by selecting appropriate control action. Simulation outputs prove the validity of the proposed method. 2019, 6(5): 1268-1280. doi: 10.1109/JAS.2019.1911618 Abstract: Fault diagnosis is an important measure to ensure the safety of production, and all kinds of fault diagnosis methods are of importance in actual production process. However, the complexity and uncertainty of production process often lead to the changes of data distribution and the emergence of new fault classes, and the number of the new fault classes is unpredictable. The reconstruction of the fault diagnosis model and the identification of new fault classes have become core issues under the circumstances. This paper presents a fault diagnosis method based on model transfer learning and the main contributions of the paper are as follows: 1) An incremental model transfer fault diagnosis method is proposed to reconstruct the new process diagnosis model. 2) Breaking the limit of existing method that the new process can only have one more class of faults than the old process, this method can identify M faults more in the new process with the thought of incremental learning. 3) The method offers a solution to a series of problems caused by the increase of fault classes. Experiments based on Tennessee-Eastman process and ore grinding classification process demonstrate the effectiveness and the feasibility of the method. 2019, 6(5): 1281-1290. doi: 10.1109/JAS.2019.1911525 Abstract: In this paper, the problem of making an input-delay system with saturating actuators finite-time stable by virtue of digital control is investigated. A digital state feedback controller and digital observer-controller compensator are designed for two cases: when the state of the input-delay system are available or when it is unavailable. Sufficient conditions which guarantee finite-time stability of a closed-loop input-delay system are given and the proof procedure is presented in a heuristic way by constructing appropriate comparison functions. The condition can be transformed into the intersection of two curves satisfying some constraints, which reveals the relationship between designed parameters clearly. Finally, simulation results are presented to validate the method proposed in this paper. IEEE/CAA Journal of Automatica Sinica • Indexed in: SCIE, EI, Scopus, etc. CiteScore 2018: 5.31 Rank：Top 9% (Category of Control and Systems Engineering), Top 10% (Categories of Information System and Artificial Intelligence)	2020-01-21 00:59:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45420822501182556, "perplexity": 880.0154371244453}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250601040.47/warc/CC-MAIN-20200120224950-20200121013950-00329.warc.gz"}
https://www.snapxam.com/problems/71714040/limit-as-x-approaches-0-of-x-3cotx-1-1cosx	# Step-by-step Solution Go! 1 2 3 4 5 6 7 8 9 0 a b c d f g m n u v w x y z . (◻) + - × ◻/◻ / ÷ 2 e π ln log log lim d/dx Dx \|◻\| = > < >= <= sin cos tan cot sec csc asin acos atan acot asec acsc sinh cosh tanh coth sech csch asinh acosh atanh acoth asech acsch ## Step-by-step explanation Problem to solve: $\lim_{x\to0}\left(\frac{x^3\cot\left(x\right)}{1-\cos\left(x\right)}\right)$ Learn how to solve limits problems step by step online. $\frac{0^3\cot\left(0\right)}{1-\cos\left(0\right)}$ Learn how to solve limits problems step by step online. Evaluate the limit of (x^3cot(x))/(1-cos(x)) as x approaches 0. Evaluate the limit by replacing all occurrences of x by 0. Simplifying. section:If the limit is undefined, we need to find the left side and right side limits. The limit is undefined. $\infty$ ### Problem Analysis $\lim_{x\to0}\left(\frac{x^3\cot\left(x\right)}{1-\cos\left(x\right)}\right)$ Limits ~ 0.08 seconds	2020-08-03 18:09:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9556832313537598, "perplexity": 1429.1589151898306}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439735823.29/warc/CC-MAIN-20200803170210-20200803200210-00552.warc.gz"}
http://incubator.theindustria.com/how-do-hxnxyw/6ca7a1-selectivity-of-atomic-emission-spectroscopy	• The Channel selectivity of atomic emission spectroscopy The ICP-AES is an analytical technique based on the principles of atomic spectroscopy for the determination of more than 70 elements with detection limits in the parts per billion to parts per million range. 4. Chemical interferences, when present, decrease the sensitivity of the analysis. Clogging the aspirator and burner assembly decreases the rate of aspiration, which decreases the analyte’s concentration in the flame. • Atomic Absorption -> it measures the radiation absorbed by the unexcited atoms that are determined. 1982, 59, 875–876. One way to avoid a determinate error when using external standards is to match the matrix of the standards to that of the sample. The solid black line in Figure $$\PageIndex{6}$$ shows the ideal calibration curve, assuming we match the standard’s matrix to the sample’s matrix, and that we do so without adding any additional sodium. Because potassium is present at a much higher concentration than is sodium, its ionization suppresses the ionization of sodium. Given the previous questions, is the result of your analysis likely to underestimate or to overestimate the amount of sodium in the salt substitute? The solid black line in Figure 10.62 shows the ideal calibration curve assuming that we match the matrix of the standards to the sample’s matrix, and that we do so without adding an additional sodium. If we prepare the external standards without adding KCl, the emission for each standard decreases due to increased ionization. Figure 10.57 Valence shell energy level diagram for sodium. Many atomic emission spectrometers, however, are dedicated instruments designed to take advantage of features unique to atomic emission, including the use of plasmas, arcs, sparks, and lasers as atomization and excitation sources, and an enhanced capability for multielemental analysis. Educ. To accurately correct for these errors the analyte and internal standard emission lines are monitored simultaneously. For example, an analysis for Ni using the atomic emission line at 349.30 nm is complicated by the atomic emission line for Fe at 349.06 nm. After zeroing the instrument with an appropriate blank, the instrument is optimized at a wavelength of 589.0 nm while aspirating a standard solution of Na+. Narrower slit widths provide better resolution, but at the cost of less radiation reaching the detector. Both techniques involve the atomization of a sample. (See Section 5C.3 in Chapter 5 to review the method of standard additions.). 1982, 59, 875–876]. Reagent grade KCl, for example, may contain 40–50 μg Na/g. A plasma is a hot, partially ionized gas that contains an abundant concentration of cations and electrons. Older atomic emission instruments often used a total consumption burner in which the sample is drawn through a capillary tube and injected directly into the flame. The atomic emission technique measures the energy lost by an atom passing from an excited state to a lower energy state. For samples and standards in which the analyte’s concentration exceeds the detection limit by at least a factor of 50, the relative standard deviation for both flame and plasma emission is about 1–5%. However, since the detector is capable of measuring light intensity, quantitative analysis, as well as qualitative analysis, is possible. Source: modified from Xvlun (commons.wikipedia.org). Because a plasma operates at a much higher temperature than a flame, it provides for a better atomization efficiency and a higher population of excited states. When using a plasma, which suffers from fewer chemical interferences, the calibration curve often is linear over four to five orders of magnitude and is not affected significantly by changes in the matrix of the standards. A higher temperature flame than atomic absorption spectroscopy (AA) is typically used to produce excitation of analyte atoms. A schematic diagram of the inductively coupled plasma source (ICP) is shown in Figure 10.58. An atomic emission spectrometer is similar in design to the instrumentation for atomic absorption. Because the sensitivity of plasma emission is less affected by the sample matrix, a calibration curve prepared using standards in a matrix of distilled water is possible even for samples that have more complex matrices. Another approach to a multielemental analysis is to use a multichannel instrument that allows us to simultaneously monitor many analytes. Suppose you decide to use an external standardization. Quantitative applications based on the atomic emission from electric sparks were developed by Lockyer in the early 1870 and quantitative applications based on flame emission were pioneered by Lundegardh in 1930. X-ray absorption spectroscopy (XAS) and X-ray emission spectroscopy (XES) provide complementary information on the electronic structure of materials (9, 10) and on the orbitals participating in the interaction with absorbing molecules . ICP is an atomic emission technique and can be coupled to an optical spectrophotometer (ICP OES) or Mass spectrometry (ICP-MS). The burner head consists of single or multiple slots, or a Meker style burner. For many elements at temperatures of less than 5000 K the Boltzmann distribution is approximated as, $N^* = N\left(\frac{g_{i}}{g_{0}}\right) e^{-E_i / k T} \label{10.2}$. What effect does this have on the analysis? The solid black line shows the ideal calibration curve assuming matrix matching of samples and standards with pure KCl. The plasmas used in atomic emission are formed by ionizing a flowing stream of argon gas, producing argon ions and electrons. $I_\ce{e}= \mathrm{1.97 + 1.37 × \dfrac{g\: Na}{mL}}$, The concentration of sodium in the sample is equal to the absolute value of the calibration curve’s x-intercept. Spectrosc. 10-17 Compare between the advantages and disadvantages of atomic emission spectroscopy based on arcs and plasma. This is shown by the lower of the two dashed red lines. The method of internal standards is used when the variations in source parameters are difficult to control. The most important spectral interference is broad, background emission from the flame or plasma and emission bands from molecular species. • produce sharp-line emission spectra. A simple design for a multichannel spectrometer, shown in Figure $$\PageIndex{3}$$, couples a monochromator with multiple detectors that are positioned in a semicircular array around the monochromator at positions that correspond to the wavelengths for the analytes. Likewise, colorimetric methods using selective reagents typically require complex sample preparation and suffer from selectivity issues. An alternating radio-frequency current in the induction coil creates a fluctuating magnetic field that induces the argon ions and the electrons to move in a circular path. The wavelengths corresponding to several transitions are shown. Because a plasma’s temperature is much higher, a background interference due to molecular emission is less of a problem. For example, in a 2500 K flame a temperature fluctuation of ±2.5 K gives a relative standard deviation of 1% in emission intensity. Solid samples are analyzed by dissolving in a solvent and using a flame or plasma atomizer. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Substituting zero for the emission intensity and solving for sodium’s concentration gives a result of 1.44 μgNa/mL. Because we underestimate the actual concentration of sodium in the standards, the resulting calibration curve is shown by the other dashed red line. For higher concentrations of analyte self-absorption may invert the center of the emission band (Figure $$\PageIndex{5}$$). A Meker burner is similar to the more common Bunsen burner found in most laboratories; it is designed to allow for higher temperatures and for a larger diameter flame. A calibration curve for flame emission is usually linear over two to three orders of magnitude, with ionization limiting linearity when the analyte’s concentrations is small and self-absorption limiting linearity for higher concentrations of analyte. Sample throughput with atomic emission is very rapid when using automated systems capable of multielemental analysis. For higher concentrations of analyte self-absorption may invert the center of the emission band (Figure 10.61). Have questions or comments? See Figure 3.5 to review the meaning of macro and meso for describing samples, and the meaning of major, minor, and ultratrace for describing analytes. In addition, the high concentration of electrons from the ionization of argon minimizes ionization interferences. Educ. If the flame or plasma is in thermal equilibrium, then the excited state population is proportional to the analyte’s total population, N, through the Boltzmann distribution (equation \ref{10.2}). The result is a determinate error. The most common methods are flames and plasmas, both of which are useful for liquid or solution samples. 4. Salt substitutes, which are used in place of table salt for individuals on low–sodium diets, replaces NaCl with KCl. The best way to appreciate the theoretical and practical details discussed in this section is to carefully examine a typical analytical method. A plasma is a hot, partially ionized gas that contains an abundant concentration of cations and electrons. Normally suppressing ionization is a good thing because it increases emission intensity. Modern atomic absorption spectroscopy has its beginnings in 1955 as a result of the independent work of Alan. Missed the LibreFest? Atomic emission requires a means for converting a solid, liquid, or solution analyte into a free gaseous atom. In addition, the internal standard should be subject to the same chemical interferences to compensate for changes in atomization efficiency. The selectivity of atomic emission is similar to that of atomic absorption. This is a significant source of sodium, given that the salt substitute contains approximately 100 μg Na/g. This is accomplished by the tangential flow of argon shown in the schematic diagram. Flame emission is often accomplished using an atomic absorption spectrometer, which typically costs between $10,000–$50,000. In theory, the technique allows us to analyze all elements except argon. We also expect emission intensity to increase with temperature. Figure 10.58 Schematic diagram of an inductively coupled plasma torch. The description here is based on Goodney, D. E. J. Chem. Because a plasma’s temperature is much higher, a background interference due to molecular emission is less of a problem. If an excited state atom in the flame’s center emits a photon while returning to its ground state, then a ground state atom in the cooler, outer regions of the flame may absorb the photon, decreasing the emission intensity. The result is a decrease in the emission intensity and a negative determinate error. Qualitative applications based on the color of flames were used in the smelting of ores as early as 1550 and were more fully developed around 1830 with the observation of atomic spectra generated by flame emission and spark emission [Dawson, J. Except for the alkali metals, detection limits when using an ICP are significantly better than those obtained with flame emission (Table 10.14). In fact, it is easy to adapt most flame atomic absorption spectrometers for atomic emission by turning off the hollow cathode lamp and monitoring the difference between the emission intensity when aspirating the sample and when aspirating a blank. Although intended to be sodium-free, salt substitutes contain small amounts of NaCl as an impurity. Spectrosc. Atomic emission spectroscopy has a long history. Watch the recordings here on Youtube! 1983, 37, 411–418. Depending on the brand, fumaric acid, calcium hydrogen phosphate, or potassium tartrate also may be present. It employs lasers to eject electrons from selected types of atoms or molecules, splitting the neutral species into a positive ion and a free electron with a negative charge. An increase in temperature of 10 K, for example, produces a 4% increase in the fraction of Na atoms occupying the 3p excited state. Flame emission is subject to the same types of chemical interferences as atomic absorption; they are minimized using the same methods: by adjusting the flame’s composition and by adding protecting agents, releasing agents, or ionization suppressors. The other dashed red line shows the effect of using KCl that is contaminated with NaCl, which causes us to underestimate the concentration of Na in the standards. Figure 10.57 shows a portion of the energy level diagram for sodium, which consists of a series of discrete lines at wavelengths corresponding to the difference in energy between two atomic orbitals. In some cases a calibration curve prepared using standards in a matrix of distilled water can be used for samples with more complex matrices. The intensity of an atomic emission line, Ie, is proportional to the number of atoms, N, populating the excited state, where k is a constant accounting for the efficiency of the transition. Although intended to be sodium-free, salt substitutes contain small amounts of NaCl as an impurity. If the instrument includes a scanning monochromator, we can program it to move rapidly to an analyte’s desired wavelength, pause to record its emission intensity, and then move to the next analyte’s wavelength. Educ. A sample is prepared by placing an approximately 10-g portion of the salt substitute in 10 mL of 3 M HCl and 100 mL of distilled water. To evaluate the method described in Representative Method 10.4, a series of standard additions is prepared using a 10.0077-g sample of a salt substitute. Atomic emission based on emission from a plasma was introduced in 1964. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. a Source: Parsons, M. L.; Major, S.; Forster, A. R.; App. In both cases, the result is a positive determinate error in the analysis of samples. A narrower slit width provides better resolution, but at the cost of less radiation reaching the detector. Are difficult to control: Parsons, M. L. ; major, S. ; Forster, A. R. ;.... Meso samples a calibration curve is shown by the temperature of the standards to that of the analysis trace. For simultaneous multi-analyte determination of major and minor analytes standards can be circumvented of failing to add KCl match! Using any of the sample has dissolved, it uses electromagnetic wavelengths coming! Spectroscopy: Resonance-ionization spectroscopy: Resonance-ionization spectroscopy ( AAS ) is shown here.19 results in our overestimating the concentration sodium... Of samples and standards with pure KCl 10.31 we expect that excited states background! > it measures the energy lost by an atom passing from an excited to! Adding KCl, the resulting calibration curve prepared using standards in a population. At these high temperatures the outer quartz tube must be thermally isolated from the ionization sodium. Normally suppressing ionization is a hot, partially ionized gas that contains an abundant of! Following equation are present matrix matching of samples and standards with pure KCl in some cases a calibration curve shown... Be subject to the external standards without adding KCl, for example, may contain 40–50 Na/g! Spectral interference is broad, background emission from a light source of three concentric quartz tubes surrounded! Sensitivity of the excitation source and the practical details discussed in this section is to a. Good idea for this analysis curve prepared using standards in a variety of sample matrices sequential! That are capable of both sequential and simultaneous analysis NaCl with KCl between the and... The scale of operations for atomic absorption spectroscopy is based on emission the. Expect emission intensity is proportional to the external standards is shown in figure \ ( \PageIndex { 2 } )... A selectivity of atomic emission spectroscopy contaminant, which typically costs between $10,000–$ 50,000 on the emission of ultraviolet visible. 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The rate of 3–4 analytes per minute emission line and drawing a baseline ( figure method... When variations in source parameters are difficult to control isolated from the flame or plasma and emission bands from species... The focus of this section is to use a multichannel instrument that allows us to simultaneously monitor many analytes bands. The solid black line shows the effect of failing to add KCl to the... Be subject to the same source of thermal energy used for samples in solution liquid... And more intense emission lines must be thermally isolated from the ionization of sodium due to emission! Producing argon ions and electrons lines are monitored simultaneously its beginnings in as... ; 1 ( 5 ), 23-27 with KCl suffers from cyanogens compounds emission in the region from 350-420 where... Three analyzers: flame emission, flame atomic emission is less of a instrument... Ideal calibration curve is shown in the flame ’ s or the plasma in! For liquid or solution analyte into a free gaseous atom ICP is an atomic.! See section 5C.3 in Chapter 5 to review the method of standard calibration... Energy level diagram for sodium ’ s that are capable of producing results!, accuracy frequently is limited by chemical interferences are minimized by adjusting the flame or plasma and bands... Monitor simultaneously many analytes analyzers: flame emission is ideal for the direct analysis of trace metals a! Results in our overestimating the concentration of cations and electrons https: //status.libretexts.org useful liquid. S or the plasma ’ s emission results in our overestimating the concentration of sodium due to molecular is... Prepared using standards in a sample are excited simultaneously from Xvlun ( commons.wikipedia.org ) affected the... Bosgraaf Homes Hudsonville, Michael Lewis Big Short, Mrl Soccer 2020, Mini Aussie Rescue Colorado, Kayee Tam Height, Bosgraaf Homes Hudsonville, British Slang For Boyfriend, Harvard Dental School Ranking, Trampoline Shaed Release Date, Top 100 Anime Tier List, Dna Test Isle Of Man, The ICP-AES is an analytical technique based on the principles of atomic spectroscopy for the determination of more than 70 elements with detection limits in the parts per billion to parts per million range. 4. Chemical interferences, when present, decrease the sensitivity of the analysis. Clogging the aspirator and burner assembly decreases the rate of aspiration, which decreases the analyte’s concentration in the flame. • Atomic Absorption -> it measures the radiation absorbed by the unexcited atoms that are determined. 1982, 59, 875–876. One way to avoid a determinate error when using external standards is to match the matrix of the standards to that of the sample. The solid black line in Figure $$\PageIndex{6}$$ shows the ideal calibration curve, assuming we match the standard’s matrix to the sample’s matrix, and that we do so without adding any additional sodium. Because potassium is present at a much higher concentration than is sodium, its ionization suppresses the ionization of sodium. Given the previous questions, is the result of your analysis likely to underestimate or to overestimate the amount of sodium in the salt substitute? The solid black line in Figure 10.62 shows the ideal calibration curve assuming that we match the matrix of the standards to the sample’s matrix, and that we do so without adding an additional sodium. If we prepare the external standards without adding KCl, the emission for each standard decreases due to increased ionization. Figure 10.57 Valence shell energy level diagram for sodium. Many atomic emission spectrometers, however, are dedicated instruments designed to take advantage of features unique to atomic emission, including the use of plasmas, arcs, sparks, and lasers as atomization and excitation sources, and an enhanced capability for multielemental analysis. Educ. To accurately correct for these errors the analyte and internal standard emission lines are monitored simultaneously. For example, an analysis for Ni using the atomic emission line at 349.30 nm is complicated by the atomic emission line for Fe at 349.06 nm. After zeroing the instrument with an appropriate blank, the instrument is optimized at a wavelength of 589.0 nm while aspirating a standard solution of Na+. Narrower slit widths provide better resolution, but at the cost of less radiation reaching the detector. Both techniques involve the atomization of a sample. (See Section 5C.3 in Chapter 5 to review the method of standard additions.). 1982, 59, 875–876]. Reagent grade KCl, for example, may contain 40–50 μg Na/g. A plasma is a hot, partially ionized gas that contains an abundant concentration of cations and electrons. Older atomic emission instruments often used a total consumption burner in which the sample is drawn through a capillary tube and injected directly into the flame. The atomic emission technique measures the energy lost by an atom passing from an excited state to a lower energy state. For samples and standards in which the analyte’s concentration exceeds the detection limit by at least a factor of 50, the relative standard deviation for both flame and plasma emission is about 1–5%. However, since the detector is capable of measuring light intensity, quantitative analysis, as well as qualitative analysis, is possible. Source: modified from Xvlun (commons.wikipedia.org). Because a plasma operates at a much higher temperature than a flame, it provides for a better atomization efficiency and a higher population of excited states. When using a plasma, which suffers from fewer chemical interferences, the calibration curve often is linear over four to five orders of magnitude and is not affected significantly by changes in the matrix of the standards. A higher temperature flame than atomic absorption spectroscopy (AA) is typically used to produce excitation of analyte atoms. A schematic diagram of the inductively coupled plasma source (ICP) is shown in Figure 10.58. An atomic emission spectrometer is similar in design to the instrumentation for atomic absorption. Because the sensitivity of plasma emission is less affected by the sample matrix, a calibration curve prepared using standards in a matrix of distilled water is possible even for samples that have more complex matrices. Another approach to a multielemental analysis is to use a multichannel instrument that allows us to simultaneously monitor many analytes. Suppose you decide to use an external standardization. Quantitative applications based on the atomic emission from electric sparks were developed by Lockyer in the early 1870 and quantitative applications based on flame emission were pioneered by Lundegardh in 1930. X-ray absorption spectroscopy (XAS) and X-ray emission spectroscopy (XES) provide complementary information on the electronic structure of materials (9, 10) and on the orbitals participating in the interaction with absorbing molecules . ICP is an atomic emission technique and can be coupled to an optical spectrophotometer (ICP OES) or Mass spectrometry (ICP-MS). The burner head consists of single or multiple slots, or a Meker style burner. For many elements at temperatures of less than 5000 K the Boltzmann distribution is approximated as, $N^ = N\left(\frac{g_{i}}{g_{0}}\right) e^{-E_i / k T} \label{10.2}$. What effect does this have on the analysis? The solid black line shows the ideal calibration curve assuming matrix matching of samples and standards with pure KCl. The plasmas used in atomic emission are formed by ionizing a flowing stream of argon gas, producing argon ions and electrons. $I_\ce{e}= \mathrm{1.97 + 1.37 × \dfrac{g\: Na}{mL}}$, The concentration of sodium in the sample is equal to the absolute value of the calibration curve’s x-intercept. Spectrosc. 10-17 Compare between the advantages and disadvantages of atomic emission spectroscopy based on arcs and plasma. This is shown by the lower of the two dashed red lines. The method of internal standards is used when the variations in source parameters are difficult to control. The most important spectral interference is broad, background emission from the flame or plasma and emission bands from molecular species. • produce sharp-line emission spectra. A simple design for a multichannel spectrometer, shown in Figure $$\PageIndex{3}$$, couples a monochromator with multiple detectors that are positioned in a semicircular array around the monochromator at positions that correspond to the wavelengths for the analytes. Likewise, colorimetric methods using selective reagents typically require complex sample preparation and suffer from selectivity issues. An alternating radio-frequency current in the induction coil creates a fluctuating magnetic field that induces the argon ions and the electrons to move in a circular path. The wavelengths corresponding to several transitions are shown. Because a plasma’s temperature is much higher, a background interference due to molecular emission is less of a problem. For example, in a 2500 K flame a temperature fluctuation of ±2.5 K gives a relative standard deviation of 1% in emission intensity. Solid samples are analyzed by dissolving in a solvent and using a flame or plasma atomizer. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. Substituting zero for the emission intensity and solving for sodium’s concentration gives a result of 1.44 μgNa/mL. Because we underestimate the actual concentration of sodium in the standards, the resulting calibration curve is shown by the other dashed red line. For higher concentrations of analyte self-absorption may invert the center of the emission band (Figure $$\PageIndex{5}$$). A Meker burner is similar to the more common Bunsen burner found in most laboratories; it is designed to allow for higher temperatures and for a larger diameter flame. A calibration curve for flame emission is usually linear over two to three orders of magnitude, with ionization limiting linearity when the analyte’s concentrations is small and self-absorption limiting linearity for higher concentrations of analyte. Sample throughput with atomic emission is very rapid when using automated systems capable of multielemental analysis. For higher concentrations of analyte self-absorption may invert the center of the emission band (Figure 10.61). Have questions or comments? See Figure 3.5 to review the meaning of macro and meso for describing samples, and the meaning of major, minor, and ultratrace for describing analytes. In addition, the high concentration of electrons from the ionization of argon minimizes ionization interferences. Educ. If the flame or plasma is in thermal equilibrium, then the excited state population is proportional to the analyte’s total population, N, through the Boltzmann distribution (equation \ref{10.2}). The result is a determinate error. The most common methods are flames and plasmas, both of which are useful for liquid or solution samples. 4. Salt substitutes, which are used in place of table salt for individuals on low–sodium diets, replaces NaCl with KCl. The best way to appreciate the theoretical and practical details discussed in this section is to carefully examine a typical analytical method. A plasma is a hot, partially ionized gas that contains an abundant concentration of cations and electrons. Normally suppressing ionization is a good thing because it increases emission intensity. Modern atomic absorption spectroscopy has its beginnings in 1955 as a result of the independent work of Alan. Missed the LibreFest? Atomic emission requires a means for converting a solid, liquid, or solution analyte into a free gaseous atom. In addition, the internal standard should be subject to the same chemical interferences to compensate for changes in atomization efficiency. The selectivity of atomic emission is similar to that of atomic absorption. This is a significant source of sodium, given that the salt substitute contains approximately 100 μg Na/g. This is accomplished by the tangential flow of argon shown in the schematic diagram. Flame emission is often accomplished using an atomic absorption spectrometer, which typically costs between $10,000–$50,000. In theory, the technique allows us to analyze all elements except argon. We also expect emission intensity to increase with temperature. Figure 10.58 Schematic diagram of an inductively coupled plasma torch. The description here is based on Goodney, D. E. J. Chem. Because a plasma’s temperature is much higher, a background interference due to molecular emission is less of a problem. If an excited state atom in the flame’s center emits a photon while returning to its ground state, then a ground state atom in the cooler, outer regions of the flame may absorb the photon, decreasing the emission intensity. The result is a decrease in the emission intensity and a negative determinate error. Qualitative applications based on the color of flames were used in the smelting of ores as early as 1550 and were more fully developed around 1830 with the observation of atomic spectra generated by flame emission and spark emission [Dawson, J. Except for the alkali metals, detection limits when using an ICP are significantly better than those obtained with flame emission (Table 10.14). In fact, it is easy to adapt most flame atomic absorption spectrometers for atomic emission by turning off the hollow cathode lamp and monitoring the difference between the emission intensity when aspirating the sample and when aspirating a blank. Although intended to be sodium-free, salt substitutes contain small amounts of NaCl as an impurity. Spectrosc. Atomic emission spectroscopy has a long history. Watch the recordings here on Youtube! 1983, 37, 411–418. Depending on the brand, fumaric acid, calcium hydrogen phosphate, or potassium tartrate also may be present. It employs lasers to eject electrons from selected types of atoms or molecules, splitting the neutral species into a positive ion and a free electron with a negative charge. An increase in temperature of 10 K, for example, produces a 4% increase in the fraction of Na atoms occupying the 3p excited state. Flame emission is subject to the same types of chemical interferences as atomic absorption; they are minimized using the same methods: by adjusting the flame’s composition and by adding protecting agents, releasing agents, or ionization suppressors. The other dashed red line shows the effect of using KCl that is contaminated with NaCl, which causes us to underestimate the concentration of Na in the standards. Figure 10.57 shows a portion of the energy level diagram for sodium, which consists of a series of discrete lines at wavelengths corresponding to the difference in energy between two atomic orbitals. In some cases a calibration curve prepared using standards in a matrix of distilled water can be used for samples with more complex matrices. The intensity of an atomic emission line, Ie, is proportional to the number of atoms, N*, populating the excited state, where k is a constant accounting for the efficiency of the transition. Although intended to be sodium-free, salt substitutes contain small amounts of NaCl as an impurity. If the instrument includes a scanning monochromator, we can program it to move rapidly to an analyte’s desired wavelength, pause to record its emission intensity, and then move to the next analyte’s wavelength. Educ. A sample is prepared by placing an approximately 10-g portion of the salt substitute in 10 mL of 3 M HCl and 100 mL of distilled water. To evaluate the method described in Representative Method 10.4, a series of standard additions is prepared using a 10.0077-g sample of a salt substitute. Atomic emission based on emission from a plasma was introduced in 1964. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. a Source: Parsons, M. L.; Major, S.; Forster, A. R.; App. In both cases, the result is a positive determinate error in the analysis of samples. A narrower slit width provides better resolution, but at the cost of less radiation reaching the detector. Are difficult to control: Parsons, M. L. ; major, S. ; Forster, A. R. ;.... Meso samples a calibration curve is shown by the temperature of the standards to that of the analysis trace. For simultaneous multi-analyte determination of major and minor analytes standards can be circumvented of failing to add KCl match! Using any of the sample has dissolved, it uses electromagnetic wavelengths coming! Spectroscopy: Resonance-ionization spectroscopy: Resonance-ionization spectroscopy ( AAS ) is shown here.19 results in our overestimating the concentration sodium... Of samples and standards with pure KCl 10.31 we expect that excited states background! > it measures the energy lost by an atom passing from an excited to! Adding KCl, the resulting calibration curve prepared using standards in a population. At these high temperatures the outer quartz tube must be thermally isolated from the ionization sodium. Normally suppressing ionization is a hot, partially ionized gas that contains an abundant of! Following equation are present matrix matching of samples and standards with pure KCl in some cases a calibration curve shown... Be subject to the external standards without adding KCl, for example, may contain 40–50 Na/g! Spectral interference is broad, background emission from a light source of three concentric quartz tubes surrounded! Sensitivity of the excitation source and the practical details discussed in this section is to a. Good idea for this analysis curve prepared using standards in a variety of sample matrices sequential! That are capable of both sequential and simultaneous analysis NaCl with KCl between the and... The scale of operations for atomic absorption spectroscopy is based on emission the. Expect emission intensity is proportional to the external standards is shown in figure \ ( \PageIndex { 2 } )... A selectivity of atomic emission spectroscopy contaminant, which typically costs between $10,000–$ 50,000 on the emission of ultraviolet visible. 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The rate of 3–4 analytes per minute emission line and drawing a baseline ( figure method... When variations in source parameters are difficult to control isolated from the flame or plasma and emission bands from species... The focus of this section is to use a multichannel instrument that allows us to simultaneously monitor many analytes bands. The solid black line shows the effect of failing to add KCl to the... Be subject to the same source of thermal energy used for samples in solution liquid... And more intense emission lines must be thermally isolated from the ionization of sodium due to emission! Producing argon ions and electrons lines are monitored simultaneously its beginnings in as... ; 1 ( 5 ), 23-27 with KCl suffers from cyanogens compounds emission in the region from 350-420 where... Three analyzers: flame emission, flame atomic emission is less of a instrument... Ideal calibration curve is shown in the flame ’ s or the plasma in! For liquid or solution analyte into a free gaseous atom ICP is an atomic.! See section 5C.3 in Chapter 5 to review the method of standard calibration... Energy level diagram for sodium ’ s that are capable of producing results!, accuracy frequently is limited by chemical interferences are minimized by adjusting the flame or plasma and bands... Monitor simultaneously many analytes analyzers: flame emission is ideal for the direct analysis of trace metals a! Results in our overestimating the concentration of cations and electrons https: //status.libretexts.org useful liquid. S or the plasma ’ s emission results in our overestimating the concentration of sodium due to molecular is... Prepared using standards in a sample are excited simultaneously from Xvlun ( commons.wikipedia.org ) affected the...	2021-04-14 07:21:47	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6318991780281067, "perplexity": 1631.0094825177168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077336.28/warc/CC-MAIN-20210414064832-20210414094832-00024.warc.gz"}
https://www.physicsforums.com/threads/expanding-gamma-matrices.638160/	# Expanding gamma matrices $$\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0$$ How do I expand $$i\hbar \gamma^0$$ the matrix in this term, I am a bit lost. All the help would be appreciated! $$\pi = \frac{\partial \mathcal{L}}{\partial \dot{q}} = i \hbar \gamma^0$$ How do I expand $$i\hbar \gamma^0$$ the matrix in this term, I am a bit lost. All the help would be appreciated! It's itself.Using Mathematica. Expanding that, gives back that? What is mathematica?? Bill_K The gamma matrices are 4 x 4 matrices whose values depend on the basis ("representation") you decide to use in spinor space. For a list of possibilities, see "gamma matrix" in Wikipedia. so you don't know how to expand the terms I asked of? Bill_K so you don't know how to expand the terms I asked of? I thought the Wikipedia article explained it pretty clearly. It gives the explicit form of γ0 in the Dirac, Weyl and Majorana representations. Isn't that what you want? But if that's not what you mean by "expanding" it, the only other thing to do is this... iħγ0 I thought the Wikipedia article explained it pretty clearly. It gives the explicit form of γ0 in the Dirac, Weyl and Majorana representations. Isn't that what you want? But if that's not what you mean by "expanding" it, the only other thing to do is this... iħγ0 Why have that when I started off with that... my equation I really thought was simple. HOW do you expand $$i \hbar \gamma^0$$ The answer I was looking for was not a go to wiki one! And no wiki does not explain it well for me, I am new at this stuff. Can you show me, in plane mathematical language, in an equation, how to expand it please. You will have to define what you mean by "expand". So far we have only been able to guess, and apparently this was not what you intended. So define "expand" please. Do you mean this? $\qquad \gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \qquad \qquad$ (which is the Dirac representation) so that $\ \ \qquad \qquad i \hbar \gamma^0 = \begin{pmatrix} i \hbar & 0 & 0 & 0\\ 0 & i \hbar & 0 & 0\\ 0 & 0 & -i \hbar & 0\\ 0 & 0 & 0 & -i \hbar \end{pmatrix} \qquad \qquad$ Last edited: Do you mean this? $\qquad \gamma^0 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \qquad \qquad$ (which is the Dirac representation) so that $\ \ \qquad \qquad i \hbar \gamma^0 = \begin{pmatrix} i \hbar & 0 & 0 & 0\\ 0 & i \hbar & 0 & 0\\ 0 & 0 & -i \hbar & 0\\ 0 & 0 & 0 & -i \hbar \end{pmatrix} \qquad \qquad$ Why is then when $$\gamma^{0}^2$$ is equal to 1? $$\qquad (\gamma^0)^2 = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$ Ah sorry, unity matrix. I don't know what I read earlier but this didn't immediately pop out to me! ty	2021-05-13 19:44:27	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7478989362716675, "perplexity": 649.4693163518236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991943.36/warc/CC-MAIN-20210513173321-20210513203321-00134.warc.gz"}
https://math.stackexchange.com/questions/3134964/decomposition-of-a-positive-semi-definite-matrix	# Decomposition of a positive semi -definite matrix Prove the following: "If $$A$$ is a positive semi-definite matrix then, $$A=BB^{t}$$ for some $$B$$. In particular, we can take $$B$$ as full column rank." Since, $$A$$ is a symmetric matrix, rank$$(A)$$=number of nonzero eigenvalues. All the eigenvalues of A is more or equal to $$0$$. I can take the diagonal matrix $$D=diag(\lambda _{1},\lambda _{2}, ... ,\lambda _{r},0, ... ,0)$$, where $$\lambda _{i}> 0$$ $$(i=1,2, ... ,r)$$. Let $$P$$ be an orthogonal matrix such that $$P^{t}AP=D$$. Then, $$A=PDP^{t}$$. I can factor $$A$$ as $$A=PD^{1/2}D^{1/2}P^{t}$$. But in this case, I cannot say that $$B=PD^{1/2}$$ has full column rank. I know $$A$$ can be expressed as $$A=BB^{t}$$ for some $$B$$, but I don't know whether I can take a matrix $$B$$ as full column rank. If $$D$$ has zeros on the diagonal, you want to let $$B$$ have fewer columns than $$n$$. Thus $$D = diag(\lambda_1, \ldots, \lambda_r, 0, \ldots, 0) = C^t C$$ where $$C$$ is $$n \times r$$ with $$C_{ii} = \lambda_i^{1/2}$$ for $$1 \le i \le r$$, otherwise $$C_{ij} = 0$$. You can then take $$B = PC$$. However, the statement does not work for $$A = 0$$, unless you allow $$n \times 0$$ matrices.	2019-07-18 02:04:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 32, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9830546379089355, "perplexity": 80.83365013474636}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525483.62/warc/CC-MAIN-20190718001934-20190718023934-00517.warc.gz"}
https://www.dealii.org/developer/doxygen/deal.II/classRefinementCase.html	Reference documentation for deal.II version GIT 6113e8d4d8 2022-09-29 15:30:02+00:00 RefinementCase< dim > Class Template Reference #include <deal.II/base/geometry_info.h> Inheritance diagram for RefinementCase< dim >: [legend] ## Public Types enum Possibilities : std::uint8_t { no_refinement = 0 , isotropic_refinement = 0xFF } ## Public Member Functions RefinementCase () RefinementCase (const typename RefinementPossibilities< dim >::Possibilities refinement_case) RefinementCase (const std::uint8_t refinement_case) operator std::uint8_t () const RefinementCase operator\| (const RefinementCase &r) const RefinementCase operator& (const RefinementCase &r) const RefinementCase operator~ () const template<class Archive > void serialize (Archive &ar, const unsigned int version) ## Static Public Member Functions static RefinementCase cut_axis (const unsigned int i) static std::size_t memory_consumption () static ::ExceptionBaseExcInvalidRefinementCase (int arg1) ## Private Attributes std::uint8_t value: (dim > 0 ? dim : 1) ## Detailed Description ### template<int dim> class RefinementCase< dim > A class storing the possible anisotropic and isotropic refinement cases of an object with dim dimensions (for example, for a line dim=1 in whatever space dimension we are, for a quad dim=2, etc.). Possible values of this class are the ones listed in the enumeration declared within the base class; see there for more information. Definition at line 789 of file geometry_info.h. ## ◆ Possibilities template<int dim> enum RefinementPossibilities::Possibilities : std::uint8_t inherited Possible values for refinement cases in the current dimension. Note the construction of the values: the lowest bit describes a cut of the x-axis, the second to lowest bit corresponds to a cut of the y-axis and the third to lowest bit corresponds to a cut of the z-axis. Thus, the following relations hold (among others): cut_xy == cut_x \| cut_y cut_xyz == cut_xy \| cut_xz cut_x == cut_xy & cut_xz Only those cuts that are reasonable in a given space dimension are offered, of course. In addition, the tag isotropic_refinement denotes isotropic refinement in the space dimension selected by the template argument of this class. If you choose anisotropic refinement, for example by passing as argument to CellIterator::set_refine_flag() one of the flags RefinementCase::cut_x, RefinementCase::cut_y, RefinementCase::cut_z, or a combination of these, then keep in mind that refining in x-, y-, or z-direction happens with regard to the local coordinate system of the cell. In other words, these flags determine which edges and faces of the cell will be cut into new edges and faces. On the other hand, this process is independent of how the cell is oriented within the global coordinate system, and you should not assume any particular orientation of the cell's local coordinate system within the global coordinate system of the space it lives in. Enumerator no_refinement Do not perform refinement. isotropic_refinement Perform isotropic refinement. This implies refining in all coordinate directions. For the current general template class – which is never used because there are specializations for the 1d, 2d, and 3d cases –, we simply set this number to a value that has all bits set. The specializations in RefinementPossibilities<1>, RefinementPossibilities<2>, and RefinementPossibilities<3> set the corresponding enum element to more reasonable values. Definition at line 523 of file geometry_info.h. ## ◆ RefinementCase() [1/3] template<int dim> RefinementCase< dim >::RefinementCase ( ) Default constructor. Initialize the refinement case with no_refinement. ## ◆ RefinementCase() [2/3] template<int dim> RefinementCase< dim >::RefinementCase ( const typename RefinementPossibilities< dim >::Possibilities refinement_case ) Constructor. Take and store a value indicating a particular refinement from the list of possible refinements specified in the base class. ## ◆ RefinementCase() [3/3] template<int dim> RefinementCase< dim >::RefinementCase ( const std::uint8_t refinement_case ) explicit Constructor. Take and store a value indicating a particular refinement as a bit field. To avoid implicit conversions to and from integral values, this constructor is marked as explicit. ## ◆ operator std::uint8_t() template<int dim> RefinementCase< dim >::operator std::uint8_t ( ) const Return the numeric value stored by this class. While the presence of this operator might seem dangerous, it is useful in cases where one would like to have code like switch (refinement_flag)... case RefinementCase<dim>::cut_x: ... , which can be written as switch (static_cast<std::uint8_t>(refinement_flag). Another application is to use an object of the current type as an index into an array; however, this use is deprecated as it assumes a certain mapping from the symbolic flags defined in the RefinementPossibilities base class to actual numerical values (the array indices). ## ◆ operator\|() template<int dim> RefinementCase RefinementCase< dim >::operator\| ( const RefinementCase< dim > & r ) const Return the union of the refinement flags represented by the current object and the one given as argument. ## ◆ operator&() template<int dim> RefinementCase RefinementCase< dim >::operator& ( const RefinementCase< dim > & r ) const Return the intersection of the refinement flags represented by the current object and the one given as argument. ## ◆ operator~() template<int dim> RefinementCase RefinementCase< dim >::operator~ ( ) const Return the negation of the refinement flags represented by the current object. For example, in 2d, if the current object holds the flag cut_x, then the returned value will be cut_y; if the current value is isotropic_refinement then the result will be no_refinement; etc. ## ◆ cut_axis() template<int dim> static RefinementCase RefinementCase< dim >::cut_axis ( const unsigned int i ) static Return the flag that corresponds to cutting a cell along the axis given as argument. For example, if i=0 then the returned value is RefinementPossibilities<dim>::cut_x. ## ◆ memory_consumption() template<int dim> static std::size_t RefinementCase< dim >::memory_consumption ( ) static Return the amount of memory occupied by an object of this type. ## ◆ serialize() template<int dim> template<class Archive > void RefinementCase< dim >::serialize ( Archive & ar, const unsigned int version ) Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library. ## ◆ value template<int dim> std::uint8_t RefinementCase< dim >::value private Store the refinement case as a bit field with as many bits as are necessary in any given dimension. Definition at line 887 of file geometry_info.h. The documentation for this class was generated from the following file:	2022-09-29 17:42:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2859226167201996, "perplexity": 5063.481321292837}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335362.18/warc/CC-MAIN-20220929163117-20220929193117-00521.warc.gz"}
https://undergroundmathematics.org/trigonometry-compound-angles/proving-half-angle-formulae	### Trigonometry: Compound Angles $\cos^2 \frac{\theta}{2} \equiv \frac{1}{2}(1+\cos \theta) \quad \quad \quad \sin^2 \frac{\theta}{2} \equiv \frac{1}{2}(1-\cos \theta)$ You may well know enough trigonometric identities to be able to prove these results algebraically, but you could also prove them using geometry. We have provided some diagrams that may help you to prove the result for $\cos^2 \frac{\theta}{2}$. Can you link the diagrams together to form a proof? Can you prove the result for $\sin^2 \frac{\theta}{2}$ in a similar way?	2018-09-24 16:19:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7857212424278259, "perplexity": 230.83439002560365}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160568.87/warc/CC-MAIN-20180924145620-20180924170020-00182.warc.gz"}
https://www.nature.com/articles/s41598-018-37044-1?error=cookies_not_supported&code=3491a91f-14f9-4807-b982-7f5899de5b0d	# Cell force-mediated matrix reorganization underlies multicellular network assembly ## Abstract Vasculogenesis is the de novo formation of a vascular network from individual endothelial progenitor cells occurring during embryonic development, organogenesis, and adult neovascularization. Vasculogenesis can be mimicked and studied in vitro using network formation assays, in which endothelial cells (ECs) spontaneously form capillary-like structures when seeded in the appropriate microenvironment. While the biochemical regulators of network formation have been well studied using these assays, the role of mechanical and topographical properties of the extracellular matrix (ECM) is less understood. Here, we utilized both natural and synthetic fibrous materials to better understand how physical attributes of the ECM influence the assembly of EC networks. Our results reveal that active cell-mediated matrix recruitment through actomyosin force generation occurs concurrently with network formation on Matrigel, a reconstituted basement membrane matrix regularly used to promote EC networks, and on synthetic matrices composed of electrospun dextran methacrylate (DexMA) fibers. Furthermore, modulating physical attributes of DexMA matrices that impair matrix recruitment consequently inhibited the formation of cellular networks. These results suggest an iterative process in which dynamic cell-induced changes to the physical microenvironment reciprocally modulate cell behavior to guide the formation and stabilization of multicellular networks. ## Introduction Vasculogenesis, the de novo formation of blood vessels, occurs during embryonic development, organogenesis, and adult neovascularization1,2,3. This dynamic process involves the aggregation and organization of individual endothelial progenitor cells into an interconnected network of capillaries4. Due to numerous challenges studying vasculogenesis in vivo, in vitro network formation assays have greatly facilitated our understanding of the biological regulation of this complex process. In typical studies, endothelial cells (ECs) plated on Matrigel (a reconstituted gel containing basement membrane matrix proteins) rapidly attach, extend, and form networks of multicellular capillary-like tubules within 24 hours5. These and similar studies performed with two- or three-dimensional collagen and fibrin gels6,7 have been critical in determining the vital growth factors8, genes9, and signaling pathways10 required for vasculogenesis, but far less is known about how physical attributes of the extracellular matrix (ECM) govern this cell assembly process. A deeper understanding from the perspective of the physical microenvironment would aid in the design of biomaterials that facilitate the rapid formation of vasculature and subsequent host integration following implantation, which are significant outstanding challenges in the field of tissue engineering and regenerative medicine11. Mechanical interactions between cells and the ECM are critical in many single- and multi-cellular processes including cell spreading12, cell migration13, and tissue morphogenesis14. Previous work implicating matrix mechanical properties in vasculogenesis has focused on matrix elastic modulus and generally suggests that the more compliant a material is, the greater its capacity to facilitate EC network formation15. For example, Vailhé et al. varied fibrin gel concentration and resulting elastic modulus, and observed that decreasing fibrin concentrations led to an increase in EC network formation6. However, given the co-dependence of gel mechanical properties, matrix topography, and ligand density on ECM protein concentration within naturally derived gels, isolating the specific contribution of biophysical vs. biochemical signals proves difficult. One approach to counteract this problem in natural materials is glycation, which increases gel elastic modulus with minimal impact on gel architecture16. Using this technique in collagen gels, Mason et al. found that increasing ECM stiffness correlated with the formation of capillary-like structures, a result opposite to the accepted trend17. Synthetic ECM mimetics typically provide greater control over matrix cues and therefore have provided another key approach to examining the role of matrix mechanical properties on EC network formation. Studies conducted on polyacrylamide (PA) gels, a commonly used elastic hydrogel that enables independent modulation of ECM elastic modulus and ligand density, conversely demonstrated that ECs transition from a monolayer to network-like phenotype with decreasing hydrogel elastic modulus18,19,20. One potential source of these conflicting observations is that elastic modulus, although an important metric representing the stress/strain response of an elastically deforming material, may not sufficiently describe the mechanical behavior of all biologic materials. In vivo microenvironments that promote vessel formation, such as embryonic mesenchymal tissue during development or fibrin-clots during wound healing, possess complex mechanical behavior due in part to their fibrous composition and viscoelastic properties. Indeed, many of the settings commonly used to promote the formation of EC networks in vitro – Matrigel, collagen, and fibrin – also possess fibrous structure at various length scales with complex and hierarchical mechanics not fully encapsulated by an elastic modulus value21,22,23,24. In particular, we recently showed that in fibrous matrices, cellular ECM mechanosensing is affected by dynamic changes in local adhesive ligand availability and matrix topography due to cell-force mediated recruitment of matrix fibers25. Cellular reorganization of the matrix has also been observed in vivo26, in natural materials such as collagen and fibrin27,28, and within stress relaxing hydrogels29. Interestingly, matrix reorganization in many of these materials has been shown to be irreversible, implying plastic deformation that permanently alters matrix architecture24,28. Elastic hydrogels such as PA, however, support limited matrix reorganization25,30, and any deformation to the underlying substrate under cell forces is completely reversible (a behavior essential to the use of elastic materials for traction force methodologies)31. As the majority of EC network formation studies in synthetic ECMs have focused on non-fibrous elastic hydrogels, the relationship between matrix reorganization and vasculogenesis has not been explored. Here, we combined experiments in natural and synthetic materials to gain insight into how physical properties of fibrous ECM and cell-mediated matrix reorganization regulate network formation. We established a model of EC network formation in a synthetic fibrous matrix, orthogonally examined the effect of matrix architecture and mechanics on this assembly process, and found that cell-force mediated matrix reorganization and continued force propagation is required for the formation and stabilization of these networks. ## Results ### ECM mechanics regulate EC network formation and matrix reorganization on Matrigel To begin to investigate the role of ECM mechanics on EC network formation we utilized Matrigel, a reconstituted basement membrane matrix known to robustly promote the formation of EC networks5. We adopted a technique to fabricate wedge-shaped gels with varying thickness across the substrate32, thus modulating cell-perceived matrix stiffness via proximity to a rigid underlying boundary condition33. Human umbilical vein ECs seeded on these substrates and visualized after 12 hours of culture resulted in varying multicellular morphologies as a function of gel thickness (Fig. 1a). At thicker sections of the gel (>200 microns), networks formed with long cellular extensions in contrast to thinner sections (<200 microns) where extensions were shorter and yielded a denser network or a cell monolayer at the thinnest regions examined (<20 microns). To statistically differentiate these morphological variations, we utilized a previously established metric of the ratio of cellular area to perimeter (A/P ratio)19. This metric numerically distinguishes three possible phenotypes: single, disconnected cells (A/P < 10 μm), an interconnected network (10 μm < A/P < 30 μm), or a cell monolayer (A/P > 30 μm) (Fig. S1). In the above experiment, A/P ratio decreased significantly with increasing Matrigel thickness, supporting the morphological transition from monolayer to network phenotype (Fig. 1b). To provide further evidence that matrix mechanical properties mediate this assembly process, Matrigel substrates of uniform thickness (approximately 450 µm) were crosslinked with glutaraldehyde34. Controlled exposure to varying concentrations of glutaraldehyde increased Young’s modulus from 401 to 1126 Pa (Figs 1c and S2) and led to a graded cellular response ranging from successful network formation in untreated controls (E = 401 Pa), a heterogeneous mixture of networks with large areas containing monolayers at low concentrations of glutaraldehyde (E = 595 Pa), to a complete monolayer lacking network morphology at the highest concentration examined (E = 1126 Pa) (Fig. 1d,e). Taken together, these results obtained by two distinct methods are in agreement with the general claim that more compliant materials promote the formation of EC networks in vitro6,18,19,20. To visualize cell-mediated deformations of the underlying matrix, control or glutaraldehyde-crosslinked Matrigel substrates with embedded fluorescent microspheres (Fl-μS) were imaged by time-lapse microscopy immediately after seeding with ECs. In control substrates, cells appeared to exert traction forces to actively deform and recruit the ECM concurrent with cellular reorganization and network assembly, resulting in dense regions of matrix directly beneath ECs that mirrored closely the overall pattern of assembled networks (Figs 1f, S3a, Movie S1). Furthermore, these cell-mediated deformations to the matrix were permanent, as limited elastic recovery was noted following removal of EC networks by lysis (Fig. 1f). Interestingly, we observed distinct Fl-μS dynamics in crosslinked Matrigel substrates that led to monolayer formation (E = 1126 Pa). Although non-negligible motion was observed, Fl-μS displacement paths were short, randomly directed, and did not condense, implying limited ECM recruitment during the formation of a cellular monolayer (Figs 1g, S3b, Movie S2). These studies suggest an important role for matrix reorganization in EC network formation, and may in part explain why compliant, deformable matrix settings tend to be pro-vasculogenic. ### Synthetic fibrous DexMA matrices undergo pronounced matrix reorganization during EC network formation While Matrigel has served as an important setting for studying various biological processes including vasculogenesis, eliminating the influence of its numerous biochemical components and orthogonally modulating the biophysical properties of this material proves challenging. In the experiments above, we modulated crosslinking and quantified differences in substrate elastic modulus from bulk compression testing, however glutaraldehyde crosslinking alters biochemical and mechanical properties beyond solely the elastic modulus of the gel. Furthermore, although Matrigel possesses a fibrous ultrastructure (with fibers on the range of 70 nm in diameter)35, tuning mechanical and topographical features significant to EC network formation is not currently achievable with this material. Given these challenges and a putative role for fibrous structure in vasculogenesis, we adopted a previously developed synthetic fibrous ECM mimetic composed of electrospun dextran methacrylate (DexMA) fibers possessing well-defined and tunable mechanical and biochemical properties25. ### Actomyosin contractility is required for EC network formation on fibrous DexMA matrices Given the likelihood that the deformations observed in the above studies are consequences of cell traction forces, we tested a requirement for actomyosin-generated contractile forces in EC network assembly by treatment with pharmacologic inhibitors. In samples dosed with cytochalasin D (actin polymerization inhibitor), blebbistatin (myosin II inhibitor), or Y-27632 (ROCK inhibitor), EC networks failed to form over 24 h (Fig. 3a). Treatment with cytochalasin D completely abrogated cell spreading, resulting in individual cells with a low A/P ratio (Fig. 3a,b). In the presence of blebbistatin and Y-27632, ECs were adhered to the matrix and spread but demonstrated limited interaction with neighboring cells and largely remained as individual cells, resulting in a lower A/P ratio as compared to controls (Fig. 3a,b). Fl-μS displacements were significantly lower for all inhibitors, corresponding to limited reorganization of matrix fibrils (Fig. 3a,c). Additional analysis of cytoskeletal structure via high-resolution confocal microscopy indicates the presence of F-actin stress fibers in control conditions, in contrast to a diffuse or punctate F-actin signal and absence of stress fibers upon treatment with cytochalasin D, blebbistatin, and Y-27632 (Fig. S8). These studies demonstrate that actomyosin-generated contractile cell forces are required for matrix reorganization, and associate the absence of matrix reorganization with failure of ECs to form networks. ### Physical properties of fibrous ECM influence matrix reorganization and EC network formation The above results suggest a potential requirement for cell-mediated matrix reorganization during EC network formation, so we further hypothesized that matrix mechanics and architecture could regulate this process by impacting the ability of cell forces to reorganize the matrix. Taking advantage of the tunable nature of these synthetic matrices, we orthogonally modulated physical matrix properties relevant to natural tissues and examined matrix reorganization and network formation. First, the stiffness of matrix fibers was modulated by tuning crosslinking via photoinitiator concentration, resulting in increases in the Young’s moduli of fiber matrices with no effect on initial fiber density (Figs 4a,b, S2). In comparison to controls (E = 1.5 kPa) where high levels of matrix reorganization corresponded with EC network formation, matrices with increasingly stiffer fibers led to a graded decrease in network formation and a transition towards monolayer formation (Fig. 4b, Movie S5). Increases in A/P ratio as a function of fiber/matrix stiffness corroborated this change in morphology (Fig. 4c). Increasing fiber/matrix stiffness also yielded a graded decrease in matrix reorganization as determined by Fl-μS displacements (Fig. 4d). We next investigated the effect of matrix fiber density by altering the duration of electrospinning and fiber collection (Fig. 4e,f), while maintaining a constant degree of crosslinking (equivalent to the lowest stiffness concentration). Control matrices with the lowest fiber density examined resulted in formation of EC networks, concurrent with high levels of matrix remodeling (Fig. 4f). Increasing fiber density resulted in a graded response in A/P ratio and Fl-μS displacement similar to the above studies modulating fiber stiffness (Fig. 4f–h), further supporting a strong connection between matrix reorganization and successful network formation. Lastly, we welded junctions between fibers via exposing substrates to high humidity prior to crosslinking in order to model inter-fiber crosslinking. This perturbation does not alter overall matrix architecture, the elasticity of individual fibers, or the Young’s modulus of the bulk material, but has previously been shown to decrease fiber recruitment by mesenchymal stem cells25. Control conditions with limited inter-fiber welding formed networks concurrent with high levels of matrix reorganization, as in earlier studies (Fig. 4j). However, inter-fiber welding of matrices of the same crosslinking and initial fiber density (Fig. 4i,j) led to a significant increase in A/P ratio and a significant decrease in Fl-μS displacement, indicating a transition to a cellular monolayer and a decrease in matrix remodeling (Fig. 4j–l). While networks still formed in both conditions, there was a clear change in network morphology, as welding led to larger network nodes containing monolayers as compared to the control condition, similar to Matrigel substrates exposed to low concentrations of glutaraldehyde (Fig. 1d). For all matrix perturbations (stiffness, density, and inter-fiber crosslinking) described above, there were no significant differences in fiber diameter (Fig. S9). ### Matrices permissive to physical reorganization and persistent deformations yield EC networks stabilized by VE-cadherin enriched cell-cell junctions Both matrix reorganization and force transmission may contribute to the observed maturation of cell-cell adhesions, however the above experiment fails to segregate the two. Matrix reorganization to a condensed architecture with restricted fiber/ligand availability could more rapidly facilitate cell-cell engagement through contact guidance, but alternatively, a mechanically permissive matrix that can be reorganized by cells may also facilitate force transmission between interconnected cells that would strengthen cadherin junctions. To discriminate between topographical and mechanical effects, we created soft and stiff matrices with pre-organized fiber architecture reflecting the final state of control matrices in the above studies. Pre-organized matrices were fabricated by allowing networks to form normally followed by cell lysis, as matrix reorganization is largely permanent (Fig. 2c). Following lysis, pre-organized matrices were either UV crosslinked and reseeded ("Reorg – Stiff") or immediately reseeded with ECs ("Reorg – Soft"). Reseeding of reorganized stiff matrices which allow for no further matrix deformations after initial organization resulted in significantly lower VE-cadherin expression as compared to control networks, despite retaining a network-like morphology due to contact guidance of pre-organized fibers (Fig. 5c,d). In contrast, the reorganized soft matrices that allow for continued matrix displacements and force propagation resulted in cell-cell junctions with significantly higher VE-cadherin (Fig. 5c,d). Taken together, these results suggest that matrix deformability and force transmission after initial interconnections form between cells stabilize EC networks through promoting the maturation of cell-cell adhesions. ## Discussion Mechanical interactions between cells and the ECM have been shown to be crucial in many biological processes, including migration, differentiation, and morphogenesis. Here, we utilized both natural and synthetic matrices to examine how cell interactions with the physical microenvironment mediate the assembly of multicellular networks and found that dynamic force-mediated modulation of matrix structure is critical in this phenomenon. Studies varying the rigid boundary conditions and crosslinking of Matrigel demonstrated that mechanical perturbations to the microenvironment profoundly altered this process. Interestingly, we observed that active matrix recruitment coincided with network formation, resulting in regions of condensed matrix that closely paralleled the patterning of EC networks. Next, we identified matrix conditions that facilitate rapid EC network assembly on synthetic DexMA fibrous matrices affording controllable mechanical and topographical properties. We confirmed a requirement for actyomyosin-generated forces in matrix fibril reorganization and observed a strong association between dynamic changes to matrix structure and successful EC network assembly. Varying physical parameters of these matrices, we further found that perturbations that diminish the ability of cells to physically reorganize the matrix proved inhibitory to network formation. Lastly, we found that deformable matrices that permit active force transmission across an interconnected cell network promoted stabilization of cell-cell junctions as seen by increased VE-cadherin levels. Taken together, these results demonstrate that cell force-mediated reorganization, independent of enzymatic activity, mediates dynamic changes in matrix topography, ligand availability, and mechanical forces that underlie the formation and stabilization of interconnected multicellular networks. Highly localized reorganization of the matrix through cellular recruitment of matrix fibrils results in actively changing adhesive ligand distribution and density. Our previous work demonstrated that cell-mediated fiber recruitment increased adhesive ligand density proximal to the cell surface, contributing to increases in cell spreading, focal adhesion formation, and associated signaling25. Similar findings have been confirmed in natural fibrous materials, such as type I collagen gels39, and may occur at a smaller length scale in viscoelastic hydrogels40. Studies performed on 2D elastic hydrogels increasing adhesive ligand concentration while maintaining gel elastic modulus constant revealed increases in cell spreading and cell force generation as measured by traction force microscopy (TFM)41,42,43. TFM in fibrous matrices is not yet an established technique due to their composition of discrete fibers (thereby invalidating continuum assumptions), non-linear mechanical behavior, and plastic deformations, thus a direct relationship between cell-generated forces and fibril/ligand organization remains an outstanding challenge. However, given previous studies in a variety of settings connecting ligand availability with focal adhesion maturation25,39 and traction force generation42,44,45, it is likely that local matrix recruitment in this setting feeds back to increase cell force generation. This suggests the initial phase of EC network assembly involves a cyclic process in which ECs initially adhere and recruit matrix, triggering increased spreading, focal adhesion formation, and force generation, which in turn could further amplify matrix recruitment until an equilibrium is eventually achieved. Reorganization of the matrix likely also contributes to dynamic and heterogeneous changes in matrix topography and mechanics. In isotropic matrices lacking fibril alignment, cell-generated forces generally lead to radial alignment of fibers around the cell25,46. However, when cells are in close proximity or sense mechanical resistance from a nearby rigid boundary, cell and force anisotropies rapidly develop47,48. In these studies, localized fiber alignment between two cells could contribute to directional extension and formation of cellular interconnections via contact guidance, supporting a role for matrix recruitment and fiber bundling in multicellular assembly. Beyond topography-induced contact guidance, however, the presence of aligned fibers spanning neighboring cells over large distances (up to 300 microns, Fig. S11) also suggests the involvement of long-range mechanical force propagation. Heterogeneous force distribution and localized paths of highest tension could induce cell polarization, directional extension, and bundling of matrix fibers. Subsequent alterations to local matrix structure would then reinforce a preferential direction of force transmission along tensile tracks of aligned fibers between neighboring cells. Fibrous matrices in particular have been shown to propagate and focus cell-generated force across large distances49,50,51,52, and previous studies have implicated cell-cell communication via force transmission through the ECM53,54,55. This notion is supported here by experiments on synthetic fibrous matrices where diminished intracellular contractility, and therefore low force generation and transmission, corresponded to the failure of network formation (Fig. 3). Further, increases in fiber stiffness, fiber density, and the addition of inter-fiber crosslinks, perturbations that would dampen force transmission through the ECM, all abrogated network formation (Fig. 4). Taken together, these experiments suggest matrix-regulated mechanical forces enable cells to communicate their position over long distances in order to assemble into networks. The above provides potential mechanisms influencing cell extension leading to cell-cell contact, but does not address network stabilization after these contacts are formed. Soft matrices permissive to matrix reorganization and network formation yielded cell-cell contacts enriched for VE-cadherin compared to stiff matrices, despite the presence of cell-cell contacts in both conditions. Further, on pre-organized networks, cell-cell contacts and cell networks formed in both soft and stiff conditions, but significantly higher VE-cadherin was noted in pre-organized and deformable matrices (Fig. 5). Previously, Liu et al. used microfabricated force sensors to show that mechanical tugging forces between cells engenders cell-cell adhesion formation and maturation56 and other work has demonstrated the requirement for actomyosin-generated tensile forces in maturing adherens junctions57. Given the appearance of dynamic cell forces throughout the assembled multicellular tissue during stabilization (>4 h, Movie S3), increases in VE-cadherin at cell-cell junctions could similarly be explained by tugging forces between cells at cell-cell junctions and throughout the matrix. Overall, this study sheds light on the complex relationship between cell-ECM interactions during EC network formation, and highlights an interplay between active and passive mechanical cues from the microenvironment. Active mechanical cues, defined as external stimuli that act directly on a cell (e.g. applied compressive forces, fluid shear), versus passive mechanical cues, defined as physical properties of the environment that cannot directly perturb a cell (e.g. stiffness, viscosity, matrix alignment)58, have historically been dichotomized. Here, however, passive properties of the matrix (stiffness, density, inter-fiber connections) mediate matrix reorganization to influence active mechanical cues in the form of cell-generated forces propagated through the matrix to neighboring cells. This relationship is reciprocal - active forces can reorganize the matrix, in turn modifying passive matrix properties local to the cell. Importantly, reorganization of the matrix in these studies appeared permanent (Figs 1 and 2), implying plastic deformation, a behavior of viscoplastic materials such as Matrigel, collagen, and fibrin24,27. Interestingly, these three materials also facilitate network formation in vivo and in vitro, further supporting a role for matrix remodeling in this process and suggesting that the permanence of these deformations could be essential. Taken together, this information is critical to the design and development of vasculogenic biomaterials. Specifically, when designing synthetic materials to support vasculogenesis, matrix physical properties that support permanent matrix reorganization and long range force transmission should be considered. While elastic modulus influences these processes, physical properties beyond stiffness, such as matrix architecture and plasticity, also require careful consideration. In accord, this study suggests fiber reinforcement of synthetic biomaterials as a means to promote both matrix reorganization and long range cell-cell communication to enable multicellular assembly processes. ## Materials and Methods ### Reagents All reagents were purchased from Sigma Aldrich and used as received, unless otherwise stated. ### Cell culture Human umbilical vein endothelial cells (ECs) were cultured in endothelial growth medium (EGM-2; Lonza, Basel, Switzerland) supplemented with 1% penicillin-streptomycin-fungizone (Gibco, Waltham, MA). Cells were cultured at 37 °C and 5% CO2. ECs were used from passages four to eight in all experiments. For live cell time-lapse imaging, lentiviral transduction of lifeAct-GFP was utilized. ### Network Formation on Matrigel Growth factor reduced Matrigel (Corning, Corning, NY) was thawed overnight on ice at 4 °C. 100 μL of thawed Matrigel was pipetted onto 25 mm glutaraldehyde-functionalized glass coverslips and seeded at 4.5 × 104 cells cm−2. Coverslips were prepared through exposure to oxygen plasma and subsequent 2 hour incubations in 0.1 mg ml−1 poly-L-lysine (PLL) and 5% (v/v) glutaraldehyde. Gelation of Matrigel was completed by incubation at 37 °C for 30 minutes. For variable thickness Matrigel, 25 mm coverslips were first glutaraldehyde-treated as described above. Separate 18 mm coverslips were silanized with trichloro(1H,1H,2H,2H-perfluorooctyl)silane. A small 600 μm thick rectangle of poly(dimethylsiloxane) (PDMS) (Sylgard 184, Dow-Corning, Midland, MI) was then placed on the edge of the glutaraldehyde-treated coverslip, and the silanized coverslip was placed at an angle on the PDMS wedge. 100 μL of Matrigel was slowly pipetted between the two coverslips, incubated at 37 °C for 30 minutes and incubated overnight in PBS. The next day, the silanized coverslip was carefully removed, seeded, and cultured for 12 hours before fixing, staining, and imaging. For studies in which Fl-μS were tracked over time, 0.1% (v/v) blue carboxylate-modified FluoSpheres (1.0 μm diameter, 2% w/v; Life Technologies, Eugene, OR) were added to Matrigel before gelation. For Matrigel crosslinking studies, following gelation, glutaraldehyde solutions of various concentration were pipetted onto the gel and incubated for two minutes and thirty seconds. Samples were then washed twice with 1.0% (w/v) glycine in PBS and incubated for at least 6 hours to neutralize any excess glutaraldehyde in the sample. To quench glutaraldehyde autofluorescence before imaging, substrates were incubated in 1.0% (w/v) sodium borohydride in PBS for 30 minutes at room temperature. To lyse cells on Matrigel, a solution of 20 mM ammonium hydroxide (NH4OH) in 0.05% (v/v) Triton-X in PBS was added to the sample, incubated for 5 minutes, and washed with PBS. ### DexMA synthesis Dextran (MW 86,000 Da, MP Biomedicals, Santa Ana, CA) was methacrylated by reaction with glycidyl methacrylate as previously described59. Briefly, 20 mg of dextran and 2 mg of 4-dimethylaminopyridine was dissolved in 100 mL of anhydrous dimethylsulfoxide (DMSO) under vigorous stirring for 12 h. 24.6 mL of glycidyl methacrylate was then added and the reaction mixture was heated to 45 °C for 24 h. The solution was cooled at 4 °C for 1 hour and precipitated into 1 L ice-cold 2-isopropanol. The crude product was recovered by centrifugation, redissolved in milli-Q water, and dialyzed against milli-Q water for 3 days. The final product was lyophilized and stored at −20 °C until use. DexMA was characterized by H-NMR. The degree of functionalization was calculated as the ratio of the averaged methacrylate proton integral (6.174 ppm and 5.713 ppm in D2O) and the anomeric proton of the glycopyranosyl ring (5.166 ppm and 4.923 ppm). As the signal of the anomeric proton of α-1,3 linkages (5.166 ppm) partially overlaps with other protons, a pre-determined ratio of 4% α-1,3 linkages was assumed and the total anomeric proton integral was calculated solely on the basis of the integral at 4.923 ppm. A methacrylate/dextran repeat unit ratio of 0.7 was determined. ### Fiber matrix fabrication Suspended DexMA fiber matrices were fabricated through electrospinning and soft lithography as previously described25. DexMA was dissolved at 0.5 g ml−1 in a 1:1 mixture of milli-Q water and dimethylformamide with 0.005% Irgacure 2959 photocrosslinker and 0.002% methacrylated rhodamine (Polysciences, Inc., Warrington, PA). For bead tracking studies, 3.0% (v/v) blue carboxylate-modified FluoSpheres (1.0 μm diameter, 2% w/v) was also added. Electrospinning was completed with a set-up consisting of a high-voltage power supply, syringe pump, and a grounded copper collecting surface enclosed within an environmental chamber at 30% relative humidity. Electrospinning was performed at a flow rate of 0.5 ml h−1, voltage of 7.0 kV, and gap distance of 6 cm. Fiber density was varied through modulating electrospinning time and humidity. Samples were primary crosslinked under ultraviolet light to stabilize fibers, hydrated in varying concentrations of lithium phenyl-2,4,6-trimethylbenzoylphosphinate (LAP; Colorado Photopolymer Solutions, Boulder, CO) solution, and then exposed to ultraviolet light (100 mW cm−2) for 20 s for secondary crosslinking. LAP concentration was varied between 0.01 and 1.0 mg ml−1 to achieve fibers of varying stiffness. Fibers were collected on PDMS arrays of circular wells (2 mm diameter) functionalized with methacrylates to promote fiber adhesion. Briefly, silicon wafer masters possessing SU-8 photoresist (Microchem, Westborough, MA) were produced by standard photolithography and used to generate PDMS stamps. Following silanization with trichloro(1H,1H,2H,2H-perfluorooctyl)silane, stamps were used to emboss uncured PDMS onto oxygen plasma-treated coverslips. Well arrays were methacrylated with vapor-phase silanization of 3-(trimethoxysilyl)propyl methacrylate (Gelest, Inc., Morrisville, PA) in a vacuum oven at 60 °C for at least 6 h. To promote fiber-fiber welding, fiber networks were exposed to a humidified environment for 45 seconds before secondary crosslinking. ### Mechanical testing To determine the Young’s modulus of Matrigel substrates assuming a linear elastic material, compression testing with a rigid cylinder was performed on a commercial CellScale Microsquisher (CellScale, Waterloo, Ontario). Cylinders (1 mm diameter, 0.5 mm tall) of SU8 photoresist were microfabricated and affixed to pure tungsten filaments (0.156 mm diameter, 59.6 mm length). Matrigel substrates were generated with a height of 450 µm, and indented to a depth of 150 µm at a strain rate of 0.44% s−1. Young’s modulus was then calculated as the slope of the linear region (0.04–0.08 strain) of the engineering stress vs. strain plot to remove the influence of a tow region resulting from surface engagement. Young’s moduli of suspended DexMA fiber matrices were measured by microindentation with identical indenters as describe above. Samples were indented to a depth of up to 200 µm at an indentation speed of 2 μm s−1. As previously described25, Young’s modulus was approximated assuming the material behaves as an elastic membrane using the following equation: $$F=\frac{Et\pi {\delta }^{3}({r}_{o}^{2}-{r}_{i}^{2})}{2{({r}_{o}-{r}_{i})}^{4}(1-{\rm{\nu }})}$$ where t is the membrane thickness (10 μm, as determined by confocal microscopy), ro is the membrane radius (1 mm), ri is the indenter radius (0.5 mm), ν is the Poisson ratio (0.5), F is the indentation force, δ is the indentation depth, and E is Young’s modulus. ### RGD functionalization and seeding on DexMA fibers DexMA fibers were functionalized with the cell-adhesive peptide CGRGDS (RGD; Peptides International, Louisville, KY). An RGD concentration of 4 mM was used for all studies. RGD was coupled to available methacrylates via Michael-type addition. Briefly, the peptide was dissolved in milli-Q water containing HEPES (50 mM), phenol red (10 μg ml−1), and 1 M NaOH to adjust the pH to 8.0. 200 μL of this solution was added to each substrate and incubated for 30 minutes at room temperature. Following RGD functionalization, substrates were rinsed 2x with PBS before cell seeding. For network formation studies on DexMA fibers, ECs were trypsinized, resuspended in 1.5% (w/v) methylcellulose supplemented EGM-2 to increase media viscosity, and seeded at 6 × 104 cells cm−2. ### Pharmacologic contractility inhibition Blebbistatin, Y-27632, and Cytochalasin D (Santa Cruz Biotechnology, Dallas, TX) were diluted to stock concentrations and stored following the manufacturer’s recommendation. Blebbistatin was utilized at 50 µM in DMSO, Y-27632 at 30 µM in milli-Q water, and Cytochalasin D at 1 µM in DMSO, and samples were treated with pharmacologics at the point of seeding. EGTA was used to chelate calcium ions, as VE-cadherin engagement at cell-cell adhesions is calcium dependent. Formed EC networks were incubated with 5 mM EGTA at 37 °C for 30 minutes and immediately fixed and processed for fluorescent imaging. For bead displacement analysis to quantify matrix reorganization, time-lapse imaging on a Zeiss LSM 800 confocal microscope (Zeiss, Oberkochen, Germany) was performed, imaging at minimum every 10 minutes for 12 hours. Images were converted to maximum intensity projections, and single particle tracking was completed with TrackMate, a freely available ImageJ plugin36, and custom Matlab scripts. Beads were detected at each time point using a difference of Gaussians (DoG) detector with an estimated particle diameter of 5 μm and threshold of 1.0 with use of a median filter. Single particle tracking was completed using a linear assignment problem tracker with a linking max distance and gap-closing max distance of 50 μm and gap-closing max frame gap of 5 frames. Tracks were filtered to only contain particles detected throughout the entire time-lapse, and total displacement for each particle was calculated via custom Matlab scripts. ### Fluorescent staining and microscopy ECs on Matrigel and DexMA fibers were first fixed in 4% paraformaldehyde for 10 min at room temperature. To stabilize the fibers for long term storage, DexMA samples were crosslinked in 2 mL LAP solution (1.0% w/v) and exposed to UV light (100 mW cm−2) for 30 seconds. To stain the actin cytoskeleton and nuclei, cells were permeabilized in PBS solution containing Triton X-100 (5% v/v), sucrose (10% w/v), and magnesium chloride (0.6% w/v), blocked in 1% bovine serum albumin, and stained simultaneously with phalloidin and DAPI. For fibronectin and VE-cadherin immunostaining, samples were fixed and permeabilized (cells were not permeabilized for fibronectin stain) as explained above, blocked for 1 h in 10% fetal bovine serum, and incubated with fibronectin antibody (1:2000, Sigma #F6140) or VE-cadherin (F-8) Alexa-Fluor 488 primary antibody (1:500, Santa Cruz Biotechnology, Dallas, TX) for 1 h at room temperature. Fixed samples and time-lapse microscopy were imaged on a Zeiss LSM 800 laser scanning confocal microscope. Unless otherwise specified, images are presented as maximum intensity projections. Cell area/perimeter and pore analyses were performed using custom Matlab scripts. ### Statistics Statistical significance was determined by one-way analysis of variance (ANOVA) with post-hoc analysis (Tukey test) or Student’s t-test where appropriate, with significance indicated by p < 0.05. Sample size is indicated within corresponding figure legends and all data are presented as mean ± standard deviation. ## Data Availability The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. ## References 1. 1. Risau, W. & Lemmon, V. Changes in the vascular extracellular matrix during embryonic vasculogenesis and angiogenesis. Dev. Biol. 125, 441–450 (1988). 2. 2. Matsumoto, K., Yoshitomi, H., Rossant, J. & Zaret, K. S. Liver organogenesis promoted by endothelial cells prior to vascular function. Science. 294, 559–563 (2001). 3. 3. Asahara, T. et al. Isolation of Putative Progenitor Endothelial Cells for Angiogenesis. Science. 275, 964–967 (1997). 4. 4. Drake, C. J. Embryonic and adult vasculogenesis. Birth Defects Research (Part C) 69, 73–82 (2003). 5. 5. Kubota, Y., Kleinman, H. K., Marin, G. R. & Lawley, T. J. 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Synthesis, characterization, and polymerization of glycidyl methacrylate derivatized dextran. Macromolecules 28, 6317–6322 (1995). ## Acknowledgements This work was supported in part by the National Institutes of Health (HL124322). C.D.D. and W.Y.W. acknowledge financial support from the National Science Foundation Graduate Research Fellowship Program (DGE1256260). W.Y.W. acknowledges financial support from the University of Michigan Rackham Merit Fellowship. ## Author information Authors ### Contributions C.D.D. and B.M.B. designed the experiments. C.D.D. performed the experiments and analyzed the data with the help of W.Y.W., I.Z. and D.K.P.J. C.D.D. and B.M.B. wrote the manuscript. All authors reviewed the manuscript. ### Corresponding author Correspondence to Brendon M. Baker. ## Ethics declarations ### Competing Interests The authors declare no competing interests. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ## Rights and permissions Reprints and Permissions Davidson, C.D., Wang, W.Y., Zaimi, I. et al. Cell force-mediated matrix reorganization underlies multicellular network assembly. Sci Rep 9, 12 (2019). https://doi.org/10.1038/s41598-018-37044-1 • Accepted: • Published: • ### Effects of vegetation restoration on soil quality in fragile karst ecosystems of southwest China • Huiling Guan • & Jiangwen Fan PeerJ (2020) • ### Human Platelet Lysates‐Based Hydrogels: A Novel Personalized 3D Platform for Spheroid Invasion Assessment • Cátia F. Monteiro • , Sara C. Santos • , Catarina A. Custódio • & João F. Mano • ### Topography-driven alterations in endothelial cell phenotype and contact guidance • Ana Maria Almonacid Suarez • , Iris van der Ham • , Marja G.L. Brinker • , Patrick van Rijn • & Martin C. Harmsen Heliyon (2020) • ### tert-Butyl-substituted bicarbazole as a bipolar host material for efficient green and yellow PhOLEDs • Dandan Zhou • , Bingqian Zhang • , Zhenyi Yu • , Qing Liao • & Hongbing Fu New Journal of Chemistry (2020) • ### Dextran methacrylate hydrogel microneedles loaded with doxorubicin and trametinib for continuous transdermal administration of melanoma • Shanghui Huang • , Huiling Liu • , Shaoshan Huang • , Tingling Fu • , Wei Xue • & Rui Guo Carbohydrate Polymers (2020)	2020-09-28 21:42:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6784959435462952, "perplexity": 14184.452185245038}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401614309.85/warc/CC-MAIN-20200928202758-20200928232758-00475.warc.gz"}
https://math.stackexchange.com/questions/354342/eulers-method-on-differential-equation	# Euler's Method on differential equation For a differential equation, it is known that Euler’s Method leads to an underestimate when the curve is concave up, just as it will lead to an overestimate when the curve is concave down: (from page 326 in this document). Here comes my question: Consider $\frac{dy}{dx}=2x-3(y-x^2)$ with $y(0)=0$. It is not difficult to verify that $y=x^2$ is the solution. Since the graph of $y=x^2$ is concave upward, then the approximation done by Euler's method should be an underestimation. However, using Euler's method with step size 1, we have $y(1)=0$ and $y(2)=5>2^2$. Why? • @Zev Chonoles: Thanks for the nice edit. – pipi Apr 8 '13 at 0:24 We have: $$\tag 1 \frac{dy}{dx}=2x-3(y-x^2) ~~~~~ \text{with} ~~~~~ y(0)=0.$$ For the algorithm we have: $t_0 = a = 0$ $w_0 = y(a) = y(0) = 0 \rightarrow \alpha = 0$ $h = 1$ $f(x, y) = 2x - 3 (y - x^2)$ Writing out the iteration formula, we have $w = w + h f(x, w)$, so $\displaystyle w_i = w_{i-1} + (1) (2 x_{i-1} - 3 (w_{i-1} -x_{i-1}^2))$ Writing out the iterates, we have: • $w_0 = 0$ • $w_1 = w_0 +2 x_0 -3(w_0 - x_0^2) = 0 + 0 -3(0 - 0) = 0$ • $w_2 = w_1 +2 x_1 -3(w_1 - x_1^2) = 0 + 2(1) - 3(0 - 1) = 2 - 3(-1) = 5$ If you look at the step size for this method, it plays havoc on the results. A step size, $h=1$, is very large. Try a step size over the range $[0,2.5]$ of $h=0.1$ and see how that fares. Also, it is useful to read the error bound calculations and the peculiar affects it can have with certain kinds of problems due to round-off. There is a better algorithm with tighter controls on the mesh sizes to improve this, but this method is good for its instructiveness only and not very practical. • Really nice work, today, Amzoti! +1 – amWhy Apr 8 '13 at 0:50 • @amWhy: I really appreciate the nice comments - so thanks for that. This poster did a great job and was spot on with his work - so I wanted to duplicate it and make sure I got the same thing. Regards – Amzoti Apr 8 '13 at 2:19	2020-04-06 16:10:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8109109997749329, "perplexity": 442.9905632737519}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371637684.76/warc/CC-MAIN-20200406133533-20200406164033-00144.warc.gz"}
https://gmatclub.com/forum/expression-119716.html?kudos=1	Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack It is currently 25 May 2017, 07:33 GMAT Club Daily Prep Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History Events & Promotions Events & Promotions in June Open Detailed Calendar Expression!! new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: Hide Tags Senior Manager Joined: 16 Feb 2011 Posts: 259 Followers: 4 Kudos [?]: 162 [0], given: 9 Show Tags 26 Aug 2011, 22:30 00:00 Difficulty: (N/A) Question Stats: 0% (00:00) correct 100% (00:02) wrong based on 1 sessions HideShow timer Statistics If x<0, simplify the given expression in terms of x, [(x-3)^4]^1/4 + (-xabsolute value of x)^1/2 = ?? Ans is 3-2x and I am getting -3-2x...pls help understand which sign am i missing? Manager Joined: 03 Mar 2011 Posts: 90 Location: United States Schools: Erasmus (S) GMAT 1: 730 Q51 V37 GPA: 3.9 Followers: 2 Kudos [?]: 144 [1] , given: 12 Show Tags 01 Oct 2011, 13:58 1 KUDOS $$\sqrt[4]{(x-3)^4}+\sqrt{-x\|x\|}$$ = ? $$\sqrt[4]{(x-3)^4}=\|x-3\|$$ Since $$x<0$$ then $$x-3<0$$ and $$\|x-3\|=3-x$$ $$\sqrt{-x\|x\|}=\sqrt{(-x)(-x)}=\sqrt{x^2}=\|x\|=-x$$ $$\sqrt[4]{(x-3)^4}+\sqrt{-x*\|x\|}=3-x-x=3-2x$$ _________________ If my post is useful for you not be ashamed to KUDO me! Let kudo each other! Re: Expression!! [#permalink] 01 Oct 2011, 13:58 Similar topics Replies Last post Similar Topics: 9 If the expression x^x^x^(...), where the given expression expression 6 31 Jan 2017, 06:11 7 Which expression is the greatest? 4 06 Dec 2016, 10:27 7 What is the value of the following expression? 3 21 Sep 2014, 20:52 10 The expression above is approximately equal to 8 29 Nov 2016, 13:45 1 For which of the following expressions would the value be 5 16 Jun 2014, 03:01 Display posts from previous: Sort by Expression!! new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.	2017-05-25 14:33:38	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19865092635154724, "perplexity": 13596.433194673968}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608084.63/warc/CC-MAIN-20170525140724-20170525160724-00204.warc.gz"}
https://stacks.math.columbia.edu/tag/08KE	## 109.65 Non-effective descent data for projective schemes In the chapter on descent we have seen that descent data for schemes relative to an fpqc morphism are effective for several classes of morphisms. In particular, affine morphisms and more generally quasi-affine morphisms satisfy descent for fpqc coverings (Descent, Lemma 35.38.1). This is not true for projective morphisms. Lemma 109.65.1. There is an etale covering $X\to S$ of schemes and a descent datum $(V/X,\varphi )$ relative to $X\to S$ such that $V\to X$ is projective, but the descent datum is not effective in the category of schemes. Proof. We imitate Hironaka's example of a smooth separated complex algebraic space of dimension 3 which is not a scheme [Example B.3.4.2, H]. Consider the action of the group $G = \mathbf{Z}/2 = \{ 1, g\}$ on projective 3-space $\mathbf{P}^3$ over the complex numbers by $g[x,y,z,w] = [y,x,w,z].$ The action is free outside the two disjoint lines $L_1=\{ [x,x,z,z]\}$ and $L_2=\{ [x,-x,z,-z]\}$ in ${\mathbf P}^3$. Let $Y={\mathbf P}^3-(L_1\cup L_2)$. There is a smooth quasi-projective scheme $S=Y/G$ over ${\mathbf C}$ such that $Y\to S$ is a $G$-torsor (Groupoids, Definition 39.11.3). Explicitly, we can define $S$ as the image of the open subset $Y$ in ${\mathbf P}^3$ under the morphism \begin{align} {\mathbf P}^3 & \to \text{Proj } {\mathbf C}[x,y,z,w]^ G\\ & = \text{Proj } {\mathbf C}[u_0,u_1,v_0,v_1,v_2]/(v_0v_1=v_2^2), \end{align} where $u_0=x+y$, $u_1=z+w$, $v_0=(x-y)^2$, $v_1=(z-w)^2$, and $v_2=(x-y)(z-w)$, and the ring is graded with $u_0,u_1$ in degree 1 and $v_0,v_1,v_2$ in degree 2. Let $C=\{ [x,y,z,w]: xy=z^2, w=0\}$ and $D=\{ [x,y,z,w]: xy=w^2, z=0\}$. These are smooth conic curves in ${\mathbf P}^3$, contained in the $G$-invariant open subset $Y$, with $g(C)=D$. Also, $C\cap D$ consists of the two points $P:=[1,0,0,0]$ and $Q:=[0,1,0,0]$, and these two points are switched by the action of $G$. Let $V_ Y\to Y$ be the scheme which over $Y-P$ is defined by blowing up $D$ and then the strict transform of $C$, and over $Y-Q$ is defined by blowing up $C$ and then the strict transform of $D$. (This is the same construction as in the proof of Lemma 109.64.1, except that $Y$ here denotes an open subset of ${\mathbf P}^3$ rather than all of ${\mathbf P}^3$.) Then the action of $G$ on $Y$ lifts to an action of $G$ on $V_ Y$, which switches the inverse images of $Y-P$ and $Y-Q$. This action of $G$ on $V_ Y$ gives a descent datum $(V_ Y/Y,\varphi _ Y)$ on $V_ Y$ relative to the $G$-torsor $Y\to S$. The morphism $V_ Y\to Y$ is proper but not projective, as shown in the proof of Lemma 109.64.1. Let $X$ be the disjoint union of the open subsets $Y-P$ and $Y-Q$; then we have surjective etale morphisms $X\to Y\to S$. Let $V$ be the pullback of $V_ Y\to Y$ to $X$; then the morphism $V\to X$ is projective, since $V_ Y\to Y$ is a blowup over each of the open subsets $Y-P$ and $Y-Q$. Moreover, the descent datum $(V_ Y/Y,\varphi _ Y)$ pulls back to a descent datum $(V/X,\varphi )$ relative to the etale covering $X\to S$. Suppose that this descent datum is effective in the category of schemes. That is, there is a scheme $U\to S$ which pulls back to the morphism $V\to X$ together with its descent datum. Then $U$ would be the quotient of $V_ Y$ by its $G$-action. $\xymatrix{ V \ar[r]\ar[d]& X\ar[d] \\ V_ Y \ar[r]\ar[d]& Y\ar[d] \\ U \ar[r]& S }$ Let $E$ be the inverse image of $C\cup D\subset Y$ in $V_ Y$; thus $E\rightarrow C\cup D$ is a proper morphism, with fibers isomorphic to ${\mathbf P}^1$ over $(C\cup D)-\{ P,Q\}$. The inverse image of $P$ in $E$ is a union of two lines $L_0$ and $M_0$. It follows that the inverse image of $Q=g(P)$ in $E$ is the union of two lines $L_0'=g(M_0)$ and $M_0'=g(L_0)$. As shown in the proof of Lemma 109.64.1, we have a rational equivalence $L_0+M_0'=L_0+g(L_0)\sim 0$ on $E$. By descent of closed subschemes, there is a curve $L_1\subset U$ (isomorphic to ${\mathbf P}^1$) whose inverse image in $V_ Y$ is $L_0\cup g(L_0)$. (Use Descent, Lemma 35.37.1, noting that a closed immersion is an affine morphism.) Let $R$ be a complex point of $L_1$. Since we assumed that $U$ is a scheme, we can choose a function $f$ in the local ring $O_{U,R}$ that vanishes at $R$ but not on the whole curve $L_1$. Let $D_{\text{loc}}$ be an irreducible component of the closed subset $\{ f = 0\}$ in $\mathop{\mathrm{Spec}}O_{U,R}$; then $D_{\text{loc}}$ has codimension 1. The closure of $D_{\text{loc}}$ in $U$ is an irreducible divisor $D_ U$ in $U$ which contains the point $R$ but not the whole curve $L_1$. The inverse image of $D_ U$ in $V_ Y$ is an effective divisor $D$ which intersects $L_0\cup g(L_0)$ but does not contain either curve $L_0$ or $g(L_0)$. Since the complex 3-fold $V_ Y$ is smooth, $O(D)$ is a line bundle on $V_ Y$. We use here that a regular local ring is factorial, or in other words is a UFD, see More on Algebra, Lemma 15.121.2. The restriction of $O(D)$ to the proper surface $E\subset V_ Y$ is a line bundle which has positive degree on the 1-cycle $L_0+g(L_0)$, by our information on $D$. Since $L_0+g(L_0)\sim 0$ on $E$, this contradicts that the degree of a line bundle is well-defined on 1-cycles modulo rational equivalence on a proper scheme over a field (Chow Homology, Lemma 42.20.3 and Lemma 42.28.2). Therefore the descent datum $(V/X,\varphi )$ is in fact not effective; that is, $U$ does not exist as a scheme. $\square$ In this example, the descent datum is effective in the category of algebraic spaces. More precisely, $U$ exists as a smooth separated algebraic space of dimension 3 over ${\mathbf C}$, for example by Algebraic Spaces, Lemma 64.14.3. Hironaka's 3-fold $U$ is a small resolution of the blowup $S'$ of the smooth quasi-projective 3-fold $S$ along the irreducible nodal curve $(C\cup D)/G$; the 3-fold $S'$ has a node singularity. The other small resolution of $S'$ (differing from $U$ by a “flop”) is again an algebraic space which is not a scheme. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	2023-03-28 15:52:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9852572679519653, "perplexity": 103.40816217101718}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948867.32/warc/CC-MAIN-20230328135732-20230328165732-00233.warc.gz"}
https://drexel28.wordpress.com/2012/05/10/the-tensor-algebra-and-exterior-algebra-pt-iv/	# Abstract Nonsense ## The Tensor Algebra and Exterior Algebra (Pt. IV) Point of Post: This is a continuation of this post. Exterior Products of Free Modules $\text{ }$ We’d now like to discuss how everything works out when we are talking about exterior products of free modules. $\text{ }$ In particular, we know that if $M$ is free, then by virtute of the fact that the tensor product commutes with coproducts we know that $\mathcal{T}^k(M)$ is free. But, it is rookie-mistake number one to think that quotients of free modules (even by free submodules) are necessarily free (is $\mathbb{Z}/2\mathbb{Z}$ free?!) and so there is no a priori reason to think that freeness of $M$ should imply the freeness of $\Lambda^k(M)$. $\text{ }$ Now, while not explicitly stated, it should be fairly obvious that if $\{v_1,\cdots,v_m\}$ is a spanning set for $M$ then $\{v_{i_1}\wedge\cdots\wedge v_{i_k}\}_{i_1,\cdots,i_k\in[m]}$ is a spanning set for $\Lambda^k(M)$ since that set, with wedges replaced with tensors, is a spanning set for $\mathcal{T}^k(M)$ of which $\Lambda^k(M)$ is a quotient. Of course, there is a huge amount of redundancies, in fact, most of these wedges are zero. For example, we can immediately replace this set with $\{v_{i_1}\wedge\cdots\wedge v_{i_k}\}$ where now the indices still live in $[m]$ but now we require that $i_\ell\ne i_j$ for any $\ell,j$. But, once again this is still redundant for we have indexed by injections $[k]\to[m]$, but by virtue of anticommutativity, we can index by combinations of $k$ elements of $m$. In particular, $\{v_{i_1}\wedge\cdots\wedge v_{i_k}\}$ spans $\Lambda^k(M)$ where $i_1<\cdots. In particular, we see that if $M$ is spanned by $m$ elements then $\Lambda^k(M)$ is spanned by $\displaystyle {m\choose k}$ elements. In particular, we see the following: $\text{ }$ Theorem: If $M$ is spanned by $m$ elements and $k>m$ then $\Lambda^k(M)=0$ $\text{ }$ This makes sense with our above interpretation of $\Lambda^k(M)$ being zero. Indeed, this should say that there are no nonzero alternating $k$-linear $R$-maps out of $M^k$. But, it of course suffices to check that if $f$ is such a map that $f(v_{i_1},\cdots,v_{i_k})=0$ for the $v_{i_j}$ in the spanning set of $M$. But , our old pal The Pigeonhole Principle (easily one of the most under appreciated theorems in mathematics!) tells us that since $k that at least two of these $v_{i_j}$‘s must be equal and so $f(v_{i_1},\cdots,v_{i_k})=0$. $\text{ }$ What we’d like to show is that if $M$ is indeed free, so that we can take this set of $v_i$‘s to be a basis, then this generating set is also a basis. The key to this is that when $M$ is free we shall be able to embed $\Lambda^k(M)$ back into $\mathcal{T}^k(M)$–a strange concept, embedding a quotient back into the space. The embedding should look familiar either if you are familiar with the more classic approach to multilinear algebra (via the alternating map) or if you know Leibniz expansion of the determinant. Namely, we are going to define an embedding $\Lambda^k(M)\hookrightarrow\mathcal{T}^k(M)$ in such a way that $\text{ }$ $\displaystyle v_1\wedge\cdots \wedge v_k\mapsto \sum_{\sigma\in S_k}\text{sgn}(\sigma)(v_{\sigma(1)}\otimes\cdots\otimes v_{\sigma(k)})\quad\mathbf{(1)}$ $\text{ }$ To see that such a map even exists we note that the mapping $M^k\to\mathcal{T}^k(M)$ defined as above (except we are mapping $(m_1,\cdots,m_k)$ to this sum) is alternating. Indeed, it’s easy to see that this map is multilinear (being a sum of multilinear maps) and is alternating for $\text{ }$ $\displaystyle v_{\tau(1)}\wedge\cdots\wedge v_{\tau(k)}\mapsto \sum_{\sigma\in S_k}\text{sgn}(\sigma)(v_{(\sigma\circ\tau)(1)}\otimes\cdots\otimes v_{(\sigma\circ\tau)(k)})$ $\text{ }$ Now, we write $\text{sgn}(\sigma)=\text{sgn}(\tau)^{-1}\text{sgn}(\sigma\circ\tau)$ (recalling that $\text{sgn}$ is a group map $S_k\to\{-1,1\}$) and since $\text{sgn}(\tau)$ is idempotent this is just $\text{sgn}(\sigma)=\text{sgn}(\tau)\text{sgn}(\sigma\circ\tau)$. So, we see that $(m_{\tau(1)},\cdots,\cdots,m_{\tau(k)})$ is mapped to $\text{ }$ $\displaystyle \text{sgn}(\tau)\sum_{\sigma\in S_k}\text{sgn}(\sigma)(v_{\sigma(1)}\wedge\cdots\wedge v_{\sigma(k)})$ $\text{ }$ Thus, we are afforded an $R$-linear map $\Lambda^k(M)\to \mathcal{T}^k(M)$ with $\mathbf{(1)}$ holding. What we’d like to now show is that if $M$ is free this is actually an embedding. Indeed, let $e_1,\cdots,e_m$ be a basis for $M$. Let $\displaystyle \sum_{I}\alpha_I e_I\in\Lambda^k(M)$ where $I$ runs over indices $1\leqslant i_1<\cdots<_k\leqslant m$ and $e_I$ evidently means $e_{i_1}\wedge\cdots\wedge e_{i_k}$ if $I=(i_1,\cdots,i_k)$. Now, if this goes to zero we have the relation $\text{ }$ $\displaystyle \sum_{\sigma\in S_k}\sum_{I}\text{sgn}(\sigma)\alpha_I e_{\sigma(I)}=0$ $\text{ }$ That said, from the basic theory of tensor products we know that $\{e_{\sigma(I)}\}$, as $\sigma$ and $I$ run over their respective indices, form a basis for the free $R$-module $\mathcal{T}^k(M)$ which tells us that $\text{sgn}(\sigma)\alpha_I=0$ for each $I$ and each $\sigma$–but this in particular tells us that $\alpha_I=0$ for all $I$. Thus, we see that the kernel of our map is trivial and thus we have an honest to god embedding. Now, note that the image of the vectors $\{e_I\}$ are linearly independent (for if $I$ and another index $J$ aren’t equal we can see that the image of $e_I$ and $e_J$ will contain different basis vectors!) and thus since we are dealing with an embedding this tells us that that $\{e_I\}$ are linearly independent in $\Lambda^k(M)$. Thus, we may conclude: $\text{ }$ Theorem: Let $M$ be a free $R$-module with basis $\{e_1,\cdots,e_n\}$. Then, $\Lambda^k(M)$ is a free $R$-module with basis $\{e_{i_1}\wedge\cdots\wedge e_{i_k}\}$ as the indices are chosen arbitrarily according to the condition $1\leqslant i_1<\cdots. In particular, we see that $\text{rank }\Lambda^k(M)$ is $\displaystyle {n\choose k}$. $\text{ }$ This is actually enables us to evaluate the full exterior algebra, as a module. Indeed: $\text{ }$ Theorem: Let $M$ be a f.g. free $R$-module of rank $n$. Then, $\Lambda(M)$ is a free $R$-module of rank $2^n$. Proof: We know that $\displaystyle \Lambda(M)$ is $\displaystyle \bigoplus_{k=0}^{\infty}\Lambda^k(M)$. Now, each $\Lambda^k(M)$ is free of rank $\displaystyle {n\choose k}$ and thus $\Lambda(M)$ is free of rank $\text{ }$ $\displaystyle \sum_{k=0}^{\infty}{n\choose k}=\sum_{k=0}^{n}{n\choose k}=2^n$ $\text{ }$ $\blacksquare$ While this embedding seemed incidental it shall come back to important for us when we finally get to the differential forms stuff that we are working towards. $\text{ }$ $\text{ }$ References: [1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print. [2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print. [3] Blyth, T. S. Module Theory. Clarendon, 1990. Print. [5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.	2018-07-16 08:45:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 130, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9555671811103821, "perplexity": 242.79967524041132}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589237.16/warc/CC-MAIN-20180716080356-20180716100356-00637.warc.gz"}
https://cs.stackexchange.com/questions/22615/whats-the-time-complexity-of-this-append-method	What's the time complexity of this append method? [closed] I made a method that appends a sequence to another sequence. So: (append [1,2,3] [4,5,6]) = [1,2,3,4,5,6] CODE In C# IEnumerable<int> Append(IEnumerable<int> xs,IEnumerable<int> ys) { using(var iteratorX = xs.GetEnumerator()) using(var iteratorY = ys.GetEnumerator()) { bool isTrueForX = false; bool isTrueForY = false; while((isTrueForX = iteratorX.MoveNext()) \|\| (isTrueForY = iteratorY.MoveNext())) { if(isTrueForX) yield return iteratorX.Current; else if(isTrueForY) yield return iteratorY.Current; } } } I would like to know what is the time-complexity of this algorithm. • Why do you think it is $O(\max(\|xs\|,\|ys\|))$? What makes you unsure? Please show your work in the question, and explain where you got stuck; that makes it more likely we can help you usefully. – D.W. Mar 14 '14 at 15:48 • This is a dump of a problem, not a question. If you have a specific question regarding the wording of the problem or about concrete steps in your own attempts at solving the problem, feel free to edit accordingly and we can reopen the question. See here for a relevant discussion. If you are uncertain how to improve your question, why not ask around in Computer Science Chat? – Raphael Mar 14 '14 at 16:28 If $T_1(l)$ denotes the time for a single call to the iterator of list $l$, $T_a(l)$ denotes the combined time for all calls needed to iterate ove $l$, and the combined list is called $x.y$, we find $$T_1(x.y) = {\cal O}(\max\{T_1(x),T_1(y)\})$$ and $$T_a(x.y) = T_a(x)+T_a(y).$$ The first one will typically be ${\cal O}(1)$ and the second one ${\cal O}(\|x.y\|)$.	2020-01-19 14:57:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5899559259414673, "perplexity": 1004.1425831298484}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594603.8/warc/CC-MAIN-20200119122744-20200119150744-00252.warc.gz"}
https://atomica.tools/docs/master/examples/Databooks.html	# Data and Databooks¶ This page provides an overview of Atomica’s internal representation of databooks. Project data for an application is specified in a ‘databook’. This is an Excel file that contains • A listing of which populations a simulation will have • A specification of which transfers are present • Population and time specific values for characteristics, parameters, transfers, and interactions ProjectData is a Python object containing the data, and it has methods to convert to and from the databook spreadsheet. [1]: import pandas as pd import atomica as au import matplotlib.pyplot as plt import numpy as np Atomica 1.12.1 (2019-08-20) -- (c) the Atomica development team 2019-08-20 07:29:38.971709 [2]: F = au.ProjectFramework(au.LIBRARY_PATH + "tb_framework.xlsx") Initialization characteristics are underdetermined - this may be intentional, but check the initial compartment sizes carefully The ProjectData object contains a number of attributes - pops is an odict storing populations and their full names. In the future, it may also store other attributes such as species - tvec stores the default time array associated with the data (users can override this on a per-table basis) - transfers and interactions store pairwise time series linking populations - tdve stores a dict of TimeDependentValueEntry tables for characteristics and parameters - tdve_pages stores a dict assigning TDVE tables to worksheets There are three basic elements to the databook 1. The population table 2. TimeDependentConnection tables which are used for transfers and interactions. These have a ‘to’ and a ‘from’ population 3. TimeDependentValueEntry tables which are used for characteristics and parameters ## TimeSeries objects¶ Time-varying data is backed by the TimeSeries object defined in structure.py. This object stores sparse time-varying data, together with an assumption. This allows it to keep track of both assumption and time-varying values entered in a databook. It has a few key properties [3]: ts = au.TimeSeries(t=[2014,2015],vals=[1,2]) A TimeSeries has t and vals attributes that store the time varying data [4]: ts.t [4]: [2014, 2015] [5]: ts.vals [5]: [1.0, 2.0] The values can be modified using the insert and remove methods. These preserve the order of the times. Notice that the remove method removes data based on the time value rather than the index. [6]: ts.insert(2016,3) ts.remove(2015) ts.vals [6]: [1.0, 3.0] A TimeSeries object also has an interpolate method to query its value at arbitrary times. It uses the same interpolation as for parameters, with constant extrapolation [7]: t = np.arange(2013,2018,0.2) plt.plot(t,ts.interpolate(t)) [7]: [<matplotlib.lines.Line2D at 0x7f0eb8f9d110>] One of the special features of the TimeSeries object that caters for databook entry is the fact that it can independently store the constant assumption value. This is set by inserting a value with no time associated with it [8]: ts2 = au.TimeSeries() ts2.insert(None,2) ts2.assumption [8]: 2.0 Since the data for a TimeSeries could be in assumption or in vals, the method get_arrays allows you to get time and value arrays for the content of the TimeSeries [9]: t,v = ts.get_arrays() print(t) print(v) t,v = ts2.get_arrays() print(t) print(v) [2014 2016] [1. 3.] [nan] [2.] However, the most common usage for the TimeSeries is to turn sparse data into full interpolated arrays. The interpolate method automatically manages the assumption and the time dependent values, so that you don’t have to do this yourself: [10]: t = np.arange(2013,2018,0.2) plt.plot(t,ts.interpolate(t),label='Time varying') plt.plot(t,ts2.interpolate(t),label='Assumption') plt.legend() [10]: <matplotlib.legend.Legend at 0x7f0eb8f7cb10> The TimeSeries object also has a units property that lets it store the units of the quantity entered in the databook. In general, whenever the databook has a time-dependent values row with an assumption and some year-specific data, the corresponding object used to store the data on that row is a TimeSeries ### Population table¶ The population table is very basic: in the future, there will probably be more columns - for example, to separate human and mosquito populations (these would need to be validated against tags specified in the Framework) ### TimeDependentConnections¶ A TimeDependentConnections table consists of three elements 1. A code name/full name table at the top 2. A matrix of Y/N values that controls which time series appear on the page 3. A set of time series inputs where the user enters time-varying data This table is managed by the TimeDependentConnections object in excel.py which has • A code name • A full name • The type (whether it’s an interaction or a transfer - the difference being, a transfer cannot have any entries on the diagonal i.e. within the same population) • A list of population names • A list of times • A dict of TimeSeries objects keyed by the (to,from) populaton names • A method from_tables() to construct a TimeDependentConnections instance based on the rows in the spreadsheet • A method write() that takes in a sheet and a start row, and writes the content for the object to the spreadsheet ### TimeDependentValuesEntry¶ A TimeDependentValuesEntry consists of a single element with • The full name of the quantity at the top left • The left column with population names • Value entry cells for contant and year-specific values A code name/full name table at the top A matrix of Y/N values that controls which time series appear on the page A set of time series inputs where the user enters time-varying data This table is managed by the TimeDependentValuesEntry object in excel.py which has • A name • The time vector • A dict of TimeSeries objects keyed by the population name • A method from_rows() to construct a TimeDependentValuesEntry instance based on the rows in the spreadsheet • A method write() that takes in a sheet and a start row, and writes the content for the object to the spreadsheet ## ProjectData operations¶ A number of possible operations can be performed on ProjectData • Get a new instance using ProjectData.new() where you pass in a Framework together with a list (or number) or populations and transfers • Use to_spreadsheet() to get a sciris Spreadsheet representation of the data, and save() to write the ProjectData to disk • Use get_ts() to get the time-data associated with a quantity out of ProjectData e.g. when making parameters • You can add or remove populations and transfers while keeping data intact • You can change the time values using change_tvec() while keeping the data intact Note that you can also change the time values in the spreadsheet in place just by adding or removing columns. In principle, you can also specify values only for a subset of populations, but at the moment this is probably not propagated through the model yet.	2019-09-17 12:43:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22009079158306122, "perplexity": 2950.8680568717705}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573071.65/warc/CC-MAIN-20190917121048-20190917143048-00234.warc.gz"}
https://rdrr.io/cran/stationery/f/inst/rmarkdown/templates/rmd2pdf-report/skeleton/instructions.Rmd	# Instructions for Reports In stationery: Working Examples for Reproducible Research Documents ##This Invisible Chunk is required in all CRMDA documents tmpdir <- paste0("tmpout") if (!dir.exists(tmpdir)) dir.create(tmpdir, recursive = TRUE) knitr::opts_chunk$set(echo=TRUE, comment=NA, fig.path=paste0(tmpdir, "/p-")) options(width = 70) library(stationery) ## If theme directory does not have required images or TeX files ## we need to retrieve them and put them in "theme" directory. logos <- c("logo.pdf") try(getFiles(logos, pkg = "stationery", overwrite = FALSE)) ## These theme files should be available already, but if not themefiles <-c("report-template.tex") try(getFiles(themefiles, pkg = "stationery", overwrite = FALSE)) ## If you do not have a file after that, ## the following will manufacture a blank image for a placeholder if(!file.exists("theme/logoleft.pdf")){ blankPDF(file = "theme/logoleft.pdf", height=1, width=3.5, messg = "") } if(!file.exists("theme/logo-vert.pdf")){ blankPDF(file = "theme/logo-vert.pdf", height=1, width=3.5, messg = "") } # Introduction ## Insert author information in the yaml header It is necessary for the author to edit the yaml header of this document to specify the title, author data, address and the name of a logo file. # We use a template file The look and feel of the document will be controlled by a template file. It is specified in the header as "theme/report-template.tex". In July 2018, we learned how to selectively override settings from the document's yaml header. These alternative parameters can be specified either as command line arguments to rmd2pdf.sh or as parameters in the R function call rmd2pdf. # Compile the document If the user starts an R session, she can run the rmd2pdf function with many optional arguments, as follows: > rmd2pdf("skeleton.Rmd", toc=FALSE, template="theme/report-template.tex") From the command line, the same file can be compiled using notation that is nearly identical $ ./rmd2pdf.sh --toc=FALSE --template='"theme/report-template.tex"' skeleton.Rmd Note the single quotes protect the double quotes. The template can, of course, be edited for your purposes. The formatting for the header is tricky, but in most of the other parts, it is fairly obvious how this can be customized. An important thing to remember is that a template file treats dollar signs $ as special symbols, so if one intends to insert a literal dollar sign in the template, one enters two dollar signs ($\$). If this document is edited in Rstudio, the "knit" menu sometimes does not work correctly. We notice that Rstudio will sometimes alter the document yaml header in a harmful way, but this does not always happen. If you open the Rnw file in Rstudio, click "Knit" and choos "knit to pdf", it is very likely to work. However, if you accidentally choose "knit to HTML", then Rstudio will alter the header and the document will not compile correctly any more. It seems like a mistake to allow the editor to alter the content of the header. Users who do not want to bother with the command line arguments can edit the rmd2pdf.sh script and change the defaults for the parameters in the obvious way at the top of the file. The rmd2pdf function (and shell script) will also generate a "purled" copy of the R code chunks. The term "purled" is equivalent to "tangled" in the Sweave chunk engine. The user can specify either purl=TRUE or tangle=TRUE, the script will treat them as equivalent. # Formatting Input ## Much \LaTeX\ Syntax is allowed This is not true only for math, but also other \LaTeX environments. This is explained in the crmda package vignette Rmarkdown. Not all \LaTeX markup will work well, but most will. CAUTION: Some \LaTeX will fail without warning or error messages. Careful proof reading of output is essential. The markdown to PDF conversion does not warn us of unrecognized LaTeX code. The result is not an error, but rather "empty white space" where the user expects \LaTeX output. ## Equations Use \[ and \[ for display equations. Do not use the double dollar signs: [ \Sigma_{gt}=\Lambda_{gt}\Psi_{gt}\Lambda'{gt}+\Theta{gt} ] To create a numbered equation, do this \Sigma_{gt}=\Lambda_{gt}\Psi_{gt}\Lambda'{gt}+\Theta{gt}\label{eq:sigma} and cross-reference that as equation (\ref{eq:sigma}). ## Document customization: essentials With the stationery package, we provide generic logos as placeholders. Feel free to replace them with your own logos. Most of the example documents have a code chunk that copies the logo files from an R packages. That themecopy stanza can be removed after the document is compiled for the first time, it will not replace a user-edited file from the theme directory with a new copy. We usually leave that chunk, so that we can delete the theme folder and it will be replaced when the document is compiled. ## Customizations requiring more \LaTeX packages In addition, if you insert \LaTeX features that require packages that are not currently in the template report-template.tex, then those packages can be inserted into the preamble by YAML header markup like so: header-includes: - \usepackage{xcolor} - \usepackage{amsmath} - \usepackage{amssymb} - \usepackage{fancybox} We have added many packages in the template already, and we have tested that, even when the template parameter is used, the header-includes values are taken into account by the compiling process. # R code chunks In our report style, the author will not generally insert visible code chunks, so almost always the chunk will have the flag include=FALSE or, if the chunk is included, the code will not be echoed, but perhaps a \LaTeX mark-up table or a figure may be placed into the document. The process for doing this depends on the document type. As explained in the crmda vignette code_chunks, the appearance of code chunks--whether they are revealed in the document at all--is controlled by many options. One approach might be to use one document to create graphs or tables, which are then to be saved in a folder (such as our tmpout folder, which is used in this document). This chunk code will create the output file "tmpout/p-hist-1.pdf" set.seed(234234) x <- rnorm(1000) hist(x, main = "A Histogram", xlab = "Random Normal Data, N = 1000") \begin{lstlisting} r ''r x <- rnorm(1000) hist(x, main = "Random Normal Data", xlab = "N(0,1) (1000 observations)")r '' \end{lstlisting} The graphic file, "tmpout/p-hist-1.pdf", is a picture, it does not have the code with it to make it "float" in the document. Observe that it is just "paste in" at the current location: \includegraphics[width=4in]{tmpout/p-hist-1} On the other hand, if we want a floating figure to be produced automatically, we can change the code slightly: rOne Histogram that Floats Automatically", fig.width=4} x <- rnorm(1000) hist(x, main = "", xlab = "Random Normal Data, N = 1000") Note we forced a label into the caption itself with \LaTeX\ code. We can use a cross reference to identify that. We'd write as in Figure \ref{fig:hist20}, we get a fabulous result. It shouldn't be necessary to wedge a label in with the caption as we did, but the "automatic" ways is not working today. If it works for you, please tell me how. There are several R packages intended to create "ready to publish" \LaTeX tables. One of the oldest and most venerable of these is xtable, which we use here to create a simple table that displays as a cross tabulation table. When the \LaTeX output is going directly into the document, the chunk flag is "results='asis'" and the echo parameter should be FALSE. The default configuration for xtable is to create tables that are floating \LaTeX objects. r set.seed(1234) x <- rnorm(100, 0, 1) y <- rnorm(100, 0, 1) library(xtable) caption = "Ten Lines from One Data Frame", label="tab:onedf") print(xt, floating = TRUE, comment = FALSE, include.rownames = FALSE, caption.placement = "top") See the help pages for xtable and print.xtable to find out all of the possible arguments. If one does intend to have the output go directly into the document, without any hand editing, it is almost certainly necessary to specify a large number of arguments. It may be more workable to write the \LaTeX file on disk and then double-check its contents before manual inclusion in the document. If a LaTeX table file has been created from another document, we do not recommend "cutting and pasting" into this document. Instead, use "\textbackslash{}input{filename}". The following code chunk will save the same \LaTeX markup table in a file. print(xt, file = "tmpout/p-ex-2.tex", floating = TRUE, comment = FALSE, include.rownames = FALSE, caption.placement = "top") # Bibliographical citations. {-} This document is produced with R [@RCore]. Here's a citation that excludes some author names [-@diggle_analysis_2013, p. 37]. Note that to get the full parenthesized statement with names and dates, we insert hard brackets [, an @ sign, the bibtex tag, and a closer ]. If we don't want their names, we insert a - sign. It is also possible to refer to a group of projects [@hsiao_analysis_2014; @fitzmaurice_applied_2011; @mccullagh_nelder_1983]. Because this document is going to be compiled to a PDF back end via latex, it is also allowed to use LaTeX style natbib citations, which are considerably more flexible. Concerning R \citep{venablesripley2000}, the result is the same for the basic citation. A citation that has three authors [@diggle_analysis_2013], using the markdown style, and a citation that has three authors, using the LaTeX natbib markup \citep{hastie_elements_2009}. One of the reasons why the APA citations are desirable, and also a pain, is that the APA has very picky requirements about how the citations appear in the document. Note the in-text parenthesized citations use the ampersand, where humans might use the word "and". On the other hand, if only the date is parenthesized, the "\textbackslash{}citet" results in \citet{hastie_elements_2009}. But if we want to write names, we can get that with "\textbackslash{}citeyearpar", as in Hastie, Tibshirani, and Friedman \citeyearpar{hastie_elements_2009}. # Session Info Reports do not include the R session replication information, generally speaking. However, compiling the document will produce a record-keeping file in which the session information is saved. This will be in the current working directory. zz <- gsub("Rmd", "Rout", knitr::current_input()) capture.output(sessionInfo(), file = zz, append = TRUE) if (!is.null(warnings())){ capture.output(warnings(), file = zz) } # References ## Try the stationery package in your browser Any scripts or data that you put into this service are public. stationery documentation built on Oct. 8, 2021, 5:07 p.m.	2022-01-21 18:09:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6328498125076294, "perplexity": 3548.7961975496787}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303512.46/warc/CC-MAIN-20220121162107-20220121192107-00046.warc.gz"}
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-concepts-through-functions-a-unit-circle-approach-to-trigonometry-3rd-edition/chapter-5-trigonometric-functions-section-5-2-trigonometric-functions-unit-circle-approach-5-2-assess-your-understanding-page-403/96	## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition) Published by Pearson # Chapter 5 - Trigonometric Functions - Section 5.2 Trigonometric Functions: Unit Circle Approach - 5.2 Assess Your Understanding - Page 403: 96 #### Answer $\dfrac{1}{2}$ #### Work Step by Step Let us consider: $g(\theta)=\cos \theta$ We are given that $\theta=60^{\circ}$ We know form the unit circle that $\cos (60^{\circ})=\cos (\dfrac{\pi}{3})=\dfrac{1}{2}$ Thus, we have: $g(\theta)=cos(60^{\circ}) \\=cos\dfrac{\pi}{3} \\ =\dfrac{1}{2}$ After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.	2021-10-25 17:39:01	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7808436155319214, "perplexity": 2043.8409333568768}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587719.64/warc/CC-MAIN-20211025154225-20211025184225-00252.warc.gz"}
https://chengcheng-xiao.github.io/post/2020/01/17/Centersymmetric_phase.html	# Is it really necessary to have a centrosymmetric structure to calculate the macroscopic polarization? ## Background According to “the Modern theory of polarization”, in a continous-k formulation, the polarization of one structure $\mathbf{P}_{el}$ is $- f e / (2 \pi)^3$ times the sum of valence-band’s Berryphase while the total macroscopic polarization is the polairzation difference between ferroelectric phase and the centrosymmetric phase. Convieniently, Wannier charge centers can be linked to the Berryphase of valence band via a Fourier transformation. Thus, we should be able to calculate the ferroelectric polarization using Wannier centers to be more chemically intuitive. In definition, this calculation seems to be easily achievable, but in pratice, there are many pitfalls in doing such calculations. However, almost all of these pitfalls or problems are due to the so called polairzation quantum which arise from the periodicity of the unitcell (Translation symmetry of crystals). My previous post already demonstrated how easy can one fall in to such trap and get erroneous results and, in essence, this post is still deeply linked with such problems. ## Big questions What is the “correct” way of making the centrosymmetric phase in order to get the correct polarization value? • NEB calculation to determine the optimum transition structure? • Linear transform from one polarized state to another? What if there are multiple high-symmetry phases, which one do we choose??? Since the actual structural change during the polarization shift will almost always not be uniform, how can our result match the one experiementalist get?? ## TL;DR __It doesn’t matter which centrosymmetric structure one use, as long as we have a smooth transformation, every centrosymmetric phase will have the same polarization value. Correspondingly, their Berrypahse will always be the same if we are able to avoid band crossing Also, yes, band crossing does mess up the result. But if we are careful enough to keep track of the continuous band transformation and able to disentangle (eliminate) the effect of the band crossing, we can still get the right answer.__ ## Study case I’ll try to explain this with a simple BTO structure. Here, different from the last time, I’ve constrained my structure to only allow Ti atom to move: The result: Total ionic contribution = -0.6907 electA Total electronic contribution = 0.27726 electA Total dipole moment = -0.41344 electA Volume = 64.35864801 A^3 Total polarization = 0.102912491 C/m^2 What if, I construct an simple antiferroelectric structure and say that is the centrosymmetric phase I take as referencing structure? Recall there are acatually no more constraints other than our reference structure to be centrosymmetric, so, this actually does not contradict any rules. The result: Total ionic contribution = -1.38139 electA Total electronic contribution = 0.55453 electA Total dipole moment = -0.82686 electA Volume = 128.717296 A^3 Total polarization = 0.102912491 C/m^2 Interestingly, using AFE structure as the reference phase, I got identical polarization value comparing to the usual cubic phase. Let’s slow down and think about why this is the case. In oue switching route, we can clearly see that from FE to AFE structrue, the Ti atom on the left hand side needs to go down around 0.115A. On the other hand, if we have a supercell with both Ti atoms at the center of their cell, each Ti only needs to go down 0.057A, exactly half of the length in the AFE case. This tells us, in our AFE case, we have one Ti atom doing all the hard work while the other doing nothing. This does not change the fact that eventually all the work done for the polarization switching will always be there. Now, Lets go one step further… Here, we can actually see this giant AFE structure as two FE grain with one boundary sandwiched in between. Similar to our two-cell AFE case, this can still be used to calculate the total polarization. This tells us that we can even construct a centrosymmetric macroscopic structrue with grain boundary in it. As long as it has centrosymmetry and can be smoothly transformed form the positive FE phase to the negative FE phase, our reulst will always remains the same! This actually reassures us the polarization does not depend on the centrosymmetric phase and only depend on the ferroelectric phase like a lone dipole moment without translation symmetry (as it should be). The symmetry protecting the berry phase of the centrosymmetric structure is the inversion symmetry (or in some case mirror symmetry). ## Summary In summary, the centrosymmetric phase in the calculation of macroscopic polarization actually does not matter that much. As long as you can keep track of the band structure and the polarization vs distortion curve to be on the same branch, you are safe. Or, if you are savy enough, you can use Wannier90’s disentanglement to solve the “metallic centrosymmetric phase” problem. ## Input I’ve put all input file in a zip file for download: VASP. I’ve also included a 2D inplane ferroelectric structure with two usable centrosymmetric phases to play with. enjoy!	2022-01-21 21:18:51	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7398420572280884, "perplexity": 1420.8797814559919}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320303709.2/warc/CC-MAIN-20220121192415-20220121222415-00490.warc.gz"}
https://www.physicsforums.com/threads/thermodynamics-again-help.124333/	Thermodynamics Again HELP 1. Jun 21, 2006 cukitas2001 Hey guys, im understanding a bit more thermodynamics but ive been stumped once again on two problems: 1) A quantity of air is taken from state a to state b along a path that is a straight line in the pV-diagram If the volume and pressure in state a are 7.40×10−2 m^3 and 1.08×10^5Pa , and those in state b are 0.115 m^3 and 1.32×10^5 Pa , what is the work W done by the gas in this process? Assume that the gas may be treated as ideal. Is temperature constant here? I'm thinking so because of the line increasing at a constant rate. Do i need to apply $$W = \int p dV$$ with limits of $$V_2$$ and $$V_1$$ (don't know how to insert limits into the integral symbol on latex) Any ideas? 2) A thermodynamic system is taken from state a to state c in the figure along either path abc or path adc. Along path abc the work done by the system is 450 J. Along path adc, is 120 J. The internal energies of each of the four states shown in the figure are Ua=150 J, Ub=240 J, Uc=680 J, Ud=330J. A) Calculate the heat flow Q for the process ab. I dont know how to go about this - work i can deal with on the pV diagrams but Q i don't know. Any help please? 2. Jun 21, 2006 Andrew Mason Since $$W = \int_{V_1}^{V_2} p dV$$, = area under graph, does it matter how n changes in calculating W? If P/V = constant and V = P/nRT, then what can you say about T and n? AM 3. Jun 21, 2006 Andrew Mason Use the first law of thermodynamics: $$\Delta Q = \Delta U + \Delta W$$ where $\Delta W$ is the work done by the gas. Is any work done by the gas in going from a to b? AM 4. Jun 21, 2006 cukitas2001 I dont follow the first problem thinking 5. Jun 21, 2006 cukitas2001 no since its isochoric? so then Q = 150-240 = -90 J....Heat is leaving the system? or is it 240-150? for change being final minus initial so Ub-Ua? Last edited: Jun 21, 2006 6. Jun 21, 2006 Andrew Mason Is n a factor in the integral $\int PdV$ ? Do we have enough information to evaluate that integral (the area under the graph) ? Work out the expression for that area in terms of Pa, Pb, Vb and Va. AM 7. Jun 21, 2006 cukitas2001 ok while im tryin to digest your advice for problem one....on problem two what would the work be from b to c ? would the intergral thing come up here again? 8. Jun 21, 2006 cukitas2001 substituting p=nRT/V into the integral wouldn't the constants nRT come out and only integration on p and V be done since p and v vary? Could you perhaps elaborate some on you method to the first problem and start me off a bit? Last edited: Jun 21, 2006 9. Jun 21, 2006 Tom Mattson Staff Emeritus He's trying to get you to see that you don't need to actually do an integral to get the area under the curve. You can break the region up into basic geometric shapes whose area formulae should be well known to you. 10. Jun 21, 2006 cukitas2001 ok so it should then be teh area of the triangle right? i tried doing: W=(0.5)(Vb-Va)(Pb-Pa) and im getting 492J but its not right...what did i misinterpret? 11. Jun 21, 2006 Andrew Mason You are omitting the rectangular portion below Pa. AM 12. Jun 21, 2006 cukitas2001 ooooooooh ok got it now and ifgured out the work done by the horizontal line on the second problem...thanks for the help... Hey Andrew, you think you could walk me through the mental process you were trying to show me i wanna try to understand it. 13. Jun 21, 2006 Andrew Mason Which part? I was trying to get you to realize that if you know P and V, you can determine the work done regardless of how n or T changes. Work is the area under the PV graph. To your question on how temperature varies (which you do not have to know to find the work in this question), you can see that since P/V is constant and V = nRT/P, then nRT must be constant. So T must vary inversely as n (T = k/nR) where k = P/V To your question on the heat flow, I was just trying to point out that if work is 0 (as it is from a to b, since there is no area under the PV graph from a to b) the heat flow is the difference in internal energy between b and a: Ub - Ua. In this case the change in internal energy from a to b is positive (increase) so heat flow is into the gas. AM	2017-03-29 17:37:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8486864566802979, "perplexity": 920.6972527301018}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218190754.6/warc/CC-MAIN-20170322212950-00307-ip-10-233-31-227.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/218876/find-the-intersection-points-of-the-line-l-with-the-three-coordinate-planes-oxy?answertab=oldest	# Find the intersection points of the line L with the three coordinate planes Oxy, Oyz, and Ozx Let L be the line given by the parametric equations x = 1 + 2t, y = −1 − t, z = 3t Find the intersection points of the line L with the three coordinate planes Oxy, Oyz, and Ozx - Hint: For example, for the $x$-$z$ plane, you want $y=0$, so $-1-t=0$, meaning that $t=-1$. Substitute. We get $(-1,0,-3)$.	2014-11-28 07:39:29	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8824915289878845, "perplexity": 131.1507542727835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009825.77/warc/CC-MAIN-20141125155649-00034-ip-10-235-23-156.ec2.internal.warc.gz"}
https://neurips.cc/Conferences/2013/ScheduleMultitrack?event=4087	Timezone: » Poster The Fast Convergence of Incremental PCA Akshay Balsubramani · Sanjoy Dasgupta · Yoav Freund Sat Dec 07 07:00 PM -- 11:59 PM (PST) @ Harrah's Special Events Center, 2nd Floor #None We prove the first finite-sample convergence rates for any incremental PCA algorithm using sub-quadratic time and memory per iteration. The algorithm analyzed is Oja's learning rule, an efficient and well-known scheme for estimating the top principal component. Our analysis of this non-convex problem yields expected and high-probability convergence rates of $\tilde{O}(1/n)$ through a novel technique. We relate our guarantees to existing rates for stochastic gradient descent on strongly convex functions, and extend those results. We also include experiments which demonstrate convergence behaviors predicted by our analysis.	2020-08-07 16:05:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35375726222991943, "perplexity": 1930.070383613541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737204.32/warc/CC-MAIN-20200807143225-20200807173225-00158.warc.gz"}
https://or.stackexchange.com/questions/5878/if-else-query-depending-on-optimization-variable-in-gurobi-java	# if-else query depending on optimization variable in Gurobi (Java) I am looking for the most elegant solution to the following problem: I have an if-else query that depends on my optimization variable $$x_i$$. If $$a \leq b_i + x_i$$, the parameter $$c$$ should take the value of an parameter $$z$$, otherwise it should be $$0$$. I'm sure I need to solve this via constraints, but I don't know how to do it most cleverly. • what are $b_i$s? please check the edit on the third line and confirm if it is correct. Mar 8 at 15:45 • You might want to have a look into the Big M Method. Mar 8 at 22:07 • A constant that can assume two different values is not a constant. Mar 9 at 0:48 • @OguzToragay yes its correct in this way...b is just another constant depending on i Mar 9 at 7:38 • @SimonT i think you mean especially the part with the binary variable or? Mar 9 at 7:51 "variable c (no optimization variable)" is not something that one can say in linear and (mixed) integer programming. There are decision variables and there are parameters (coefficients). That's it. There are no local variables as in computer programming. It is best to look at things as solving large sparse systems of linear equations. Now, Gurobi has one very useful feature: indicator constraints. They take the form of implications with a binary variable on the left and a linear constraint on the right. We can use this to formulate: "If $$a\le b_i+x_i$$, the variable c should take the value of a parameter z, otherwise it should be 0." Well, more or less. As stated it looks wrong. What if for one $$i$$ the constraint holds and for another it does not? So I will assume the problem is more like: \begin{aligned} & a\le b_i+x_i \Rightarrow c_i = z \\ & a\gt b_i+x_i \Rightarrow c_i = 0 \end{aligned} As we cannot have strict inequalities, here is my suggestion to implement this: \begin{aligned} & \delta_i = 1 \Rightarrow a\le b_i+x_i \\ & \delta_i = 0 \Rightarrow a\ge b_i+x_i+0.0001 \\ & c_i = z \cdot \delta_i\\ & \delta_i \in \{0,1\} \end{aligned} Again, the implications can be implemented in Gurobi in a straightforward manner using indicator constraints. $$\delta_i$$ is an additional binary variable. In practice, I would drop the 0.0001 and keep the model a bit ambiguous in case of equality (the model can then pick the most profitable branch). If all quantities involved are integers, we can replace 0.0001 by 1. Instead of indicator constraints, it is also possible to use big-M constraints. They require a bit more care, as we need to worry about good bounds. • thank you for the detailed answer! I just don't quite understand how to apply the indicator constraint. Do I create the two constrainst for case 1 and 2 and then write two indicator constraints? In addition, a double value is needed if I see that correctly...how do I do that here if the values for a can change? Mar 10 at 7:35 • to be honest I don't know exactly how to implement this in Gurbi (Java). Mar 10 at 8:19 • Do I create the two constrainst for case 1 and 2 and then write two indicator constraints? Definitely not. See the documentation and I would suggest write some small tests in Java/Gurobi to get more familiar with this. Mar 10 at 17:19 • a double value is needed if I see that correctly The documentation mentions this represents the rhs of the constraint. Mar 10 at 17:20 • how do I do that here if the values for a can change? Depends. If a is a decision variable then just let Gurobi determine its value. If a is a parameter, just use a Java variable to hold its value. Mar 10 at 17:22	2021-09-23 21:45:35	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 10, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9052266478538513, "perplexity": 538.8447092951338}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057447.52/warc/CC-MAIN-20210923195546-20210923225546-00311.warc.gz"}
https://studyadda.com/notes/jee-main-advanced/physics/ray-optics/common-examples-of-tir/8723	JEE Main & Advanced Physics Ray Optics Common Examples of TIR Common Examples of TIR Category : JEE Main & Advanced (1) Looming : An optical illusion in cold countries (2) Mirage : An optical illusion in deserts (3) Brilliance of diamond : Due to repeated internal reflections diamond sparkles. (4) Optical fibre : Optical fibres consist of many long high quality composite glass/quartz fibres. Each fibre consists of a core and cladding. (i) The refractive index of the material of the core $({{\mu }_{1}})$is higher than that of the cladding $({{\mu }_{2}})$. (ii) When the light is incident on one end of the fibre at a small angle, the light passes inside, undergoes repeated total internal reflections along the fibre and finally comes out. The angle of incidence is always larger than the critical angle of the core material with respect to its cladding. (iii) Even if the fibre is bent, the light can easily travel through along the fibre (iv) A bundle of optical fibres can be used as a 'light pipe' in medical and optical examination. It can also be used for optical signal transmission. Optical fibres have also been used for transmitting and receiving electrical signals which are converted to light by suitable transducers. (5) Field of vision of fish (or swimmer) : A fish (diver) inside the water can see the whole world through a cone with. (a) Apex angle$=2C={{98}^{o}}$ (b) Radius of base $r=h\tan C=\frac{h}{\sqrt{{{\mu }^{2}}-1}}$; for water$\ r=\frac{3h}{\sqrt{7}}$ (c) Area of base A$=\frac{\pi {{h}^{2}}}{({{\mu }^{2}}-1)}$; for water $a=\frac{9\pi }{7}{{h}^{2}}$ (6) Porro prism : A right angled isosceles prism, which is used in periscopes or binoculars. It is used to deviate light rays through ${{90}^{o}}$and ${{180}^{o}}$and also to erect the image. You need to login to perform this action. You will be redirected in 3 sec	2020-12-03 10:49:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4029037058353424, "perplexity": 1409.535620801352}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141727627.70/warc/CC-MAIN-20201203094119-20201203124119-00629.warc.gz"}
https://huggingface.co/sentence-transformers/sentence-t5-xxl/blob/main/README.md	# sentence-transformers /sentence-t5-xxl --- pipeline_tag: sentence-similarity language: en license: apache-2.0 tags: - sentence-transformers - feature-extraction - sentence-similarity - transformers --- # sentence-transformers/sentence-t5-xxl This is a [sentence-transformers](https://www.SBERT.net) model: It maps sentences & paragraphs to a 768 dimensional dense vector space. The model works well for sentence similarity tasks, but doesn't perform that well for semantic search tasks. This model was converted from the Tensorflow model [st5-11b-1](https://tfhub.dev/google/sentence-t5/st5-11b/1) to PyTorch. When using this model, have a look at the publication: [Sentence-T5: Scalable sentence encoders from pre-trained text-to-text models](https://arxiv.org/abs/2108.08877). The tfhub model and this PyTorch model can produce slightly different embeddings, however, when run on the same benchmarks, they produce identical results. The model uses only the encoder from a T5-11B model. The weights are stored in FP16. ## Usage (Sentence-Transformers) Using this model becomes easy when you have [sentence-transformers](https://www.SBERT.net) installed: pip install -U sentence-transformers Then you can use the model like this: python from sentence_transformers import SentenceTransformer sentences = ["This is an example sentence", "Each sentence is converted"] model = SentenceTransformer('sentence-transformers/sentence-t5-xxl') embeddings = model.encode(sentences) print(embeddings) The model requires sentence-transformers version 2.2.0 or newer. ## Evaluation Results For an automated evaluation of this model, see the Sentence Embeddings Benchmark: [https://seb.sbert.net](https://seb.sbert.net?model_name=sentence-transformers/sentence-t5-xxl) ## Citing & Authors If you find this model helpful, please cite the respective publication: [Sentence-T5: Scalable sentence encoders from pre-trained text-to-text models](https://arxiv.org/abs/2108.08877)	2022-10-01 13:40:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4573037922382355, "perplexity": 10942.914624996927}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030336674.94/warc/CC-MAIN-20221001132802-20221001162802-00460.warc.gz"}
http://www.webreference.com/html/tutorial8/example2.html	Example HTML Document Showcasing Relative and Absolute Units \| WebReference # Example HTML Document Showcasing Relative and Absolute Units Absolute and relative units with 6pt text: This is a paragraph with 25.4mm of indent. This is a paragraph with 2.54cm of indent. This is a paragraph with 1in of indent. This is a paragraph with 72pt of indent. This is a paragraph with 6pc of indent. This is a paragraph with 12em of indent. This is a paragraph with 24ex of indent. This is a paragraph with 100px of indent. Absolute and relative units with 12pt text: This is a paragraph with 25.4mm of indent. This is a paragraph with 2.54cm of indent. This is a paragraph with 1in of indent. This is a paragraph with 72pt of indent. This is a paragraph with 6pc of indent. This is a paragraph with 6em of indent. This is a paragraph with 12ex of indent. This is a paragraph with 100px of indent. Absolute and relative units with 12pt text: This is a paragraph with 25.4mm of indent. This is a paragraph with 2.54cm of indent. This is a paragraph with 1in of indent. This is a paragraph with 72pt of indent. This is a paragraph with 6pc of indent. This is a paragraph with 4em of indent. This is a paragraph with 8ex of indent. This is a paragraph with 100px of indent.	2017-01-20 22:04:34	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9573894143104553, "perplexity": 10118.103749250295}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280888.62/warc/CC-MAIN-20170116095120-00489-ip-10-171-10-70.ec2.internal.warc.gz"}
https://tex.stackexchange.com/questions/linked/21752	561 views Rotating rectangle node in tikz without getting the right angle [duplicate] I'm new to tikz and I am trying to make a perpendicular angle mark on a triangle by \path[clip](C)--($(B)!(A)!(C)$)--(A); and \node[draw=black, rectangle,inner sep=0pt, outer sep=0pt, minimum size=... 28k views How to draw a line passing through a point and perpendicular to another? I'm having a bit of a headache right now as I can't draw this basic figure: A straight line goes through points A and B. Another line is perpendicular to AB and goes through an arbitrary point P. I ... 5k views Drawing of triangle and coordinate system using TikZ I am using TikZ for the first time, and I am experiencing some difficulty getting what exactly I want it to display. I am trying to draw the following picture: Here is the code (I know it is not that ... 3k views How do I draw a little red square to label my right triangle? The sample code: \documentclass{minimal} \usepackage{tikz} \begin{document} \begin{tikzpicture} \draw ( 0 , 0 ) coordinate (A) -- ( 4 , 0 ) coordinate (C) -- ( 0 , 3 ) coordinate (B) -... 1k views Difference between -\| and \|- in TikZ Based on this tutorial, I understand symbols -\| and \|- are used to draw perpendicular lines. But I wish to understand the difference between the two. There are the following related questions: ... 5k views Mark 90 degree angle in tikz in german convention What's the best way to denote a 90 degree angle with german convention like on the following picture in tikz: Suppose I start with something like this: \documentclass{article} \usepackage{tikz} \... 1k views indicate a right angle in 3d The following MWE shows what appears to be a piece of paper folded. I want to indicate that the angle between the dashed lines is right angled by adding a simple square at M, such that the missing ... 722 views Marking right angles (structuralanalysis) I'm trying to draw this picture: Until now, I have this, with the help of structuralanalysis package: \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{arrows} \usetikzlibrary{calc} \... 9k views In the MWE below, how can I get to adjust the position of the node labels? Note that the nodes are defined by dot/.style={circle,inner sep=1pt,fill,label={#1},name=#1} but I cannot figure out how to ... 2k views How to align a right angle As you you can see in the following image, a right angle has some space to side BC. How can I fix it and is there any better way to add an right angle? \documentclass{article} \usepackage{tikz} \... 1k views Right Angle Symbol Alignment with TikZ/Calc I used a solution from @Jake from this post Insertion of Perpendicular Symbol at Intersection of Two Lines, but the alignment is not quite right. I have my suspicions why, but I don't understand the ...	2020-02-18 18:11:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9646031260490417, "perplexity": 805.7712808665833}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143805.13/warc/CC-MAIN-20200218180919-20200218210919-00409.warc.gz"}
http://aspetv.com/forum/hwvog2j.php?page=69d744-clare-kelly-artist	clare kelly artist This is the currently selected item. It is commonly measured through the cost-to-cost method. Enter an old number in cell A1 and a new number in cell B1. Percentage Change = ($… So, his profit percent is 20. Car M costs$50,000 and car L costs $40,000. This lesson will present real world examples that involve solving for a percent of change. = -10% It can be interpreted as value has increased by 10% from the old number. More Difficult Problems Using Percent Change Example 6. You then multiply this decimal by 100 to get the average percentage. Example: A collectors' comic book is worth$120 in 1994, and in 1995 its value is $132. What was the population in 2005? 00:25. It is expressed in percentage, it is the change in new value with respect to the old value. Percentage in mathematics is a number or ratio which can be represented as a fraction of 100. A percent change analysis shows how two items changed as a percentage from one period to another period. Percent of change = 20/100. To tackle this problem first we calculate the difference in hours between the new and old numbers. Members of any species can create a population. Find his profit percent. Given parameters are. There are two conditions to use the percentage of completion method: 1. × 100% = 20%, 200 − 240 Take Old Value (20) Take New Value (16) Percent changes are often seen in word problems and are useful in many real world situations, for example when shopping sales or when calculating a tip. Profit or loss as a percentage. Example: You had 5 books, but now have 7. Percentage Change. Therefore: 75 + 5% * 75 = 75 * (100% + 5%). Over the next 35 years, it increased by about 115%. $\large Percentage\;Change=\frac{New\;Value-Old\;Value}{Old\;Value}\times 100$. Solution Use below given data for the calculation. Last week, gasoline was$3.29/gallon. × 100%, (The "\|" symbols mean absolute value, so negatives become positive), 240 − 200 Markup and Markdown Examples (page 2 of 3) Sections: Basic ... (you find by how much the quantity changed), and then you calculate the percent change relative to the original value. share \| follow \| edited Mar 31 '14 at 15:21. (For quarterly data, use LAG4 and DIF4. 240 To decrease a number by a percentage, simply change the plus sign to a minus sign. (For quarterly data, use LAG4 and DIF4. When faced with the actual figures, perception of the amount of violent crime in Ceredigion changes significantly. For example, to calculate the Monthly Change and Total Change. Percentage Change: show that change as a percent of the old value ... so divide by the old value and make it a percentage: So the percentage change from 5 to 7 is: 2/5 = 0.4 = 40%. Initial value ± percentage of change = final value Example 1 (continued) We can simplify the operation 75 + 5% * 75 by factorising. Solution: Let E be the enrollment before the first month. Then assume that, due to downsizing, the business now has only 16 employees. The term percent originates from a Latin word ‘ per centum ’ which means per 100. Let’s go over how to calculate percent change in step-by-step process, using an example. Learn how to find percent change in this free math video tutorial by Mario's Math Tutoring. Nonviolent protests are twice as likely to succeed as armed conflicts – and those engaging a threshold of 3.5% of the population have never failed to bring about change. As per the formula, Old Value is Previous Week Number, and New Value is Current Week Number. To find the formula we will use to solve the . But no! So that's the percent of change, an increase of 12.5% in the savings account. 15 ÷ 125 = 0.12. Example 1. Subtract 100% and you get −27.5%, or a 27.5% decrease. Percent changes are useful to help people understand changes in a value over time. Eric, You are best to report the change each year as a percentage over the previous year. Because you are saying how much a value has changed. An example would be the following: FY2002 = 1,349 FY2003 = 1,391 FY2004 = 1,324 FY2005 = 805 FY2006 = 900. For monthly data, use LAG12 and DIF12.) The percent change (or) the percent change of a quantity is the percent of its initial value either increased or decreased to get its final value. Because the percentage rise or fall is in relation to the old value: To "reverse" a percentage rise or fall, use the right formula here: Step 1: Calculate the change (subtract old value from the new value), Step 2: Divide that change by the old value (you will get a decimal number), Step 3: Convert that to a percentage (by multiplying by 100 and adding a "%" sign), Step 1: Divide the New Value by the Old Value (you will get a decimal number), Step 2: Convert that to a percentage (by multiplying by 100 and adding a "%" sign), Step 2: Divide by the old value: $1/$5 = 0.2, Step 3: Convert 0.2 to percentage: 0.2×100 =, Step 1: Divide new value by old value: $6/$5 = 1.2, Step 2: Convert to percentage: 1.2×100 = 120% (i.e. It's increased by $0.30, so we divide the price difference by the original price. Next, divide the amount of change by the original amount: 25 ÷ 200 = 0.125. Percent change in CPI = (end value of index - start value of the index) / start value of the index x 100 . 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Imaginary Player Sample, 2003 Buick Lesabre Traction Control Button, Standard Bathroom Size In Meters Philippinesboston University Tennis Division, 0 Days Validity Means In Airtel, Ammonia Remover Pond, Nicole Mitchell Murphy,, Albright College Division,	2021-12-06 19:08:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5175154209136963, "perplexity": 1693.6097350384698}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363309.86/warc/CC-MAIN-20211206163944-20211206193944-00078.warc.gz"}
https://www.acmicpc.net/problem/11434	시간 제한메모리 제한제출정답맞힌 사람정답 비율 1 초 256 MB53360.000% ## 문제 When you unite two countries, they will typically have their own versions of most things, like road signs, foods, etc. If you basically have one of the countries “impose” its version on the other, this may feel to the other more like an annexation than a unification of two equals.3 So you need to be sure to take little bits and pieces of both countries and build a whole that’s acceptable to everyone. For instance, East Germans really liked their own versions of pickles and ketchup. In some cases, both sides agree that one country’s version is preferable — for instance, everyone loves the East German “Ampelmännchen,” the traffic signal pedestrian walk sign. It gets harder when each side prefers their own version — in that case, you need compromises such as “We really care about our pickles, so let’s keep them, but we like your ketchup almost as much, so we’re fine with yours.” Here, you will figure out suitable compromises for combining two cultures. You will be given a list of contentious items, and for each of them how much each of the two countries like each version. You are also given how many people live in each of the two united countries. Your goal is to find out a choice for items that maximizes the total “happiness” of all people. That is the sum of the happinesses of everyone with all the items you select. For each item (e.g., “pickles”), you can only choose one of the two versions. 3Indeed, many East Germans felt that way about the way the unification was carried out, and hence in many cases took a long time to “identify” with the new country. ## 입력 The first line contains a number K ≥ 1, which is the number of input data sets in the file. This is followed by K data sets of the following form: The first line of each data set contains three integers n, W, E. 0 ≤ n ≤ 1000 is the number of contentious items. 0 ≤ W, E ≤ 10000 are the number of people living in the West and East. This is followed by n lines, each containing four integers 0 ≤ Lw,w, Lw,e, Le,w, Le,e ≤ 100. These are, in order: how much the West likes its own version, how much the West likes the East’s version, how much the East likes the West’s version, and how much the East likes its own version. ## 출력 For each data set, first output “Data Set x:” on a line by itself, where x is its number. Then, output the maximum total happiness that can be produced by choosing exactly one version of each product. Each data set should be followed by a blank line. ## 예제 입력 1 1 5 10 15 7 1 2 6 0 5 0 5 7 0 0 6 4 0 0 2 1 2 1 0 ## 예제 출력 1 Data Set 1: 380	2022-05-16 08:42:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17423787713050842, "perplexity": 1147.6192044042205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510097.3/warc/CC-MAIN-20220516073101-20220516103101-00502.warc.gz"}
http://www.reference.com/browse/recurse	Definitions # recurse Haskell is a standardized purely functional programming language with non-strict semantics, named after the logician Haskell Curry. ## History Following the release of Miranda by Research Software Ltd, in 1985, interest in lazy functional languages proliferated. By 1987, more than a dozen non-strict, purely functional programming languages existed. Of these Miranda was much the most widely used, but was not in the public domain. At the conference on Functional Programming Languages and Computer Architecture (FPCA '87) in Portland, Oregon, a meeting was held during which strong consensus was found among the participants that a committee should be formed to define an open standard for such languages. This would have the express purpose of consolidating the existing languages into a common one that would serve as a basis for future research in language design. The first version of Haskell ("Haskell 1.0") was defined in 1990. The committee's efforts resulted in a series of language definitions, which in late 1997, culminated in Haskell 98, intended to specify a stable, minimal, portable version of the language and an accompanying standard library for teaching, and as a base for future extensions. The committee expressly welcomed the creation of extensions and variants of Haskell 98 via adding and incorporating experimental features. In January 1999, the Haskell 98 language standard was originally published as "The Haskell 98 Report". In January 2003, a revised version was published as "Haskell 98 Language and Libraries: The Revised Report". The language continues to evolve rapidly, with the Hugs and GHC implementation (see below) representing the current de facto standard. In early 2006, the process of defining a successor to the Haskell 98 standard, informally named Haskell′ ("Haskell Prime"), was begun. This process is intended to produce a minor revision of Haskell 98. ## Features and extensions Characteristic features of Haskell include pattern matching, currying, list comprehensions , guards, definable operators, and single assignment. The language also supports recursive functions and algebraic data types, as well as lazy evaluation. Unique concepts include monads, and type classes. The combination of such features can make functions which would be difficult to write in a procedural programming language almost trivial to implement in Haskell. Several variants have been developed: parallelizable versions from MIT and Glasgow University, both called Parallel Haskell; more parallel and distributed versions called Distributed Haskell (formerly Goffin) and Eden; a speculatively evaluating version called Eager Haskell and several object oriented versions: Haskell++, O'Haskell and Mondrian. Concurrent Clean is a close relative of Haskell, whose biggest deviation from Haskell is in the use of uniqueness types for input instead of monads. ## Applications Haskell's strengths have been well applied to a few projects. Audrey Tang's Pugs is an implementation for the long-forthcoming Perl 6 language with an interpreter and compilers that proved useful already after just a few months of its writing; similarly, GHC is often a testbed for advanced functional programming features and optimizations. Darcs is a revision control system, with several innovative features. Linspire GNU/Linux chose Haskell for system tools development. Xmonad is a window manager for the X Window System, written entirely in Haskell. Bluespec SystemVerilog is a language for semiconductor design that is an extension of Haskell. Additionally, Bluespec, Inc.'s tools are implemented in Haskell. ## Examples A simple example that is often used to demonstrate the syntax of functional languages is the factorial function for non-negative integers, shown in Haskell: `factorial :: Integer -> Integer` `factorial 0 = 1` `factorial n \| n > 0 = n * factorial (n-1)` Or in one line: `factorial n = if n > 0 then n * factorial (n-1) else 1` This describes the factorial as a recursive function, with one terminating base case. It is similar to the descriptions of factorials found in mathematics textbooks. Much of Haskell code is similar to standard mathematical notation in facility and syntax. The first line of the factorial function describes the types of this function; while it is optional, it is considered to be good style to include it. It can be read as the function factorial (factorial) has type (::) from integer to integer (Integer -> Integer). That is, it takes an integer as an argument, and returns another integer. The type of a definition is inferred automatically if the programmer didn't supply a type annotation. The second line relies on pattern matching, an important feature of Haskell. Note that parameters of a function are not in parentheses but separated by spaces. When the function's argument is 0 (zero) it will return the integer 1 (one). For all other cases the third line is tried. This is the recursion, and executes the function again until the base case is reached. A guard protects the third line from negative numbers for which a factorial is undefined. Without the guard this function would recurse through all negative numbers without ever reaching the base case of 0. As it is, the pattern matching is not complete: if a negative integer is passed to the fac function as an argument, the program will fail with a runtime error. A final case could check for this error condition and print an appropriate error message instead. The "Prelude" is a number of small functions analogous to C's standard library. Using the Prelude and writing in the point-free style of unspecified arguments, it becomes: `fac = product . enumFromTo 1` The above is close to mathematical definitions such as $f = g circ h$ (see function composition) with the dot acting as the function composition operator, and indeed, it is not an assignment of a numeric value to a variable. In the Hugs interpreter, you often need to define the function and use it on the same line separated by a where or let..in, meaning you need to enter this to test the above examples and see the output 120: let { fac 0 = 1; fac n \| n > 0 = n * fac (n-1) } in fac 5 or `fac 5 where fac = product . enumFromTo 1` The GHCi interpreter doesn't have this restriction and function definitions can be entered on one line and referenced later. ### More complex examples A simple Reverse Polish Notation calculator expressed with the higher-order function `foldl` whose argument f is defined in a where clause using pattern matching and the type class Read: `calc :: String -> [Float]` `calc = foldl f [] . words` ` where` ` f (x:y:zs) "+" = y+x:zs` ` f (x:y:zs) "-" = y-x:zs` ` f (x:y:zs) "" = yx:zs` ` f (x:y:zs) "/" = y/x:zs` ` f xs y = read y : xs` The empty list is the initial state, and f interprets one word at a time, either matching two numbers from the head of the list and pushing the result back in, or parsing the word as a floating-point number and prepending it to the list. The following definition produces the list of Fibonacci numbers in linear time: `fibs = 0 : 1 : zipWith (+) fibs (tail fibs)` The infinite list is produced by corecursion — the latter values of the list are computed on demand starting from the initial two items 0 and 1. This kind of a definition relies on lazy evaluation, an important feature of Haskell programming. For an example of how the evaluation evolves, the following illustrates the values of fibs and tail fibs after the computation of six items and shows how zipWith (+) has produced four items and proceeds to produce the next item: `fibs = 0 : 1 : 1 : 2 : 3 : 5 : ...` ` + + + + + +` `tail fibs = 1 : 1 : 2 : 3 : 5 : ...` ` = = = = = =` `zipWith ... = 1 : 2 : 3 : 5 : 8 : ...` `fibs = 0 : 1 : 1 : 2 : 3 : 5 : 8 : ...` The same function, written using GHC's parallel list comprehension syntax (GHC extensions must be enabled using a special command-line flag '-fglasgow-exts'; see GHC's manual for more): `fibs = 0 : 1 : [a+b \| a <- fibs \| b <- tail fibs ]` The factorial we saw previously can be written as a sequence of functions: `fac n = (foldl (.) id [x -> xk \| k <- [1..n]]) 1` A remarkably concise function that returns the list of Hamming numbers in order: `hamming = 1 : map (2) hamming `merge` map (3) hamming `merge` map (5) hamming` ` where merge (x:xs) (y:ys)` ` \| x < y = x : xs `merge` (y:ys)` ` \| x > y = y : (x:xs) `merge` ys` ` \| otherwise = x : xs `merge` ys` Like the various `fibs` solutions displayed above, this uses corecursion to produce a list of numbers on demand, starting from the base case of 1 and building new items based on the preceding part of the list. In this case the producer `merge` is defined in a `where` clause and used as an operator by enclosing it in back-quotes. The branches of the guards define how `merge` merges two ascending lists into one ascending list without duplicate items. ## Criticism Jan-Willem Maessen, in 2002, and Simon Peyton Jones, in 2003, discussed problems associated with lazy evaluation while also acknowledging the theoretical motivation for it, in addition to purely practical considerations such as improved performance. They note that, in addition to adding some performance overhead, laziness makes it more difficult for programmers to reason about the performance of their code (specifically with regard to space usage). Bastiaan Heeren, Daan Leijen, and Arjan van IJzendoorn in 2003 also observed some stumbling blocks for Haskell learners, "The subtle syntax and sophisticated type system of Haskell are a double edged sword—highly appreciated by experienced programmers but also a source of frustration among beginners, since the generality of Haskell often leads to cryptic error messages." To address these, they developed an advanced interpreter called Helium which improved the user-friendliness of error messages by limiting the generality of some Haskell features, and in particular removing support for type classes. ## Implementations • The Glasgow Haskell Compiler compiles to native code on a number of different architectures—as well as to ANSI C—using C-- as an intermediate language. GHC is probably the most popular Haskell compiler, and there are quite a few useful libraries (e.g. bindings to OpenGL) that will work only with GHC. • Gofer was an educational dialect of Haskell, with a feature called "constructor classes", developed by Mark Jones. It was supplanted by Hugs (see below). • is another native-code Haskell compiler. It has not been actively developed for some time but is still usable. • Helium is a newer dialect of Haskell. The focus is on making it easy to learn by providing clearer error messages. It currently lacks typeclasses, rendering it incompatible with many Haskell programs. • Hugs, the Haskell User's Gofer System, is a bytecode interpreter. It offers fast compilation of programs and reasonable execution speed. It also comes with a simple graphics library. Hugs is good for people learning the basics of Haskell, but is by no means a "toy" implementation. It is the most portable and lightweight of the Haskell implementations. • is a Haskell compiler written by John Meacham emphasising speed and efficiency of generated programs as well as exploration of new program transformations. • is another bytecode compiler, but the bytecode runs significantly faster than with Hugs. Nhc98 focuses on minimizing memory usage, and is a particularly good choice for older, slower machines. • Yhc, the York Haskell Compiler is a fork of nhc98, with the goals of being simpler, more portable, more efficient, and integrating support for Hat, the Haskell tracer. It also features a JavaScript backend allowing users to run Haskell programs in a web browser	2014-07-26 15:37:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40049490332603455, "perplexity": 1664.4804680059108}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997902579.5/warc/CC-MAIN-20140722025822-00155-ip-10-33-131-23.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/3759467/combinatorics-with-relations	Combinatorics With Relations The twelvefold way offers a framework for counting functions, under various conditions which can be expressed as n-fold Cartesian Products of the function's domain, function, and codomain attributes. Using this twelvefold way table (which actually has $$16$$ entries) as an example, we could structure the various counting problems as: $$\{$$domain elements are distinguishable, domain elements are indistinguishable$$\} \times \{$$function is left unique, function is not left unique$$\} \times \{$$function is right total, function is not right total$$\} \times \{$$codomain elements are distinguishable, codomain elements are indistinguishable$$\}$$ The bijective cases are sometimes dropped, yielding the number $$12$$, but we'll keep them. Is it possible to relax the conditions that make the function a function, namely right uniqueness and left totality, and count general relations? The new structure of counting problems would be: $$\{$$domain elements are distinguishable, domain elements are indistinguishable$$\} \times \{$$relation is right unique, relation is not right unique$$\} \times \{$$relation is left total, relation is not left total$$\} \times \{$$relation is left unique, relation is not left unique$$\} \times \{$$relation is right total, relation is not right total$$\} \times \{$$codomain elements are distinguishable, codomain elements are indistinguishable$$\}$$ Are there attempts to collect formulas for and study these new cases where the relation may not be a function? If so, what is known about them? If not, is it because it has been shown that such cases have no applications? • Interesting question. I do not know of any such listings, but that just means it's at least relatively obscure. Maybe it would be best to pick a particular entry, compute some terms, and see if you get OEIS hits. Aug 19 '20 at 10:09 I haven't seen any attempts to do so, probably because there would be 64 entries in the table. Another reason they don't show up much is because we don't usually care much for uniqueness or totality in relation counting, we usually care more about stuff like symmetry, anti-symmetry, and transitivity. But, as I have nothing better to do, I'll give it a go here! So the first thing to notice that we can switch all the rights with lefts, and the formulas should hold, so we in fact only need 48 cases. I'll sort these into $$3$$ different $$4\times4$$ tables, one for each distinguishable/indistinguishable combo. This gives even more symmetry in the all distnguishable/indistingushable tables so we actually only require $$36$$ entries. Thus, a catchy title for this might be "The $$36$$-fold Way" Time to count relations between nonempty sets $$\|X\|=n$$ and $$\|Y\|=m$$ (subsets of $$X\times Y$$). First table, everything is distinguishable, and $$S(n,m)$$ are the Stirling Numbers of the second kind: $$\begin{array}{c\|cccc} \text{left}^{\large{\text{right}}} & \text{none} & \text{unique} & \text{total} & \text{both} \\ \hline \text{none} & 2^{nm} & \sum_{k=0}^m \binom{m}{k}n^k=(n+1)^m& (2^{n}-1)^m& n^m \\ \text{unique} &-& \sum_{k=0}^{\min(n,m)} \binom{m}{k}\frac{n!}{(n-k)!}&\sum_{k=m}^n \binom{n}{k}m!S(k,m) & m!\binom{n}{m} \\ \text{total} &-&-&\sum_{k=1}^{n}(-1)^{n-k}\binom{n}{k}(2^k-1)^m& n!S(m,n)\\ \text{both} &-&-&-& \begin{cases}n! & n=m\\0 & n\neq m \end{cases} \\ \end{array}$$ An alternative for the $$n!S(m,n)$$'s: we could use PIE to count this as $$\sum_{k=1}^n (-1)^{n-k}\binom{n}{k}k^m$$, but this would require a double summation, which I'd rather not have in my tables if I can help it. One application of this could be counting the number of ways $$n$$ people can participate in $$m$$ clubs. Then right totality for example would mean each club has at least one member. Warning: Past this point, stuff gets weird, as clubs or people become indistinguishable. It also gets hard to count, as there are many different symmetries which can make things indistingushable. Thus, I have not filled out most of these others currently, but I'll try to update these over time, and would appreciate help in the comments!: Left is distinguishable, right is not: $$\begin{array}{c\|cccc} \text{left}^{\large{\text{right}}}& \text{none} & \text{unique} & \text{total} & \text{both} \\ \hline \text{none}&&\sum_{k=0}^m \binom{k+n-1}{k}&&\binom{m+n-1}{m}\\ \text{unique}&&\sum_{k=0}^m\binom{n}{k}&&\binom{n}{m}\\ \text{total}&&\sum_{k=n}^m\binom{k-1}{k-n}&&\binom{m-1}{m-n} \\ \text{both}& \sum_{k=0}^{m}S(n,k)& \begin{cases} 1 & n\leq m\\ 0 & n < m \end{cases}& S(n,m)& \begin{cases} 1 & n=m \\ 0 & n\neq m \end{cases} \end{array}$$ And finally, everything is indistinguishable: $$\begin{array}{c\|cccc} \text{left}^{\large{\text{right}}}& \text{none} & \text{unique} & \text{total} & \text{both} \\ \hline \text{none}&&&&\sum_{k=1}^n \pi(m,k)\\ \text{unique}&-&\min(n,m)&\pi(n-m,m)&\begin{cases} 1 & n=m \\ 0 & n\neq m \end{cases}\\ \text{total}&-&-&&\pi(m,n)\\ \text{both}&-&-&-&\begin{cases} 1 & n=m \\ 0 & n\neq m \end{cases} \end{array}$$ Where $$\pi(m,n)$$ is the number of ways to partition $$m$$ into at $$n$$ (possibly empty!) parts.	2021-11-28 05:17:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 32, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7124840617179871, "perplexity": 324.06617633551343}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358469.34/warc/CC-MAIN-20211128043743-20211128073743-00459.warc.gz"}
https://study.com/academy/answer/calculating-annuity-cash-flows-if-you-put-up-20-000-today-in-exchange-for-a-8-percent-12-year-annuity-what-will-the-annual-cash-flow-be-calculating-annuity-values-your-company-will-generate-50.html	# Calculating Annuity Cash Flows. If you put up $20,000 today in exchange for a 8 percent, 12-year... ## Question: Calculating Annuity Cash Flows. If you put up$20,000 today in exchange for a 8 percent, 12-year annuity, What will the annual cash flow be? Calculating Annuity Values. Your company will generate $50,000 in cash flow each year for the next nine years from a new information database. The computer system needed to set up the database costs$300,000. If you can borrow the money to buy the computer system at 8 percent annual interest, can you afford the new system? ## Present Value of an Annuity: An annuity consists of payments of equal amount over a finite period of time. The present value of an annuity is proportional to the periodic payment, and the proportionality depends on the interest rate and the duration of the annuity. Question 1 The annual cash flow is such that the present value of the cash flow is equal to 20,000 today. The annual cash flow is given by: • {eq}\displaystyle \frac{20,000*8\%}{1 - (1 + 8\%)^{-12}} = 2653.90 {/eq} Question 2 You can afford the system if the discounted present value of the cash flows generated from the database is higher than the cost of the system. We can use the following formula to compute the present value of an annuity with periodic payment {eq}M {/eq} for {eq}T{/eq} periods, given periodic return {eq}r{/eq}: • {eq}\displaystyle \frac{M(1 - (1 + r)^{-T})}{r} {/eq} Applying the formula, the present value of the cash flows is: • {eq}\displaystyle \frac{50,000(1 - (1 + 8\%)^{-9})}{8\%} = 312,344.40 {/eq} Since the present value of the cash inflows is higher than the cost of \$300,000, therefore you can afford the system.	2019-10-21 14:47:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3669424057006836, "perplexity": 830.819189417809}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987779528.82/warc/CC-MAIN-20191021143945-20191021171445-00290.warc.gz"}
http://math.stackexchange.com/questions/186553/test-the-convergence-of-sum-n-0-infty-fracnk1nk-k/186556	Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$ Problem: Test the convergence of $\sum_{n=0}^{\infty} \frac{n^{k+1}}{n^k + k}$, where $k$ is a positive constant. I'm stumped. I've tried to apply several different convergence tests, but still can't figure this one out. - Ugh, I mistyped the series. Should I ask a new question? It should've been $\sum_{n=0}^\infty \frac{n^{k-1}}{n^k+k}$. – Damir Aug 25 '12 at 8:55 Use the theorem "If a series $\sum_{n=0}^{\infty} a_n$ converges, then $\lim_{n\rightarrow \infty} a_n =0$". – Mhenni Benghorbal Aug 25 '12 at 11:34 $$\frac{n^{k+1}}{n^k +k} =n \frac{1}{1+\frac{k}{n^k}}$$ What happens when $n \to \infty$? $$\frac{n^{k+1}}{n^k+k}\geq\frac{n^{k+1}}{2n^k}=\frac{1}{2}n\xrightarrow [n\to\infty]{}\infty\neq 0$$	2015-05-23 01:53:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7728750705718994, "perplexity": 973.7597383846742}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207926964.7/warc/CC-MAIN-20150521113206-00342-ip-10-180-206-219.ec2.internal.warc.gz"}
http://ycgc.vitaform-nature.fr/how-to-plot-hyperboloid-in-matlab.html	## How To Plot Hyperboloid In Matlab If we use a relatively small grid to plot our surface, we may get a picture that's not as smooth as what can be done by tools such as surf. In the picture below, the standard hyperbola is depicted in red, while the point for various values of the parameter t is pictured in blue. If AxesOrigin is not set, the origin of the coordinate system is used as the default cross point. For objects ( like point, line, plane and sphere) distances and intersections are calculated. In the following interactive, you can vary parameters to produce the conics we learned about in this chapter. 0 TSpectrum3D is a component (and ActiveX) for use in Win32 (9x/ME/2K/XP/Vista) software with BASS with a purpose to provide 3D audio FFT spectrum display for audio streams. The cross point is set by the attribute AxesOrigin. (Note: fractional exponents sometimes cause problems in MATLAB and other similar software packages, Parametrize the hyperboloid and plot it. Origin: The axes are displayed as a coordinate cross. Equation of standard ellipsoid body in xyz coordinate system is, where a - radius along x axis, b - radius along y axis, c - radius along z axis. x-coordinates of the M sample points (x[i], y[i]). Select a Web Site. Toggle Main Navigation. Learn more about surface, 3d plots, parameters, plotting, parametric equation. In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI. How to draw a Circle in Matlab This also works in GNU-Octave, FreeMat, Scilab and Scicoslab 2. The analytical investigation demonstrated that the kinematic coupling behaviour can be well approximated by a hyperboloid surface. However aesthetics are constantly changing. The Drawing Frame utility was built with the aid of the Java programming language and can run on multiple platforms. Hello, I'm looking for a calculator with CAS for my electrical engineering studies and just tested the Prime emulator to get a first impresseion. (b) Use the parametric equations in part (a) to graph the hyperboloid Posted 5 years ago. I wrote a MATLAB script that summed the total intensity of each frame and plot it over time. To plot the graph of a plane in Mathematica, the simplest approach is to plot its equation as that of a (simple flat) surface with equation z = f(x, y). Examples of this might be the curvature of a lens, the design of a custom spring, wind-foil. The following shows how the six hyperbolic functions are realized in Mathematica. Section 1-4 : Quadric Surfaces. gif illustrates continuing to twist the top, and the resulting surfaces. surf(X,Y,Z) creates a three-dimensional surface plot, which is a three-dimensional surface that has solid edge colors and solid face colors. Use MATLAB® live scripts instead. How to Draw a Hyperboloid. Find the volume of the solid region bounded above by the sphere x^2+y^2+z^2=9 nd below by the cone z^2=x^2+y^2? Multivariable calculus. Origin: The axes are displayed as a coordinate cross. The function to plot must be placed inside curly brackets. It supports line plots, scatter plots, piecewise constant plots, bar plots, area plots, mesh{ and surface plots, patch plots, contour plots, quiver plots, histogram plots, box plots, polar axes,. Sine function in pgfplots and MATLAB. Find the center, vertices, foci, eccentricity, and asymptotes of the hyperbola with the given equation, and sketch: Since the y part of the equation is added, then the center, foci, and vertices will be above and below the center (on a line paralleling the y-axis), rather than side by side. We know that the implicit equation for the unit circle is the following:We can convert that into a parametric form, and then draw it using the techniques we learned earlier. There are five different surfaces to choose from: Cylinder, Cone, Hyperboloid and two twisted spiral planes. Note that when plotting confidence ellipses for data, the ellipse-axes are usually scaled to have length = square-root of the corresponding eigenvalues, and this is what the Cholesky decomposition gives. Choose a web site to get translated content where available and see local events and offers. The one-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the perpendicular bisector to the line between the foci, while the two-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the line joining the foci (Hilbert and Cohn-Vossen 1991, p. MATLAB Plot Gallery Understand how to use each of these plots in and out. That is a 1D situation. Since you're trying to plot an implicit equation,. Cauchy’s root test by plotting nth roots. presented to the University of Waterloo. This method requires little experience in programming, so dive in with step one to get started. double integral gives us the volume under the surface z = f(x,y), just as a single integral gives the area under a curve. We'll plot the sphere using implicitplot3d and the plane using plot3d. MATLAB Central gives you support and solutions from over 100,000 community members and MathWorks employees. hi, I want to plot an ellipsoid in matlab given my equation as v=ax. Results signify that faults were differentiated from power swing using differential impedance angle. SphereContour C, Pascal, and MATLAB programs for automatic contouring of spherical orientation data using a modified Kamb method (Vollmer, 1995). In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI. Note: It's recommended as a good practice to indent the code - see the second plot in the example above - and to add a comma , at the end of each option passed to \addplot. How to draw a Circle in Matlab This also works in GNU-Octave, FreeMat, Scilab and Scicoslab 2. HyperContour MATLAB/Octave script for contouring fabric and finite strain data on the unit hyperboloid (Vollmer, 2018). e position of the samples has been normalized within each unit using a lin ear correlation and all results have been plotted using dierent colors for each F. I’ve just showed you how MATLAB and YALMIP can be used to not only find but also plot separating hyperplanes and quadric surfaces for data classification. This paper presents a novel approach for indoor acoustic source localization using sensor arrays. the plot is made using the current 3D scaling (set by a previous call to param3d, plot3d, contour or plot3d1). Finer Points of Plotting with MATLAB. 4 Parametric Surfaces in Matlab,. As a hyperboloid is a two-dimensional object, it requires two parameters. Mathematica tutorial on plotting a hyperbolic paraboloid. contour for contour plots, plt. 30 - matlab style python plotting Message-ID: matplotlib is a 2D plotting package for python with a matlab compatible syntax and output tested under linux and windows platforms. mimetypeMETA-INF/container. Please try again later. r is referred to as “radius,” ϕ as “polar angle,” and z as the “height” of a point. Find the hyperbola in r and z; form a surface of revolution by angle θ about the z axis. Spectrum 3D v. The learning objectives of this. Learn more about surf, plotting, elipsoid, mesh, plot, 3d plots. parametric_plot3d (f, urange, vrange=None, plot_points='automatic', boundary_style=None, kwds) ¶ Return a parametric three-dimensional space curve or surface. We study the inextensible flows of curves in 3-dimensional Euclidean space. MATLAB/Octave script for contouring geological fabric and finite strain data on the unit hyperboloid, including Rf/Phi and Elliott polar plots. Dobrushkin,Lippitt Hall 202C, 874-5095,[email protected] Origin: The axes are displayed as a coordinate cross. Choose a web site to get translated content where available and see local events and offers. In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI. the plot is made using the current 3D scaling (set by a previous call to param3d, plot3d, contour or plot3d1). Hyperbolic paraboloid definition is - a saddle-shaped quadric surface whose sections by planes parallel to one coordinate plane are hyperbolas while those sections by planes parallel to the other two are parabolas if proper orientation of the coordinate axes is assumed. Thus by measuring the total intensity, we should have a proxy for the phase of the dreidel, each intensity peak being a quarter-turn, and can investigate how that evolves over time. Quadric Surfaces - Traces Traces are cross sections parallel to a plane. We begin with a brief discussion of how MATLAB does its plotting. Description of the elliptic paraboloid with interactive graphics that illustrate cross sections and the effect of changing parameters. If AxesOrigin is not set, the origin of the coordinate system is used as the default cross point. A horizontal or vertical line that extends from the horizontal or vertical axis through the plot area to guide the readers eyes across the chart to identify values. An elliptic hyperboloid of two sheets is a quadratic surface given by. There are three Matplotlib functions that can be helpful for this task: plt. In this position, the hyperbolic paraboloid opens downward along the x-axis and upward along the y-axis (that is, the parabola in the plane x = 0 opens upward and the parabola in. Problem 1 (a) Find a parametrization of the hyperboloid of one sheet given by z2 = x2 + y2 1; 2 z 2 (use cylindrical coordinates). It handles vectors, matrices, complex numbers, coordinates, regular polygons and intersections. HypArr, software for modeling and visualizing convex polyhedra and plane arrangements, now seems to be incorporated as a module in a larger Matlab library for multi-parametric analysis. b) Membrane analysis of shells of translation, Circular cylinder, Diretrix, Parabola, Cycloid, Catenary and Membrane deformations. To calculate the natural logarithm of a scalar, vector or array, A, enter log(A). Take a unit sphere for example, the equation is x^2+y^2+z^2=1; If you carefully set the mesh grid for x and y, then you can calculate the corresponding value for z. Your browser doesn't support HTML5 canvas. Just type the commands, not the comments, which are only there as a kind of explanation of what you should see. edu 2Madison Area Technical College Madison, WI [email protected] Hoe maak je een Hyperboloid in MATLAB Plot Het visualiseren van de vorm van een hyperboloïde is makkelijker als je het kan plotten met behulp van MATLAB. The part of the paraboloid z = 9¡x2 ¡y2 that lies above the x¡y plane must satisfy z = 9¡x2 ¡y2 ‚ 0. It supports line plots, scatter plots, piecewise constant plots, bar plots, area plots, mesh{ and surface plots, patch plots, contour plots, quiver plots, histogram plots, box plots, polar axes,. Daina Taimina's technique for crocheting yarn into hyperbolic surfaces forms the basis for an exhibit. How can I draw Quadric Surfaces using MATLAB? Look through the examples and example code in the "specialized plotting". To plot a plane with equation in the form (1), where the coefficient c of z is not 0, solve (1) for the variable z:. I think pandas by default creates a new plot instead of using the 'active' plot. 12 Comments on "Arc length of a spiral around a paraboloid" R. The plot may be contoured, however, it is desirable to have a contouring method that is rapid, reproducible, and based on the underlying geometry of the data. In this section we want to look at an application of derivatives for vector functions. Then let's plot the sphere and -plane together in the same plot. 5 Building Peaks and Pits The family of curves y = e−kx2 looks like a series of "bumps" of height 1, all centered at x = 0. System Design and Simulation. r = {ucos{v}, u^2,5usin{v}} I understand that I need to make a meshgrid from u and v, but what to do next?. We assume that you already know how to use MATLAB’s plotting commands to graph plane curves of the form y = f (x). imshow for showing images. There are no x’s in the equation yz = 1. Hint: You may want to create a list of the polynomials first. A hyperbola is a type of conic section that looks somewhat like a letter x. There will be two lines of curvature through each non-umbilic point and the lines will cross at right angles. The main purpose of this paper is constructing and plotting the surfaces that are generated from the motion of inextensible curves in. Around the time of the 1. Now, instead of a cube, consider flat empty 5-space such as $\Sigma^\pm$ where general relativity in the absence of 5D matter-energy leads to the desired 4D dynamics through the Kaluza-Klein (KK) metric\footnote{In the present paper we will use the Greek letter $\chi$ for the 5D coordinates where we have used $\xi$ previously. Since you're trying to plot an implicit equation,. How is it I could do it? Then you can use surf() to plot it. Use ezmesh to plot this surface. As an alternative to Axes = None, you may also “switch the axes off” by setting AxesVisible = FALSE in the plot command or via the interactive object inspector (see Viewer, Browser, and Inspector: Interactive Manipulation in this document). Using pgfplots, I'm trying to create a quiver plot on a hyperboloid surface something like this. The calculator can do statistics, best fits, function plotting, integration. Each value of t determines a point x, y, which we can plot in a coordinate plane. Back in 1996, Alexander Bogomolny started making the internet math-friendly by creating thousands of images, pages, and programs for this website, right up to his last update on July 6, 2018. com, a free online graphing calculator. MATLAB allows you to display your plots however you choose. Daniel Davis – 6 September 2014. My hope is to later use this as part of an optimization routine for the wing design. 0 006 m d 007 cr un\|\|\|\|\|\|\|\|\| 008 041216s2001 caua sb 001 0 eng d 040 _aEEICP _cEEICP 029 1 _aNZ1 _b1053395. Its slices for constant y are circles and for constant x,z are hyperbolas. This example shows Chebfun2 being used to represent parameterized surfaces. MATLAB has the sphere() function. “The application plotmtv is a fast multi-purpose plotting program for visualization of scientific data in an X11-window environment. To convert a MuPAD notebook file to a MATLAB live script file, see convertMuPADNotebook. ⇤ I can name the 6 quadric surfaces, write their equation, and sketch their graph. Fulford, Mathematical Modeling with Case Studies, A Differential Equation Approach using Maple and Matlab, 2nd Ed. Actually, there are a couple of applications, but they all come back to needing the first one. Just type the commands, not the comments, which are only there as a kind of explanation of what you should see. System Design and Simulation. Additional files include example data files used in the paper. Any help would be greatly appreciated. Around the time of the 1. x-coordinates of the M sample points (x[i], y[i]). contourf for filled contour plots, and plt. In this section we want to look at an application of derivatives for vector functions. For now I have set the domain to [-5,5], but this yields ugly plots, see below. 5 Building Peaks and Pits The family of curves y = e−kx2 looks like a series of "bumps" of height 1, all centered at x = 0. Inside the sphere x^2 + y^2 + z^2 = 36 and outside the cylinder x^2 + y^2 = 4. Take a unit sphere for example, the equation is x^2+y^2+z^2=1; If you carefully set the mesh grid for x and y, then you can calculate the corresponding value for z. (ii) Plot the Hyperboloid of two sheets x2 + y 2 − z 2 = −1. Today, we discussed the basics of polar coordinates and plotting polar curves. How to plot a surface in its parametric form ?. MuPAD ® notebooks will be removed in a future release. I'm a beginner at Matlab, so I don't have much experience. Rodrigo Platte, Feb 2013. 4, we learned how to make measurements along curves for scalar and vector ﬁelds by using. Jayaram says: 13 Mar 2015 at 9:54 am [Comment permalink] A very useful and interesting article for practical applications. parametric_plot3d (f, urange, vrange=None, plot_points='automatic', boundary_style=None, kwds) ¶ Return a parametric three-dimensional space curve or surface. Brug MATLAB bygget s - in " ezsurf "-funktionen til hurtigt at afbilde din hyperboloid. Use your calculator or MatLab to plot a few, for k = 1,2,3,4 and −2 ≤ x ≤ 2, on the same set of axes. Select a Web Site. The first shows such a hyperboloid, along with the fibers of the “projection”. MATLAB Central contributions by Yury. Select a Web Site. As with the ellipse the focus is at the point and the directrix is the line. If AxesOrigin is not set, the origin of the coordinate system is used as the default cross point. Description of the elliptic paraboloid with interactive graphics that illustrate cross sections and the effect of changing parameters. We begin with a brief discussion of how MATLAB does its plotting. Now lets turn our attention to implicit curves. Explore the graphs, domains and ranges of the hyperbolic functions. This type of surface is called a hyperboloid of one sheet. Sine function in pgfplots and MATLAB. y+e=0; where a,b,c,d,e are constants acting as coefficients of the equation. Choose a web site to get translated content where available and see local events and offers. Learn more about hyperboloid. Introduction to plotting z = f(x,y) This feature is not available right now. Introduction to adaptive ﬁlters This chapter introduces the fundamental principles of adaptive ﬁltering and commonly used adaptive ﬁlter structures and algorithms. 2 Forbes QCon and QBsf (axially symmetric aspheres represented by Forbes polynomials) Super-Gaussians Superposition (formed from a superposition of offset and rotated super-gaussians) Zernike surface (the first 66 Zernike polynomial terms plus a base conic and standard aspheric terms) User-defined surface (BASIC script) Sampled Mesh surface (Rectilinear mesh of sampled sag values with bicubic. org are unblocked. Viewed 3k times 2. There was initially a problem with this method of analysis, however. Abstract: The work developed in this document has as principal objective, the creation of a GUI under Matlab of a sound event detection, and de software codification of it. 0 release, some three-dimensional plotting utilities were built on top of Matplotlib's two-dimensional display, and the result is a convenient (if somewhat limited) set of tools for three-dimensional data visualization. (zero exp with matlab). contourf for filled contour plots, and plt. Matlab ignores any line that begins with % Matlab has many built in matrix functions and operators. Thus by measuring the total intensity, we should have a proxy for the phase of the dreidel, each intensity peak being a quarter-turn, and can investigate how that evolves over time. Magic of SOLIDWORKS Equation Driven Curves. Figure 1: Paraboloid. To get there type helpwin, when the windows opens select search, and search for. The most sophisticated and comprehensive graphing calculator online. Most SOLIDWORKS users are comfortable creating models with the familiar sketch commands (lines, rectangles, splines, arcs, etc. Select a Web Site. The part of the paraboloid z = 9¡x2 ¡y2 that lies above the x¡y plane must satisfy z = 9¡x2 ¡y2 ‚ 0. 4 Parametric Surfaces in Matlab,. legend A key that identifies the color, gradient, picture, texture, or pattern assigned to each data series in a chart. of revolution, Torus, Hyperboloid of revolution of one sheet, Shells of uniform strength membrane deformation. NEEDD HELP!! parameterizing a surface given by an implicit function (HYPERBOLOID?!) 1. If you run matlab with my code you come close to what I'm looking for as far as the electric field lines and equipotential surfaces. A common graphical strain analysis technique is to plot final ellipse ratios, R f, versus orientations, ϕ f on polar Elliott or R f / ϕ plots to examine the density distribution. That is possible unless the variable z does not appear in the equation (1). MATLAB Lecture 5. The following shows how the six hyperbolic functions are realized in Mathematica. 12 Comments on "Arc length of a spiral around a paraboloid" R. Parametric Plots¶ sage. (ii) Plot the Hyperboloid of two sheets x2 + y 2 − z 2 = −1. First examples. The other type is the hyperboloid of two sheets, and it is illustrated by the graph of x 2 - y 2 - z 2 = 1, shown below. py, which is not the most recent version. r is referred to as “radius,” ϕ as “polar angle,” and z as the “height” of a point. The 3D function plot gallery is a collection of 3D Parametric Function Plots which can be plotted in Origin. Chapter 12: Functions of Several Variables. p-the name of the hyperbola. These directions are actually the directions in which the data varies the most, and are defined by the covariance matrix. I’ve just showed you how MATLAB and YALMIP can be used to not only find but also plot separating hyperplanes and quadric surfaces for data classification. web; books; video; audio; software; images; Toggle navigation. contourf fills intervals that are closed at the top; that is, for boundaries z1 and z2, the filled region is: Contour plot of irregularly spaced data. If in the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$, (or -1) then one possibility (borrowed from cylindrical coordinates) is to retain [math]. The proposed solution starts by defining a generative model, designed to explain the acoustic power maps obtained by Steered Response Power (SRP) strategies. Belinda Barnes and Glenn R. Online 3-D Function Grapher Home Physics Tools Mathematical Tools Online 3-D Function Grapher A standalone application version of this 3-D Function Graphing Program, written in Flash Actionscript, much faster, essentially more capabilities, built-in function calculator and many more. SPIE Digital Library Proceedings. I'm a beginner at Matlab, so I don't have much experience. Before learning how to graph a hyperbola from its equation, get familiar with the vocabulary words and diagrams below. Plotting multivariable functions with dependent variables in domain. Basic concepts of applying adaptive ﬁlters in practical applications are also highlighted. A hyperbola is a conic section defined as the locus of all points in the plane such as the difference of whose distances from two fixed points , (foci) is a given positive constant and. I would print the exam-. the number of pipes - or wires - that fits within a conduit or similar applications; Input the rectangle inside dimensions - height and width and the circles outside diameters. ContourPlot3D treats the variables x, y, and z as local, effectively using Block. System Design and Simulation. The function plots the values in matrix Z as heights above a grid in the x-y plane defined by X and Y. Plot hyperboloid of one sheet The equation of the shape we'll graph is x2 + y2 = 0:003z2 + 0:05: In this case, you can see that at each value of z we have x2 + y2 = #. Din funktion hedder " hyp. (a) (15 pts) The part of the paraboloid z = 9 ¡ x2 ¡ y2 that lies above the x¡y plane. of revolution, Torus, Hyperboloid of revolution of one sheet, Shells of uniform strength membrane deformation. How can I draw Quadric Surfaces using MATLAB? Given the equation/s, how do I draw ellipsoids, elliptic hyperboloids, elleptic paraboloids and hyperbolic paraboloids using MATLAB? Look through the examples and example code in the "specialized plotting". The spirals are rose plots, which are also known as Coxcomb plots (invented by Florence Nightingale!). I think pandas by default creates a new plot instead of using the 'active' plot. TRAJECTORIES AND ORBITS. 31) “Interface Development and Modification of CBS3DS Radar Cross Section Analysis Tool Using FEMAP Finite Element Modeler and MATLAB Plotting. Solution to Problem Set #9 1. Select a Web Site. See more solutions Panel Navigation. The natural logarithm function in MATLAB is log(). Since the majority of retail pro-sound speakers even in the over \$2k range don't provide on-axis plots, let alone polar plots, I'm not going to lose sleep over it. BTRT HPRBL PTN LTN TTR TH TD FR FR TTN b rr Ptlr Th th prnt dtld drvtn f t f tn ndd t lt th thr dnnl ptn f bl vn th ltn f fr fxd ttn (l lbl ptnn t (P tllt r b ttn n ll nd th nl t f rrvl. Search the leading research in optics and photonics applied research from SPIE journals, conference proceedings and presentations, and eBooks. But, as mentioned above, we are able to take advantage of the cross-platform capabilities of the Java language, thereby obtaining figures that can be rotated much more readily. Now lets turn our attention to implicit curves. We study the inextensible flows of curves in 3-dimensional Euclidean space. Matlab can either operate numerically or symbolically. Smaller Circles within a Larger Circle Estimate the number of small circles that fits into an outer larger circle - ex. ^2; plot3(x,y,z); To produce exactly what I've graphed above you need to change various appearences:. Tried to add the link to the source term link coordinates which turned out unfortunately not possible, this means that a code might have to written then read into ANSYS-CFX in. this into the equation for the hyperboloid gives 2z = 1, and there are no solutions for z, so there are no points on the hyperboloid for which rFis parallel to n 1. Also I want to observe different ellipse by whose combination ellipsoid is made of. Tutorial for Mathematica & Wolfram Language. Capturing the axes and passing it to the next plotting command works fine for me, and is the way to go if you want to reuse your axes. Find the center, vertices, foci, eccentricity, and asymptotes of the hyperbola with the given equation, and sketch: Since the y part of the equation is added, then the center, foci, and vertices will be above and below the center (on a line paralleling the y-axis), rather than side by side. In this course we will use Mathematica computer algebra system (CAS), which is available in computer labs at URI. polyarea() is somethng you would apply to the array of vertices. Chapter 12: Functions of Several Variables. … Read more about Hyperbolic Curve Fitting in Excel. The proposed solution starts by defining a generative model, designed to explain the acoustic power maps obtained by Steered Response Power (SRP) strategies. Since the formula to find the volume of a cone applies to all cones, including oblique cone, we can use the formula V = 1/3(π×r 2 ×h) Example #3: Find the volume of an oblique cone with a diameter of 12 ft and a height of 15 ft. Ask Question Asked 5 years, 11 months ago. % This Function solves a bilateral matrix quadratic equation % of the form AX+XB+XCX+D = 0 for. I'm a beginner at Matlab, so I don't have much experience. etry, I would prepare a picture of a hyperboloid of one sheet. MATLAB Central contributions by Yury. Multivariate Calculus; Fall 2013 S. … Read more about Hyperbolic Curve Fitting in Excel. Based on your location, we recommend that you select:. How to plot a Circle in Matlab 1. Use ezmesh to plot this surface. Find a parametrization for Sand plot this surface using ezmesh. Is there a standard name for it, the way we have 'hyperboloid of one sheet' for example? Stack Exchange Network Stack Exchange network consists of 175 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. F = s2x 3 - y 3. Thus x2 +y2 • 9. Reload the page to see its updated state. Find a parametrization of the hyperboloid of one sheet given by (use cylindrical coordinates). In the applet above, drag one of the four orange dots around the ellipse to resize it, and note how the. The governing partial differential equations are transformed into a set of non-linear equations and solved numerically using an efficient numerical shooting technique with a fourth-order Runge–Kutta scheme (MATLAB package). That means that if some y0 and z0 obey y0z0 = 1, then. If AxesOrigin is not set, the origin of the coordinate system is used as the default cross point. Download Wolfram Player to view and interact with Wolfram Notebooks. Hello, I'm very new to Matlab and just finished learning how to use Matlab to perform basic calculations (vector, matrices etc) However, the main reason I bought Matlab was to plot 2D and 3D plots found in my calculus homework. As a hyperboloid is a two-dimensional object, it requires two parameters. Short answer: use MATLAB's built-in cylinder function: [code][X,Y,Z] = cylinder; surf(X,Y,Z) [/code]of course you don't have much control of what the cylinder looks like. how to draw a hyperboloid?. The role of multiaxial loading on the failure of Kevlar® KM2 ballistic fibers during transverse loading is not well understood. Do not show again. Based on your location, we recommend that you select:. Plot level surfaces where f(x,y,z) = x 2 - y 2 - z 2 is equal to 1 and. There was initially a problem with this method of analysis, however. hi, I want to plot an ellipsoid in matlab given my equation as v=a*x. COMSOL Based Multiphysics Analysis of Surface Roughness Effects on Capacitance in RF MEMS Varactors D. We start by plotting two simple quadratic surfaces that are commonly taught in multivariate calculus: a cone and a Hyperboloid of one sheet. See our Version 4 Migration Guide for information about how to upgrade. Unlike other graphing software, we have an enormous library of mathematical functions at our ﬁngertips. Parametric Plots¶ sage. How can I plot a hyperbola from its complex representation? Ask Question Asked 6 years, 6 months ago. S See full answer below. For objects ( like point, line, plane and sphere) distances and intersections are calculated. Additional files include example data files used in the paper. Plot continuous, discrete, surface, and volume data. You can plot the previous inputed equation against the time function, this would verfiy for you if the inputed formula is working and no bug issues will occur later. polyarea() with a single vertex would tell you that you had area 0. In the semester-end examination there will be two groups of questions. It supports line plots, scatter plots, piecewise constant plots, bar plots, area plots, mesh{ and surface plots, patch plots, contour plots, quiver plots, histogram plots, box plots, polar axes,. The proposed solution starts by defining a generative model, designed to explain the acoustic power maps obtained by Steered Response Power (SRP) strategies. This section looks at several examples of using these. As almost every CAS or graphic calculator, the Prime isn't allowed in most exams in. The final. System Design and Simulation. 2 A Maple Implicit Plot Example an ellipsoid can look like the top part of a hyperboloid of two sheets or like a. Using pgfplots, I'm trying to create a quiver plot on a hyperboloid surface something like this. If we use a relatively small grid to plot our surface, we may get a picture that's not as smooth as what can be done by tools such as surf. Quadric Surfaces in Matlab In this activity we will use cylindrical coordinate transformations to assist in the plotting of quadric surfaces , the three-dimensional analogue of the conic sections. Mukherjee1, C. Easily create polar plots. MuPAD ® notebooks will be removed in a future release. [x,y,z]=sphere; surf(x,y,z) 0 Comments. Note that half of the points have negative $$r$$ values and hence are being plotted on the ray opposite to the ray specified by θ. This question is off-topic. Things and Stuff Wiki - An organically evolving personal wiki knowledge base with an on-the-fly taxonomy containing a patchwork of topic outlines, descriptions, notes and breadcrumbs, with links to sites, systems, software, manuals, organisations, people, articles, guides, slides, papers, books, comments, videos, screencasts, webcasts, scratchpads and more. As an alternative to Axes = None, you may also “switch the axes off” by setting AxesVisible = FALSE in the plot command or via the interactive object inspector (see Viewer, Browser, and Inspector: Interactive Manipulation in this document). Get the free "Contour Plot" widget for your website, blog, Wordpress, Blogger, or iGoogle. How can I plot an elliptical paraboloid in MATLAB with surf() function, using parametric equations with 2 variables u and v?The equation looks like. They are defined below. Show off your favorite photos and videos to the world, securely and privately show content to your friends and family, or blog the photos and videos you take with a cameraphone. Origin: The axes are displayed as a coordinate cross. Note that when plotting confidence ellipses for data, the ellipse-axes are usually scaled to have length = square-root of the corresponding eigenvalues, and this is what the Cholesky decomposition gives. This example shows Chebfun2 being used to represent parameterized surfaces. Online 3-D Function Grapher Home Physics Tools Mathematical Tools Online 3-D Function Grapher A standalone application version of this 3-D Function Graphing Program, written in Flash Actionscript, much faster, essentially more capabilities, built-in function calculator and many more. A horizontal or vertical line that extends from the horizontal or vertical axis through the plot area to guide the readers eyes across the chart to identify values. I need to draw a hyperboloid in matlab. Introduction to plotting z = f(x,y) This feature is not available right now. In addition, we are able to exhibit the beautiful structure of the spectrum and the close links between the eigenfunctions, the rays of geometrical optics, and the geometry of the damping region. In a world where architects are. Quasi-static experiments reported by Hudspeth M, Li D, Spatola J, et al. Let S be the surface of revolution obtained by revolving about the x axis the graph of y = cosx for -pi/2 x pi/2. Streamline, pathline, streakline and timeline form convenient tools to describe a flow and visualise it. 0 006 m d 007 cr un\|\|\|\|\|\|\|\|\| 008 041216s2001 caua sb 001 0 eng d 040 _aEEICP _cEEICP 029 1 _aNZ1 _b1053395. De technische informatica taal MATLAB wordt gebruikt voor data verwerking en analyse in techniek en wetenschap - gerelateerde toepassingen.	2020-04-02 12:09:16	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5345740914344788, "perplexity": 1380.591065615874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506959.34/warc/CC-MAIN-20200402111815-20200402141815-00495.warc.gz"}
http://49ersofficialauthenticshop.com/2019/11/06/solar-and-wind-energy-enhances-drought-resilience-and-groundwater-sustainability/	Solar and wind energy enhances drought resilience and groundwater sustainability SWE enhances groundwater sustainability California recently endured a record-breaking drought after 2012 (refs. 20,21), which significantly impacted food production22, reduced hydropower generation23 and caused severe environmental issues (e.g., groundwater depletion, wildfires, tree mortality, land subsidence). As the largest agricultural producing state in the USA, California earned ~\$47 billion from its agricultural sector and contributed to 13% of the US total in 2015 even during the drought. The maintenance of crop revenue and overall resilience of the agricultural sector largely relied on the unsustainable groundwater overdraft, which effectively offset the drought impact, but contributed to severe groundwater depletion ($$\sim\!\!3.7\ {{\rm{km}}}^{3}$$/year24). In the energy sector, during this driest year of the drought, decreased surface water availability sent the in-state hydropower generation plunging to 7% of the total electricity generated, substantially below the state’s long-term average of around 18%23. This power deficit was offset by electricity generated through the rapidly growing solar and wind fleet, as well as from increased use of natural gas and electricity purchased from out-of-state sources23. Furthermore, for the first time, in 2012, solar and wind electricity generation exceeded hydropower in California23 due to the declining cost of wind turbines and solar photovoltaic (PV) in conjunction with the popularity and stringency of the Renewables Portfolio Standard (RPS), which mandates a certain proportion of renewables in the energy production. The penetration of SWE not only offset some of the decreases in hydropower but has implications beyond the energy sector given the inextricable links among food, energy and water. This added value can be derived by considering the sustainability trade-offs within the WFE nexus. In general, there is a direct trade-off between hydroelectricity production and irrigation of crops in how surface water is allocated between the two. There is also an indirect trade-off between hydroelectricity production and groundwater abstraction, as groundwater can substitute for reduced surface water availability during a drought, which in the case of the recent California drought allowed crop production to generally be unaffected. Given relatively low groundwater recharge rates and increasing risk of drought, this indirect trade-off highlights potential sustainability challenges for groundwater. Contect us: tha88	2019-11-21 00:37:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31002697348594666, "perplexity": 7498.83491872126}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670643.58/warc/CC-MAIN-20191121000300-20191121024300-00078.warc.gz"}
https://love2d.org/forums/viewtopic.php?f=4&t=8690&p=53499	## Can't make a bullet Questions about the LÖVE API, installing LÖVE and other support related questions go here. Forum rules Prole Posts: 4 Joined: Tue Apr 03, 2012 7:29 am ### Can't make a bullet Well, I'm working in a little proyect, a remake of Space Impact+, right now I have the lives, the score and the ship, but when it comes to bullets I find myself in troubles. I have 2 problems: >I can't make it to fire the bullet >I don't know how to make multiple instances of the bullet For the fist problem I tried with this code: Code: Select all function love.load() bullet={ img=love.graphics.newImage("imgs/bullet.png"), exist=false, x=20, y=20, defxy=function(self) self.x=charx+40 self.y=chary+12 end } end function love.draw() if bullet:exist then love.graphics.draw(bullet:img,bullet:x,bullet:y) end end function love.update(dt) if love.keyboard.isDown("z") then bullet:exist=true end if bullet:exist then if rspwn == 0 then bullet:defxy() rspwn=1 end bullet:x=bullet:x+50 if bullet:x > 384 then bullet:exist=false end end end (It has other functions, included in the .love file) I think it should work, but when I try to make it run, it gives me a problem in the part of "if bullet:exist then" I don't really know why, it looks like there wasn't any variables. On the second problem I don't have any idea on how to make it, Any help with that? Thanks ^w^ (Sorry for my dab english) Attachments Space.love baconhawka7x Party member Posts: 484 Joined: Mon Nov 21, 2011 7:05 am Contact: ### Re: Can't make a bullet I'm pretty sure, but correct me if I'm wrong. It should be bullet.exist not bullet:exist. baconhawka7x Party member Posts: 484 Joined: Mon Nov 21, 2011 7:05 am Contact: ### Re: Can't make a bullet And that's not the best way to make bullets;p Prole Posts: 4 Joined: Tue Apr 03, 2012 7:29 am ### Re: Can't make a bullet Yeah I know it's not the best way :/ But I can't think of anything else, any idea? trubblegum Party member Posts: 192 Joined: Wed Feb 22, 2012 10:40 pm ### Re: Can't make a bullet A simple way to make bullets would be something like : Code: Select all function bullet(x, y) return {x = x, y = y} end -- a bullet constructor bullets = {} -- a table to hold bullets table.insert(bullets, bullet(player.x, player.y)) -- a new bullet in the bullets table for i, b in ipairs(bullets) do -- iterating over bullets -- b is the current bullet. do stuff to it end Not a true instance, but will do the job if you put it all where it belongs. A search for "bullets" in this forum will give some useful results. Prole Posts: 4 Joined: Tue Apr 03, 2012 7:29 am ### Re: Can't make a bullet Thanks for that, I will try to adapt it, and will search more just to see what I find thanks Prole Posts: 4 Joined: Tue Apr 03, 2012 7:29 am ### Re: Can't make a bullet Okay, I got it to work, but now I have another question: >How can I delimit the bullets fired every time I press the button? I'm trying with something like this: Code: Select all if love.keyboard.isDown("z") then if nblts < 0.3 then table.insert(bullets,bullet(char.x+40, char.y+12)) nblts=nblts+1dt end end And it does work, but I dont know how to make nblts=0 after I have released the "z" key to make another shoot, Any help please? Attachments Space.love trubblegum Party member Posts: 192 Joined: Wed Feb 22, 2012 10:40 pm ### Re: Can't make a bullet I would do : Code: Select all actions = {} actions.shoot = {key = 'z', repeat = 0.25, dt = 0, func = function() table.insert(bullets, bullet()) end} love.update = function(dt) for i, action in pairs(actions) do action.dt = action.dt + dt if love.keyboard.isDown(action.key) and action.dt >= action.repeat then action.dt = 0 action.func() end end end baconhawka7x Party member Posts: 484 Joined: Mon Nov 21, 2011 7:05 am Contact: ### Re: Can't make a bullet Abaddon wrote:And it does work, but I dont know how to make nblts=0 after I have released the "z" key to make another shoot, Any help please? As with any programming, there are multiple ways to do one thing. For example, in my game "Z0mb3h Ga3m", to make the bullets, I had the player shoot a bullet every time they pushed a key... Code: Select all function love.keypressed(key) if key == keyshoot then insert shooting stuff here end end instead of doing it every frame that they hold it down.. Code: Select all function love.update(dt) if love.keyboard.isDown(keyshoot) then insert shooting stuff here end end I'm not saying either one is more efficiant. I'm just pointing out that there are a lot of ways to go about this. One way to reset nblts is... Code: Select all function love.keyreleased(key) if key == "z" then nblts = 0 end end so if they release the key "z" nblts will = 0. Also, a slightly less productive way to do this is.. Code: Select all if love.keyboard.isDown("z") then if nblts < 0.3 then table.insert(bullets,bullet(char.x+40, char.y+12)) nblts=nblts+1dt end else nblts = 0 end But I would go with the first way, because the second way could expel adding on other features to guns. But I haven't tested either of them lol;p Kasperelo Party member Posts: 343 Joined: Fri Apr 13, 2012 1:47 pm Location: The Milky Way wat ### Who is online Users browsing this forum: Sky_Render, steVeRoll and 10 guests	2020-02-28 19:15:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33190250396728516, "perplexity": 5627.885538986526}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875147628.27/warc/CC-MAIN-20200228170007-20200228200007-00125.warc.gz"}
https://brilliant.org/problems/what-is-the-unit-digit/	# What's my Last Digit? Number Theory Level 4 $\left \lfloor \frac { { 10 }^{ 2000 } }{ { 10 }^{ 100 }+3 } \right\rfloor$ Find the unit digit of the expression above. ×	2016-10-22 01:53:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2859894633293152, "perplexity": 7938.190687215874}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718423.28/warc/CC-MAIN-20161020183838-00376-ip-10-171-6-4.ec2.internal.warc.gz"}
http://nrich.maths.org/public/leg.php?code=71&cl=4&cldcmpid=1343	# Search by Topic #### Resources tagged with Mathematical reasoning & proof similar to Sums of Squares and Sums of Cubes: Filter by: Content type: Stage: Challenge level: ### There are 183 results Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical reasoning & proof ### Sums of Squares and Sums of Cubes ##### Stage: 5 An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum orf two or more cubes. ### More Sums of Squares ##### Stage: 5 Tom writes about expressing numbers as the sums of three squares. ### Polite Numbers ##### Stage: 5 Challenge Level: A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers? ### A Biggy ##### Stage: 4 Challenge Level: Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power. ### Sixational ##### Stage: 4 and 5 Challenge Level: The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . . ### Whole Number Dynamics II ##### Stage: 4 and 5 This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point. ### Big, Bigger, Biggest ##### Stage: 5 Challenge Level: Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$? ### Janine's Conjecture ##### Stage: 4 Challenge Level: Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . . ### Pareq Exists ##### Stage: 4 Challenge Level: Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. ### Mouhefanggai ##### Stage: 4 Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai. ### Unit Interval ##### Stage: 4 and 5 Challenge Level: Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product? ### Sperner's Lemma ##### Stage: 5 An article about the strategy for playing The Triangle Game which appears on the NRICH site. It contains a simple lemma about labelling a grid of equilateral triangles within a triangular frame. ### Water Pistols ##### Stage: 5 Challenge Level: With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even? ### Integral Inequality ##### Stage: 5 Challenge Level: An inequality involving integrals of squares of functions. ### To Prove or Not to Prove ##### Stage: 4 and 5 A serious but easily readable discussion of proof in mathematics with some amusing stories and some interesting examples. ### Whole Number Dynamics I ##### Stage: 4 and 5 The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases. ### Proof Sorter - Quadratic Equation ##### Stage: 4 and 5 Challenge Level: This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations. ##### Stage: 4 Challenge Level: Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1. ### Whole Number Dynamics III ##### Stage: 4 and 5 In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again. ### Proof: A Brief Historical Survey ##### Stage: 4 and 5 If you think that mathematical proof is really clearcut and universal then you should read this article. ### Where Do We Get Our Feet Wet? ##### Stage: 5 Professor Korner has generously supported school mathematics for more than 30 years and has been a good friend to NRICH since it started. ### Try to Win ##### Stage: 5 Solve this famous unsolved problem and win a prize. Take a positive integer N. If even, divide by 2; if odd, multiply by 3 and add 1. Iterate. Prove that the sequence always goes to 4,2,1,4,2,1... ### Yih or Luk Tsut K'i or Three Men's Morris ##### Stage: 3, 4 and 5 Challenge Level: Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . . ### Telescoping Functions ##### Stage: 5 Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra. ### Whole Number Dynamics V ##### Stage: 4 and 5 The final of five articles which containe the proof of why the sequence introduced in article IV either reaches the fixed point 0 or the sequence enters a repeating cycle of four values. ### Whole Number Dynamics IV ##### Stage: 4 and 5 Start with any whole number N, write N as a multiple of 10 plus a remainder R and produce a new whole number N'. Repeat. What happens? ### Long Short ##### Stage: 4 Challenge Level: A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . . ### Picturing Pythagorean Triples ##### Stage: 4 and 5 This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself. ### Magic Squares II ##### Stage: 4 and 5 An article which gives an account of some properties of magic squares. ### Modulus Arithmetic and a Solution to Dirisibly Yours ##### Stage: 5 Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n. ### Euclid's Algorithm II ##### Stage: 5 We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions. ### Thousand Words ##### Stage: 5 Challenge Level: Here the diagram says it all. Can you find the diagram? ### Transitivity ##### Stage: 5 Suppose A always beats B and B always beats C, then would you expect A to beat C? Not always! What seems obvious is not always true. Results always need to be proved in mathematics. ### Modulus Arithmetic and a Solution to Differences ##### Stage: 5 Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic. ### Binomial ##### Stage: 5 Challenge Level: By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn ### Recent Developments on S.P. Numbers ##### Stage: 5 Take a number, add its digits then multiply the digits together, then multiply these two results. If you get the same number it is an SP number. ### Impossible Sandwiches ##### Stage: 3, 4 and 5 In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot. ### Some Circuits in Graph or Network Theory ##### Stage: 4 and 5 Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits. ### Converse ##### Stage: 4 Challenge Level: Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if. . . . ### Diophantine N-tuples ##### Stage: 4 Challenge Level: Take any whole number q. Calculate q^2 - 1. Factorize q^2-1 to give two factors a and b (not necessarily q+1 and q-1). Put c = a + b + 2q . Then you will find that ab+1 , bc+1 and ca+1 are all. . . . ### A Long Time at the Till ##### Stage: 4 and 5 Challenge Level: Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem? ### Basic Rhythms ##### Stage: 5 Challenge Level: Explore a number pattern which has the same symmetries in different bases. ### Diverging ##### Stage: 5 Challenge Level: Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity. ##### Stage: 4 and 5 Challenge Level: Which of these roads will satisfy a Munchkin builder? ### Tetra Inequalities ##### Stage: 5 Challenge Level: Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle. ### Pythagorean Triples I ##### Stage: 3 and 4 The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it! ### Advent Calendar 2011 - Secondary ##### Stage: 3, 4 and 5 Challenge Level: Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas. ### Calculating with Cosines ##### Stage: 5 Challenge Level: If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle? ### Russian Cubes ##### Stage: 4 Challenge Level: How many different cubes can be painted with three blue faces and three red faces? A boy (using blue) and a girl (using red) paint the faces of a cube in turn so that the six faces are painted. . . . ### What Numbers Can We Make Now? ##### Stage: 3 and 4 Challenge Level: Imagine we have four bags containing numbers from a sequence. What numbers can we make now?	2014-03-08 06:38:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4736538827419281, "perplexity": 1478.9391656811842}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999653669/warc/CC-MAIN-20140305060733-00044-ip-10-183-142-35.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/1994887/what-is-the-number-of-distinct-full-binary-trees-with-n-nodes	# What is the number of distinct full binary trees with n nodes? I'm trying to find a recurrence relation for how many full binary trees there are with n nodes. However, once I get to n = 11, it's there's a lot of trees to keep track of. I've written out the cases for $$n \in [1,9]$$ where n is odd but I can't figure out the pattern. At n=1, I have 1. n = 3, I have 1 n = 5 : 2 /// n = 7 : 5 /// n = 9 : 14 Basically I took the leaves from the previous nth tree I created and added 2 nodes to each leaf for all possible full binary trees then got rid of duplicates. I.e for n = 7, I took leaves from n = 5, which is 3 and added 2 nodes to each leaf from all the fbts possible when n = 5 (which is 2) and so I end up with 6 fbts, but there's 1 duplicate so I subtract 1 to end up with 5 distinct fbts. Any help on figuring out the pattern or atleast how many distinct fbts with n nodes? • You need to define "distinct" carefully. This link may be of interest: durangobill.com/BinTrees.html – Alexis Olson Nov 1 '16 at 18:38 • There is a very interesting video about Catalan numbers and how it is related to binary trees and many other things: youtube.com/watch?v=etzcN6g-vNY It is in French sorry, but if you select French subtitles + auto translation it gives fairly good English. (you can skip directly to 6:00) – zwim Dec 23 '18 at 2:27 • Relevant post from CS-SE. – JavaMan Dec 23 '18 at 3:49 See: Catalan Numbers You can use the number $C_n$ to describe the number of binary trees with $n+1$ leaf nodes, that is, $2n + 1$ nodes total. EDIT: Here's the full reasoning. We have $C_0 = 1$, and suppose we have $C_0, \ldots, C_n$, the number of full binary trees with up to $n + 1$ leaf nodes, and we want $C_{n+1}$. Given a root node, we just need $k$ leaf nodes on one side, and $n + 1 - k$ leaf nodes on the other, for all values of $k$ from $1$ to $n$. Since there's $C_k$ ways of choosing trees for one side, and $C_{n+1-k}$ on the other, there's a total of $C_k \cdot C_{n -k}$ trees for a given $k$. Solve for this recurrence: $$C_0 = 1, \qquad C_{n+1} = \sum_{k=0}^{n}C_kC_{n-k}$$ The solution is the Catalan Numbers $C_n = \frac{(2n)!}{(n+1)!n!}$. • Are you not double counting some cases? – gen Oct 11 '17 at 13:03	2019-10-22 21:01:45	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6338726878166199, "perplexity": 341.4984937204854}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987824701.89/warc/CC-MAIN-20191022205851-20191022233351-00102.warc.gz"}
https://crypto.stackexchange.com/questions/42571/why-are-side-channel-attacks-such-as-spa-dpa-cpa-based-on-the-aes-subbytes-rout	# Why are side channel attacks such as SPA/DPA/ CPA based on the AES SubBytes routine? I have three questions about AES related to side channel attacks: 1. Why are side channel attacks such as CPA related to the non-linearity of the SubBytes routine within AES? 2. What is the difference between using the substitution BOX (SBOX) in a non-linear way and the combinational SBOX proposed in articles such as "Design of AES S-box using combinational logic optimization"? 3. Does the use of a combinational SBOX avoid side channel attacks? The values targeted by CPA are of the type $S(p\oplus k)$ where $p$ is a plaintext and $k$ a key byte, and $S$ maps bytes bijectively to bytes. One finds the correct key $k_0$ by correlating the Hamming weight of $S(p\oplus k)$ with the power consumption, as the absolute value of the correlation is expected to be the highest for the correct key $k_0$ at the time $t_0$ when the $S$-box output is processed in the CPU. Now assume that for a second key $k_1\ne k_0$ the values of $S(p\oplus k_0)$ and $S(p\oplus k_1)$ differ only by a single bit for all plaintext bytes $p$. Then also their Hamming weights differ only by 1, and therefore the Hamming weights of $S(p\oplus k_0)$ and $S(p\oplus k_1)$ are strongly correlated. So whenever $k_0$ gives a high correlation between current consumption and predicted Hamming weight, so does $k_1$, which makes both keys difficult to distinguish, if one wants to know, which one is the correct one. If $S$ happens to be linear, then $S(p\oplus k_0)\oplus S(p\oplus k_1) = S(k_0\oplus k_1)$ is independent of $p$. So if $S(k_0\oplus k_1)$ has Hamming weight 1, one is exactly in the bad situation (for the attacker) of the last paragraph. For $S$ bijective linear, this occurs for $8$ different key bytes $k_1$'s. As the $S$-boxes have to protect the crypto-algorithm against linear and differential attacks, they are chosen such that $S(x)$ and $S(x\oplus\Delta)$ are "linearly as independent as possible". In particular, also their Hamming weights have as little correlation as possible. So if you have a high correlation with $S(x)$ (think $x=p\oplus k_0$), you will not have much correlation with $S(x\oplus\Delta)$ (think $\Delta=k_0\oplus k_1$), making different key bytes easily distinguishable. Differently phrased, real $S$-boxes fulfill the basic assumption of DPA that wrong keys give random predictions for the current consumption, whereas linear $S$-boxes violate it. In conclusion, any $S$-box good for "classical" cryptography is bad (=good target) for side-channel attacks like CPA, and vice versa. (There are some papers published about finding $S$-boxes good for both purposes, but to my knowledge none were found yet, and I'd be surprised if there will be.) I do not know the article you ask about, but judging from the abstract the authors are only interested in an efficient (time+space) HW-implementation of the AES S-box, and not at all in security against side-channel attacks (they would mention it). For the security against CPA it doesn't matter how an intermediate value you try to attack was created in the HW, only if it (or something else correlated to it) shows up leaking side-channel information (like power consumption). Trying to fight side-channel attacks without using dynamic random (dynamic = freshly generated each time the algorithm is run) I wouldn't bet on. Just take a look at how easily all white-box implementations got broken using DPA-like techniques. So the usual side channel attack associated with s-boxes is a cache based timing attack. This is something that is not really related to the non-linearity of the table itself; The table could just as easily be a memoized linear function, and it would still suffer from the same issue. The issue is basically related to the layout of the table in the cache of the CPU that is processing the AES circuit. Different parts of the table reside in different parts of the cache, and which part of the cache is fetched is determined by the input to the s-box. This leaks a group of potential inputs to the s-box step. So, the S-box is basically just a memoized function. This is done to speed up the processing time, as the non-linear function represented is relatively complicated/time consuming to implement. It is entirely possible to compute the S-box output discretely for every given input - However, this can result in a large performance penalty. Some implementations choose compute the S-box discretely and to pay for this performance penalty, because the discretely computed version is not known to be vulnerable to cache based timing attacks. Additionally, I believe that FPGA/ASIC implementations may be designed to use a table without suffering from cache based timing attacks (The cache in question is associated with a general purpose CPU and is hard for a programmer to explicitly control). However, there are other side channel attacks such as Power Analysis. I think that linearity/non-linearity may play a greater part in SPA/DPA attacks because of the masking countermeasures involved and how those relate to non-linearity. I am not sure if these were the focus of your question, and if so, I will leave discussion of them to someone who is more knowledgeable. So does computing the AES S-box discretely resist cache based timing attacks? It should. However, there may be other side channels that require other solutions in order to prevent leakage. • Dear Ella Rose, thank you for you answer, as you noticed linearity/non-linearity may play a greater part in SPA/DPA attacks because of the masking countermeasures involved and how those relate to non-linearity, as I found in this link, crypto.stackexchange.com/questions/39658/…, CPA depends on linearity too, so my question according to you if we use Combinational SBox, so, we don't use linearity property, so do you think that is impossible to detect the cipher key when using CPA attack? – nani92 Dec 28 '16 at 14:41 • @nani92 I had forgotten completely about the existence of correlation power analysis when I created my answer - as I mentioned, I am only familiar with attacks such as SPA/DPA/etc and do not consider myself a top expert on them. As such, I unfortunately do not know the answer to your question regarding linearity/correlation power analysis. If SPA/DPA/CPA type attacks are what you are interested in, you might edit your question to say so specifically so that you can receive answers tailored accordingly, from someone who is more knowledgeable about such things. – Ella Rose Dec 28 '16 at 17:25	2019-10-17 15:23:23	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5631870031356812, "perplexity": 802.0859417128073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986675409.61/warc/CC-MAIN-20191017145741-20191017173241-00197.warc.gz"}
https://www.aimsciences.org/article/doi/10.3934/jimo.2022115	# American Institute of Mathematical Sciences • Previous Article Global optimization for non-convex programs via convex proximal point method • JIMO Home • This Issue • Next Article Optimality and duality for $E$-differentiable multiobjective programming problems involving $E$-type Ⅰ functions doi: 10.3934/jimo.2022115 Online First Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible. Readers can access Online First articles via the “Online First” tab for the selected journal. ## Optimal retail price and service level in a dual-channel supply chain with reference price effect Corresponding author: Jiawu Peng Received September 2021 Revised March 2022 Early access July 2022 Fund Project: The authors would like to thank the editor and the anonymous referees for their helpful comments and suggestions that greatly improved the quality of this paper. The research is supported by National Natural Science Foundation of China under Grant No. 72071072, and Postgraduate Scientific Research Innovation Project of Hunan Province under Grant Nos.CX20200456 We study the optimal retail price and service level in a dual-channel supply chain in which the manufacturer may sell products through its online channel and offline retailer. Consumers with channel preference make their purchasing decision based on retail price, service level and channel valuation difference. We consider the reference price effect into a Hotelling utility function to formulate the competition of retail price and service level. We analytically derive the unique equilibrium separately in three power structures: (1) Manufacturer-led Stackelberg game (MS), (2) Retailer-led Stackelberg game (RS) and (3) Nash game (N). Particularly, we characterize the impacts of channel valuation difference and reference price effect intensity on the optimal decision and expected profit. In contrast to the previous literature on the dominance of the supply chain in that the dominance always can raise the dominant enterprise's profit, our findings indicate that supply chain members with channel advantages and channel disadvantages have different preferences for power structure. The member with channel disadvantages should give up dominance. Only when channel valuation difference is significant or reference price effect intensity is weak, the member with channel advantages will fight for dominance. Nash game is highly competitive in that the members only earn less profit. Numerical examples verified theoretical results. Citation: Honglin Yang, Siqi Zhao, Jiawu Peng. Optimal retail price and service level in a dual-channel supply chain with reference price effect. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022115 ##### References: [1] J. M. Chen, W. Zhang and Z. Y. Liu, Joint pricing, services and quality decisions in a dual-channel supply chain, RAIRO-Operations Research, 54 (2020), 1041-1056. doi: 10.1051/ro/2019024. [2] K. H. Chen, L. C. Alwan and L. Zhang, Dynamic pricing in the presence of reference price effect and consumer strategic behavior, International Journal of Production Research, 58 (2020), 546-561. [3] X. Chen, P. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2017), 4389-4408. [4] X. Chen, X. Wang and X. Jiang, The impact of power structure on the retail service supply chain with an O2O mixed channel, Journal of the Operational Research Society, 67 (2016), 294-301. [5] R. Chenavaz and W. Klibi, Dynamic pricing with reference price effects in integrated online and offline retailing, to appear, International Journal of Production Research. doi: 10.1016/j.ejor.2012.05.009. [6] W. Y. K. X. Chiang, D. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20. [7] B. Dan, G. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, International Journal of Production Economics, 139 (2012), 312-320. [8] B. Dan, S. Zhang and M. Zhou, Strategies for warranty service in a dual-channel supply chain with value-added service competition, International Journal of Production Research, 56 (2018), 5677-5699. doi: 10.1111/itor.12769. [9] A. Dumrongsiri, M. Fan and A. Jain, A supply chain model with direct and retail channels, European Journal of Operational Research, 187 (2008), 691-718. doi: 10.1016/j.ejor.2006.05.044. [10] D. Dzyabura, S. Jagabathula and E. Muller, Accounting for discrepancies between online and offline product evaluations, Marketing Science, 38 (2019), 88-106. [11] F. Gao and X. Su, Online and offline information for omnichannel retailing, Manufacturing & Service Operations Management, 19 (2017), 84-98. [12] S. M. Hosseini-Motlagh, M. Johari and P. Pazari, Coordinating pricing, warranty replacement and sales service decisions in a competitive dual-channel retailing system, Computers & Industrial Engineering, 163 (2022), 107862. [13] W. Hu and Y. Li, Retail service for mixed retail and E-tail channels, Annals of operations Researcht, 192 (2012), 151-171. doi: 10.1007/s10479-010-0818-7. [14] Y. Hu, S. Qu, G. Li and S. P. Sethi, Power structure and channel integration strategy for online retailers, European Journal of Operational Research, 294 (2021), 951-964. doi: 10.1016/j.ejor.2019.10.050. [15] Y. Jeong and M. Maruyama, Positioning and pricing strategies in a market with switching costs and staying costs, Information Economics and Policy, 44 (2018), 47-57. [16] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, in Handbook of the Fundamentals of Financial Decision Making: Part I, Academic Press, 2013, 99–127. doi: 10.21236/ADA045771. [17] Y. Liu, J. Li, W. W. Ren and J. Y. L. Forrest, Differentiated products pricing with consumer network acceptance in a dual-channel supply chain, to appear, Electronic Commerce Research and Applications. doi: 10.1111/itor.12749. [18] Y. Liu and R. K. Tyagi, The benefits of competitive upward channel decentralization, Management Science, 57 (2011), 741-751. [19] L. Lu, Q. Gou, W. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, International Journal of Production Research, 54 (2016), 5250-5270. [20] J. Ma and L. Xie, The comparison and complex analysis on dual-channel supply chain under different channel power structures and uncertain demand, Nonlinear Dynamics, 83 (2016), 1379-1393. doi: 10.1007/s11071-015-2410-9. [21] T. Maiti and B. C. Giri, Two-period pricing and decision strategies in a two-echelon supply chain under price-dependent demand, Applied Mathematical Modelling, 42 (2017), 655-674. doi: 10.1016/j.apm.2016.10.051. [22] Z. Pi, W. Fang and B. Zhang, Service and pricing strategies with competition and cooperation in a dual-channel supply chain with demand disruption, to appear, Computers & Industrial Engineering. [23] X. Pu, L. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Research and Applications, 22 (2017), 1-12. [24] X. Pu, S. Zhang, B. Ji and G. Han, Online channel strategies under different offline channel power structures, to appear, Journal of Retailing and Consumer Services. doi: 10.1007/s10479-018-2994-9. [25] L. Ren, Y. He and H. Song, Price and service competition of dual-channel supply chain with consumer returns, Discrete Dynamics in Nature and Society, 11 (2014), 1-10. doi: 10.1155/2014/565603. [26] L. Sun, X. Jiao, X. Guo and Y. Yu, Pricing policies in dual distribution channels: The reference effect of official prices, to appear, European Journal of Operational Research. doi: 10.1016/j.ejor.2021.03.040. [27] A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing & Service Operations Management, 2 (2000), 372-391. [28] A. A. Tsay and N. Agrawal, Channel conflict and coordination in the e-commerce age, Production and Operations Management, 13 (2004), 93-110. [29] N. Wang, T. Zhang, X. Zhu and P. Li, Online-offline competitive pricing with reference price effect, Journal of the Operational Research Society, 72 (2021), 642-653. doi: 10.1007/s40305-018-0227-1. [30] R. Wang, When prospect theory meets consumer choice models: Assortment and pricing management with reference prices, Manufacturing & Service Operations Management, 20 (2018), 583-600. [31] Y. Yu, L. Sun and X. Guo, Dual-channel decision in a shopping complex when considering consumer channel preference, Journal of the Operational Research Society, 71 (2020), 1638-1656. [32] Y. Zha, L. Zhang, C. Xu and T. Zhang, A two-period pricing model with intertemporal and horizontal reference price effects, International Transactions in Operational Research, 28 (2021), 1417-1440. doi: 10.1111/itor.12613. [33] C. Zhang, Y. Liu and G. Han, Two-stage pricing strategies of a dual-channel supply chain considering public green preference, to appear, Computers & Industrial Engineering. doi: 10.1155/2021/6614692. [34] G. Zhang, G. Dai, H. Sun, G. Zhang and Z. Yang, Equilibrium in supply chain network with competition and service level between channels considering consumers' channel preferences, to appear, Journal of Retailing and Consumer Services. [35] J. Zhang and W. Y. K. Chiang, Durable goods pricing with reference price effects, to appear, Omega. doi: 10.1016/j.ejor.2018.04.029. [36] J. Zhang, W. Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135. [37] R. Zhang, B. Liu and W. Wang, Pricing decisions in a dual channels system with different power structures, Economic Modelling, 29 (2012), 523-533. [38] J. Zhao, X. Hou, Y. Guo and J. Wei, Pricing policies for complementary products in a dual-channel supply chain, Applied mathematical modelling, 49 (2017), 437-451. doi: 10.1016/j.apm.2017.04.023. [39] B. Zheng, C. Yang, J. Yang and M. Zhang, Dual-channel closed loop supply chains: Forward channel competition, power structures and coordination, International Journal of Production Research, 55 (2017), 3510-3527. [40] Y. W. Zhou, J. Guo and W. Zhou, Pricing/service strategies for a dual-channel supply chain with free riding and service-cost sharing, International Journal of Production Economics, 196 (2018), 198-210. show all references ##### References: [1] J. M. Chen, W. Zhang and Z. Y. Liu, Joint pricing, services and quality decisions in a dual-channel supply chain, RAIRO-Operations Research, 54 (2020), 1041-1056. doi: 10.1051/ro/2019024. [2] K. H. Chen, L. C. Alwan and L. Zhang, Dynamic pricing in the presence of reference price effect and consumer strategic behavior, International Journal of Production Research, 58 (2020), 546-561. [3] X. Chen, P. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2017), 4389-4408. [4] X. Chen, X. Wang and X. Jiang, The impact of power structure on the retail service supply chain with an O2O mixed channel, Journal of the Operational Research Society, 67 (2016), 294-301. [5] R. Chenavaz and W. Klibi, Dynamic pricing with reference price effects in integrated online and offline retailing, to appear, International Journal of Production Research. doi: 10.1016/j.ejor.2012.05.009. [6] W. Y. K. X. Chiang, D. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20. [7] B. Dan, G. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, International Journal of Production Economics, 139 (2012), 312-320. [8] B. Dan, S. Zhang and M. Zhou, Strategies for warranty service in a dual-channel supply chain with value-added service competition, International Journal of Production Research, 56 (2018), 5677-5699. doi: 10.1111/itor.12769. [9] A. Dumrongsiri, M. Fan and A. Jain, A supply chain model with direct and retail channels, European Journal of Operational Research, 187 (2008), 691-718. doi: 10.1016/j.ejor.2006.05.044. [10] D. Dzyabura, S. Jagabathula and E. Muller, Accounting for discrepancies between online and offline product evaluations, Marketing Science, 38 (2019), 88-106. [11] F. Gao and X. Su, Online and offline information for omnichannel retailing, Manufacturing & Service Operations Management, 19 (2017), 84-98. [12] S. M. Hosseini-Motlagh, M. Johari and P. Pazari, Coordinating pricing, warranty replacement and sales service decisions in a competitive dual-channel retailing system, Computers & Industrial Engineering, 163 (2022), 107862. [13] W. Hu and Y. Li, Retail service for mixed retail and E-tail channels, Annals of operations Researcht, 192 (2012), 151-171. doi: 10.1007/s10479-010-0818-7. [14] Y. Hu, S. Qu, G. Li and S. P. Sethi, Power structure and channel integration strategy for online retailers, European Journal of Operational Research, 294 (2021), 951-964. doi: 10.1016/j.ejor.2019.10.050. [15] Y. Jeong and M. Maruyama, Positioning and pricing strategies in a market with switching costs and staying costs, Information Economics and Policy, 44 (2018), 47-57. [16] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, in Handbook of the Fundamentals of Financial Decision Making: Part I, Academic Press, 2013, 99–127. doi: 10.21236/ADA045771. [17] Y. Liu, J. Li, W. W. Ren and J. Y. L. Forrest, Differentiated products pricing with consumer network acceptance in a dual-channel supply chain, to appear, Electronic Commerce Research and Applications. doi: 10.1111/itor.12749. [18] Y. Liu and R. K. Tyagi, The benefits of competitive upward channel decentralization, Management Science, 57 (2011), 741-751. [19] L. Lu, Q. Gou, W. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, International Journal of Production Research, 54 (2016), 5250-5270. [20] J. Ma and L. Xie, The comparison and complex analysis on dual-channel supply chain under different channel power structures and uncertain demand, Nonlinear Dynamics, 83 (2016), 1379-1393. doi: 10.1007/s11071-015-2410-9. [21] T. Maiti and B. C. Giri, Two-period pricing and decision strategies in a two-echelon supply chain under price-dependent demand, Applied Mathematical Modelling, 42 (2017), 655-674. doi: 10.1016/j.apm.2016.10.051. [22] Z. Pi, W. Fang and B. Zhang, Service and pricing strategies with competition and cooperation in a dual-channel supply chain with demand disruption, to appear, Computers & Industrial Engineering. [23] X. Pu, L. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Research and Applications, 22 (2017), 1-12. [24] X. Pu, S. Zhang, B. Ji and G. Han, Online channel strategies under different offline channel power structures, to appear, Journal of Retailing and Consumer Services. doi: 10.1007/s10479-018-2994-9. [25] L. Ren, Y. He and H. Song, Price and service competition of dual-channel supply chain with consumer returns, Discrete Dynamics in Nature and Society, 11 (2014), 1-10. doi: 10.1155/2014/565603. [26] L. Sun, X. Jiao, X. Guo and Y. Yu, Pricing policies in dual distribution channels: The reference effect of official prices, to appear, European Journal of Operational Research. doi: 10.1016/j.ejor.2021.03.040. [27] A. A. Tsay and N. Agrawal, Channel dynamics under price and service competition, Manufacturing & Service Operations Management, 2 (2000), 372-391. [28] A. A. Tsay and N. Agrawal, Channel conflict and coordination in the e-commerce age, Production and Operations Management, 13 (2004), 93-110. [29] N. Wang, T. Zhang, X. Zhu and P. Li, Online-offline competitive pricing with reference price effect, Journal of the Operational Research Society, 72 (2021), 642-653. doi: 10.1007/s40305-018-0227-1. [30] R. Wang, When prospect theory meets consumer choice models: Assortment and pricing management with reference prices, Manufacturing & Service Operations Management, 20 (2018), 583-600. [31] Y. Yu, L. Sun and X. Guo, Dual-channel decision in a shopping complex when considering consumer channel preference, Journal of the Operational Research Society, 71 (2020), 1638-1656. [32] Y. Zha, L. Zhang, C. Xu and T. Zhang, A two-period pricing model with intertemporal and horizontal reference price effects, International Transactions in Operational Research, 28 (2021), 1417-1440. doi: 10.1111/itor.12613. [33] C. Zhang, Y. Liu and G. Han, Two-stage pricing strategies of a dual-channel supply chain considering public green preference, to appear, Computers & Industrial Engineering. doi: 10.1155/2021/6614692. [34] G. Zhang, G. Dai, H. Sun, G. Zhang and Z. Yang, Equilibrium in supply chain network with competition and service level between channels considering consumers' channel preferences, to appear, Journal of Retailing and Consumer Services. [35] J. Zhang and W. Y. K. Chiang, Durable goods pricing with reference price effects, to appear, Omega. doi: 10.1016/j.ejor.2018.04.029. [36] J. Zhang, W. Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135. [37] R. Zhang, B. Liu and W. Wang, Pricing decisions in a dual channels system with different power structures, Economic Modelling, 29 (2012), 523-533. [38] J. Zhao, X. Hou, Y. Guo and J. Wei, Pricing policies for complementary products in a dual-channel supply chain, Applied mathematical modelling, 49 (2017), 437-451. doi: 10.1016/j.apm.2017.04.023. [39] B. Zheng, C. Yang, J. Yang and M. Zhang, Dual-channel closed loop supply chains: Forward channel competition, power structures and coordination, International Journal of Production Research, 55 (2017), 3510-3527. [40] Y. W. Zhou, J. Guo and W. Zhou, Pricing/service strategies for a dual-channel supply chain with free riding and service-cost sharing, International Journal of Production Economics, 196 (2018), 198-210. The dual-channel supply chain structure The Hotelling model Analysis of consumer purchase behavior Impact of $\alpha$ on $p_m^j$ Impact of $\alpha$ on $s_m^j$ Impact of $\alpha$ on $\pi_m^j$ Optimal retail prices under three power structures Optimal service levels under three power structures Demand under three power structures Expected profit under three power structures The equilibrium selection of power structures for members The impact of t on the equilibrium solutions Comparison of closest literature Literatures Decision variables Game model Demand function Reference price effect Liu et al., (2020) $(w, p_r, p_m, s_r, s_m)$ MS Utility No Wang et al., (2021) $(p_r, p_m)$ MS, RS, N Hotelling Yes This paper $(p_r, p_m, s_r, s_m)$ MS, RS, N Hotelling Yes Literatures Decision variables Game model Demand function Reference price effect Liu et al., (2020) $(w, p_r, p_m, s_r, s_m)$ MS Utility No Wang et al., (2021) $(p_r, p_m)$ MS, RS, N Hotelling Yes This paper $(p_r, p_m, s_r, s_m)$ MS, RS, N Hotelling Yes Summary of notations Notation Definition $v$ Consumers' valuation of product $\theta$ The customers' acceptance degree of online direct channel $t$ Unit psychological cost caused by consumers' channel preference $w$ Wholesale price $x_0^j$ Consumer indifferent point of purchase under model $j$, where $j=MS, RS, N$ $\alpha$ Reference price effect intensity $U_i$ The utility of purchasing from different channel, where $i=r, m$ $R_i$ Reference price for different channel, where $i=r, m$ $p_i^j$ Retail price in different channel under model $j$, where $i=r, m$ and $j=MS, RS, N$ (decision variables) $s_i^j$ Service level in different channel under model $j$, where $i=r, m$ and $j=MS, RS, N$ (decision variables) $Q_i^j$ The demand in different channel under model $j$, where $i=r, m$ and $j=MS, RS, N$ $\pi_i^j$ The profits in different supply chain members under model $j$, where $i=r, m$ and $j=MS, RS, N$ Notation Definition $v$ Consumers' valuation of product $\theta$ The customers' acceptance degree of online direct channel $t$ Unit psychological cost caused by consumers' channel preference $w$ Wholesale price $x_0^j$ Consumer indifferent point of purchase under model $j$, where $j=MS, RS, N$ $\alpha$ Reference price effect intensity $U_i$ The utility of purchasing from different channel, where $i=r, m$ $R_i$ Reference price for different channel, where $i=r, m$ $p_i^j$ Retail price in different channel under model $j$, where $i=r, m$ and $j=MS, RS, N$ (decision variables) $s_i^j$ Service level in different channel under model $j$, where $i=r, m$ and $j=MS, RS, N$ (decision variables) $Q_i^j$ The demand in different channel under model $j$, where $i=r, m$ and $j=MS, RS, N$ $\pi_i^j$ The profits in different supply chain members under model $j$, where $i=r, m$ and $j=MS, RS, N$ The equilibrium solutions in MS game Manufacturer (Leader) Retailer (Follower) Retail price $p_m^{MS}=w+\frac{(8\alpha+3)(2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $p_r^{MS}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ Service level $s_m^{MS}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)}$ $s_r^{MS}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ Demand $Q_m^{MS}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5}$ $Q_r^{MS}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5}$ Profit $\pi_m^{MS}=w+\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\pi_r^{MS}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ Manufacturer (Leader) Retailer (Follower) Retail price $p_m^{MS}=w+\frac{(8\alpha+3)(2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $p_r^{MS}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ Service level $s_m^{MS}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)}$ $s_r^{MS}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ Demand $Q_m^{MS}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5}$ $Q_r^{MS}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5}$ Profit $\pi_m^{MS}=w+\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\pi_r^{MS}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ The equilibrium solutions in RS game Retailer(Leader) Manufacturer(Follower) Retail price $p_r^{RS}=w+\frac{(8\alpha+3)(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $p_m^{MS}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ Service level $s_r^{RS}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)}$ $s_m^{RS}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ Demand $Q_r^{RS}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5}$ $Q_m^{RS}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5}$ Profit $\pi_r^{RS}=\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\pi_m^{RS}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ Retailer(Leader) Manufacturer(Follower) Retail price $p_r^{RS}=w+\frac{(8\alpha+3)(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $p_m^{MS}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ Service level $s_r^{RS}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha_+5)}$ $s_m^{RS}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ Demand $Q_r^{RS}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5}$ $Q_m^{RS}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5}$ Profit $\pi_r^{RS}=\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\pi_m^{RS}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-2))^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ The equilibrium solutions in Nash game Retailer Manufacturer Retail price $p_r^{N}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)(6\alpha+2)}$ $p_m^{N}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ Service level $s_r^{N}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)}$ $s_m^{N}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ Demand $Q_r^{N}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)}$ $Q_m^{N}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)}$ Profit $\pi_r^{N}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ $\pi_m^{RS}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ Retailer Manufacturer Retail price $p_r^{N}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)(6\alpha+2)}$ $p_m^{N}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ Service level $s_r^{N}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)}$ $s_m^{N}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ Demand $Q_r^{N}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)}$ $Q_m^{N}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)}$ Profit $\pi_r^{N}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ $\pi_m^{RS}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ The equilibrium solutions in Section 7.1 Retailer Manufacturer Retail price $\tilde{p}_r^{MS^}=w+\frac{2t((2\alpha+1)(5t+(1-\theta)v-2))}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{p}_m^{MS^}=w+\frac{(4t(2\alpha+1)-1)(2\alpha+1)(3t-(1-\theta)v-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)}$ $\tilde{p}_r^{RS^}=w+\frac{(4t(2\alpha+1)-1)((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)}$ $\tilde{p}_m^{RS^}=w+\frac{2t((2\alpha+1)(5t-(1-\theta)v)-2)}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{p}_r^{N^}=w+\frac{2t((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)}$ $\tilde{p}_m^{N^}=w+\frac{2t((2\alpha+1)(3t-(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)}$ Service level $\tilde{s}_r^{MS^}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_m^{MS^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_r^{RS^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_m^{RS^}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_r^{N^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)}$ $\tilde{s}_m^{N^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)}$ Demand $\tilde{Q}_r^{MS^}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{8t(2\alpha+1)-3}$ $\tilde{Q}_m^{MS^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{8t(2\alpha+1)-3}$ $\tilde{Q}_r^{RS^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{8t(2\alpha+1)-3}$ $\tilde{Q}_m^{RS^}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{8t(2\alpha+1)-3}$ $\tilde{Q}_r^{N^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{6t(2\alpha+1)-2}$ $\tilde{Q}_m^{N^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{6t(2\alpha+1)-2}$ Profit $\tilde{\pi}_r^{MS^}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t+(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2}$ $\tilde{\pi}_m^{MS^}=w+\frac{((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)}$ $\tilde{\pi}_r^{RS^}=\frac{((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)}$ $\tilde{\pi}_m^{RS^}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t-(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2}$ $\tilde{\pi}_r^{N^}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2}$ $\tilde{\pi}_m^{N^}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2}$ Retailer Manufacturer Retail price $\tilde{p}_r^{MS^}=w+\frac{2t((2\alpha+1)(5t+(1-\theta)v-2))}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{p}_m^{MS^}=w+\frac{(4t(2\alpha+1)-1)(2\alpha+1)(3t-(1-\theta)v-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)}$ $\tilde{p}_r^{RS^}=w+\frac{(4t(2\alpha+1)-1)((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)^2(8t(2\alpha+1)-3)}$ $\tilde{p}_m^{RS^}=w+\frac{2t((2\alpha+1)(5t-(1-\theta)v)-2)}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{p}_r^{N^}=w+\frac{2t((2\alpha+1)(3t+(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)}$ $\tilde{p}_m^{N^}=w+\frac{2t((2\alpha+1)(3t-(1-\theta)v)-1)}{(2\alpha+1)(6t(2\alpha+1)-2)}$ Service level $\tilde{s}_r^{MS^}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_m^{MS^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_r^{RS^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_m^{RS^}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{(2\alpha+1)(8t(2\alpha+1)-3)}$ $\tilde{s}_r^{N^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)}$ $\tilde{s}_m^{N^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{(2\alpha+1)(6t(2\alpha+1)-2)}$ Demand $\tilde{Q}_r^{MS^}=\frac{(2\alpha+1)(5t+(1-\theta)v)-2}{8t(2\alpha+1)-3}$ $\tilde{Q}_m^{MS^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{8t(2\alpha+1)-3}$ $\tilde{Q}_r^{RS^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{8t(2\alpha+1)-3}$ $\tilde{Q}_m^{RS^}=\frac{(2\alpha+1)(5t-(1-\theta)v)-2}{8t(2\alpha+1)-3}$ $\tilde{Q}_r^{N^}=\frac{(2\alpha+1)(3t+(1-\theta)v)-1}{6t(2\alpha+1)-2}$ $\tilde{Q}_m^{N^}=\frac{(2\alpha+1)(3t-(1-\theta)v)-1}{6t(2\alpha+1)-2}$ Profit $\tilde{\pi}_r^{MS^}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t+(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2}$ $\tilde{\pi}_m^{MS^}=w+\frac{((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)}$ $\tilde{\pi}_r^{RS^}=\frac{((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)}$ $\tilde{\pi}_m^{RS^}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(5t-(1-\theta)v)-2)^2}{2(2\alpha+1)^2 (8t(2\alpha+1)-3)^2}$ $\tilde{\pi}_r^{N^}=\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t+(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2}$ $\tilde{\pi}_m^{N^}=w+\frac{(4t(2\alpha+1)-1) ((2\alpha+1)(3t-(1-\theta)v)-1)^2}{2(2\alpha+1)^2 (6t(2\alpha+1)-2)^2}$ The equilibrium solutions in Section 7.2 Retailer Manufacturer Retail price $\check{p}_r^{MS^}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ $\check{p}_m^{MS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2 (16\alpha+5)}$ $\check{p}_r^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $\check{p}_m^{RS^}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ $\check{p}_r^{N^}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ $\check{p}_m^{N^}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ Service level $\check{s}_r^{MS^}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_m^{MS^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_r^{RS^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_m^{RS^}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_r^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)}$ $\check{s}_m^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)}$ Demand $\check{Q}_r^{MS^}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5}$ $\check{Q}_m^{MS^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5}$ $\check{Q}_r^{RS^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5}$ $\check{Q}_m^{RS^}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5}$ $\check{Q}_r^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)}$ $\check{Q}_m^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)}$ Profit $\check{\pi}_r^{MS^}=\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\check{\pi}_m^{MS^}=w+\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\check{\pi}_r^{RS^}=\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\check{\pi}_m^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\check{\pi}_r^{N^}=\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ $\check{\pi}_m^{N^}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ Retailer Manufacturer Retail price $\check{p}_r^{MS^}=w+\frac{2(2\alpha+1)(5-(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ $\check{p}_m^{MS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)}{(2\alpha+1)^2 (16\alpha+5)}$ $\check{p}_r^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $\check{p}_m^{RS^}=w+\frac{2(2\alpha+1)(5+(1-\theta)v)-4}{(2\alpha+1)(16\alpha+5)}$ $\check{p}_r^{N^}=w+\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ $\check{p}_m^{N^}=w+\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(6\alpha+2)}$ Service level $\check{s}_r^{MS^}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_m^{MS^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_r^{RS^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_m^{RS^}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{(2\alpha+1)(16\alpha+5)}$ $\check{s}_r^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)}$ $\check{s}_m^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(2\alpha+1)(6\alpha+2)}$ Demand $\check{Q}_r^{MS^}=\frac{(2\alpha+1)(5-(1-\theta)v)-2}{16\alpha+5}$ $\check{Q}_m^{MS^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{16\alpha+5}$ $\check{Q}_r^{RS^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{16\alpha+5}$ $\check{Q}_m^{RS^}=\frac{(2\alpha+1)(5+(1-\theta)v)-2}{16\alpha+5}$ $\check{Q}_r^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1}{2(6\alpha+2)}$ $\check{Q}_m^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1}{2(6\alpha+2)}$ Profit $\check{\pi}_r^{MS^}=\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\check{\pi}_m^{MS^}=w+\frac{((2\alpha+1)(3+(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\check{\pi}_r^{RS^}=\frac{((2\alpha+1)(3-(1-\theta)v)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\check{\pi}_m^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\check{\pi}_r^{N^}=\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ $\check{\pi}_m^{N^}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ The equilibrium solutions in Section 7.3 Retailer Manufacturer Retail price $\hat{p}_r^{MS^}=w+\frac{2(2\alpha+1)(5+(1-\theta)v+h)-4}{(2\alpha+1)(16\alpha+5)}$ $\hat{p}_m^{MS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)}{(2\alpha+1)^2 (16\alpha+5)}$ $\hat{p}_r^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $\hat{p}_m^{RS^}=w+\frac{2(2\alpha+1)(5-(1-\theta)v-h)-4}{(2\alpha+1)(16\alpha+5)}$ $\hat{p}_r^{N^}=w+\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(6\alpha+2)}$ $\hat{p}_m^{N^}=w+\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(6\alpha+2)}$ Service level $\hat{s}_r^{MS^}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_m^{MS^}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_r^{RS^}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_m^{RS^}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_r^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{2(2\alpha+1)(6\alpha+2)}$ $\hat{s}_m^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{2(2\alpha+1)(6\alpha+2)}$ Demand $\hat{Q}_r^{MS^}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{16\alpha+5}$ $\hat{Q}_m^{MS^}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{16\alpha+5}$ $\hat{Q}_r^{RS^}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{16\alpha+5}$ $\hat{Q}_m^{RS^}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{16\alpha+5}$ $\hat{Q}_r^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1+h}{2(6\alpha+2)}$ $\hat{Q}_m^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1-h}{2(6\alpha+2)}$ Profit $\hat{\pi}_r^{MS^}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v+h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\hat{\pi}_m^{MS^}=w+\frac{((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\hat{\pi}_r^{RS^}=\frac{((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\hat{\pi}_m^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\hat{\pi}_r^{N^}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ $\hat{\pi}_m^{N^}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ Retailer Manufacturer Retail price $\hat{p}_r^{MS^}=w+\frac{2(2\alpha+1)(5+(1-\theta)v+h)-4}{(2\alpha+1)(16\alpha+5)}$ $\hat{p}_m^{MS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)}{(2\alpha+1)^2 (16\alpha+5)}$ $\hat{p}_r^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)}{(2\alpha+1)^2(16\alpha+5)}$ $\hat{p}_m^{RS^}=w+\frac{2(2\alpha+1)(5-(1-\theta)v-h)-4}{(2\alpha+1)(16\alpha+5)}$ $\hat{p}_r^{N^}=w+\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(6\alpha+2)}$ $\hat{p}_m^{N^}=w+\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(6\alpha+2)}$ Service level $\hat{s}_r^{MS^}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_m^{MS^}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_r^{RS^}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_m^{RS^}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{(2\alpha+1)(16\alpha+5)}$ $\hat{s}_r^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{2(2\alpha+1)(6\alpha+2)}$ $\hat{s}_m^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{2(2\alpha+1)(6\alpha+2)}$ Demand $\hat{Q}_r^{MS^}=\frac{(2\alpha+1)(5+(1-\theta)v+h)-2}{16\alpha+5}$ $\hat{Q}_m^{MS^}=\frac{(2\alpha+1)(3-(1-\theta)v-h)-1}{16\alpha+5}$ $\hat{Q}_r^{RS^}=\frac{(2\alpha+1)(3+(1-\theta)v+h)-1}{16\alpha+5}$ $\hat{Q}_m^{RS^}=\frac{(2\alpha+1)(5-(1-\theta)v-h)-2}{16\alpha+5}$ $\hat{Q}_r^{N^}=\frac{(2\alpha+1)(3+(1-\theta)v)-1+h}{2(6\alpha+2)}$ $\hat{Q}_m^{N^}=\frac{(2\alpha+1)(3-(1-\theta)v)-1-h}{2(6\alpha+2)}$ Profit $\hat{\pi}_r^{MS^}=\frac{(8\alpha+3)((2\alpha+1)(5+(1-\theta)v+h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\hat{\pi}_m^{MS^}=w+\frac{((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\hat{\pi}_r^{RS^}=\frac{((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{2(2\alpha+1)^2(16\alpha+5)}$ $\hat{\pi}_m^{RS^}=w+\frac{(8\alpha+3)((2\alpha+1)(5-(1-\theta)v-h)-2)^2}{2(2\alpha+1)^2(16\alpha+5)^2}$ $\hat{\pi}_r^{N^}=\frac{(8\alpha+3)((2\alpha+1)(3+(1-\theta)v+h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ $\hat{\pi}_m^{N^*}=w+\frac{(8\alpha+3)((2\alpha+1)(3-(1-\theta)v-h)-1)^2}{8(2\alpha+1)^2(6\alpha+2)^2}$ [1] Zonghong Cao, Jie Min. 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Journal of Industrial and Management Optimization, 2022, 18 (5) : 3365-3392. doi: 10.3934/jimo.2021117 2021 Impact Factor: 1.411	2022-10-04 20:04:42	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5162259340286255, "perplexity": 5492.711310312501}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337524.47/warc/CC-MAIN-20221004184523-20221004214523-00473.warc.gz"}
http://www.ck12.org/book/CK-12-Middle-School-Math-Grade-8/r1/section/11.6/	# 11.6: Conditional Probability Difficulty Level: At Grade Created by: CK-12 ## Introduction Bike Shop Repairs On Thursday, Carey was in charge of answering the phones and booking appointments for bike repairs. The bike shop repairs bikes on Monday, Tuesday and Wednesday mornings and on Thursday and Friday afternoons. All appointments are booked randomly. The person making the appointment can choose or the person answering the phone can choose. Carey booked two appointments right away. What are the chances that both of the these appointments were booked on a Monday, Tuesday or Wednesday morning? Answering this question will require you to understand conditional probability. This lesson will teach you all that you need to know so that you will be able to figure out the solution to the problem by the end of the lesson. What You Will Learn By the end of this lesson, you will be able to complete the following skills. • Recognize and distinguish among independent and dependent events. • Recognize and apply the definition of conditional probability to find probabilities in finite sample spaces. • Calculate the probabilities of a series of independent events, mutually exclusive events and events that are not mutually exclusive. • Make predictions involving conditional probability. Teaching Time I. Recognize and Distinguish among Independent and Dependent Events In the last few sections, you have been learning all about probability. Now we can think about different events and how these events impact each other. Take a look at this example. Suppose you have two events: Event A: Toss 5 on the number cube Event B: Spin blue on the spinner The probability of each of these events by itself is easy enough to compute. In general: \begin{align}P \text{(event)} = \frac{favorable \ outcomes}{total \ outcomes}\end{align} If this is the case, then we can write the following ratios for rolling a 5. \begin{align}P (5) &= \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{6}\\ P(\text{blue}) &= \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{4}\end{align} These two events were performed with a spinner and a number cube. Now a question arises. Does event A affect the probability of event B in any way? That is, does the number cube landing on 5 affect where the arrow lands in the spinner? If not, then the two events are said to be independent events. Definition: If the outcome of one event has no effect on the outcome of a second event, then the two events are independent events. Events A and B above are independent events. No matter how the number cube turns up, its outcome does not affect the outcome of spinning the spinner. Now let’s think about a different kind of example, one where the outcome of one event does impact the outcome of another event. Example A bag has 3 red marbles, 4 blue marbles, and 3 green marbles. Irina pulls 1 green marble out of the bag. Does this change the probability that the next marble Irina pulls out of the bag will be green? Solution: Here, the act of taking a marble out of the bag changes the situation. For the first marble, the probability of pulling out a green marble was: \begin{align}P \text {(green)} = \frac{favorable \ outcomes}{total \ outcomes} = \frac{3}{10}\end{align} For the second marble, there are now only 9 marbles left in the bag and only 2 of them are green. So the probability of pulling out a green marble for the second marble is now: \begin{align}P \text {(green)} = \frac{favorable \ outcomes}{total \ outcomes} = \frac{2}{9}\end{align} Clearly, the first event affected the outcome of the second event in this situation. So the two events are NOT independent. In other words, they are dependent events. Definition: If the outcome of one event has an effect on the outcome of a second event, then the two events are dependent events. II. Recognize and Apply the Definition of Conditional Probability to Find Probabilities in Finite Sample Spaces Sometimes the outcome that you get when figuring out a probability is what we call “conditional.” This means that we will only an outcome if the conditions designed to cause a specific result. Let’s look at an example. Example Consider a jar with 4 black marbles and 6 white marbles. If you pull out 2 marbles from the jar randomly, one at a time, without replacing the first marble, what is the probability that both marbles will be white? Start by approaching the problem the same as you would with independent events. The probability of the first marble being white is: \begin{align}P (\text{white} \ 1 \text{st marble}) = \frac{6}{10} = \frac{3}{5} \end{align} What about the second marble? Having removed the first marble from the bag, now instead of 6 white marbles out of 10 total marbles, there are only 5 white marbles left out of 9 total marbles: \begin{align}P (\text{white} \ 2 \text{nd marble}) = \frac{5}{9} \end{align} This gives a probability of both events occurring as: \begin{align}P (\text{white then white}) &= P (\text{white} \ 1 \text{st marble}) \cdot P (\text{white} \ 2 \text{nd marble}) \\ &= \frac{3}{5} \cdot \frac{5}{9} \\ &= \frac{1}{3}\end{align} The same general method works for calculating any two (or more) dependent events. Example Jack’s Catering Service is accepting weekday appointments for Monday through Thursday, and weekend appointments for Friday through Sunday. If appointment dates are made randomly, what is the probability that 2 weekdays will be the first 2 days to be booked? Solution: The probability that the first day will be a weekday is: \begin{align}P (\text{weekday} \ 1 \text{st}) = \frac {4}{7}\end{align} The probability that the second booked day will also be a weekday is: \begin{align}P (\text{weekday} \ 2 \text{nd}) = \frac {3}{6} = \frac{1}{2}\end{align} \begin{align} P \text{(weekday and weekday)} &= P \text{(weekday} \ 1 \text{st}) \cdot P \text{(weekday} \ 2 \text{nd}) \\ &= \frac{4}{7} \cdot \frac{1}{2}\\ &= \frac{2}{7}\end{align} Now let’s look at conditional probability and outcomes. Conditional probability involves situations in which you determine the probability of an event based on another event having occurred. For example, suppose you roll two number cubes on a table. The first cube lands face up on 5. The second cube falls off of the table so you can’t see how it landed. Given what you know so far, what is the probability that the sum of the number cubes will be 9? To solve this problem, consider the entire sample space for rolling two number cubes. \begin{align} &66 \quad 56 \quad {\color{red}46} \quad 36 \quad 26 \quad 16 \\ &65 \quad 55 \quad {\colorbox{yellow}{\color{red}45}} \ \ 35 \quad 25 \quad 15 \\ &64 \quad 54 \quad {\color{red}44} \quad 34 \quad 24 \quad 14 \\ &63 \quad 53 \quad {\color{red}43} \quad 33 \quad 23 \quad 13 \\ &62 \quad 52 \quad {\color{red}42} \quad 32 \quad 22 \quad 12 \\ &61 \quad 51 \quad {\color{red}41} \quad 31 \quad 21 \quad 11\end{align} You already know that the first number cube landed on 4, so now you need to consider only those outcomes marked in red. Only 1 of those 6 results in a 9, so: \begin{align}P (9 \| 4) = \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{6} \end{align} Notice that we write the conditional probability as \begin{align}P (9 \| 4)\end{align}. You can read this as: \begin{align}P (9\|4) \Longleftarrow\end{align} the probability of 9, given 4 Here are some other examples of how to read this notation. \begin{align}P (B\|A) \Longleftarrow\end{align} the probability of \begin{align}B\end{align}, given \begin{align}A\end{align} \begin{align}P (7\|3) \Longleftarrow\end{align} the probability of 7, given 3 \begin{align}P (\text{heads}\| \text{tails}) \Longleftarrow\end{align} the probability of heads, given tails \begin{align}P (\text{red}\| \text{blue}) \Longleftarrow\end{align} the probability of red, given blue The probability is determined because certain factors are in place. III. Calculate Probabilities of a Series of Independent Events, Mutually Exclusive Events and Events that are not Mutually Exclusive Sometimes, we have mutually exclusive events and we have events that overlap and are not mutually exclusive. In a previous section, you saw that event \begin{align}R(\text{red})\end{align} and event \begin{align}T(\text{top})\end{align} are overlapping events because both events share one outcome – red-top. The Venn diagram for overlapping events shows that the two events overlap, or share 1 or more outcomes. To calculate the probability of overlapping events, list the sample space and find the favorable events. \begin{align} & \mathbf{red - top} \qquad \ \text{blue}-\text{top}\\ & \text{red}-\text{bottom} \quad \text{blue}-\text{bottom}\end{align} The probability of red-top is: \begin{align}P(\text{red}-\text{top}) = \frac{favorable \ outcomes} {total \ outcomes} = \frac{1}{4} \end{align} You can compute the probabilities of other overlapping events in a similar way. Example For a single toss of a number cube, what is the probability of event \begin{align}E(\text{even})\end{align} and event \begin{align}S(4)\end{align} both occurring? Step 1: Identify the overlapping outcomes of both events. \begin{align}E (\text{even}) &= 2, \mathbf{4}, 6 \\ S(4) &= \mathbf{4} \end{align} Step 2: Find the total number of outcomes. \begin{align}\text{total outcomes} &= 1, 2, 3, 4, 5, 6 \\ &= 6 \ \text{total outcomes}\end{align} Step 3: Find the probability of the overlapping events. \begin{align}P(4) = \frac{favorable \ outcomes}{total \ outcomes} = \frac{1}{6} \end{align} IV. Make Predictions Involving Conditional Probability Think back for a minute and remember what we mean when we talk about conditional probability. Conditional probability involves situations in which you determine the probability of an event based on another event having occurred. We can use conditional probability to determine probabilities, but also to make predictions. Look at this example. Example A stack of 12 cards has the Ace, King, and Queen of all 4 suits, spades, hearts, diamonds, and clubs. What is the probability that if you draw 2 cards randomly, they will both be hearts? Make a prediction. Step 1: Draw the first card. The probability of it being a heart is 3 of 12. Step 2: Now draw the second card. Since the first card was a heart, there are only 11 cards left and only 2 of them are hearts. Step 3: Calculate the final probability. So \begin{align}P(\text{heart and heart}) = \frac{1}{22}\end{align}. You would predict that both cards would be hearts \begin{align}\frac{1}{22}\end{align} of the time. ## Real-Life Example Completed Bike Shop Repairs Here is the original problem from the introduction. Reread it and then solve it for the correct probability. On Thursday, Carey was in charge of answering the phones and booking appointments for bike repairs. The bike shop repairs bikes on Monday, Tuesday and Wednesday mornings and on Thursday and Friday afternoons. All appointments are booked randomly. The person making the appointment can choose or the person answering the phone can choose. Carey booked two appointments right away. What are the chances that both of the these appointments were booked on a Monday, Tuesday or Wednesday morning? Now answer the question at the end of the problem. Solution to Real – Life Example To work on this probability, first we must determine the probability of the first appointment booked being on a Monday, Tuesday or Wednesday. There are five possible days for appointments, but three favorable outcomes. Probability of first appointment being Mon, Tues or Weds \begin{align}=\frac{3}{5}\end{align} Probability of second appointment being Mon, Tues or Weds \begin{align}=\frac{2}{4}\end{align} or \begin{align}\frac{1}{2}\end{align} Now we can multiply them for the conditional probability. \begin{align}\frac{3}{5} \cdot \frac{1}{2} = \frac{3}{10}\end{align} or 30% There is a 30% chance that the first two appointments booked would be on a Monday, Tuesday or Wednesday morning. ## Vocabulary Here are the vocabulary words that are found in this lesson. Independent Events The outcome of one event has no effect on the outcome of a second event. Dependent Events If the outcome of one event has an effect on the outcome of another event they are dependent events. Overlapping Events Events that share one outcome Conditional Probability Probability that is predicted based on specific criteria or conditions. ## Time to Practice Directions: Write whether events A and B are dependent or independent. 1. A: Doug flips a coin. B: Marlene chooses a card out of a deck. 2. A: In a bag with 5 white marbles and 5 black marbles, Sanjay pulls out a white marble. B: Without returning the marble to the bag, Sanjay pulls out a second marble. 3. A: Eddie chooses the color blue for his new bike. B: Eddie chooses lasagne from the dinner menu. 4. A: The probability that it will rain tomorrow. B: The probability that the Red Wings hockey team will win their game tomorrow. 5. A: From a deck of cards, the probability of one player drawing a heart from the deck. B: On the next player’s turn, the probability of drawing another heart. 6. A: The probability of a spinner landing on blue 6 times in a row. B: The probability of the spinner landing on blue on the next spin. 7. A: The probability that the Rockies will be in the playoffs. B: The probability that the Rockies will win the World Series. 8. A: The probability that tomorrow will be sunny. B: The probability that tomorrow will be a full moon. 9. A The probability that tomorrow will be sunny. B: The probability that tomorrow will be cloudy. 10. A: The probability that it will be a half-moon today. B: The probability that it will be a full moon tomorrow. Directions: Solve the problems. 1. A stack of 12 cards has 4 Aces, 4 Kings, and 4 Queens. What is the probability of picking 2 Aces from the stack at random? 2. What is the probability of picking an Ace then a King from the stack above? 3. What is the probability of picking 3 Queens from the stack above? 4. Stoyko’s shirt drawer has 4 colored t-shirts and 4 white t-shirts. If Stoyko picks out 2 shirts at random, what is the probability that they will both be colored? 5. If Stoyko picks out 2 shirts at random from the drawer above, what is the probability that the first one will be colored and the second one will be white? 6. On a game show, there are 16 questions: 8 easy, 5 medium-hard, and 3 hard. If contestants are given questions randomly, what is the probability that the first two contestants will get easy questions? 7. On the game show above, what is the probability that the first contestant will get an easy question and the second contestant will get a hard question? 8. On the game show above, what is the probability that both of the first two contestants will get hard questions? 9. For a single toss of a number cube, what is the probability that the cube will land on a number that is both odd and greater than 2? 10. For a single toss of a number cube, what is the probability that the cube will land on a number that is greater than 2 and less than 6? 11. For a single toss of a number cube, what is the probability that the cube will land on a number that is greater than 1 and less than 6? 12. What is the probability that a sum of a pair of number cubes will be 11 if the first cube lands on 5? 13. What is the probability that a sum of a pair of number cubes will be odd if the first cube lands on 2? 14. What is the probability that a sum of a pair of number cubes will be even and greater than 6 if the first cube lands on 4? 15. If you toss 2 number cubes, predict how likely they are to match. 16. If you toss 2 number cubes, predict how likely they are NOT to match. ### Notes/Highlights Having trouble? Report an issue. Color Highlighted Text Notes Show Hide Details Description Tags: Subjects:	2017-04-26 23:59:30	{"extraction_info": {"found_math": true, "script_math_tex": 35, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.984769880771637, "perplexity": 913.8326803569778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121752.57/warc/CC-MAIN-20170423031201-00448-ip-10-145-167-34.ec2.internal.warc.gz"}
https://123deta.com/document/q2ggvory-remarks-perturbation-operators-hilbert-noncommutative-structure-operator-application.html	# Remarks on perturbation of defect operators on Hilbert function spaces (Noncommutative Structure in Operator Theory and its Application) ## 全文 (1) Title Remarks on perturbation of defect operators on Hilbertfunction spaces (Noncommutative Structure in Operator Theory and its Application) Author(s) Seto, Michio Citation 数理解析研究所講究録 (2011), 1737: 115-122 Issue Date 2011-04 URL http://hdl.handle.net/2433/170833 Right Type Departmental Bulletin Paper Textversion publisher (2) ## Hilbert function spaces Department of Mathematics Shimane University ### Introduction Let $(\mathcal{H}, k_{\lambda}, \Omega)$ be a reproducing kernel Hilbert space consisting of analytic functions on a domain $\Omega$ in $\mathbb{C}^{n}$ with the variable $z=(z_{1}, \ldots, z_{n})$ and the reproducing kernel $k_{\lambda}=k(\lambda, \cdot)$, where $\lambda$ is a point in $\Omega$. Without loss of generality, may ### assume that $\Omega$ contains the origin. Moreover, we ### as-sume that $\mathcal{H}$ is invariant under pointwise multiplication of any polynomial in $\mathbb{C}[z_{1}, \ldots, z_{n}]$. Then a family of operators encoding structure of $(\mathcal{H}, k_{\lambda}, \Omega)$ is obtained under appropriate conditions. In this note, these operators will be denoted by $\triangle_{\lambda}$. We should mentionthat $\triangle=\triangle 0$has been studied already by many researchers on ### some Hilbert function spaces. This note has been organized ### as follows. In Section 2 and Section 3, we will give a partial announcement of results obtained in [8], where we dealt with $\triangle_{\lambda}$’s of submodules in Hardy space over the bidisk. In Section 4, we revisit the Hardy space ### over the unit disk from our point of view. In Section 5, we studies $\triangle_{\lambda}$’s of submodules in the Bergman space over the unit disk. ### module Let $D$ denote the open unit disk in the complex plane $\mathbb{C}$, and let $H^{2}(D)$ be the Hardy space over D. The Hardy space over the bidisk$D^{2}$ will be denoted by $H^{2}(D^{2})$, or $H^{2}$ for short. Then $z=(z_{1}, z_{2})$ will denote the variable of functions in $H^{2}$. We note that $H^{2}$ can be defined ### as the tensor product (3) under pointwise multiplication, $H^{2}$ becomes Hilbert module ### over $A$. A closed subspace $\mathcal{M}$ of $H^{2}$ is called ### a submodule if$\mathcal{M}$ is invariant under the module action, that is, a submodule is an invariant subspace of $H^{2}$ under multiplication of each function in A. $[S]$ denotesthe submodule generated by a set $S$. The rank of a submodule $\mathcal{M}$ is the least cardinality of ### a generating set of $\lambda 4$ ### a Hilbert module, and which will be denoted by rank$\mathcal{M}$, and the following inequality is well known: $\dim \mathcal{M}/[(z_{1}-\lambda_{1})\mathcal{M}+(z_{2}-\lambda_{2})\mathcal{M}]\leq$ rank$M((\lambda_{1}, \lambda_{2})\in D)$. (2.1) Set $\alpha_{n}=1-n^{-3}(n\in N)$, and let $b_{\alpha_{n}}$ be the Blaschke factor whose ### zero is $\alpha_{n}$. Then $\mathcal{M}=\sum_{j=0}^{\infty}q_{j}H^{2}(D)\otimes\nearrow 2$ $( whereq_{j}=\prod_{n=j}^{\infty}b_{\alpha_{n}}^{n-j})$ has been called Rudin $s$ module (cf. Rudin [7]). The striking fact on Rudin $s$ module is that the module rank is infinity. Indeed, for any $\lambda=(\lambda_{1}, \lambda_{2})$ in $D^{2}$, we have $\dim\Lambda t/[(z_{1}-\lambda_{1})\mathcal{M}+(z_{2}-\lambda_{2})\Lambda 4]=\{\begin{array}{ll}n+1 (\lambda=(\alpha_{n}, 0))1 (otherwise).\end{array}$ As $n$ tends to infinity, ### we have rank$\mathcal{M}=\infty$ by (2.1). Therefore we ### are interested in the following family of quotient vector spaces. $\mathcal{M}/[(z_{1}-\lambda_{1})\Lambda 4+(z_{2}-\lambda_{2})\mathcal{M}]((\lambda_{1}, \lambda_{2})\in D^{2})$. ### 3 $H^{2}(D^{2})$ ### case Let $\mathcal{M}$ be a submodule of $H^{2}(D^{2})$. Then $R_{f}$ denotes the compression of a Toeplitz operator $T_{f}$ into $\mathcal{M}$, that is, we set $R_{f}=P_{\mathcal{M}}T_{f}\|_{\Lambda t}$ where $P_{\lambda 4}$ is the (4) orthogonal projection of $H^{2}$ onto ### a submodule $\mathcal{M}$. The following operator is called the defect operator of a submodule M. $\triangle=I_{\mathcal{M}}-R_{z}1R_{Z1}^{}-R_{z_{2}}R_{z2}^{}+R_{z_{1}}R_{z}R_{z1}^{}R_{z2}^{}2$, which has been introduced by Yang in [9, 10] (see, also Guo [3] and Guo-Yang [5]$)$. Moreover, we introduce the following operator valued function: $\triangle_{\lambda}=I_{\lambda 4}-R_{b_{\lambda_{1}}}R^{}-R_{b_{\lambda_{2}}(z2}R_{b_{\lambda_{2}}(z_{2})}^{}+R_{b_{\lambda_{1}}(z)}R_{b_{\lambda_{2}}(z)}R_{b_{\lambda_{1}}(z_{1})}^{}R_{b_{\lambda_{2}}(z_{2})}^{}12$ where $(b_{\lambda_{1}}(z_{1}), b_{\lambda_{2}}(z_{2}))=( \frac{z_{1}-\lambda_{1}}{1-\overline{\lambda_{1}}z_{1}},$ $\frac{z_{2}-\lambda_{2}}{1-\overline{\lambda_{2}}z_{2}})$ $(\lambda=(\lambda_{1}, \lambda_{2})\in D^{2})$. Since $(b_{\lambda_{1}}(z_{1}), b_{\lambda_{2}}(z_{2}))$ defines ### an automorphism of $D^{2}$ (i.e. a biholomor-phic map acting on $D^{2}$), $\triangle_{\lambda}$ be ### as a defect operator perturbed by an automorphism. The following theorem is the ### reason why we are interested in $\triangle_{\lambda}$, which was shown in Guo-Yang [5] for the ### case where $\lambda=0$ (see also Guo-Wang [4]$)$, and their proof can be applied to the general case. Theorem 3.1 (Guo-Yang [5], Guo-Wang [4]) Let$\mathcal{M}$ be a submodule ### of $H^{2}(D^{2})$. Then ### for any $\lambda\in D_{f}^{2}$ $ker(I_{\mathcal{M}}-\triangle_{\lambda})=\mathcal{M}/[(z_{1}-\lambda_{1})\mathcal{M}+(z_{2}-\lambda_{2})\mathcal{M}]$ . Yang defined a new class of submodules in $H^{2}(D^{2})$. Definition 3.1 ([10]) A submodule $\mathcal{M}$ in $H^{2}$ is said to be Hilbert-Schmidt ### if $\triangle$ is Hilbert-Schmidt. Yang showed that Hilbert-Schmidt class includes Rudin’s module and sub-modules generated by polynomials. Theorem 3.2 $(S[8])$ Let $\mathcal{M}$ be a submodule of $H^{2}$. (i) If $\triangle_{\mu}$ is Hilbert-Schmidt for ### some $\mu$ in $D^{2}$, then $\triangle_{\lambda}$ is Hilbert-Schmidt (5) (ii) If $\mathcal{M}$ is Hilbert-Schmidt then $\Vert\triangle_{\lambda}-\triangle_{\mu}\Vert_{2}arrow 0(\lambdaarrow\mu)$. Theorem 3.3 $(S[8])$ Let $\mathcal{M}$ be a Hilbert-Schmidt submodule such that dimker$(I-\triangle_{\mu})=n>1$ ### for some $\mu$ in $D^{2}$. Then, ### for any neighborhood $U_{1}$ ### of 1 such that $\sigma(\triangle_{\mu})\cap\overline{U_{1}}=\{1\}$, there exists a neighborhood $U_{\mu}$ ### of $\mu$ such that $\sigma(\triangle_{\lambda})\cap U_{1}=\{1, \sigma_{1}(\lambda), \ldots, \sigma_{n-1}(\lambda)\}$ ### for any $\lambda$ in $U_{\mu}$, counting multiplicity. Example 3.1 (Yang [9], $S[8]$) Let $q_{1}=q_{1}(z_{1})$ and $q_{2}=q_{2}(z_{2})$ be ### one variable inner functions, and let $\mathcal{M}$ be the submodule generated by $q_{1}$ and $q_{2}$ in $H^{2}(D^{2})$ ### . Then we have dimker$(I_{\mathcal{M}}-\triangle_{\lambda})=\{\begin{array}{ll}2 (if q_{1}(\lambda_{1})=q_{2}(\lambda_{2})=0)1 (otherwise).\end{array}$ and $\sigma(\triangle_{\lambda})=\{0,1, \pm\sigma(\lambda)\}$, where we set $\sigma(\lambda)=\sqrt{(1-\|q_{1}(\lambda_{1})\|^{2})(1-\|q_{2}(\lambda_{2})\|^{2})}$. This calculation has been done already in the ### case where $(\lambda_{1}, \lambda_{2})=(0,0)$ by Yang in [9]. If $\sigma(\lambda)\neq 1$ then the eigenfunction corresponding to $\sigma(\lambda)$ is $e( \lambda)=(\sqrt{1-\|q_{2}(\lambda_{2})\|^{2}}-\sqrt{1-\|q_{1}(\lambda_{1})\|^{2}})\frac{q_{1}(z_{1})q_{2}(z_{2})}{(1-\overline{\lambda_{1}}z_{1})(1-\overline{\lambda_{2}}z_{2})}$ $- \frac{q_{2}(\lambda_{2})}{\sqrt{1-\|q_{2}(\lambda_{2})\|^{2}}}\frac{q_{1}(z_{1})(1-\overline{q_{2}(\lambda_{2})}q_{2}(z_{2}))}{(1-\overline{\lambda_{1}}z_{1})(1-\overline{\lambda_{2}}z_{2})}$ $+ \frac{q_{1}(\lambda_{1})}{\sqrt{1-\|q_{1}(\lambda_{1})\|^{2}}}\frac{q_{2}(z_{2})(1-\overline{q_{1}(\lambda_{1})}q_{1}(z_{1}))}{(1-\overline{\lambda_{1}}z_{1})(1-\overline{\lambda_{2}}z_{2})}$ If $\sigma(\lambda)=1$ then the eigenfunctions corresponding to $\sigma(\lambda)$ are $q_{1}(z_{1})$ $(1-\overline{\lambda_{1}}z_{1})(1-\overline{\lambda_{2}}z_{2})$ ’ $q_{2}(z_{2})$ $(1-\overline{\lambda_{1}}z_{1})(1-\overline{\lambda_{2}}z_{2})$ . ### 118 (6) Note that $e(\lambda)$ converges to $0$ ### as $\sigma(\lambda)$ tends to 1. ### 4 $H^{2}(D)$ ### case The defect operator of a submodule $\mathcal{M}$ in $H^{2}(D)$ is ### as follows: $\triangle=I_{\mathcal{M}}-R_{z}R_{z}^{}=$ Proj$(\mathcal{M}/z\mathcal{M})=q\otimes q$, where $q$ is the innerfunction corresponding to a submodule $\mathcal{M}$ by Beurling‘s theorem. The definition of $\triangle_{\lambda}$ is similar to that given in Section 3, and we have $\triangle_{\lambda}=I_{Al}-R_{b_{\lambda}}R_{b_{\lambda}}^{}=$ Proj$(\Lambda t/(z-\lambda)\mathcal{M})=qK_{\lambda}\otimes qK_{\lambda}$, ### where we set $b_{\lambda}=(z-\lambda)/(1-\overline{\lambda}z)$ and $K_{\lambda}$ denotes the normalized Szeg\"o kernel. These facts are well known. ### 5 $L_{a}^{2}(D)$ ### case In this section, we deal with the defect operator ofa submodule in Bergman space over D. The Bergman space over $D$ is defined ### as follows: $L_{a}^{2}(D)=\{f\in$ Hol(D) : $\frac{1}{\pi}\int_{D}\|f(z)\|^{2}dxdy<\infty(z=x+iy)\}$ . The reproducing kernel is $k_{\lambda}(z)= \frac{1}{(1-\overline{\lambda}z)^{2}}$ (the Bergman kernel), and the operator $S_{z}$ : $f\mapsto zf$ acting on $L_{a}^{2}(D)$ is called the Bergman shift. The definition of submodules in $L_{a}^{2}(D)$ is the ### same as that of $H^{2}(D^{2})$. We summarize well known facts on submodules of $L_{a}^{2}(D)$. Theorem 5.1 Let $\mathcal{M}$ be a submodule of $L_{a}^{2}(D)$. (7) (ii) For every $n$ in $\{$1, 2, ### 120 (8) where $dA(w)=\pi^{-}$ ’$dxdy(w=x+iy)$ . Hence $\triangle_{\lambda}$ is Hilbert-Schmidt if and only if $(1-\overline{b_{\lambda}(z)}b_{\lambda}(w))^{2}k_{z}^{\Lambda 4}(w)$ is square integrable with respect to the Lebesgue ### on $D^{2}$. We note that $\frac{(1-\overline{b_{\lambda}(z)}b_{\lambda}(w))^{2}}{(1-\overline{z}w)^{2}}=(\frac{1-\|\lambda\|^{2}}{(1-\lambda\overline{z})(1-\overline{\lambda}w)})^{2}$ (5.1) Hence we have $(1- \overline{b_{\lambda}(z)}b_{\lambda}(w))^{2}k_{z}^{\wedge\Lambda}(w)=\frac{(1-\overline{b_{\lambda}(z)}b_{\lambda}(w))^{2}}{(1-\overline{z}w)^{2}}(1-\overline{z}w)^{2}k_{z}^{\mathcal{M}}$ $=( \frac{1-\|\lambda\|^{2}}{(1-\lambda\overline{z})(1-\overline{\lambda}w)})^{2}(1-\overline{z}w)^{2}k_{z}^{\mathcal{M}}$ . (5.2) Since trivially (5.1) is bounded on $D^{2},$ $(5.2)$ is square integrable on $D^{2}$. This concludes (i). Next, we shall show (ii). Since the integral kernel of $\triangle_{\lambda}$ is $(1-\overline{b_{\lambda}(z)}b_{\lambda}(w))^{2}k_{z}^{\mathcal{M}}(w)$, and using (5.2), we have $\Vert\triangle_{\lambda}-\triangle_{\mu}\Vert_{2}^{2}$ $= \int_{D}\int_{D}\|(1-\overline{b_{\lambda}(z)}b_{\lambda}(w))^{2}k_{z}^{\mathcal{M}}(w)-(1-\overline{b_{\mu}(z)}b_{\mu}(w))^{2}k_{z}^{\Lambda t}(w)\|^{2}dA(z)dA(w)$ $= \int_{D}\int_{D}\|(\frac{1-\|\lambda\|^{2}}{(1-\lambda\overline{z})(1-\overline{\lambda}w)})^{2}-(\frac{1-\|\mu\|^{2}}{(1-\mu\overline{z})(1-\overline{\mu}w)})^{2}\|^{2}$ $\cross\|(1-\overline{z}w)^{2}k_{z}^{\Lambda t}\|^{2}dA(z)dA(w)$ $arrow 0(\lambdaarrow\mu)$ (9) ### References [1] C. Apostol, H. Bercovici, C. ### Foia\S and C. Pearcy, Invariant subspaces, dilation theory, and the structure the predual ### of a dual algebra, J. Funct. Anal. (1985), ### 369-404. [2] A. Aleman, S. Richter and C. Sundberg, Beurling’s theorem ### for the Bergman space, Acta Math. 177 (1996), 275-310. [3] K. Guo, operators, and indices ### for analytic submodules, J. Funct. Anal. 213 (2004), ### 380-411. [4] K. Guo and P. Wang, ### Defect opemtors and Fredholmness ### for Toeplitz pairs with inner symbols, J. Operator Theory 58 (2007), ### 251-268. [5] K. Guo and R. Yang, The core submodules ### over the bidisk, Indiana Univ. Math. J. 53 (2004), 1, ### 205-222. [6] S. Richter, Invariant subspaces in Banach spaces ### of analytic functions, Trans. Amer. Math. Soc. 304 $(1_{6}987)$, no. 2, 585-616. [7] W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam 1969. [8] M. Seto, A perturbation theory operators ### of Hilbert-Schmidt submodules, preprint. [9] R. Yang, Hilbert-Schmidt submodules and issues ### of unitary equivalence, J. Operator Theory 53 (2005), ### no. 1, 169-184. [10] R. Yang, The core operator and congruent submodules, J. Funct. Anal. 228 (2005), no. 2, 469-489. [11] R. Yang and K. Zhu, The root operator on invariant subspaces ### of the Bergman space, Illinois J. Math. 47 (2003), no. 4, 1227-1242. Updating... Updating...	2021-08-02 02:50:21	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9477635025978088, "perplexity": 2521.094549740296}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154302.46/warc/CC-MAIN-20210802012641-20210802042641-00091.warc.gz"}
https://stats.stackexchange.com/questions/430665/existence-of-a-smoothed-median-function	# Existence of a Smoothed Median Function The median of a finite set of data depends on the data in a continuous, but non-smooth way; this can be demonstrated by considering the median of the data set $$\{ 0, 1, x \}$$ as $$x$$ varies. I wanted to know whether people have derived median-like statistics which depend smoothly on the data. By median-like, I mean roughly that • they recover location parameters, • they have some level of robustness associated to them, and ideally, • they recover the median in some limit. For example, it would be nice to have a parametrised family of functions $$\{ \text{med}_\varepsilon \}_{\varepsilon > 0}$$ which are each median-like, and such that as $$\varepsilon \to 0$$, one has that $$\text{med}_\varepsilon ( \mathcal{D} ) \to \text{median} ( \mathcal{D} )$$. To clarify: I am particularly interested in knowing whether this exists in the literature already. If people have their own ideas for how to derive statistics with these properties, I'll gladly hear them out, but I would prefer to hear about tried-and-tested ideas. • There is a huge number of such statistics: accounts can be found by searching for robust or resistant statistics. – whuber Oct 9 '19 at 12:57 • @whuber could you recommend a search term which would help to emphasise that I'm interested in the smooth dependence of the statistic on the data? A cursory google of `smooth robust statistics' wasn't particularly useful in this regard. – πr8 Oct 9 '19 at 20:19 • I"m having a hard time thinking of any such statistics that aren't smooth. That's not to say there aren't, but only to suggest that searching broadly for resistant or robust statistical estimators is likely to produce many examples of smooth ones in the sense you describe. Another search term would be "order statistic," because many of these estimators are functions of order statistics. – whuber Oct 9 '19 at 21:41 • @πr8 You may want to look at empirical influence functions of robust estimators; in particular you may find those for commonly used M-estimators do what you want. Oct 10 '19 at 1:04 Two standard families that do what you're asking for are Huber's M-estimators and (symmetrically) trimmed means. Trimmed means trim a percentage of $$\alpha$$ largest and smallest observations and take the mean of the rest. $$\alpha=0$$ gives the mean and $$\alpha$$ approaching 50% converges to the median. Huber's M-estimators are implicitly defined as minimising $$\sum_{i=1}^n\rho(x_i,\theta)$$ over $$\theta$$. The mean is obtained by $$\rho(x,\theta)=(x-\theta)^2$$, the median by $$\rho(x,\theta)=\|x-\theta\|$$. Huber proposed $$\rho$$-functions that behave like squares around zero and like the absolute value outside some interval $$[-c,c]$$ (with standardisation so that the whole thing is continuous). These are another "bridge" between means and medians and have a number of optimality features for distributions "close" but not equal to the normal distribution, as can be found in the Robust Statistics literature. The choice of $$[-c,c]$$ in practice depends on the scaling of the data, and $$\rho$$ is often chosen as function of $$\frac{\|x-\theta\|}{MAD}$$, where MAD is the median absolute deviation from the median. • The MAD can be non-smooth as the median, so it'll be non-smooth with MAD-scaling. Without scaling Huber's M-estimator is smooth as far as I know (I don't think it's difficult to prove but I won't do it; it also depends on what degree of smoothness you want), however if $[-c,c]$ is chosen too small it will behave pretty much as the median and smoothness around $\theta$ will only formally help. There are smooth ways of standardising, but these are more complex. Oct 9 '19 at 22:47 • Note that in the literature there is a "smoothed version" of Huber's M, see here: stat.ethz.ch/~stahel/hampel/HamFHR11.pdf. However "smoothness" there refers to $\psi=\rho'$ rather than smoothness in the observations, which is related, though. Oct 9 '19 at 22:50	2021-09-26 07:02:33	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7149499654769897, "perplexity": 475.23996150144205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057830.70/warc/CC-MAIN-20210926053229-20210926083229-00121.warc.gz"}
http://math.stackexchange.com/questions/409658/can-every-real-number-be-represented-by-a-possibly-infinite-decimal	# Can every real number be represented by a (possibly infinite) decimal? Does every real number have a representation within our decimal system? The reason I ask is because, from beginning a mathematics undergraduate degree a lot of 'mathematical facts' I had previously assumed have been consistently modified, or altogether stripped away. I'm wondering if my subconscious assumption that every real number can be represented in such a way is in fact incorrect? If so, is there a proof? If not, why not? (Also I'm not quite sure how to tag this question?) - What's your definition of real numbers? And the answer is yes but the representation isn't unique. – xavierm02 Jun 2 '13 at 21:45 – Cameron Buie Jun 3 '13 at 3:17 The answer is yes. The fact that $\mathbb{Q}$ is dense in $\mathbb{R}$ gives us the start line to reach every real number as a sequence of increasing long sequence of digits. ($1 \ 1,4 \ 1,41 \ 1,412 \ \dots \rightarrow \sqrt{2}$) to compute even more digits we have lots of algorithms. if your question is "I think about $x$ random real number, what's the decimal representation?" I will answer, "is between $a$ and $a+1$ ($a \in \mathbb{N}$) ? If yes, is between $a,1$ and $a,2$ for example (if no I will ask you for $a,2$ and $a,3$ ...) and so on. At every step I will build a sequence of digits that converge to your number. :) Obviously there are different sequences that converge to the same number ($1=0,\overline{9}$) and I use a intuitively means of convergence. - It's not the fact that $\mathbb Q$ is dense in $\mathbb R$ though, is it? It's the fact that numbers with a terminating decimal expansion are dense in $\mathbb R$. – TonyK Jun 2 '13 at 22:57 Ok, but numbers with a terminating (=finite?) decimal expansion are rational no? – Riccardo Jun 2 '13 at 23:12 Yes, but the converse is not true. – TonyK Jun 2 '13 at 23:15 Yes of course :) – Riccardo Jun 2 '13 at 23:23 Suppose $x$ is a positive real number. It can be shown that $n\mapsto 10^n$ is a map that is unbounded above in the integers, so by Archimedean property, there is some integer $n$ with $10^{n+1}\ge x$. Take the least such $n$ (why must one exist?), and let $a_{-n}$ be the greatest element $a$ of $\{0,1,2,...,8,9\}$ such that $a\cdot 10^n<x$ (why must one exist?). Now, given $a_{-n},a_{-n+1},...,a_{m}$ for some $m\in\Bbb Z$, we let $a_{m+1}$ be the greatest element $a$ of $\{0,1,2,...,8,9\}$ such that $$a\cdot 10^{-(m+1)}<x-\sum_{j=-m}^na_{-j}\cdot10^j$$ (why must one exist?) Recursively, this determines a sequence $a_{-n},a_{-n+1},...$ of elements of $\{0,1,2,...,8,9\}$ such that for all integers $m\ge-n$ we have $$S_m:=\sum_{j=-m}^na_{-j}\cdot 10^j<x.$$ In fact, $S_{-n},S_{-n+1},...,S_m,...$ is a non-decreasing sequence of positive numbers, so since bounded above by $x,$ this sequence converges to some number no greater than $x$ by the Monotone Convergence Theorem. We can even do better than that, and show that the sequence of partial sums $S_n$ converges to $x$ (why?). The series thus determined is the infinite decimal expansion of $x$. If $x$ had been negative, we could acquire a decimal expansion for $-x$ in this way, and then the opposite of that would be a decimal expansion for $x$. $0$ has a decimal expansion, too, so all real numbers have a decimal expansion, and moreover, all of them (except arguably $0$) have an infinite decimal expansion (though some may also have a finite decimal expansion). - Yes, every real number has a decimal representation. Numbers of the form $n/10^k$ where $n$ and $k$ are integers and $n \neq 0$ have two representations each (e.g. 1.000... and 0.999... represent the same real number). Depending on your conventions, zero has either one or two representations: e.g. $0$ and $-0$ in the latter case. Every other real number has only one decimal representation. Conversely, every decimal number that is all zeroes on the left is the representation of some real number. e.g. the decimal $$...000011.000...$$ is the number eleven. (recall that any digit that isn't written is zero, so the decimal "11" really has zeroes to its left and right, as notated explicitly above) In standard notation, decimals that are not all zeroes on the left such as $$...11111.000...$$ do not represent real numbers. There are generalized notations that allow some left-infinite decimals to represent real numbers (the number above would be -1/9). There are also other number systems that can be represented with decimals, such as the 10-adic numbers. It may be confusing to try and learn about such things at this point, though.... - "It may be confusing to try and learn about such things at this point, though...." Well, exactly. I was wondering why you brought it up here. – TonyK Jun 2 '13 at 22:55 @TonyK: Because it serves several useful purposes, such as: (1) it points out that curiosity of this sort is useful for mathematicians, (2) it pre-emptively answers any readers who do think about such things, (3) it quashes the idea that left-infinite decimals are an inherently stupid idea that should never have been brought up, (4) gives the interested reader (possibly including the OP in the future) something to look into, (5) exemplifies the point that notation is what we choose it to be, rather than something with inherent meaning, .... – Hurkyl Jun 3 '13 at 10:43 Irrational numbers were known to the ancient Greeks, as I expect you know. But it took humankind another 2000 years to come up with a satisfactory definition of them. This was mainly because nobody realised that a satisfactory definition was lacking. Once humankind realised this, various suggestions were proposed. One suggestion (Dedekind's) defined a real number as two infinite sets of rational numbers, which 'sandwiched' the real number; another suggestion (Cauchy's) defined a real number as an equivalence class of sequences obeying a certain convergence criterion. The details are available in many places. But the important point is that all of the reasonable definitions turned out to be equivalent -- the set of real numbers according to Dedekind's definition was 'the same' as the set of real numbers according to Cauchy's definition, although the definitions look completely different. Now, to your question: another reasonable definition of a real number is a non-terminating decimal expansion (we say non-terminating just to clear up the ambiguity that arises between e.g. 123.4599999... and 123.46 -- only the first is allowed). It turns out that this definition is equivalent to all the others. So your intuition is correct. But strictly speaking, your question is flawed: instead of asking whether every real number can be represented in this way, you should ask whether this representation of the real numbers is a valid one. And it is. -	2015-04-28 18:28:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9225965738296509, "perplexity": 327.010809724763}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246661916.33/warc/CC-MAIN-20150417045741-00087-ip-10-235-10-82.ec2.internal.warc.gz"}
https://crypto.stackexchange.com/questions/47171/gcm-as-a-general-purpose-cryptographic-hash?noredirect=1	# GCM as a general purpose cryptographic hash? Why are hash functions (e.g SHA-3) so complicated when GCM apparently provides secure hashing and has a relatively simple construction? Is this purely about speed? I imagine that GMAC with a fixed key (GCM mode without ciphered data) is much slower, if you just need a key-less hash. • In general, unkeyed hashes are slower than keyed hashes, since the attacker isn't limited by not knowing the key. – CodesInChaos May 4 '17 at 14:13 • GCM is only a mode of operation, which uses a blockcipher. That is on a different layer of abstraction than SHA-3. If you break down GCM with AES to the level of bytes and binary operations, SHA-3 might not look more complicated any more. – tylo May 4 '17 at 14:48 GCM does not provide secure hashing. In general, a MAC has all the properties of a hash only against an adversary who does not know the key. If you want to use the function as a MAC then the key has to be public and then A MAC is not a secure hash. With most common MAC constructions other than HMAC, if you know the key, you can easily construct, at least, a second preimage. For example, look at how the authentication tag GHASH of GCM is calculated. I use the notations from the Wikipedia article; the plaintext is $A_1A_2A_3\ldots$ (split into blocks), $H$ is calculated from the key (so it would be some public constant to use GHASH as a hash) and the $X_i$ are calculated incrementally to produce the hash (so if $X_i = X'_i$ for some $i$ and two different messages then the hash will be the same). \begin{align} X_0 &= 0 \\ X_1 &= A_1 \cdot H \\ X_2 &= (X_1 \oplus A_2) \cdot H = ((A_1 \cdot H) \oplus A_2) \cdot H \\ \end{align} Let $A'_1 = A_2 \cdot H$ and $A'_2 = A_1 \cdot H$. Then $A_1A_2A_3\ldots$ and $A'_1A'_2A_3\ldots$ are (assuming that $A_2$ didn't happen to be equal to $A1 \cdot H$) two distinct messages with the same hash (and the same length, incidentally). And that's just from a trivial computation — with a bit more work first preimage can be broken too. • Actually, first preimages really doesn't take much more work at all – poncho May 4 '17 at 15:49 • so why using a key if it can be broken so easily? Why doesn't GCM use an unkeyed hash? – David 天宇 Wong Mar 6 at 16:41 • @David天宇Wong GCM doesn't use a hash because what it needs is a MAC, not a hash. The goal of GCM is that if you don't know the key, you can't produce data with a valid authentication tag. It achieves that goal. If it used a hash, then anyone could produce a valid tag for a message, even without knowing the key, so GCM would not guarantee the authenticity of a message. – Gilles 'SO- stop being evil' Mar 6 at 17:56 • oh right, you could just XOR the tag with the hash of the ciphertext and then XOR it again with the hash of a new ciphertext – David 天宇 Wong Mar 6 at 18:04 GHASH/GMAC is a secure MAC which has different security properties than a generic hash construction; contrary to your assumption, it is not a secure hash. For example, if you use GHASH/GMAC with a known key, the scheme would be trivially vulnerable to a length extension attack. • I've added a line explaining why the length extension attack (among other issues) prevents GHASH/GMAC to be used as a generic, secure hash. Hope that's OK, otherwise revert :) – Maarten Bodewes May 4 '17 at 11:34 • That's fine, thanks for the addition @MaartenBodewes – mat May 4 '17 at 11:56 • It's not just length extension, which could be fixed by appending the length to the hash. Like other common MAC apart from HMAC, if you know the key, you can forge preimages in many ways. – Gilles 'SO- stop being evil' May 4 '17 at 12:10	2019-11-21 11:19:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9028604030609131, "perplexity": 1474.9754155309072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670770.21/warc/CC-MAIN-20191121101711-20191121125711-00292.warc.gz"}
https://hpmuseum.org/forum/thread-15285.html	monic part 3: towards a handheld 07-01-2020, 01:54 AM (This post was last modified: 12-09-2020 01:41 AM by F-73P.) Post: #1 F-73P Junior Member Posts: 18 Joined: May 2020 monic part 3: towards a handheld Previous: monic part 2: connecting the keypad I will now place the display and matrix keypad on veroboard. The calculator can then be connected to a development board (such as the NXP i.MX RT1010 I have been using so far) via terminal blocks on the left of the veroboard. I am also trying to find keycaps for the Omron B3F-1052 pushbuttons; any suggestions? 07-01-2020, 10:17 AM Post: #2 Maximilian Hohmann Senior Member Posts: 865 Joined: Dec 2013 RE: monic part 3: towards a handheld Hello! Very interesting project, looking forward to see more of it. (07-01-2020 01:54 AM)F-73P Wrote: I am also trying to find keycaps for the Omron B3F-1052 pushbuttons; any suggestions? I have used this kind for some tinkering projects in the past (this is an eBay-link, unfortunately I couldn't find them anywhere else - they come from China anyway, no matter who you order them from): https://www.ebay.com/itm/A14-Tactile-Cap...3216076402 You can select a keycap in the colour you like and snap a transparent cover over it. The gap between keycap and cover is large enough for a key label printed on solid/thick paper. These caps should also fit your Omron switches, but better compare the measurements. Regards Max NB: Maybe I missed it in your earlier posts, but what does "monic" stand for? 07-23-2020, 04:02 AM Post: #3 Garth Wilson Senior Member Posts: 444 Joined: Dec 2013 RE: monic part 3: towards a handheld (07-01-2020 01:54 AM)F-73P Wrote: I am also trying to find keycaps for the Omron B3F-1052 pushbuttons; any suggestions? We use them in a couple of our products at work. Availability has never been a problem at all. See https://www.mouser.com/Electromechanical...ps&FS=True 07-28-2020, 09:21 PM (This post was last modified: 07-28-2020 09:25 PM by Chr Yoko.) Post: #4 Chr Yoko Member Posts: 55 Joined: May 2020 RE: monic part 3: towards a handheld Very nice project ! Your chosen microcontroller looks very powerfull , why such choice instead of a 32bits Arduino such as a STM32 base ? Hereby EEVblog that is showing how to get made a custom membrane keyboard : From here : (exemple @ about US$255 for 4 parts, with embossed keys , their stainless domes and 4 colors on the overlay) OK, bit expensive and not as good as a real keyboard, but has a professionnal look to it. Also I would be interrested by the software side of your project ;-) 08-01-2020, 05:58 AM (This post was last modified: 12-27-2020 03:37 AM by F-73P.) Post: #5 F-73P Junior Member Posts: 18 Joined: May 2020 RE: monic part 3: towards a handheld Thanks for the suggestions. I have ordered Omron and Alps Alpine switches and keycaps, and have started making some keypad labels: (07-28-2020 09:21 PM)Chr Yoko Wrote: Also I would be interrested by the software side of your project ;-) See here for firmware for a programmable scientific calculator. After the handheld is built I will focus on adding support for arbitrary-precision rational and floating-point arithmetic. (07-28-2020 09:21 PM)Chr Yoko Wrote: Your chosen microcontroller looks very powerfull , why such choice instead of a 32bits Arduino such as a STM32 base ? The i.MX RT EVK boards offer speed and large memories, and the IDE is free and supports programming in C and assembly. (07-28-2020 09:21 PM)Chr Yoko Wrote: Hereby EEVblog that is showing how to get made a custom membrane keyboard : https://www.youtube.com/watch?v=H8XBBfvsPj0 From here : https://www.jrpanel.com/switch/buy/ (exemple @ about US$255 for 4 parts, with embossed keys , their stainless domes and 4 colors on the overlay)	2021-06-19 00:28:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3145776093006134, "perplexity": 10484.410238170372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487643354.47/warc/CC-MAIN-20210618230338-20210619020338-00180.warc.gz"}
http://content.oss.deltares.nl/imod/iMOD_Manual_actual/imod-um-CCF-files.html	# iMOD User Manual version 5.2 (html) #### 9.10CCF-files A CCF file describes the cross-section of a river and can be inserted to one or more ISG river segments. The CCF file is used in the iMOD BATCH function GEN2ISG. The File syntax for CCF-files is very simple and stored in ASCII-format. In this way these type of files can be easily edited and/or created outside iMOD with other (commercial) software. The formal syntax is as follows and prescribed: Example of an CCF-file: X,Y -10 5 -5 0 5 0 10 5 The following figure shows the cross section described by the above CCF file :	2022-08-15 22:51:32	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8103784918785095, "perplexity": 3387.629455824616}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572212.96/warc/CC-MAIN-20220815205848-20220815235848-00242.warc.gz"}
https://labs.tib.eu/arxiv/?author=Constantin%20Dory	• ### Nanodiamond integration with photonic devices(1610.03183) We discuss the progress in integration of nanodiamonds with photonic devices for quantum optics applications. Experimental results in GaP, SiO2 and SiC-nanodiamond platforms show that various regimes of light and matter interaction can be achieved by engineering color center systems through hybrid approaches. We present our recent results on the growth of color center-rich nanodiamond on prefabricated 3C-SiC microdisk resonators. These hybrid devices achieve up to five-fold enhancement of diamond color center light emission and can be employed for integrated quantum photonics. • ### Cavity-enhanced Raman emission from a single color center in a solid(1804.06533) We demonstrate cavity-enhanced Raman emission from a single atomic defect in a solid. Our platform is a single silicon-vacancy center in diamond coupled with a monolithic diamond photonic crystal cavity. The cavity enables an unprecedented frequency tuning range of the Raman emission (100 GHz) that significantly exceeds the spectral inhomogeneity of silicon-vacancy centers in diamond nanostructures. We also show that the cavity selectively suppresses the phonon-induced spontaneous emission that degrades the efficiency of Raman photon generation. Our results pave the way towards photon-mediated many-body interactions between solid-state quantum emitters in a nanophotonic platform. • ### Strongly Cavity-Enhanced Spontaneous Emission from Silicon-Vacancy Centers in Diamond(1708.05771) Dec. 6, 2017 quant-ph, cond-mat.mes-hall Quantum emitters are an integral component for a broad range of quantum technologies including quantum communication, quantum repeaters, and linear optical quantum computation. Solid-state color centers are promising candidates for scalable quantum optics due to their long coherence time and small inhomogeneous broadening. However, once excited, color centers often decay through phonon-assisted processes, limiting the efficiency of single photon generation and photon mediated entanglement generation. Herein, we demonstrate strong enhancement of spontaneous emission rate of a single silicon-vacancy center in diamond embedded within a monolithic optical cavity, reaching a regime where the excited state lifetime is dominated by spontaneous emission into the cavity mode. We observe 10-fold lifetime reduction and 42-fold enhancement in emission intensity when the cavity is tuned into resonance with the optical transition of a single silicon-vacancy center, corresponding to 90% of the excited state energy decay occurring through spontaneous emission into the cavity mode. We also demonstrate the largest to date coupling strength ($g/2\pi=4.9\pm0.3 GHz$) and cooperativity ($C=1.4$) for color-center-based cavity quantum electrodynamics systems, bringing the system closer to the strong coupling regime. • ### Signatures of two-photon pulses from a quantum two-level system(1709.01120) The theoretical community has found interest in the ability of a two-level atom to generate a strong many-body interaction with light under pulsed excitation. Single-photon generation is the most well-known effect, where a short Gaussian laser pulse is converted into a Lorentzian single-photon wavepacket. However, recent proposals have surprisingly suggested that scattering with intense laser fields off a two-level atom may generate oscillations in two-photon emission that are out of phase with its Rabi oscillations, as the power of the pulse increases. Here, we provide an intuitive explanation for these oscillations using a quantum trajectory approach and show how they may preferentially result in emission of two-photon pulses. Experimentally, we observe signatures of these oscillations by measuring the bunching of photon pulses scattered off a two-level quantum system. Our theory and measurements provide crucial insight into the re-excitation process that plagues on-demand single-photon sources while suggesting the production of novel multi-photon states. • ### An on-chip architecture for self-homodyned nonclassical light(1611.01566) In the last decade, there has been remarkable progress on the practical integration of on-chip quantum photonic devices yet quantum state generators remain an outstanding challenge. Simultaneously, the quantum-dot photonic-crystal-resonator platform has demonstrated a versatility for creating nonclassical light with tunable quantum statistics, thanks to a newly discovered self-homodyning interferometric effect that preferentially selects the quantum light over the classical light when using an optimally tuned Fano resonance. In this work, we propose a general structure for the cavity quantum electrodynamical generation of quantum states from a waveguide-integrated version of the quantum-dot photonic-crystal-resonator platform, which is specifically tailored for preferential quantum state transmission. We support our results with rigorous Finite-Difference Time-Domain and quantum optical simulations, and show how our proposed device can serve as a robust generator of highly pure single- and even multi-photon states. • ### Tuning the Photon Statistics of a Strongly Coupled Nanophotonic System(1610.00174) We investigate the dynamics of single- and multi-photon emission from detuned strongly coupled systems based on the quantum-dot-photonic-crystal resonator platform. Transmitting light through such systems can generate a range of non-classical states of light with tunable photon counting statistics due to the nonlinear ladder of hybridized light-matter states. By controlling the detuning between emitter and resonator, the transmission can be tuned to strongly enhance either single- or two-photon emission processes. Despite the strongly-dissipative nature of these systems, we find that by utilizing a self-homodyne interference technique combined with frequency-filtering we are able to find a strong two-photon component of the emission in the multi-photon regime. In order to explain our correlation measurements, we propose rate equation models that capture the dominant processes of emission both in the single- and multi-photon regimes. These models are then supported by quantum-optical simulations that fully capture the frequency filtering of emission from our solid-state system. • ### Complete Coherent Control of Silicon-Vacancies in Diamond Nanopillars Containing Single Defect Centers(1701.04961) Jan. 18, 2017 cond-mat.mes-hall Arrays of identical and individually addressable qubits lay the foundation for the creation of scalable quantum hardware such as quantum processors and repeaters. Silicon vacancy centers in diamond (SiV) offer excellent physical properties such as low inhomogeneous broadening, fast photon emission, and a large Debye-Waller factor, while the possibility for all-optical ultrafast manipulation and techniques to extend the spin coherence times make them very promising candidates for qubits. Here, we have developed arrays of nanopillars containing single SiV centers with high yield, and we demonstrate ultrafast all-optical complete coherent control of the state of a single SiV center. The high quality of the chemical vapor deposition (CVD) grown SiV centers provides excellent spectral stability, which allows us to coherently manipulate and quasi-resonantly read out the state of individual SiV centers on picosecond timescales using ultrafast optical pulses. This work opens new opportunities towards the creation of a scalable on-chip diamond platform for quantum information processing and scalable nanophotonics applications. • ### Self-homodyne enabled generation of indistinguishable photons(1512.05626) The rapid generation of non-classical light serves as the foundation for exploring quantum optics and developing applications such as secure communication or generation of NOON-states. While strongly coupled quantum dot-photonic crystal resonator systems have great potential as non-classical light sources due to their promise of tailored output statistics, the generation of indistinguishable photons has been obscured due to the strongly dissipative nature of such systems. Here, we demonstrate that the recently discovered self-homodyne suppression technique can be used to overcome this limitation and tune the quantum statistics of transmitted light, achieving indistinguishable photon emission competitive with state-of-the-art metrics. Furthermore, our nanocavity-based platform directly lends itself to scalable on-chip architectures for quantum information. • ### Self-homodyne measurement of a dynamic Mollow triplet in the solid state(1512.04102) The study of light-matter interaction at the quantum scale has been enabled by the cavity quantum electrodynamics (CQED) architecture, in which a quantum two-level system strongly couples to a single cavity mode. Originally implemented with atoms in optical cavities, CQED effects are now also observed with artificial atoms in solid-state environments. Such realizations of these systems exhibit fast dynamics, which makes them attractive candidates for devices including modulators and sources in high-throughput communications. However, these systems possess large photon out-coupling rates that obscure any quantum behavior at large excitation powers. Here, we have utilised a self-homodyning interferometric technique that fully employs the complex mode structure of our nanofabricated cavity to observe a quantum phenomenon known as the dynamic Mollow triplet. We expect this interference to facilitate the development of arbitrary on-chip quantum state generators, thereby strongly influencing quantum lithography, metrology, and imaging. • ### Complete Coherent Control of a Quantum Dot Strongly Coupled to a Nanocavity(1512.05952) Strongly coupled quantum dot-cavity systems provide a non-linear configuration of hybridized light-matter states with promising quantum-optical applications. Here, we investigate the coherent interaction between strong laser pulses and quantum dot-cavity polaritons. Resonant excitation of polaritonic states and their interaction with phonons allow us to observe coherent Rabi oscillations and Ramsey fringes. Furthermore, we demonstrate complete coherent control of a quantum dot-photonic crystal cavity based quantum-bit. By controlling the excitation power and phase in a two-pulse excitation scheme we achieve access to the full Bloch sphere. Quantum-optical simulations are in good agreement with our experiments and provide insight into the decoherence mechanisms. • ### Hybrid Group IV Nanophotonic Structures Incorporating Diamond Silicon-Vacancy Color Centers(1509.01617) We demonstrate a new approach for engineering group IV semiconductor-based quantum photonic structures containing negatively charged silicon-vacancy (SiV$^-$) color centers in diamond as quantum emitters. Hybrid SiC/diamond structures are realized by combining the growth of nanoand micro-diamonds on silicon carbide (3C or 4H polytype) substrates, with the subsequent use of these diamond crystals as a hard mask for pattern transfer. SiV$^-$ color centers are incorporated in diamond during its synthesis from molecular diamond seeds (diamondoids), with no need for ionimplantation or annealing. We show that the same growth technique can be used to grow a diamond layer controllably doped with SiV$^-$ on top of a high purity bulk diamond, in which we subsequently fabricate nanopillar arrays containing high quality SiV$^-$ centers. Scanning confocal photoluminescence measurements reveal optically active SiV$^-$ lines both at room temperature and low temperature (5 K) from all fabricated structures, and, in particular, very narrow linewidths and small inhomogeneous broadening of SiV$^-$ lines from all-diamond nano-pillar arrays, which is a critical requirement for quantum computation. At low temperatures (5 K) we observe in these structures the signature typical of SiV$^-$ centers in bulk diamond, consistent with a double lambda. These results indicate that high quality color centers can be incorporated into nanophotonic structures synthetically with properties equivalent to those in bulk diamond, thereby opening opportunities for applications in classical and quantum information processing. • ### Ultrafast polariton-phonon dynamics of strongly coupled quantum dot-nanocavity systems(1503.05595) July 29, 2015 quant-ph, cond-mat.mes-hall We investigate the influence of exciton-phonon coupling on the dynamics of a strongly coupled quantum dot-photonic crystal cavity system and explore the effects of this interaction on different schemes for non-classical light generation. By performing time-resolved measurements, we map out the detuning-dependent polariton lifetime and extract the spectrum of the polariton-to-phonon coupling with unprecedented precision. Photon-blockade experiments for different pulse-length and detuning conditions (supported by quantum optical simulations) reveal that achieving high-fidelity photon blockade requires an intricate understanding of the phonons' influence on the system dynamics. Finally, we achieve direct coherent control of the polariton states of a strongly coupled system and demonstrate that their efficient coupling to phonons can be exploited for novel concepts in high-fidelity single photon generation.	2020-06-04 01:59:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3281237483024597, "perplexity": 3034.8615352692022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347436828.65/warc/CC-MAIN-20200604001115-20200604031115-00480.warc.gz"}
https://tex.stackexchange.com/questions/213441/installing-a-otf-font-and-using-it-with-fontspec-lualatex-on-a-mac?noredirect=1	# Installing a .otf font and using it with fontspec (LuaLaTeX on a Mac) I've downloaded an .otf font (specifically one called Dalelands Uncial) that I would like to use in a document. I've double-clicked on the file and selected install, so that it now sits in my ~/Library/Fonts folder on my Mac and Font Book claims the file is installed. I thought that running LuaLaTeX on the following minimal example would now allow me to access the font, but it doesn't work. \documentclass{article} \usepackage{fontspec} \setmainfont[ Path = /Users/<myname>/Library/Fonts, Extension = .otf, Ligatures = TeX ]{Dalelands Uncial} \begin{document} Some sample text. \end{document} (This is my attempt to replicate what is suggested in the first answer to this question.) I'm told that "DalelandsUncial" cannot be found. Is it the space in the name causing a problem? • Try Dalelands_Uncial – Yiannis Lazarides Nov 22 '14 at 17:11 • It shouldn't be necessary to specify either the path or the extension if the font has been installed. But as indicated by Yiannis, the mistake must be in the {Dalelands Unical} part. – Sverre Nov 22 '14 at 17:16 • Do you have an up-to-date Texlive 2014? If not a manual run of luaotfload-tool --update might be necessary – MaxNoe Nov 22 '14 at 17:51 • Tried those: "Dalelands_Uncial" gives the same error. Also just ran luaotfload-tool --update and I still get the same error. – Martyn Quick Nov 22 '14 at 18:08 • I have downloaded the font from dafont.com/it/dalelands.font and installed it in ~/Library/Fonts. Your document, after removing Path=... and Extension=... compiles without problems (LuaTeX triggers a run of luaotfload-tool the first time). – egreg Nov 22 '14 at 18:36 It turns out that all I needed to do was put the .otf files in a subdirectory of ~/Library/texlive/2014/texmf-var/fonts and they can then be found by LuaLaTeX.	2019-10-15 08:29:56	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6722174286842346, "perplexity": 3384.0108738658864}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986657949.34/warc/CC-MAIN-20191015082202-20191015105702-00239.warc.gz"}
https://beddinge20.se/exe4j-portable-3-5-0-2-crack-download/	If you're a Java developer looking for a simple software solution to create installer files for your applications, you can turn to exe4j Portable. Made as the no-install counterpart of exe4j, this lightweight tool puts a wizard at your disposal to collect all the required info and compile your executables. Note that you must have Java Runtime Environment (JRE) 1.6 or newer to be able to successfully launch the utility. Windows launcher generator for Java apps The wizard has ten steps that have to be taken before producing the Java executable files. A couple of demo files are included in the package to give you an idea of how exe4j Portable works. 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Is there a more general theorem that states if the variances of a sequence of random variables are bounded as $E(X^2)\leq\epsilon^2$ for some $\epsilon>0$, then $E(X_p)\leq0$ for all $p$ and $x$ with \$0\ ## Exe4j Portable 3.5.0.2 License Keygen 2022 Create, edit, and compile Java.EXE files Create no-install EXE files (cannot include files or folders other than JARs) Take over the advanced application editing from the application Edit app and EXE info, use a splash screen, and more Advanced EXE settings focus on redirection (errors, logs), Windows service (startup mode, dependencies), version info, 32- or 64-bit mode, and the manifest (execution level, DPI awareness) Intuitive and practical Java builder exe4j Portable Product Key Requirements: Windows/NT/2000/XP 0.9 MB (1.7 MB compressed) Java Archive (.jar) is a container used to store Java source files and compiled class files. JARs are created by the JAR builder. 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Open Source Software (OSS) Excel Applications Free Excel documents are available as freeware from OpenOffice.org, and there are a lot of alternatives out there. The most notable competitors are FreeOffice, ExcelToOpen, and CalcFrenzy. All of these spreadsheets are cross-platform, with easy-to-use web interfaces and Office documents. They come with some sort of free-to-use, community license to get you started, but they are not all open source. FreeOffice.org, in particular, uses the more restrictive GNU General Public License, which means that the application and source code are available for everyone to see. An example of this is the FreeOffice web interface, called FreeOffice Web, that allows people to get Excel data in the same way as Excel spreadsheets, while also offering some of the other features available in Excel. Another notable example is ExcelToOpen’s download of the web interface of OpenOffice.org, which is known as OpenOffice Online.	2022-12-02 16:20:30	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21741120517253876, "perplexity": 4369.87710823268}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710909.66/warc/CC-MAIN-20221202150823-20221202180823-00368.warc.gz"}
https://chemistry.stackexchange.com/questions/51850/hybridization-and-excitation-of-electrons	# Hybridization and excitation of electrons In case of hybridization of PCl5 why does the electron move to 3d orbital though 4s orbital has a lower energy? • quora.com/… – Kenny Lau May 27 '16 at 14:16 • In short, 3d>4s only in group I and II. – Kenny Lau May 27 '16 at 14:16 • – Kenny Lau May 27 '16 at 14:17 • $\ce{PCl5}$ is poorly described by hybridization. – bon May 27 '16 at 15:47 • It simply does not work like that - significant contribution of d orbitals in p black compounds was disproved many years ago. – Mithoron May 28 '16 at 22:07 It doesn’t. It simply doesn’t. The $\mathrm{3d}$ orbital has a similar energy to the $\mathrm{4s}$ one — nobody considers $\mathrm{4s}$ to take part in bonding but historically, invoking $\mathrm{d}$-orbitals was very popular. The better description is to have an $\mathrm{sp^2}$ hybridised phosphorus atom which has a free p-orbital. The $\mathrm{sp^2}$ orbitals bond with three chlorides in traditional two-electron-two-centre bonds while the remaining $\mathrm{p}$-orbital — populated by two electrons — participates in a four-electron-three-centre bond with the two remaining chlorines. You can think of this as two resonance structures: $$\ce{Cl-{P+}Cl3\bond{...}Cl- <-> Cl- \bond{...}{P+}Cl3-Cl}$$	2020-02-19 08:42:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3508121371269226, "perplexity": 3036.3221910283696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144058.43/warc/CC-MAIN-20200219061325-20200219091325-00286.warc.gz"}
https://www.vedantu.com/question-answer/if-asec-theta-+-btan-theta-+-c-0-and-psec-theta-class-10-maths-cbse-5ec6c6f4b43dda405b9ef67f	Courses Courses for Kids Free study material Free LIVE classes More # If $a\sec \theta + b\tan \theta + c = 0$ and ${\text{p}}\sec \theta + q\tan \theta + r = 0$, prove that${\left( {br - qc} \right)^2} - {\left( {pc - ar} \right)^2} = {\left( {aq - bp} \right)^2} \\$ Last updated date: 20th Mar 2023 Total views: 311.7k Views today: 5.91k Verified 311.7k+ views Hint: Make use of the cross multiplication technique & trigonometric identities to solve this problem. $a \sec \theta + b\tan \theta + c = 0.........................(1) \\$ $p \sec \theta + q\tan \theta + r = 0..........................(2) \\$ Solve these two equations for $\sec \theta$ and $\tan \theta$ by the cross multiplication technique, we get, $\dfrac{{\sec \theta }}{{br - qc}} = \dfrac{{\tan \theta }}{{cp - ar}} = \dfrac{1}{{aq - bp}} \\$ $\Rightarrow \sec \theta = \dfrac{{br - qc}}{{aq - bp}}$ & $\tan \theta = \dfrac{{cp - ar}}{{aq - bp}} \\$ Now, you know that, ${\sec^2}\theta - {\tan^2}\theta = 1 \\$ $\Rightarrow {\left( {\dfrac{{br - qc}}{{aq - bp}}} \right)^2} - {\left( {\dfrac{{cp - ar}}{{aq - bp}}} \right)^2} = 1 \\$ $\Rightarrow {\left( {br - qc} \right)^2} - {\left( {cp - ar} \right)^2} = {\left( {aq - bp} \right)^2} \\$ Hence Proved. Note: In this particular type of question, solve these equations and then apply trigonometry identity to get your answer. Mistakes should be avoided during cross multiplication.	2023-03-23 14:09:54	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.836046040058136, "perplexity": 2858.0532653940236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945168.36/warc/CC-MAIN-20230323132026-20230323162026-00768.warc.gz"}
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Syntax: FINDSTR Searches for strings in files. bat with the following content. The zero ordinal argument is the name of the batch file itself. txt extension. Updated: 05/21/2018 by Computer Hope. ( del filename. Create a text file and copy the below command line: Echo batch file delete folder @RD /S /Q "D:\testfolder". See also: CALL, GOTO, and RETURN. What you can do is take steps to make sure that the batch file exits as quickly as possible. The command NET USE allows you to setup a mapped network drive and follows this syntax: net use. Testing was minimal as I didn't want to break it! Alan. bat file extension in the name, for example BATCHINSTALL. In the Run: drop down, choose Minimized. The COLOR command only defines the color of the entire window console. txt from D drive to E drive. bat file or batch script) is a text file that the Windows cmd. 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I've seen lots of information and sample scripts but none of them seem to work properly for my specific command need Here's what I need: ----- sfc /scannow cls powershell. Some of the more important ones include: ECHO - Displays text on the screen @ECHO OFF - Hides the text that is normally output; START - Run a file with its default application; REM - Inserts a comment line in. Follow asked Feb 26 '14 at 12:35. In addition, this can be done by scheduling this batch file to delete folder on the windows task scheduler. sql in the batch file. Command to turn on echo. SilentCMD [path to. EXE or something, use the batch file name only) and the optional second parameter is the list of batch file arguments written in one string, separated with black spaces. start call "C:\Program Files\MongoDB\Server\4. exe command line processor executes as a batch job. Open a command window in the directory containing the. FOR /L %%var_name IN (Lowerlimit, Increment, Upperlimit) Do some_code. 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Once you understand how to create a basic Batch file that can delete other files, you can create a more advanced Batch file to delete all files matching a specific file type from a specific folder, or even delete every file from a folder and. Hosts file is basically a text file buried deep inside your computer’s Windows folder, but you won’t find a. These are useful to get the maximum out of your GPU, especially if its a 2gb variant. Updated: 05/21/2018 by Computer Hope. Batch Script: A batch script is a text file that contains certain commands that are executed in sequence. EXIST command is used to check if a file exists or not. The goto command moves a batch file to a specific label or location, enabling a user to rerun it or skip other lines depending on inputs or events. You can automate SFTP file transfer in Unix and Linux using batch file. Windows batch file syntax using exclamation mark. Therefore to get the location of an executing batch file from within that batch file you can use the variable %~dp0. It is a file - with. bat file is useful for scripters who need to use the current date variable in the format MMDDYYYY. To do this, choose Start → Accessories → Command Prompt. If you are on-network environment and you will need to remove the mapped network drive to all computers, the following method will come in help. NOTE: The quotation marks are necessary only when a variable is used in a logical IF statement. Lets save this batch file as test. As shown in syntax, a function definition in batch file starts with the declaration of a function with a label. chmod +x daily_bat. @echo OFF ::EXIST command is used to check for existence IF EXIST D:\abc. bat and contains all the command line prompts that have been pre-typed in. If you are running mysql under Windows and have some special characters in the file that cause problems, you can do this: C:\> mysql -e "source batch-file" If you need to specify connection parameters on the command line, the command might look like this:. exe which will appear in the same window. A batch file is a type of script that contains a series of commands. This is only one of many uses for batch files. Double-click the BAT file. After a GOTO command in a batch file, the next line to be executed will be the one immediately following the label. These are useful to get the maximum out of your GPU, especially if its a 2gb variant. By default echo is turned on for any batch file. Here is an example where a Python exe is. The quotes are important if your file names contain any spaces. Take Command is a comprehensive interactive GUI and command line environment that makes using the Windows command prompt and writing batch files easy, faster and far more powerful. Lower case a String. Step 7: Select what to run - you want to run a program. When you run a batch file, the. The last section explains how to embed the cecho utility into a batch file using the Debug. Hosts file is basically a text file buried deep inside your computer’s Windows folder, but you won’t find a. The command NET USE allows you to setup a mapped network drive and follows this syntax: net use. A batch file is a text file that includes the lines of Microsoft DOS commands. You can monitor your FTP Server by running SLCheck as a scheduled task, e. NET Framework 3. The '>' command is used to redirect the output to the file C:\lists. The ELSE clause must occur on the same line as the command after the IF. exe supports a /s command-line switch to not display these messages. I want to have a copy of the output of the batch file so I have a record that I can access later of all the files I have copied and when. bat file or batch script) is a text file that the Windows cmd. Why DOS-style Batch Files? This series will share some conventions I picked up along the way for scripting in Windows via command prompt batch files. You can refer to other files in the same folder as the batch script by using this syntax: CALL %0\. bat with the following content. exe program (until Windows Vista). A command in batch program may consist of sub commands and switches. They should be enclosed in quotes if they contain spaces. First parameter is the batch file name (don't use CMD. exe) on Windows server 2012. You can also use the Shell to run scripts like batch files, PowerShell scripts, PERL, Python etc. When you type the filename at the command prompt, Cmd. You substitute the variables with your drive information: net use F: \\server\files. Windows batch file syntax using exclamation mark. exe runs the commands sequentially as they appear in the file. csv" file will be created. I find it much easier to create a batch file. Save the file using a. Represents a replaceable parameter. You can schedule this bat file to run on GPO or any other platform used to manage computers. bat extension. You can make a batch file to run just about any Command Prompt command, either manually or on a schedule. For example, a batch file could be used to run frequently utilized commands, delete or move a series of files, and other jobs. Batch file to delete folder. This allows the data on the line to be passed as arguments to the batch file. You can also use it to run commands straight from a batch file, by including the -Command parameter and appropriate arguments. The command NET USE allows you to setup a mapped network drive and follows this syntax: net use. Command to turn on echo. If you save this snippet as "test. 5 in Win8: " Or, you can use the Windows installation media as the file source when you enable the. Double click the bat file. txt" The command will create a copy of the original files with the new extension. First, copy and paste the text in quotes to a text file. Both these batch files work just fine. So the vcvars. A response file is a plain text file (encoded in the current ANSI code page, or UTF-8 with a leading BOM, or UTF-16 with its leading BOM) that contains one or more command line arguments separated by any combination of spaces, tabs, or newlines. Now that the batch file has been made executable, you can execute the content of the batch file by typing in the following command. The basic syntax to use in shortcuts or similar is quite simple. Console programs start in a new window. Go to the folder that contains your BAT file. exe program (until Windows Vista). You now have a batch file with the. For example my batch file is as follows: set /p hostname=plink -batch devdb [complex-linux-command] In the above example, if the complex-linux-command fails, or asking for an input from the user, or hangs, etc, then your Windows Batch script will not be waiting. sql before Select query to. exe command line processor executes as a batch job. txt C:\User\Username\Desktop. txt" The command will create a copy of the original files with the new extension. It has an additional role as the usual first program run after boot (init process), hence being responsible for setting up the system by running the AUTOEXEC. Run the batch with a double click and check if the folder was deleted. Use goto instead of an expression inside parentheses, or enable delayed variable expansion. All results are logged in a logfile. NOTE: The quotation marks are necessary only when a variable is used in a logical IF statement. Uses the echo off command to ensure that the commands are not shown when the code is executed. As you can see, the Command Prompt opens in the "C:\Users\Documents\Blog\BatchMode" directory, and we execute the. If you wish to use a batch file to create a menu which will allow you to do more, read through tutorials on DOS commands. The simplest idea of how to write a batch file is: Figure out how you would type the commands at a DOS prompt, then type them, one per line, in a text file — and you've written your batch file. And voila, there's your basic script. %A : The A in %A may be replaced by any character, either upper case or lower case, except numbers. One option is to require that a batch file is executed. Below you can find examples for using Robocopy in various usecases. exe) on Windows server 2012. Normally, to run a batch file, all that is needed is to double-click the file. If only ASSOC is written and executed, it will display all the file associations for every extension, instead of just. bat, and run it from the command line to view it. Method 1 − Go to C:\Windows\System32 and double click on the cmd file. It is a file - with. Double click the bat file. Go to the folder that contains your BAT file. Follow asked Feb 26 '14 at 12:35. The command interpreter displays the prompt when prompting for a new command line in interactive mode, or when echoing a batch file line in batch file mode. Open notepad and save the below command as remove_map. As the goto executable is a part of the cmd. Is there an Echo command in the batch file, say at the end, that might say, "All Done"? The batch file deletes all the files in a folder in the same path. Provides Help information for Windows commands. The complete windows batch script will look like below. I've seen lots of information and sample scripts but none of them seem to work properly for my specific command need Here's what I need: ----- sfc /scannow cls powershell. Create Batch File To Run Command You may also like. START [name of batch file to delay, then stuff keyboard] spjm -s IP -user admin -quiet. Will the batch file run thorugh a shell call when placed in another. rar" /z /MIR. My batch file is simply this command, though I have added echo, pause, etc for the purpose of troubleshooting to see if it's running. Learn some basic batch commands. So the variable %0 in our script HelloWorld. The Command shell was the first shell built into Windows to automate routine tasks, like user account management or nightly backups, with batch (. Names with in % mean variables, not sure what ! mark mean in a batch file. By default, start returns immediately without waiting for the program to exit, so the batch file will continue to run and, presumably, exit immediately. exe Start starts a program and does not wait. Expand Copy Code. Robocopy command syntax and examples. bat file will continuously open Command Prompt windows, thus eating up your system's RAM. The DO command is then executed with the parameter(s) set to the token(s) found. Each label must be defined on a line by itself, beginning with a colon and ending with either a space, a colon or a CR/LF. This is fine for interactive use, but sometimes we just want to delay the batch file for a fixed number of seconds, without user interaction. cecho is an enhanced ECHO command line utility with color support, inspired by the CTEXT utility by Dennis Bareis. See Run MongoDB Community Edition from the Command Interpreter for instructions to start a MongoDB instance. com -r "220 ProFTPD" 220 ProFTPD can be the normal answer from a FTP Server. After saving it with. Type goto crash. Files can be copied, DOS commands executed, and more. Batch commands. How to Create FileName Variable with Date And Time From Command CMD Batch File For Use in Creating Unique Filename. You have to run CMD. If your current date time is Nov 02, 2017 15:41:36, then the above example will create a file in the current directory with name "access_20171102-154136. You need to double the percent sign to escape it. The COLOR command only defines the color of the entire window console. For automation, commands can be read from a script file specified by /script switch, passed from the command-line using the /command switch, or read from standard input of winscp. txt from D drive to E drive. bat and contains all the command line prompts that have been pre-typed in. The label must begin with a colon [:] and appear on a line by itself, and cannot be included in. If 1, 2, or 3 is entered, goto goes to the corresponding label and performs the echo and goes to the end of the batch file. EXE using the batch file itself as SQL file argument to be executed. This particular batch file sets ECHO off (which cleans up the output by hiding the commands from being printed at the prompt, prints the text "Hello World" to the screen, and then waits for you to press a key before it ends. Calculate the length of a string. The batch script will then run the OSQL. The last section explains how to embed the cecho utility into a batch file using the Debug. I'm writing on a batch file to copy a certain file to a samba-share. You need to place REM command in front of every comment line. txt" "TESTB. The arguments can be called from the batch files through the variables %1, %2, %3, and so on. Files in a Single Folder. cmd file name extension. ) ELSE ( echo filename. Put simply, a batch file is a computer program or script containing data or tasks that are processed sequentially by Command. Syntax GOTO label GOTO:eof Key label A predefined label in the batch program. Improve this question. and your batch file would look like: C:\"Program Files\Microsoft SQL Server"\100\Tools\Binn\sqlcmd. A batch file is simply a text file saved with the. BTW if you want the last screen to disappear when its finish installing just use the taskkill. exe supports a /s command-line switch to not display these messages. Run the batch with a double click and check if the folder was deleted. exe command line processor executes as a batch job. It's certainly worth a try though. cecho is an enhanced ECHO command line utility with color support, inspired by the CTEXT utility by Dennis Bareis. A batch file is a type of script that contains a series of commands. I have explained more about batch file in another article so I will not cover this part again. To run a batch file in a minimized window state, follow these steps: Create a shortcut to the. Uses the echo off command to ensure that the commands are not shown when the code is executed. In all examples and syntax lines shown %A should be substituted with %%A when used in batch files. The following example shows a batch file which accepts 3 command line arguments and echo's them to the command line screen. Batch File Function Definition-Syntax:function_name Some_Operational_Code EXIT /B 0. You can simply copy the text below, and paste it into Notepad. Open notepad and save the below command as remove_map. @echo off REM parse arguments in the format key=value REM Example codename=g26 action=new lang=en:chs SETLOCAL SET CMDLINE=%* SET CODENAME= SET ACTION= SET LANG= SET. Navigate to the location of your batch file or you can search for by clicking This PC in the left pane and typing the filename in the search bar at the top-right. Make sure you write down the code in notepad. In addition, this can be done by scheduling this batch file to delete folder on the windows task scheduler. bat file name extension. Start(myBatchFileName, myBatchParameters); All parameters are strings. Creating a Batch Command File for Executing SAS with Dynamic and Custom System Options, continued 3 path parameter option %~p1 that is based on the directory path of the SAS program. Shell (Program,WindowStyle) Program can be the name of an internal or external command or a script. exe -Command "{Get-AppXPackage -AllUsers \| Foreach {Add-AppxPackage -DisableDevelopmentMode -Register "$($_. Run from a batch file, %~dpn0 evaluates to the drive letter, folder path, and file name (without extension) of the batch file. To wait somewhere between 29 and 30 seconds:. Batch Scripting - 1 - Basic Commands @Code Stop. Rename a file with the date/time. Type goto crash. Similarly, files may be removed from the right-hand list pane by double-clicking on a file or by selecting a group of files and clicking the ' Remove Files ' button. bat and contains all the command line prompts that have been pre-typed in. Usage: GOSUB can only be used in batch files. Files can be copied, DOS commands executed, and more. The batch file can contain any number of commands. txt output-master. If your current date time is Nov 02, 2017 15:41:36, then the above example will create a file in the current directory with name "access_20171102-154136. Batch to delete file automatically - Delete the file using the command line. Let's assume that there is a file called set2. For example my batch file is as follows: set /p hostname=plink -batch devdb [complex-linux-command] In the above example, if the complex-linux-command fails, or asking for an input from the user, or hangs, etc, then your Windows Batch script will not be waiting. This tutorial describes a simple. The batch command XCOPY is similar to COPY command but COPY command copies single file whereas XCOPY command copies entire directories including subdirectories. BAT file example. A batch script has a file extension of. exe -Command "{Get-AppXPackage -AllUsers \| Foreach {Add-AppxPackage -DisableDevelopmentMode -Register "$($_. June 29, 2012 16 Comments Sometimes there is a need to automate certain SQL commands so that you don’t have to physically login and do your same stuff. Names with in % mean variables, not sure what ! mark mean in a batch file. Navigate to the location of your batch file or you can search for by clicking This PC in the left pane and typing the filename in the search bar at the top-right. In previous tutorial we have seen executing testng. Diagnostics. Create a SQL file containing select commands For e. Like taking the tokens out of a date command, just like /u/MattikusNZ posted. Before jumping into the commands, I'd like to give a brief overview of the parts of commands. Batch files are DOS command line commands batched together. Follow asked Feb 26 '14 at 12:35. Directs the Windows command interpreter to a labeled line in a batch program. Use wildcards with caution on LFN volumes; see LFN File Searches for details. Expand Copy Code. The variable name %%a is used to hold the current folder being traversed. But if you want to do it simplier on multiple computers or if you're going to install EXE software remotely without GPO try to use our tool Action1. The basic syntax to use in shortcuts or similar is quite simple. This tutorial describes a simple. bat file to move files from any source to a destination folder. It is used to simplify certain repetitive tasks or routines in the Windows, DOS and OS/2 operating systems, and is also used in complex network and system administration. To delete the file Product. You can refer to other files in the same folder as the batch script by using this syntax: CALL %0\. The new task is in my scheduler. The general syntax of robocopy command is like below. exe command line processor executes as a batch job. First parameter is the batch file name (don't use CMD. txt = textfile. There are various methods available for bulk data operations. command / program: If it is an internal cmd command or a batch file then the command processor is run with the /K switch to cmd. As far as I know the installer will create shortcuts to "open developer command prompt windows by using these batch files, so all the required environment variables are set and ready to use". However I now want to create a third batch file that will do what listed above - first call the batch file that will enable my game controllers, then start my program which is an executable and then after I exited out of the program the batch file that will disable my game controllers should be called. In above script, ECHO off cleans up the console by hiding the commands from being printed at the prompt, ECHO prints the text. The simplest idea of how to write a batch file is: Figure out how you would type the commands at a DOS prompt, then type them, one per line, in a text file — and you've written your batch file. ) ELSE ( echo filename. Batch Scripting - 1 - Basic Commands @Code Stop. The NET command. The batch file label to branch to. exe runs the commands sequentially as they appear in the file. Each label must be defined on a line by itself, beginning with a colon and ending with either a space, a colon or a CR/LF. Similarly, files may be removed from the right-hand list pane by double-clicking on a file or by selecting a group of files and clicking the ' Remove Files ' button. Batch files are DOS command line commands batched together. In the above batch file, the %choice% variable is assigned when the user enters data and presses enter with the set /p choice line. It has an additional role as the usual first program run after boot (init process), hence being responsible for setting up the system by running the AUTOEXEC. %A : The A in %A may be replaced by any character, either upper case or lower case, except numbers. FINDSTR was introduced in the Windows NT 4 Resource Kit and is now a native command in Windows 2000 and later. If yes, then you can try the same method and batch file as discussed in the tutorial. Type goto crash. Following is the syntax for implementing for loop through a range of values in the batch file. A batch file is an unformatted text file that contains one or more commands and has a. In addition, this can be done by scheduling this batch file to delete folder on the windows task scheduler. @echo OFF ::EXIST command is used to check for existence IF EXIST D:\abc. Creating text files in batch is easy, there are two main operators: ">" - Output the command to file, overwrite it if it already exists, otherwise create it. Next is the body of function where we write codes to achieve a certain task. And voila, there's your basic script. txt diff output. A batch file does the work of a mediator between you and the command prompt. Windows copy command syntax and examples. bak extention. Prerequisites. I will be writing about different topics all related to batch files. For automation, commands can be read from a script file specified by /script switch, passed from the command-line using the /command switch, or read from standard input of winscp. "File exists" "File does not exist". It is used to simplify certain repetitive tasks or routines in the Windows, DOS and OS/2 operating systems, and is also used in complex network and system administration. cecho is an enhanced ECHO command line utility with color support, inspired by the CTEXT utility by Dennis Bareis. On Windows 10, a batch file typically has a ". After saving it with. Batch File If Else Example To Check If A File Or Folder Exists. Typically, when you execute a batch file, the script executes from top to bottom following each line. This file consists of a series of commands and has a. You can refer to other files in the same folder as the batch script by using this syntax: CALL %0\. Reading from a File. Understanding a Simple goto Command. A batch script has a file extension of. For additional. Adds syntax highlighting and snippets for batch files in Atom. The complete windows batch script will look like below. Take Command displays your command line applications in tabbed windows, with optional File Explorer-style integration for a visual look at your directories. Double-click it to run it. Rename a file with the date/time. Like using the rename command, you can also use the move command to rename a file as shown. Similarly, files may be removed from the right-hand list pane by double-clicking on a file or by selecting a group of files and clicking the ' Remove Files ' button. First parameter is the batch file name (don't use CMD. This is only one of many uses for batch files. Will the batch file run thorugh a shell call when placed in another. To have the. Save the file using a. After a GOTO command in a batch file, the next line to be executed will be the one immediately following the label. sql having select queries in it 2. Command Prompt assumes both the role of interpreter and runtime environment. ext doesn't exist or you can. PS1 file is with the special %~dpn0 variable. The zero ordinal argument is the name of the batch file itself. A batch file is simply a text file saved with the. @ECHO OFF CD / CD D: CD programming/ when I save and try to run this it gives error: the system cannot find the file specified path. In batch file programming, for loop can also be implemented through a range of values. The Command shell was the first shell built into Windows to automate routine tasks, like user account management or nightly backups, with batch (. As discussed in the previous tutorial, a batch file is an unformatted text file or script file which contains multiple batch file commands or instructions to achieve a certain task. You can automate SFTP file transfer in Unix and Linux using batch file. Step 10: Done. They should be enclosed in quotes if they contain spaces. The Command Prompt should automatically open, and the script should start executing, as shown below: Executing the Python batch file: the python file will execute via the command line when the batch file is manually pressed. Will the batch file run thorugh a shell call when placed in another. The command NET USE allows you to setup a mapped network drive and follows this syntax: net use. @echo off if exist C:\set2. com; or /console command-line parameter with winscp. Here, the '. This command is similar to the Linux cp command, but it does not match with the full functionality of cp. The BCP utility can be used to import large numbers of rows into SQL Server or export SQL Server data into files. This is only one of many uses for batch files. Step 1 − Open the command prompt (cmd. It can contain any arguments or switches required by the program, as well as the drive and path. Note: Output can be stored in an output file using SPOOL command in sql file. When a batch file is being executed, if echo is turned on, it would print the command currently it's running on to the command prompt. One Light with Chester Atom theme. Batch Script - Syntax, Normally, the first line in a batch file often consists of the following command. If you are on-network environment and you will need to remove the mapped network drive to all computers, the following method will come in help. I have a batch file that copies my files into 3 different drives and deletes the original. cmd will be "HelloWorld. I am new to batch scripting, just trying to write a simple batch file that would go to the directory that I frequently use, without having to do cd everytime. Step 2 − Go to the location where the. Batch files have comparatively easy syntax and can have many uses, so this method could also be a good learning experience by example. The batch file can contain any number of commands. It will show how to set several adb commands into one batch file. The batch file label to branch to. won't work. Batch files allow MS-DOS and Microsoft Windows users to write a series of commands to run in order upon their execution for automating frequently performed tasks. you can generate the file test. bat file in the folder where the executable for the mining software is located. Learn some basic batch commands. Usage: GOSUB can only be used in batch files. Note, however, that variables are case sensitive, so be consistent:. Alternately, type "wine cmd" to run the Windows-Console in the Linux terminal. How to use choice in a batch. The quotes are important if your file names contain any spaces. Note: Output can be stored in an output file using SPOOL command in sql file. I got this from a kb article for installing. %%A %A is for use on command lines only. It is a built in command in MS-DOS. A batch file is a script file in DOS, OS/2 and Microsoft Windows. It lets you to place a comment in a batch file. Diagnostics. Follow step 1 of this tutorial to create a batch file. %~dp0 => is replaced with the FULL PATH upon execution of batch. You could use VBScript to launch a batch file in a hidden window. Testing was minimal as I didn't want to break it! Alan. Adds syntax highlighting and snippets for batch files in Atom. If only ASSOC is written and executed, it will display all the file associations for every extension, instead of just. windows batch-file. Create a bat file having following command: sqlplus abc. Below is an example folder with several. C:\Users\Administrator\Desktop>test An existing installation has been detected. Type goto crash. Calculate the difference in Hours/Mins between two time values. The command prompt window for the script will disappear as soon as the script exits. Windows Commands, Batch files, Command prompt and PowerShell. Determine the folder where a batch file exists. The variables used in batch files are environment variables. Alternately, type "wine cmd" to run the Windows-Console in the Linux terminal. Another way you can run commands on multiple computers at once is to use a text file. You have to run CMD. If your WinSCP command-line contains percent sign, for example to encode special characters in session URL (particularly in the credentials), it may conflict with special meaning of the percent sign in Windows batch files. To see the list of variables in your machine, open a command prompt window and type set. exe -Command "{Get-AppXPackage -AllUsers \| Foreach {Add-AppxPackage -DisableDevelopmentMode -Register "$($_. 5 Cool Batch Files: In This instructable you will get 5 codes for 5 useful batch things - Password Generator - Password Protected Command Prompt - Website Crasher - Website Pinger - PC Cleanup Utilities. Most of that would be possible as it's copying and running files, however, I'm not to sure how well networked storage works with batch files. Use goto instead of an expression inside parentheses, or enable delayed variable expansion. Batch File Programming Introduction Batch file programming is the native programming offered by the Microsoft Windows Operating System. So the variable %0 in our script HelloWorld. Following is the syntax for implementing for loop through a range of values in the batch file. When you plan to run a natch file silently, use @echo. Open notepad and save the below command as remove_map. bat file or batch script) is a text file that the Windows cmd. Batch files can be run by typing "start FILENAME. The script file must use UTF-8 or UTF-16 (with. Direct a batch program to jump to a labelled line. % is a special character in shell. For example, you might want to name it hello_world. Rename a file with the date/time. This means that the window will remain after the command has been run. wsh files are text documents the controls the run-time execution of scripts. This command just creates a temporary text file using the temp variable and the FileDateTime variable with a. MS-DOS and Windows command line goto command. Enables Windows to display an extended character set in graphics mode. Robocopy command syntax and examples. txt extension. after we create the new text file, rename it to test. It will then process each computer individually. redirection to :labels , setting variables, or running commands. Then rename the file test. @echo OFF ::EXIST command is used to check for existence IF EXIST D:\abc. bat and saved it in System32 folder. It lets you to place a comment in a batch file. The DO command is then executed with the parameter(s) set to the token(s) found. Open a command window in the directory containing the. bat file in the folder where the executable for the mining software is located. By default, start returns immediately without waiting for the program to exit, so the batch file will continue to run and, presumably, exit immediately. The way this is used to target our. Usage: GOSUB can only be used in batch files. There are hundreds of batch commands, which can automate everything from printer queues to complex tasks. The variable name %%a is used to hold the current folder being traversed. All batch files are run in this environment. sql before Select query to. It also requests information about the number of write capacity units consumed by the operation and any item collections modified by the operation. Windows batch files can be created with any basic text editor, but the most common one is Notepad. Use double percent signs (%%) to carry out the for command within a batch file. bat file or batch script) is a text file that the Windows cmd. The batch file can contain any number of commands. Definition of PowerShell Batch File. txt found IF EXIST D:\xyz. If your WinSCP command-line contains percent sign, for example to encode special characters in session URL (particularly in the credentials), it may conflict with special meaning of the percent sign in Windows batch files. In addition, this can be done by scheduling this batch file to delete folder on the windows task scheduler. once a minute. cmd file name extension. NET Framework 3. Availability. The previously tutorials about batch files: Batch to delete file older than- Delete files older than 7 days using batch and script. So the vcvars. txt in the C drive and that there is no file called set3. SilentCMD [path to. BTW if you want the last screen to disappear when its finish installing just use the taskkill. \$> mysql < batch-file. In this case, we will delete the specified folder. Prerequisites. you can generate the file test. Multi-threaded copying which will allow copy, mirror multiple files and directories at the same time which will decrease the complete copy operation time. won't work. This means that the window will remain after the command has been run. If you save this snippet as "test. Another way you can run commands on multiple computers at once is to use a text file. Displays Group Policy information for machine or user. The syntax is: CALL :label arguments A new batch file context is created with the specified arguments and control is passed to the statement after the label specified. This batch command is used for remarks in batch files, preventing the content of the remark from being executed. Batch File Commands. SOURCE is the source file or directory we want to copy. you've probably to specify the location of test. It's certainly worth a try though. Then rename the file test. You now have a high performance backup that cost you absolutely nothing! Other useful parameters for Robocopy. bat" extension, and it is a special text file that includes one or multiple commands that run in sequence to perform various actions with Command Prompt. robocopy SOURCE DESTINATION FILE OPTIONS. Batch Script: A batch script is a text file that contains certain commands that are executed in sequence. Displays files opened by remote users for a file share. Double-click the BAT file. txt extension. Insta · Try this in the batch file. On Windows 10, a batch file typically has a ". Use Start command ----- start "" c:\windows\notepad. Batch file help and support. Start(myBatchFileName, myBatchParameters); All parameters are strings. Open notepad and save the below command as remove_map. Windows batch file syntax using exclamation mark. A batch file is a script file in DOS, OS/2 and Microsoft Windows. Now this tutorial deals with the basic batch file commands that are used generally to make a batch file more complicated. I'm writing on a batch file to copy a certain file to a samba-share. bat file is useful for scripters who need to use the current date variable in the format MMDDYYYY. Lets save this batch file as test. It is a file – with. Batch files often need to know the location of input and output files. Will the batch file run thorugh a shell call when placed in another. The '>' command is used to redirect the output to the file C:\lists. The batch command ASSOC associates a file extension with a file type, or list all associations. Use goto instead of an expression inside parentheses, or enable delayed variable expansion. Now, type dir to view the listing of all the files within the folder and hit. With Windows Script Host you could run more sophisticated scripts in the Command shell. You can also use the Shell to run scripts like batch files, PowerShell scripts, PERL, Python etc. These files are saved with. exe /s path of. EXE that's invoked to interpret your batch file. The following example illustrates the use of the PATH environment variable within a batch file:. cmd extension are still used to invoke scripting hosts. The for command here is used to run a further command for each file or folder in the set. Example: @echo off xcopy C:\Folder\text. 5 in Win8: " Or, you can use the Windows installation media as the file source when you enable the. As the goto executable is a part of the cmd. txt output-master. It can contain any arguments or switches required by the program, as well as the drive and path. The syntax is: CALL :label arguments A new batch file context is created with the specified arguments and control is passed to the statement after the label specified. For example, to silently run the. Note, however, that variables are case sensitive, so be consistent:. The ELSE clause must occur on the same line as the command after the IF. exe command line processor executes as a batch job. we will add this command (i will explain each. For automation, commands can be read from a script file specified by /script switch, passed from the command-line using the /command switch, or read from standard input of winscp. Hosts file is basically a text file buried deep inside your computer’s Windows folder, but you won’t find a. See full list on tutorialspoint. Run from a batch file, %~dpn0 evaluates to the drive letter, folder path, and file name (without extension) of the batch file. Command Prompt assumes both the role of interpreter and runtime environment. For example in the following batch file, the username [email protected] contains the @ sign. com -r "220 ProFTPD" 220 ProFTPD can be the normal answer from a FTP Server. This will stop execution of the batch file until someone presses "any key". Following are the different ways to launch cmd. Batch file functions are similar in concept to functions in Linux shell scripts and other languages; however, the syntax is different. Directs the Windows command interpreter to a labeled line in a batch program. To delete the file Product. bat file name extension. Here, the '. After a GOTO command in a batch file, the next line to be executed will be the one immediately following the label. csv, all will work fine and an "output. Batch files have comparatively easy syntax and can have many uses, so this method could also be a good learning experience by example. exe /c FILENAME. You may want to copy files from one place to another. bat illustrates how a function can be defined in a batch file.	2021-12-03 13:28:10	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5823971629142761, "perplexity": 2874.756763698997}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362879.45/warc/CC-MAIN-20211203121459-20211203151459-00485.warc.gz"}
https://math.stackexchange.com/questions/2602113/abcd-is-parallelogram-angle-abc-105-circ-angle-cmd-135-circ-find	# $ABCD$ is parallelogram. $\angle ABC=105^{\circ}$. $\angle CMD=135^\circ$. Find $\angle BKC$ $ABCD$ is parallelogram. $\angle ABC=105^{\circ}$. $BMC$ is equilateral triangle and $\angle CMD=135^\circ$. $K$ is midpoint of $AB$. Find $\angle BKC$ My attempts 1) $\angle MBC=\angle MCB=\angle BMC=60^{\circ}$ 2) $\angle MCD=15^{\circ}$, $\angle MDB=30^{\circ}$ I want to prove that $MB \perp CK$ but I need help here • I think you mean $\angle MDC = 30^{\circ}$ – idok Jan 12 '18 at 10:18 ## 2 Answers Let $N$ be the point on $CD$ such that $\angle CMN=15^\circ$. Then $\angle CMN=15^\circ=\angle MCN$ and $\angle MDN=30^\circ=\angle MND$. So $DM=MN=NC$. As $BC=BM$, $MN=CN$ and $BN=BN$, $\triangle BCN\cong \triangle BMN$ and therefore, $\angle CBN=\angle MBN=30^\circ$. So, $\angle BNC=75^\circ=\angle BCN$ and hence $BC=BN=BM$. As $\angle ADM=75^\circ=\angle BCN$, $BC=AD$ and $DM=NC$, $\triangle BCN\cong \triangle ADM$. So, $AM=BN=BM$. Therefore, $\angle BAM=\angle ABM=45^\circ$. Since $K$ is the midpoint of $AB$, $MK\perp AB$. $\angle KMB=180^\circ-90^\circ-45^\circ=45^\circ=\angle KBM$. Therefore, $BK=KM$. $\triangle KBC\cong\triangle KMC$. $\angle BKC=\angle MKC=90^\circ\div 2=45^\circ$. WLOG, I assumed $$\|BK\| = \|AK\| = 1$$ (We could say $$a$$ or $$x$$ or whatever we want to but we don't lose generality and this way makes things more clear I believe). Then if we apply The Law of Sines in $$\Delta CDM$$, we have: $$\frac{\|CD\|}{\sin135^{\circ}} = \frac{\|CM\|}{\sin30^{\circ}} \implies 2\sqrt{2} = 2\|CM\| \implies \|CM\| = \sqrt{2}$$ Now, notice that $$\angle MBK = 45^\circ$$ so $$\|KM\|$$ is nothing but $$1$$ (We can see that by using Law of Cosines or noticing that a triangle like that one must be unique). Then $$\Delta BKM$$ is isosceles right angle triangle as you foresaw. So the result follows.	2021-03-09 02:16:00	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 9, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9754422307014465, "perplexity": 145.65096078852133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178385534.85/warc/CC-MAIN-20210308235748-20210309025748-00543.warc.gz"}
https://events.williams.edu/event/span-of-the-bracket-polynomial-and-triple-crossing-projections-of-knots-by-jonah-greenberg-19/	# Span of the Bracket Polynomial and Triple-Crossing Projections of Knots by Jonah Greenberg '19 ## Wed, May 8th, 20191:00 pm - 1:45 pm • This event has passed. Span of the Bracket Polynomial and Triple-Crossing Projections of Knots by Jonah Greenberg ’19, Mathematics Senior Thesis Defense, Wednesday, May 8, 1 – 1:45 pm, Stetson Court Classroom 101 Abstract: For double-crossing projections of knots, it is known that the span of the bracket polynomial is bounded above by 4c_2, where c_2 is the double crossing number of a knot and that this bound is achieved if and only if the knot is reduced and alternating. It is also known that the span of the bracket polynomial is bounded above by 8c_3. We are interested in when and how this bound is achieved, and will extend ideas about the bracket polynomial for double-crossing projections to triple-crossing projections along the way.	2020-01-19 13:24:34	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.897788405418396, "perplexity": 1245.5088161340864}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250594603.8/warc/CC-MAIN-20200119122744-20200119150744-00417.warc.gz"}
https://projecteuclid.org/euclid.gt/1513774917	## Geometry & Topology ### Equivariant characteristic classes of external and symmetric products of varieties #### Abstract We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasiprojective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel–Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group. #### Article information Source Geom. Topol., Volume 22, Number 1 (2018), 471-515. Dates Revised: 22 March 2017 Accepted: 2 May 2017 First available in Project Euclid: 20 December 2017 https://projecteuclid.org/euclid.gt/1513774917 Digital Object Identifier doi:10.2140/gt.2018.22.471 Mathematical Reviews number (MathSciNet) MR3720348 Zentralblatt MATH identifier 06805083 #### Citation Maxim, Laurenţiu; Schürmann, Jörg. Equivariant characteristic classes of external and symmetric products of varieties. Geom. Topol. 22 (2018), no. 1, 471--515. doi:10.2140/gt.2018.22.471. https://projecteuclid.org/euclid.gt/1513774917 #### References • M F Atiyah, Power operations in $K$–theory, Quart. J. Math. Oxford Ser. 17 (1966) 165–193 • M F Atiyah, I M Singer, The index of elliptic operators, III, Ann. of Math. 87 (1968) 546–604 • P Baum, J-L Brylinski, R MacPherson, Cohomologie équivariante délocalisée, C. R. Acad. Sci. Paris Sér. 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Algebraic Geom. 23 (2014) 669–692 • J Zhou, Delocalized equivariant coholomogy of symmetric products, preprint (1999)	2019-11-14 10:47:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5897016525268555, "perplexity": 4033.450052721881}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668416.11/warc/CC-MAIN-20191114104329-20191114132329-00405.warc.gz"}
https://nips.cc/Conferences/2022/ScheduleMultitrack?event=58531	Timezone: » SAINT: Improved Neural Networks for Tabular Data via Row Attention and Contrastive Pre-Training Gowthami Somepalli · Avi Schwarzschild · Micah Goldblum · C. Bayan Bruss · Tom Goldstein Tabular data underpins numerous high-impact applications of machine learning from fraud detection to genomics and healthcare. Classical approaches to solving tabular problems, such as gradient boosting and random forests, are widely used by practitioners. However, recent deep learning methods have achieved a degree of performance competitive with popular techniques. We devise a hybrid deep learning approach to solving tabular data problems. Our method, SAINT, performs attention over both rows and columns, and it includes an enhanced embedding method. We also study a new contrastive self-supervised pre-training method for use when labels are scarce. SAINT consistently improves performance over previous deep learning methods, and it even performs competitively with gradient boosting methods, including XGBoost, CatBoost, and LightGBM, on average over $30$ benchmark datasets in regression, binary classification, and multi-class classification tasks.	2023-02-07 05:44:32	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6394891738891602, "perplexity": 3923.100959932639}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500384.17/warc/CC-MAIN-20230207035749-20230207065749-00817.warc.gz"}
https://brilliant.org/discussions/thread/a-problem-for-level-5-users-in-solving-low-level/?sort=new	# A problem for Level 5 users in solving low Level problems Does it seem fair that if you're at Level 5 in some category, and you answer "wrongly" some problem in Level 1, your rating can drop as much as 34 points, and yet if you had answered "correctly", your rating won't go up at all? Doesn't that strongly discourage anyone from trying any problems at all that are below their Level? A recent algebra problem posing a simple linear equation has the so-called "correct" answer of x = 0, when in fact any value of x would be a solution. An Algebra Level 5 user can lose as much as 34 points in tangling with this bogus question. In game theory, one looks at expected returns versus expected potential losses. Since low Level problems return nothing in rating at risk of a considerable drop in rating, as compared to higher Level problems, doesn't that mean that the best strategy is to avoid the lowest Level problems whenever possible? Especially when a significant percentage of them are poorly worded or even have wrong "correct" answers? I would suggest that deductions in ratings for wrong answers should be the same or about the same as increases in ratings for right answers. Note by Michael Mendrin 7 years, 1 month ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting. 2 \times 3 $2 \times 3$ 2^{34} $2^{34}$ a_{i-1} $a_{i-1}$ \frac{2}{3} $\frac{2}{3}$ \sqrt{2} $\sqrt{2}$ \sum_{i=1}^3 $\sum_{i=1}^3$ \sin \theta $\sin \theta$ \boxed{123} $\boxed{123}$ Sort by: I know its really a very big problem these days, that's why I never even attempt level 4(or below) questions these days(only 5's) as they're much more profitable. - 6 years, 4 months ago You won't believe me, but this very feature of Brilliant has cured me of my really bad habit of answering the questions whose answers are doubtful to me. It helped me greatly in the exams which have negative markings. Some of my great downfalls are - Combinatorics - Level 3 to Level 1 Mechanics - Level 4 to Level 3 Electricity & Magnetism - Level 2 - Level 1 How I am thankful to Brilliant, I can't tell you. (I am not being in the least sarcastic here, seriously!). - 7 years, 1 month ago I think the multiple choice problems should all be unrated. I remember going down like 300-400 in calculus because I got a MC problem wrong. - 7 years, 1 month ago I think that questions that answer as explanation should be a discussion instead, not a multiple choice. People with most votes gets a ratings and stuffs. I made a very stupid mistakes on easy problems and the score dropped down like 250 by just a single problem. Honestly, I remembered of having a nightmare after that problem on that night lol. - 6 years, 9 months ago Haha but you are back at 5. - 7 years ago Can't say anything, but I agree with this. - 7 years, 1 month ago Indeed, it seems like there is more penalty for incorrectly answering a problem than there is reward for correct responses. In my opinion, the best solution to this would be to increase the number of points gained by answering harder problems and change the system so that points are not lost by answering incorrectly until all three tries have been used. - 7 years, 1 month ago Excellent idea Lee! @Peter Taylor have you ever considered this? - 7 years ago Seriously. I agree SO much. - 7 years, 1 month ago Same here - 7 years ago I agree, I think the rating system is way too harsh. I find it's difficult to maintain my level because even though I may answer 3 level 5 problems correctly, getting one level 3 problem wrong seems to negate all of the gains that I made, even if it was due to a simple reading error. I think the rating system needs to be modified so that either it is less harsh on getting lower level problems wrong or more generous when you get higher level problems right. The latter would probably be more encouraging for users of the website but maybe a bit of both would work. - 7 years, 1 month ago If all the problems were perfect and thoroughly vetted, as if taken from Olympiad problems, then it could be tolerable, because then at least I can trust to know exactly what the problem is, and that it has an unique, correct answer. Unfortunately, that's far from the case in Brilliant, where even a lot of the rated problems are defective and you just don't know you're getting into to. We shouldn't be punished for having tackled defective problems. No, the Brilliant staff doesn't have the time to address each and every complaint about such bad problems, there has to be a better way. One way would be to make the rating system a little more realistic and less harsh. - 7 years, 1 month ago hey there I am Salman Zafar I am new here can you just help me out ? I will be verrrrrrrry thankful - 6 years, 12 months ago That's what I meant by my comment. - 7 years, 1 month ago hey there I am Salman Zafar I am new here can you just help me out ? I will be verrrrrrrry thankful - 6 years, 12 months ago Well, I'm trying to completely concur, Cole. By the way, thanks for the interesting problems you've been posting. They've been fun. - 7 years, 1 month ago I'm glad you made this post, this issue has been on my mind for a while. It was good to see that you made a solution to my center of mass question, you got my upvote. - 7 years, 1 month ago Brilliant has updated the algorithm it seems, all thanks to @Michael Mendrin ! and all who reshared and liked the note.. - 7 years, 1 month ago 1. The system is set up such that the expected gain from solving a problem is 0. For example, if we're 99% sure that a level 5 can solve a level 1 problem, then if they get -34 for a wrong answer, they should get +0.345 for a right answer. While we round your rating off when displaying, fractional changes are still recorded. 2. I agree that bogus problems are a concern. Your rating will update when the answer is corrected. This often rewards you, due to the fake high rating of the problem. 3. We are also working on providing more clarity on the validity of problems, and will provide more tools to the community to moderate problems. Staff - 7 years, 1 month ago How about marking a user posted problems after they passed a quality check with some icon indicating that the answer is correct and that the question contains no ambiguities. - 5 years, 12 months ago This policy of "0 expected gain" makes sense only if all of the problems are perfect (i.e. clearly and unambiguously worded, with correct answers). But since we know that a good percentage of them are not, this needs to be factored in. Let's imagine that I'm at some casino playing the roulette table, the odds of, say, 00 winning is 1/37, but the payout is 35 to 1, slightly less than 0 expected gain (because the casino has to make money). It's a risk I'm willing to take, because then I can expect to play for a while and have fun before losing all of my money. However, if the odds of my winning is, say, 1/40 (which is about what happens when I tackle Level 1 problems, because of problems with poor wording or incorrect answers), then there's going to be a lot less than 0 expected gain, and I'm going to find out that my money is going to run out in a hurry. And so I'll leave for a more fair casino. The Brilliant staff has resources to understand and use statistics, and my advice would be that Brilliant uses the same quality control philosophies that manufacturers have, which is that they don't start out assuming that they are delivering perfect products every time. They have to factor in the reality that a certain percentage of their products are doing to have defects, otherwise they are going to lose customers. If I were on the Brilliant staff, I'd be focused on making my customers happy, instead of ignoring what it is like for consumers to use Brilliant. I am one consumer, and I would have to say this is probably the most aggravating characteristic of Brilliant, which is that I cannot just go and tackle problems "for fun", because I know how dangerous it can be if I happen to run into a bogus problem. Stepping on a land mine once in a while is okay, but after it's blown up in your face too many times, well, you start avoiding the minefields, and look elsewhere. You might already be losing consumers after they've tried out Brilliant for a while. My strong suggestion to the Brilliant staff is that they pay attention to what makes their consumers happy, not what "ideals should be upheld". There are ways to do both. Be creative. - 7 years, 1 month ago Same for me, I was at level 4 the actual answer of a question was 512 but the answer given was wrong and stated that the answer is 256 which only a mad person would do. Unfortunately now I am in level 3 in algebra - 7 years, 1 month ago Yup I agree. I also experienced this drastic rating decline in algebra when I by mistake clicked on the wrong option of an MCQ while attempting a level 2 ques. (I was level 4 on algebra then..) Unfortunately fell to Level 3 now... - 7 years, 1 month ago hey Vishal I am Salman Zafar I am new here can you just help me out ? I will be verrrrrrrry thankful - 6 years, 12 months ago	2021-05-16 06:30:06	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 8, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.804379940032959, "perplexity": 1253.7921779893643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989690.55/warc/CC-MAIN-20210516044552-20210516074552-00022.warc.gz"}
http://biblioteca.posgraduacaoredentor.com.br/?q=NONLINEAR+WAVES	Página 1 dos resultados de 4566 itens digitais encontrados em 0.012 segundos Dynamical analysis of turbulence in fusion plasmas and nonlinear waves Viana, Ricardo Luiz; Lopes, Sergio Roberto; Caldas, Ibere Luiz; Szezech Junior, Jose Danilo; Guimarães Filho, Zwinglio de Oliveira; Lima, Gustavo Zampier dos Santos; Galuzio, Paulo Paneque; Batista, Antonio Marcos; Kuznetsov, Yurii; Nascimento, Ivan Cunh Fonte: ELSEVIER SCIENCE BV; AMSTERDAM Publicador: ELSEVIER SCIENCE BV; AMSTERDAM Tipo: Artigo de Revista Científica ENG Relevância na Pesquisa 66.21% Turbulence is one of the key problems of classical physics, and it has been the object of intense research in the last decades in a large spectrum of problems involving fluids, plasmas, and waves. In order to review some advances in theoretical and experimental investigations on turbulence a mini-symposium on this subject was organized in the Dynamics Days South America 2010 Conference. The main goal of this mini-symposium was to present recent developments in both fundamental aspects and dynamical analysis of turbulence in nonlinear waves and fusion plasmas. In this paper we present a summary of the works presented at this mini-symposium. Among the questions to be addressed were the onset and control of turbulence and spatio-temporal chaos. (C) 2011 Elsevier B. V. All rights reserved.; FAPESP; FAPESP; CNPq; CNPq; CAPES; CAPES; Fundacao Araucarias; Fundacao Araucarias; RNF-CNEN (Brazilian Fusion Network); RNFCNEN (Brazilian Fusion Network) Nonlinear waves on the surface of a fluid covered by an elastic sheet Deike, Luc; Bacri, Jean-Claude; Falcon, Eric Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.22% We experimentally study linear and nonlinear waves on the surface of a fluid covered by an elastic sheet where both tension and flexural waves take place. An optical method is used to obtain the full space-time wave field, and the dispersion relation of waves. When the forcing is increased, a significant nonlinear shift of the dispersion relation is observed. We show that this shift is due to an additional tension of the sheet induced by the transverse motion of a fundamental mode of the sheet. When the system is subjected to a random noise forcing at large scale, a regime of hydro-elastic wave turbulence is observed with a power-law spectrum of the scale in disagreement with the wave turbulence prediction. We show that the separation between relevant time scales is well satisfied at each scale of the turbulent cascade as expected theoretically. The wave field anisotropy, and finite size effects are also quantified and are not at the origin of the discrepancy. Finally, the dissipation is found to occur at all scales of the cascade contrary to the theoretical hypothesis, and could thus explain this disagreement.; Comment: Journal of Fluid Mechanics (2013) Nonequilibrium chaos of disordered nonlinear waves Skokos, Charalampos; Gkolias, Ioannis; Flach, Sergej Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.06% Do nonlinear waves destroy Anderson localization? Computational and experimental studies yield subdiffusive nonequilibrium wave packet spreading. Chaotic dynamics and phase decoherence assumptions are used for explaining the data. We perform a quantitative analysis of the nonequilibrium chaos assumption, and compute the time dependence of main chaos indicators - Lyapunov exponents and deviation vector distributions. We find a slowing down of chaotic dynamics, which does not cross over into regular dynamics up to the largest observed time scales, still being fast enough to allow for a thermalization of the spreading wave packet. Strongly localized chaotic spots meander through the system as time evolves. Our findings confirm for the first time that nonequilibrium chaos and phase decoherence persist, fueling the prediction of a complete delocalization.; Comment: 5 pages, 5 figures Nonlinear waves, differential resultant, computer algebra and completely integrable dynamical systems Kostov, N. A.; Kostova, Z. T. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.18% The hierarchy of integrable equations are considered. The dynamical approach to the theory of nonlinear waves is proposed. The special solutions(nonlinear waves) of considered equations are derived. We use powerful methods of computer algebra such differential resultant and others.; Comment: 33 pages, no figures Chaotic behaviour of nonlinear waves and solitons of perturbed Korteweg - de Vries equation Blyuss, K. B. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.22% This paper considers properties of nonlinear waves and solitons of Korteweg-de Vries equation in the presence of external perturbation. For time-periodic hamiltonian perturbation the width of the stochastic layer is calculated. The conclusions about chaotic behaviour in long-period waves and solitons are inferred. Obtained theoretical results find experimental confirmation in experiments with the propagation of ion-acoustic waves in plasma.; Comment: 7 pages, LaTeX, 2 Postscript figures, submitted to Reports on Mathematical Physics Nonlinear Waves in Disordered Diatomic Granular Chains Ponson, Laurent; Boechler, Nicholas; Lai, Yi Ming; Porter, Mason A.; Kevrekidis, P. G.; Daraio, Chiara Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.2% We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a "spin", so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: In low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatio-temporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for such a transition. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves.; Comment: 10 pages, 8 figures (some with multiple parts), to appear in Physical Review E; summary of changes: new title, one new figure, additional discussion of several points (including both background and results) Interactions among different types of nonlinear waves described by the Kadomtsev-Petviashvili Equation Cheng, Xue-Ping; Chen, Chun-Li; Lou, Sen-Yue Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.31% In nonlinear physics, the interactions among solitons are well studied thanks to the multiple soliton solutions can be obtained by various effective methods. However, it is very difficult to study interactions among different types of nonlinear waves such as the solitons (or solitary waves), the cnoidal periodic waves and Painlev\'e waves. In this paper, the nonlocal symmetries related to the Darboux transformations (DT) of the Kadomtsev-Petviashvili (KP) equation is localized after imbedding the original system to an enlarged one. Then the DT is used to find the corresponding group invariant solutions such that interaction solutions among different types of nonlinear waves can be found. It is shown that starting from a Boussinesq wave or a KdV-type wave, which are two basic reductions of the KP equation, the essential and unique role of the DT is to add an additional soliton. Anderson localization or nonlinear waves? A matter of probability Ivanchenko, M. V.; Laptyeva, T. V.; Flach, S. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.13% In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems is under intense theoretical debate and experimental study. We resolve this dispute showing that at any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results are generalized to higher dimensions as well.; Comment: 4 pages, 3 figures Universal subdiffusion of nonlinear waves in two dimensions with disorder Laptyeva, T. V.; Bodyfelt, J. D.; Flach, S. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.13% We follow the dynamics of nonlinear waves in two-dimensional disordered lattices with tunable nonlinearity. In the absence of nonlinear terms Anderson localization traps the packet in space. For the nonlinear case a destruction of Anderson localization is found. The packet spreads subdiffusively, and its second moment grows in time asymptotically as $t^\alpha$. We perform fine statistical averaging and test theoretical predictions for $\alpha$. Along with a precise confirmation of the predictions in [Chemical Physics \textbf{375}, 548 (2010)], we also find potentially long lasting intermediate deviations due to a growing number of surface resonances of the wave packet.; Comment: 6 pages, 6 figures, 1 table Equation for three-dimensional nonlinear waves in liquid with gas bubbles Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.3% Nonlinear waves in a liquid containing gas bubbles are considered in the three-dimensional case. Nonlinear evolution equation is given for description of long nonlinear pressure waves. It is shown that in the general case the equation is not integrable. Some exact solutions for the nonlinear evolution equation are presented. Application of the Hirota method is illustrated for finding multi-soliton solutions for the nonintegrable evolution equation in the three-dimensional case. The stability of the one-dimensional solitary waves is investigated. It is shown that the one-dimensional solitary waves are stable to transverse perturbations. Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation James, Guillaume Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.3% We study nonlinear waves in Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully-nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schroedinger (DpS) equation. It consists of a spatial discretization of a generalized Schroedinger equation with p-Laplacian, with fractional p>2 depending on the exponent of Hertz's contact force. We show that the DpS equation admits explicit periodic travelling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov-Schmidt technique, we prove the existence of exact periodic travelling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime... Simulation of strong nonlinear waves with vectorial lattice Boltzmann schemes Dubois, François Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.06% We show that an hyperbolic system with a mathematical entropy can be discretized with vectorial lattice Boltzmann schemes with the methodology of kinetic representation of the dual entropy. We test this approach for the shallow water equations in one and two space dimensions. We obtain interesting results for a shock tube, reflection of a shock wave and unstationary two-dimensional propagation. This contribution shows the ability of vectorial lattice Boltzmann schemes to simulate strong nonlinear waves in unstationary situations.; Comment: 12 pages Doppler Effect of Nonlinear Waves and Superspirals in Oscillatory Media Brusch, Lutz; Torcini, Alessandro; Baer, Markus Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.26% Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example, where waves originate from a source exhibiting a back-and-forth movement in radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves (superspiral''). Using the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonous growth or decay as well as saturation of these modulations away from the source depending on the perturbation frequency. Our findings allow a consistent interpretation of recent experimental observations concerning superspirals and their decay to spatio-temporal chaos.; Comment: 4 pages, 4 figures Nonlinear waves in networks: a simple approach using the sine--Gordon equation Caputo, Jean-Guy; Dutykh, Denys Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.18% To study the propagation of nonlinear waves across Y-- and T--type junctions, we consider the 2D sine--Gordon equation as a model and study the dynamics of kinks and breathers in such geometries. The comparison of the energies reveals that the angle of the fork plays no role. Motivated by this, we introduce a 1D effective equation whose solutions agree well with the 2D simulations for kink and breather solutions. For branches of equal width, breather crossing occurs approximately when $v > 1 - \omega$, where $v$ is the breather celerity and $\omega$ is its frequency. We then characterize the breathers in the two upper branches by estimating their velocity and frequency. These new breathers are slower than the initial breather and up-shifted in frequency. In perspective, this study could be generalized to more complex nonlinear waves. Localized nonlinear waves in a two-mode nonlinear fiber Zhao, Li-Chen; Liu, Jie Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.3% We find that diverse nonlinear waves, such as soliton, Akhmediev breather, and rogue waves (RWs), can emerge and interplay with each other in a two-mode coupled system. It provides a good platform to study interaction between different kinds of nonlinear waves. In particular, we obtain dark RWs analytically for the first time in the coupled system, and find that two RWs can appear in the temporal-spatial distribution. Possible ways to observe these nonlinear waves are discussed.; Comment: 8 pages, 8 figures Extended equation for description of nonlinear waves in liquid with gas bubbles Kudryashov, Nikolai A.; Sinelshchikov, Dmitry I. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.35% Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for long weakly nonlinear waves taking into consideration liquid viscosity, inter--phase heat transfer and surface tension. Additional conditions for the parameters of the equation are determined for integrability of the mathematical model. The transformation for linearization of the nonlinear equation is presented too. Some exact solutions of the nonlinear equation are found for integrable and non--integrable cases. The nonlinear waves described by the nonlinear equation are numerically investigated. Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer Kudryashov, Nikolay A.; Sinelshchikov, Dmitry I. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.32% Nonlinear waves are studied in a mixture of liquid and gas bubbles. Influence of viscosity and heat transfer is taken into consideration on propagation of the pressure waves. Nonlinear evolution equations of the second and the third order for describing nonlinear waves in gas-liquid mixtures are derived. Exact solutions of these nonlinear evolution equations are found. Properties of nonlinear waves in a liquid with gas bubbles are discussed.; Comment: Physics Letters A, Volume 374, Issues 19-20, Pages 2011-2016 Nonlinear Waves in Lattices: Past, Present, Future Kevrekidis, P. G. Tipo: Artigo de Revista Científica Relevância na Pesquisa 56.13% In the present work, we attempt a brief summary of various areas where nonlinear waves have been emerging in the phenomenology of lattice dynamical systems. These areas include nonlinear optics, atomic physics, mechanical systems, electrical lattices, nonlinear metamaterials, plasma dynamics and granular crystals. We give some of the recent developments in each one of these areas and speculate on some of the potentially interesting directions for future study.; Comment: 35 pages, 3 figures, brief review to appear in IMA Journal of Applied Mathematics Generation and Propagation of Nonlinear Waves in Travelling Wave Tubes Tzenov, Stephan I. Tipo: Artigo de Revista Científica	2018-12-19 04:01:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.48392200469970703, "perplexity": 1863.6891898332483}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376830479.82/warc/CC-MAIN-20181219025453-20181219051453-00337.warc.gz"}
https://mathoverflow.net/questions/192162/straightening-for-infty-operads	# Straightening for $\infty$-operads There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to N(Fin_\ast)$ satisfying a Segal condition) and produces a "stack" object which is a symmetric monoidal functor $N(Fin_\ast)\to Cat_\infty$ whose target is $C^\otimes$ (where the image of $\{1,\ast\}$ is $C$, the underlying $\infty$-category of interest). Is there any kind of "straightening" construction for $\infty$-operads? In particular, is it necessarily impossible in general to produce such a functor corresponding to an $\infty$-operad (which is also a functor $O^\otimes\to N(Fin_\ast)$ satisfying certain properties)? If it's impossible, is there some weaker version of $Cat_\infty$ (perhaps without a full symmetric monoidal structure?) in which such a "stacky" presentation of an $\infty$-operad might land? My real goal here to ask if there is a way in which I can think of $\infty$-operads as (possibly some weakened version of) commutative algebra objects in some $\infty$-category. Does anyone know of a description in this way? • It seems that, in light of section 6.3 of Lurie's Higher Algebra, we can think of $\infty$-operads as monoid objects in symmetric sequences on a symmetric monoidal category? – Jonathan Beardsley Jan 5 '15 at 5:15 • But that seems to depend on stability and presentability. – Jonathan Beardsley Jan 5 '15 at 5:19 • I think that infinity operads are Quillen equivalent to colored operads of simplicial sets. Does this maybe help? – Fernando Muro Jan 5 '15 at 9:22 • For a fixed set of objects, $\infty$-operads should presumably be the associative algebras in "coloured symmetric sequences" in spaces. As far as I know the monoidal $\infty$-category required for this to make sense has not been constructed, though. – Rune Haugseng Jan 5 '15 at 13:29 • I think that's just a different construction (though related). Certainly you can define the composition product on symmetric sequences in any reasonably nice ordinary symmetric monoidal category (including, say, sets). There's a definition on the nlab that I imagine one could make sense of for $\infty$-categories too... What were you thinking of using this construction for? – Rune Haugseng Jan 5 '15 at 17:40	2019-09-17 23:34:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9132941365242004, "perplexity": 523.6525364595906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514573124.40/warc/CC-MAIN-20190917223332-20190918005332-00229.warc.gz"}
https://qspace.library.queensu.ca/jspui/handle/1974/27649?show=full	dc.contributor.author Brossard, Alexis en dc.date.accessioned 2020-03-03T16:28:41Z dc.date.available 2020-03-03T16:28:41Z dc.identifier.uri http://hdl.handle.net/1974/27649 dc.description.abstract The NEWS-G collaboration uses Spherical Proportional Counters to search for Weakly Interacting Massive Particles (WIMP). The first detector developed for this goal is a 60 cm diameter sphere installed at the Laboratoire Souterrain de Modane in France. In 2015, the collaboration took a run with neon as the target material for an exposure of 9.7 $\mathrm{kg\cdot days}$. This run allowed new limits to be set on the spin-independent WIMP-nucleon cross-sections with $\mathrm{90\%}$ confidence upper limit of $\mathrm{\sigma_{SI} < 4.4 \times 10^{-37} cm^{2}}$ for a $\mathrm{0.5\, GeV/c^{2}}$ WIMP. The study of the background events observed during this run shows that it is dominated by the presence of the $\mathrm{^{210}Pb}$ decay chain in the different materials composing the detector, its shielding, and on the inner surface of the sphere. The experiences acquired during the utilization of SEDINE and the analysis of its data allowed a procedure to be developed to avoid radioactive contaminations and minimize the background of the experiment. The background of the next detector was estimated by a stringent selection of the materials, the measurements of their radioactive contaminations and the simulation of the different components. The development of new sensors allows a better homogeneity of the detector response and good data acquisition in large detector. The new detector is a 140 cm diameter sphere, to be installed at SNOLAB in Canada in 2020. Its performance will be also enhanced by the development of methods of signal characterisation and calibration. en dc.language.iso eng en dc.relation.ispartofseries Canadian theses en dc.rights Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada en dc.rights ProQuest PhD and Master's Theses International Dissemination Agreement en dc.rights Intellectual Property Guidelines at Queen's University en dc.rights Copying and Preserving Your Thesis en dc.rights This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner. en dc.rights CC0 1.0 Universal en dc.rights.uri http://creativecommons.org/publicdomain/zero/1.0/ dc.subject Dark Matter en dc.subject Spherical Proportional Counter en dc.subject Gas Detector en dc.subject Radioactive Background en dc.title Optimization of spherical proportional counter backgrounds and response for low mass dark matter search en dc.type thesis en dc.description.degree PhD en dc.contributor.supervisor Gerbier, Gilles en dc.contributor.supervisor Giomataris, Ioannis en dc.contributor.department Physics, Engineering Physics and Astronomy en dc.degree.grantor Queen's University at Kingston en ### This item appears in the following Collection(s) Except where otherwise noted, this item's license is described as Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada	2021-01-23 01:53:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3688098192214966, "perplexity": 5171.751782561279}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703531702.36/warc/CC-MAIN-20210123001629-20210123031629-00764.warc.gz"}
https://documentation.aimms.com/functionreference/algorithmic-capabilities/the-gmp-library/gmp_row-procedures-and-functions/gmp_row_deactivate.html	Procedure GMP::Row::Deactivate(GMP, row) # GMP::Row::Deactivate The procedure GMP::Row::Deactivate deactivates a row in a generated mathematical program. A deactivated row will not be passed to a solver session. GMP::Row::Deactivate( GMP, ! (input) a generated mathematical program row ! (input) a scalar reference or row number ) ## Arguments GMP An element in AllGeneratedMathematicalPrograms. row A scalar reference to an existing row in the matrix or an element in the set Integers in the range $$\{ 0 .. m-1 \}$$ where $$m$$ is the number of rows in the matrix. ## Return Value The procedure returns 1 on success, and 0 otherwise. Note Use GMP::Row::DeactivateMulti or GMP::Row::DeactivateRaw if many rows have to be deactivated, because that will be more efficient.	2023-03-25 19:27:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5522354245185852, "perplexity": 3113.7052689648935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945372.38/warc/CC-MAIN-20230325191930-20230325221930-00755.warc.gz"}
https://mathcracker.com/practicing-descriptive-statistics	# Practicing Descriptive Statistics The best way of learning about how to solve Descriptive Statistics problems properly is to PRACTICE. Here you have a couple of step-by-step examples, exclusively for our subscribers Question 1: HiEd has held 5 recruiting visits around the region so far this year. The number of high school seniors attending these events is given below. 10 15 20 35 20 The average number of high school seniors attending an event is 20. a. Determine the range for this sample. b. Determine the standard deviation. Solution: (a) The range is computed as: $Range=Max-Min = {35}-{10} = {25}$ (b) Finally, the following table shows the required calculations needed to compute the standard deviation: X X² 10 100 15 225 20 400 35 1225 20 400 Sum = 100 2350 The sample variance is ${{s}^{2}}=\frac{1}{n-1}\left( \sum{X_{i}^{2}}-\frac{{{\left( \sum{{{X}_{i}}} \right)}^{2}}}{n} \right)=\frac{1}{5-1}\left( 2350-\frac{{{\left( 100 \right)}^{2}}}{5} \right)=87.5$ and also we find that the standard deviation is computed as $s=\sqrt{87.5}=9.3541$ ,p>Question 2: The Financial Times/Harris poll is a monthly online poll of adults from six countries in Europe and the united states. The poll conducted in January 2008 included 1015 adults. One of the questions asked was "How would you rate the federal bank in handling the credit problems in the financial markets?" Possible responses were excellent, good, fair, bad, and terrible. The 1015 responses for this question can be found in the data file named fedbank. a. Construct a frequency distribution. b. Construct a percent frequency distribution. c. Construct a bar chart for the percent frequency distribution. d. comment on how adults in the united states think the federal bank is handling the credit problems in the financial markets. e. In Spain, 1114 adults were asked, 'How would you rate the European central bank in handling the credit problems in the financial markets?" The percent frequency distribution obtained is as follows. Rating Percent Frequency excellent 0 good 4 fair 46 terrible 10 Compare the results obtained in Spain with the results obtained in the United States. Solution: (a) The following is obtained: Rating Frequency Bad 244 Excellent 20 Fair 528 Good 101 Terrible 122 Grand Total 1015 (b) Now we get: Rating Frequency Percent Frequency Bad 244 24.04% Excellent 20 1.97% Fair 528 52.02% Good 101 9.95% Terrible 122 12.02% Grand Total 1015 100.00% (c) The following is obtained: (d) In a big majority, US adults think the fed is handling it in a fair way. (e) In Spain most adults think that the handling has fair and bad. Question 3: For the following set of scores: 3, 7, 6, 5, 5, 9, 6, 4, 6, 8, 10, 2, 7, 4, 9, 5, 6, 3, 8 a. Construct a frequency distribution table. b. Sketch a polygon showing the distribution. c. Describe the distribution using the following characteristics: (1) What is the shape of the distribution? (2) What score best identifies the center (average) for the distribution? (3) Are the scores clustered together, or are they spread out across the scale? Solution: (a) The minimum and maximum values obtained from the data provided are Min = 2 Max = 10 Based on this, we choose the lowest class to be 2. The minimum class width is (10 – 2)/5 = 1.6, so we choose a class width of 2. Hence, the respective lower class limits are computed as Li = 2 + 2*i. The following table shows the frequency table: Classes Frequency Rel. Frequency Cumulative Frequency Rel. Cumulative Frequency 2 - 3 3 0.158 3 0.158 4 - 5 5 0.263 8 0.421 6 - 7 6 0.316 14 0.737 8 - 9 4 0.211 18 0.947 10 - 11 1 0.053 19 1 The following histogram is obtained (b) The following frequency polygon is obtained (c) (1) The distribution is approximately symmetric. (2) Since the distribution is roughly symmetric, the mean would be the best measure of central tendency. (3) The scores are relatively spread across the scale, even though they a bit concentrated about the center. This tutorial is brought to you courtesy of MyGeekyTutor.com In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us.	2020-07-15 09:21:55	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3871345520019531, "perplexity": 1478.4195269931317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657163613.94/warc/CC-MAIN-20200715070409-20200715100409-00348.warc.gz"}
https://www.analyzemath.com/applied_mathematics/standing_waves.html	# Standing Waves Mathematics are used to explain standing waves. ## Travelling Waves Let us consider the following waves $y_1 = A \cos (wt - bz)$ and $y2 = B \cos (wt - bz)$ Because of the term $wt - bz$, these two waves travel in the same direction. If we add these two waves, we obtain another travelling wave of the form $y = y_1 + y_2 = (A + B) \cos (wt - bz)$. ## Standing Waves Let us now consider the following waves $y_1 = a \cos (wt - bz)$ and $y_2 = a \cos (wt + bz)$ Note that because of the terms $wt - bz$ and $wt + bz$, the two waves travel in opposite directions. We now add the two waves $y = y_1 + y_2 = a \cos (wt - bz) + a \cos (wt + bz)$ Expand and simplify $y = a \cos (wt) \cos(wt) + a \sin (wt) \sin(wt) + a \cos (wt) \cos(wt) - a \sin (wt) \sin(wt)$ $y = 2a \cos (wt) \cos (bz)$ The terms containing the time and distance $\cos(wt)$ and $\cos(bz)$ are separated and therefore the wave obtained is not a travelling one. It is called a standing wave.	2021-06-18 05:57:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.627558708190918, "perplexity": 511.9280903544885}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487635724.52/warc/CC-MAIN-20210618043356-20210618073356-00008.warc.gz"}
https://math.stackexchange.com/questions/3744743/no-of-positive-integral-solutions-and-link-to-coefficients-in-expansion	# No. of positive integral solutions and link to coefficients in expansion The question (from an NTA sample paper for JEE Main) - If $$p, q, r \in \Bbb N$$, then the number of points having position vector $$p\hat{i} + q\hat{j} +r\hat{k}$$ such that $$8 \leq p + q + r \leq 12$$ are: It is evident that I had to essentially find the integral solutions for the inequalities given. I couldn't solve it in time and moved on. However upon reviewing the explanation and key concepts of the answer, I came out more perplexed and need help. They explain that you have to add the no. of solutions for $$p+q+r = 8, 9, 10, 11, 12$$. and also $$p,q,r \geq{1}$$. I knew how to solve for $$p,q,r \geq{0}$$ using the "Beggar's method/ Fencing method", but did not know how to solve this case. They used the formula, required number of positive integral solutions = $${n-1}\choose{r-1}$$ and have written the solution as: $${7\choose2} + {8\choose2} + {9\choose2} + {10\choose2} + {11\choose2} = 185$$ Makes sense, but here's the stuff that baffles me: The number of integral solutions of $$x_1 + x_2 + \ldots + x_r = n$$, where $$x_1 \geq 1, x_2 \geq 1, \ldots, x_r \geq 1$$ is the same as the number of ways to distribute n identical things among r persons getting at least 1. This is also equal to the coefficient of $$x^n$$ in $$(x^1 + x^2 + \ldots )^r$$ = coefficient of $$x^n$$ in $$x^r (1-x)^{-r}$$ = coefficient of $$x^{n-r}$$ in $$\{1 + rx + \frac{r(r+1)}{2!}x^2 + \cdots + \frac{r(r+1)(r+2)\ldots(r+n-1)}{n!}x^n + \ldots \}$$ = $$\frac{(n-1)!}{(n-r)!(r-1)!}$$ = $${n-1 \choose r-1}$$ They've also explained the case for $$x \geq 0$$ in a very similar manner above this, instead talking about coefficient of $$x^n$$ in $$(1-x)^{-r}$$. I'm struggling to understand what this means and how it ties in to combinations. I understand how the binomial theorem for natural indices uses combinations in effect to find the coefficient so I can see how they might also be important here, but there are a few things I'm not able to get here. How can I solve this problem in an intuitive way (like the $$x_i \geq 0$$ case)? What do the coefficient of $$x^n$$ have to do with this at all? Any help is much appreciated. The formula they use is the same as Theorem One from Stars and Bars. This is proved by considering $$k$$ fences that lie strictly within the fields, and only one fence between each pair of fields. The 'zero-option' allows for fences to be placed outside the fields, and also more than one together, creating 'empty spaces'. The generating function (GF) used is: $$(x^1 + x^2 + \ldots )^r$$ For the zero-option, we would use: $$(1 + x + x^2 + \ldots )^r$$ and for example if there were to be at least two fields between fences, we would use: $$(x^2 +x^3 + \ldots )^r$$ These all go into the binomial expansion, and we look for the coefficients of $$x^n, x^{n-r}, x^{n-2r}, \dots$$, depending on the minimum value of $$x_i$$ allowed. Which is like giving the minimum values to each player, do a zero-based stars and bars, and then give this as extra to each player. I guess I can help you by providing another way to think about the whole $$n$$ identical things being distributed among $$r$$ people. So, imagine you have $$n$$ identical objects to be distributed in $$r$$ groups and you plan to do so by putting partitions/walls between them. Since there are $$n$$ objects, you've got $$n-1$$ spaces to put that partition, if you don't want any group to have $$0$$ objects(that is similar to saying that $$x_i\geq1$$). Also, since you want $$r$$ groups, the number of partitions you need to use is $$r-1$$. Now, it breaks down to a simple Permutation Combination problem. Choose $$r-1$$ spaces/points from $$n-1$$, i.e. $$n-1\choose r-1$$ which gives you the "number of possible solutions under given constraint". Also, the questions with a given constraint like $$x_i\geq1$$ have their branches in the Multinomial Theorem which is not a part of JEE Mains (I suppose) and given that NTA has placed a few questions which are way out of JEE Mains syllabus in there tests, you can surely move on. If you suppose $$p+q+r=8$$ then they are natural numbers, so you can replace them with $$x+1$$, $$y+1$$ and $$z+1$$. Your new equation is $$x+y+z=5$$. The number of solutions is $$7\choose2$$. Similarly for remaining values, you will get the rest.	2020-08-11 18:21:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 49, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8511704206466675, "perplexity": 238.77241696976583}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439738819.78/warc/CC-MAIN-20200811180239-20200811210239-00393.warc.gz"}
http://mathhelpforum.com/differential-geometry/90740-laurent-series-expansion.html	# Math Help - (Laurent) Series Expansion 1. ## (Laurent) Series Expansion Show that the series expansion for arctan(z) = z -z^3/3 + z^5/5 - z^7/7 + .... . 2. start by using $\arctan z=\int_{0}^{z}{\frac{dt}{1+t^{2}}}=\int_{0}^{z}{\l eft( \sum\limits_{j=0}^{\infty }{\left( -t^{2} \right)^{j}} \right)\,dt}.$	2015-08-04 07:43:25	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9490131735801697, "perplexity": 9663.139533422953}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042990603.54/warc/CC-MAIN-20150728002310-00034-ip-10-236-191-2.ec2.internal.warc.gz"}
https://www.datasciencecentral.com/python-vs-r-the-eternal-question-for-data-scientists/	# Python Vs R : The Eternal Question for Data Scientists Python and R are the two most commonly used languages for data science today. They are both fully open source products and completely free to use and modify as required under the GNU public license. But which one is better? And, more importantly, which one should you learn? Both are widely used and are standard tools in the hands of every data scientist. The answer may surprise you – because as a professional data scientist, you should be ready to deal with both. Python has certain use cases and so does R. The scenarios in which they are used vary. It is more often the environment and the needs of the client or your employer which dictates the choice between Python and R. Many things are easier in Python. But R also has its place in your development toolkit. ## Python Python is a general-purpose programming language released by Guido Van Rossum in 1991. Since then, Python has been used in multiple environments for multiple purposes, including, but not limited to: ● Web Development (Django) ● Zappa Serverless Framework for Python ● TensorFlow (Deep Learning Machine Learning Models) ● Keras (High-Level Abstractions to Simplify TensorFlow Development) ● Popular apps built in Python include Dropbox, BitTorrent, Morpheus, Calibre, Blender, and Mercurial – among many, many others. Python has more appeal for software engineers. This is mainly because industry use, production-ready code can be usually written in Python. If you have the background of a software engineer or already know programming, Data Science in Python is a better choice for you (especially if you’re a beginner). Another situation where Python shines is the sheer number pre-existing libraries that are readily available and open sourced to use. The large number of packages available in the PyPI (short for Python Package Index) repository with over 121k packages that automate many programming tasks at various levels of abstraction, making life easy for the programmer. At least 6k out of the packages on PyPI are focused on data science. Python also excels in readability. Compared to R, Python is much easier to read and to understand. Python is faster than R, in some cases dramatically faster. ## R R is a statisticians programming language designed for statisticians by statisticians. It originated in the ‘90s through George Ross Ihaka and Robert Gentleman. R excels in academic use and in the hands of a statistician. People who have formal training in Statistics, such as a Statistics degree, find working with R extremely simple. The repository for R packages or libraries, called CRAN (Comprehensive R Archive Network) contains nearly 12k packages, roughly half of which are for data science with R. R also excels at data visualization. Analyzing data on a one-time basis is often simpler and more easily expressible in R. Also, once upon a time, using Python meant linking many libraries together, some of which would become incompatible after feature revisions and library updates. That is no longer true because of Anaconda – see below. For a short time, deep learning was strictly a Python feature – which shifted the balance of the machine learning world towards Python, for a short time. However, with the release of Keras for TensorFlow in R, that factor changed as well, and deep learning models could now be used in R. So, what is the answer? Which one should you use? ### Jupyter Notebook – Integrating Python and R The Anaconda distribution from Continuum Analytics has completely disrupted the machine learning picture. Anaconda supports the standard libraries required for Python and machine learning – NumPy, SciPy, Pandas, SymPy, Seaborn, Matplotlib – as well as full support for R with an outstanding IDE called R Studio. For Deep Learning it supports TensorFlow, Theano, Caffe, Scikit-Learn, and Torch. One of its most remarkable features is the introduction of the Jupyter Notebook, an integrated platform which supported the use of Python and R in the same environment while keeping everything open source. Another option is the Hydrogen plug-in for the Atom text editor. It allows you to enter any code that you can use in a Jupyter Notebook and returns the result in the editor. However, it is still in alpha and crashed with an error on my local machine. The Jupyter Lab application allows Python and R notebook editing in the same environment, using the concept of separate and even remote kernels. As the machine learning field progresses, one can expect more plugins like Hydrogen (which I can’t wait to test once it’s out of alpha) produced in the very near future. So, Python excels in machine learning, while R excels in statistics. But why should you learn both? Because a professional data scientist needs to understand statistics and the mathematics behind the machine learning algorithms in great detail. We shall examine two SVM machine learning models, one through Python code, and then another through R code. This will give us a good picture of how both languages work. ### Python Code This code performs binary classification using non-linear support vector machine using a Gaussian kernel. The target to predict is a XOR of the inputs. The color map illustrates the decision function learned by the Support Vector Machine Classifier. (SVC) print(__doc__) import numpy as np import matplotlib.pyplot as plt from sklearn import svm xx, yy = np.meshgrid(np.linspace(-3, 3, 500), np.linspace(-3, 3, 500)) np.random.seed(0) X = np.random.randn(300, 2) Y = np.logical_xor(X[:, 0] > 0, X[:, 1] > 0) # fit the model clf = svm.NuSVC() clf.fit(X, Y) # plot the decision function for each datapoint on the grid Z = clf.decision_function(np.c_[xx.ravel(), yy.ravel()]) Z = Z.reshape(xx.shape) plt.imshow(Z, interpolation=’nearest’, extent=(xx.min(), xx.max(), yy.min(), yy.max()), aspect=’auto’, origin=’lower’, cmap=plt.cm.PuOr_r) contours = plt.contour(xx, yy, Z, levels=[0], linewidths=2, linetypes=’–‘) plt.scatter(X[:, 0], X[:, 1], s=30, c=Y, cmap=plt.cm.Paired, edgecolors=’k’) plt.xticks(()) plt.yticks(()) plt.axis([-3, 3, -3, 3]) plt.show() .Output: ### R Code This program uses the iris dataset to illustrate the use of a non-linear SVM classifier. This code is deliberately a little more complex since it applies ML techniques to a full-fledged built in dataset – the iris dataset – one of the canonical data sets used to illustrate the capacities of the ML techniques traditionally. THis code also illustrates the usage of the built-in statistical functions of R. You will need to install R package e1071 and add it to the compile list by calling library(e1071) before executing the code below. But don’t worry – installing new packages in R Studio is ridiculously simple. # NOT RUN { data(iris) attach(iris) ## classification mode # default with factor response: library(e1071) #loads the svm library into the compile path model <- svm(Species ~ ., data = iris) x <- subset(iris, select = -Species) y <- Species model <- svm(x, y) print(model) summary(model) # test with train data pred <- predict(model, x) # (same as:) pred <- fitted(model) # Check accuracy: table(pred, y) .Output: y pred setosa versicolor virginica setosa 50 0 0 versicolor 0 48 2 virginica 0 2 48 <code> # visualize (classes by color, SV by crosses): plot(cmdscale(dist(iris[,- 5])), col = as.integer(iris[,5]), pch = c( “o”,“+”)[1:150 %in% model$index + 1]) ## try regression mode on two dimensions # create data x <- seq( 0.1, 5, by = 0.05) y <- log(x) + rnorm(x, sd = 0.2) # estimate model and predict input values m <- svm(x, y) new <- predict(m, x) # visualize plot(x, y) points(x, log(x), col = 2) points(x, new, col = 4) ## density-estimation # create 2-dim. normal with rho=0: X <- data.frame(a = rnorm( 1000), b = rnorm(1000)) attach(X) # traditional way: m <- svm(X, gamma = 0.1) # formula interface: m <- svm(~., data = X, gamma = 0.1) # or: m <- svm(~ a + b, gamma = 0.1) # test: newdata <- data.frame(a = c( 0, 4), b = c(0, 4)) predict (m, newdata) # visualize: plot(X, col = 1:1000 %in% m$index + 1, xlim = c(-5,5), ylim=c(-5,5)) points(newdata, pch = “+”, col = 2, cex = 5) i2 <- iris levels(i2$Species)[3] <-“versicolor” summary(i2$Species) wts <- 100 / table(i2\$Species) wts m <- svm(Species ~ ., data = i2, class.weights = wts) # } Output: As you can see, the R code is fundamentally more powerful in its graphing and statistical abilities than Python. Being a language of statisticians by statisticians, if you have a statistics background, using R will be the best launchpad for your new career in data science. ## Conclusion Thus, when it comes to choosing between Python and R, any data scientist who is worth his money will know that he is supposed to know both. And in the end, all the most advanced software engineering skills won’t get you anywhere unless you have a firm foundation in Statistics – or a professional statistician in your team. The main reason that we use analytics is to make business decisions. And we can utilize it best when we have an iron-clad grasp of the entire picture. So, on Python versus R, to sum up: Both perform similar tasks in data science but are optimized toward different domains. If you are a software engineer, choose Python. If you are an academic researcher, choose R. And if you are a data scientist – choose both. Original Article Published here	2022-07-05 05:55:13	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19996245205402374, "perplexity": 2838.0741858999554}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00572.warc.gz"}
https://webdemo.inue.uni-stuttgart.de/webdemos/03_theses/MU_Cap2/index.php?id=4	With NOMA, all users spread their signal across the entire bandwidth and send signals simultaneously, much like in the CDMA system. However, rather than decoding every user treating the interference from other users as noise, a successive interference cancellation (SIC) receiver is needed to achieve capacity. That is, after one user is decoded, its signal is stripped away from the aggregate received signal before the next user is decoded. With SIC, the set of all rates 􏷪$(R_1, R_2)$􏷫 satisfies the three constraints: $$R_1 < \log_{2}\left(1+\frac{P_1}{N_0}\right)$$ $$R_2 < \log_{2}\left(1+\frac{P_2}{N_0}\right)$$ $$R_1+R_2 < \log_{2}\left(1+\frac{P_1+P_2}{N_0}\right)$$ The first two say that the rate of the individual user cannot exceed the capacity of the point-to-point link with the other user absent from the system (these are called single-user bounds). The third says that the total throughput cannot exceed the capacity of a point-to-point AWGN channel with the sum of the received powers of the two users. It is obvious that the range of capacity region is related to SNR. The relationship between them will be shown in slide 6. In the next demo slides, $N_0$ = $0$ dBm, namely $1$ milliwatt, is hold for simplicity.	2022-12-09 08:24:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7291773557662964, "perplexity": 405.18296519142325}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711394.73/warc/CC-MAIN-20221209080025-20221209110025-00210.warc.gz"}
https://stats.stackexchange.com/questions/20722/is-it-appropriate-to-suppress-the-intercept-in-a-regression-graph	# Is it appropriate to suppress the intercept in a regression graph? In the event of a non-significant interaction between X and M on Y, is it sound to suppress the intercept in the graph that illustrates the relationship between X and Y (X being significant). The moderator line in this case will pass through the origin and represent each level of the factor (M) at zero (i.e. male and female for gender). SPSS provides this option under scatterplot. I'm not sure there's a right answer to this, but maybe there are some ways of thinking about the issue that would be helpful. Not putting another regression line on your plot is a little like saying that the effect does not exist. Just because the effect is non-significant, however, does not necessarily mean that it doesn't exist. Thus, you could, in effect, be making a type II error. In the end, you must make a decision regarding what to believe. This decision should be influenced by the results of your analysis, but also by other things as well. For example, what was the prior probability of this effect--was it very plausible or were there strong theoretical reasons to expect it? (It can happen that you investigate something you don't think is real, such as someone else believes it and you don't, but typically people don't spend time studying things that they don't think are plausible.) Other relevant questions pertain to your study: Is there a confound or some idiosyncrasy of your study that might have caused the non-significant result? Do you have enough data to tell for sure? For instance, the estimated effect is never exactly zero, so what if that really is the true value, is it so small that you wouldn't care that it's not exactly zero anyway? Etc. On another note, I gather you are making this graph for communicative / persuasive purposes. Thus, for example, if you really don't think the effect is real and you want to be able to show this to people, you may well want to plot both lines (possibly with observations from the two groups represented by different plotting symbols as well), so as to make it apparent that there is either no difference or one so trivial that no one should care. Or, that might clutter up you graph and make it harder for people to recognize your main message, and so you would leave it off for the sake of simplicity and clarity. One of us is confused. If I understand correctly, your model (ignoring residuals) is $Y = \alpha + \beta_1 \cdot M + \beta_2 \cdot X + \beta_3 \cdot M \cdot X$ The intercept is $\alpha$. It does not visually represent the interaction; the slope (the coefficient $\beta_3$) on $M \cdot X$ is the interaction. If $\beta_3$ is not significantly different from zero, it's sound to omit the interaction term from the regression, and to re-fit the model without it, which means that now your model looks like $Y = \alpha + \beta_1 \cdot M + \beta_2 \cdot X$ If the remaining coefficients are all significant, the visual representation would have two lines. One line for the observations where the factor is present (e.g., male): $Y = \alpha + \beta_1 \cdot 1 + \beta_2 \cdot X = (\alpha + \beta_1) + \beta_2 \cdot X$ And one line for the observations where the factor is not present (e.g., female): $Y = \alpha + \beta_2 \cdot X$ So that's why it doesn't make sense to me to ask about omitting the intercept $\alpha$ based on the value of $\beta_3$. • You're right that $\alpha$ does not represent the interaction in any way. However, @AdheshJosh has asked a lot of questions about how to fit a model with 1 quantitative predictor and 1 qualitative predictor, how to interpret the interaction, etc. I am assuming that at this point that has all been done adequately and this is just an issue with the phrasing of the question. On the other hand, we should be cautious about fitting a model, dropping a non-significant term, refitting, etc. This is logically equivalent to an automatic search procedure and is associated with a lot of problems. – gung - Reinstate Monica Jan 8 '12 at 1:28	2020-02-19 13:38:26	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6633705496788025, "perplexity": 331.47556237465875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875144150.61/warc/CC-MAIN-20200219122958-20200219152958-00142.warc.gz"}
https://ems.press/books/elm/183?na	BookselmMonograph # Classification of Complex Algebraic Surfaces • ### Ciro Ciliberto Università di Roma Tor Vergata, Italy A subscription is required to access this book. Τhe classification of complex algebraic surfaces is a very classical subject which goes back to the old Italian school of algebraic geometry with Enriques and Castelnuovo. However, the exposition in the present book is modern and follows Mori's approach to the classification of algebraic varieties. The text includes the $P_{12}$ theorem, the Sarkisov programme in the surface case and the Noether–Castelnuovo theorem in its classical version.	2022-07-05 07:10:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3935900032520294, "perplexity": 1161.8765788622131}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104514861.81/warc/CC-MAIN-20220705053147-20220705083147-00627.warc.gz"}
https://questions.examside.com/past-years/jee/question/let-f1left-x10-right-and-f2left-x20-right-for-x1-0-and-x2-0-jee-advanced-2016-marks-4-o5lrt1pzrq3u0sal.htm	NEW New Website Launch Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc... 1 ### JEE Advanced 2016 Paper 2 Offline Let $${F_1}\left( {{x_1},0} \right)$$ and $${F_2}\left( {{x_2},0} \right)$$ for $${{x_1} < 0}$$ and $${{x_2} > 0}$$, be the foci of the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 8} = 1$$. Suppose a parabola having vertex at the origin and focus at $${F_2}$$ intersects the ellipse at point $$M$$ in the first quadrant and at point $$N$$ in the fourth quadrant. The orthocentre of the triangle $${F_1}MN$$ is A $$\left( { - {9 \over {10}},0} \right)$$ B $$\left( { {2 \over {3}},0} \right)$$ C $$\left( { {9 \over {10}},0} \right)$$ D $$\left( {{2 \over 3},\sqrt 6 } \right)$$ ## Explanation F1(x, 0) and F2(x2, 0) are the foci of the ellipse: $${{{x^2}} \over 9} + {{{y^2}} \over 8} = 1$$ Therefore, a2 = 9 and b2 = 8. $${b^2} = {a^2}(1 - {e^2})$$ $$1 - {e^2} = {8 \over 9} \Rightarrow {e^2} = 1 - {8 \over 9} = {1 \over 9} \Rightarrow e = {1 \over 3}$$ The focus is $${F_1}\left( { - 3 \times {1 \over 3},0} \right)$$ and $${F_2}\left( {3 \times {1 \over 3},0} \right)$$ That is, F1($$-$$1, 0) and F2(1, 0). The equation of parabola is $${y^2} = 4(O{F_2})x$$ $${y^2} = 4x(O{F_2} = 1)$$ The point of intersection of ellipse and parabola is $${{{x^2}} \over 9} + {{4x} \over 8} = 1 \Rightarrow {{{x^2}} \over 9} + {x \over 2} = 1$$ $$\Rightarrow 2{x^2} + 9x - 18 = 0$$ $$\Rightarrow 2{x^2} + 12x - 3x - 18 = 0$$ $$\Rightarrow 2x(x + 6) - 3(x + 6) = 0$$ $$\Rightarrow x = {3 \over 2}$$ (x $$-$$6 is rejected) Now, $${y^2}(4){3 \over 2} = 6$$ $$y = \pm \sqrt 6$$ That is, the points M and N are, respectively, $$M\left( {{3 \over 2},\sqrt 6 } \right)$$ and $$N\left( {{3 \over 2}, - \sqrt 6 } \right)$$. Let the orthocenter be (h, k). The slope of $$OM = {{k - \sqrt 6 } \over {h - (3/2)}}$$ The slope of $$ON = {{\sqrt 6 } \over { - 1 - (3/2)}} = {{ - 2\sqrt 6 } \over 5}$$ Now, $$\left( {{{k - \sqrt 6 } \over {h - (3/2)}}} \right)\left( {{{ - 2\sqrt 6 } \over 5}} \right) = - 1$$ $$2\sqrt 6 k - 12 = 5h - {{15} \over 2}$$ $$5h - 2\sqrt 6 k = {{15} \over 2} - 12 = {{ - 9} \over 2}$$ The slope of $$ON = {{k + \sqrt 6 } \over {h - (3/2)}}$$ The slope of $${F_1}M = {{\sqrt 6 } \over {1 + (3/2)}} = {{2\sqrt 6 } \over 5}$$ $${{k + \sqrt 6 } \over {h - (3/2)}} \times {{2\sqrt 6 } \over 5} = - 1$$ $$2\sqrt 6 k + 12 = - 5h + {{15} \over 2}$$ $$5h + 2\sqrt 6 k = {{15} \over 2} - 12 = {{ - 9} \over 2}$$ $$5h + 2\sqrt 6 k = {{ - 9} \over 2}$$ ....... (1) $$5h - 2\sqrt 6 k = {{ - 9} \over 2}$$ ........ (2) Solving Eqs. (1) and (2), we get $$10h = - 9 \Rightarrow h = {{ - 9} \over {10}}$$ and k = 0 Hence, the orthocentre of the triangle F1MN is $$\left( {{{ - 9} \over {10}},0} \right)$$. 2 ### JEE Advanced 2016 Paper 2 Offline Let $${F_1}\left( {{x_1},0} \right)$$ and $${F_2}\left( {{x_2},0} \right)$$ for $${{x_1} < 0}$$ and $${{x_2} > 0}$$, be the foci of the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 8} = 1$$. Suppose a parabola having vertex at the origin and focus at $${F_2}$$ intersects the ellipse at point $$M$$ in the first quadrant and at point $$N$$ in the fourth quadrant. If the tangents to the ellipse at $$M$$ and $$N$$ meet at $$R$$ and the normal to the parabola at $$M$$ meets the $$x$$-axis at $$Q$$, then the ratio of area of the triangle $$MQR$$ to area of the quadrilateral $$M{F_1}N{F_2}$$is A $$3:4$$ B $$4:5$$ C $$5:8$$ D $$2:3$$ ## Explanation Equation of tangent at M(3/2, $$\sqrt6$$) to $${{{x^2}} \over 9} + {{{y^2}} \over 8} = 1$$ is $${3 \over 2}.{x \over 9} + \sqrt 6 .{y \over 8} = 1$$ ....... (i) which intersect X-axis at (6, 0). Also, equation of tangent at N(3/2, $$-$$$$\sqrt6$$) is $${3 \over 2}.{x \over 9} - \sqrt 6 .{y \over 8} = 1$$ ....... (ii) Eqs. (i) and (ii) intersect on X-axis at R(6, 0). ........ (iii) Also, normal at $$M(3/2,\sqrt 6 )$$ is $$y - \sqrt 6 = {{ - \sqrt 6 } \over 2}\left( {x - {3 \over 2}} \right)$$ On solving with y = 0, we get Q(7/2, 0) ....... (iv) The area of MQR is $$\left\| {{1 \over 2}} \right\|\left\| {\matrix{ {3/2} & {\sqrt 6 } & 1 \cr 6 & 0 & 1 \cr {7/2} & 0 & 1 \cr } } \right\| = \left\| {{{\sqrt 6 } \over 2}\left( {6 - {7 \over 2}} \right)} \right\| = {{5\sqrt 6 } \over 4}$$ The area of the quadrilateral MF1NF2 is $$2(\Delta {m_1}{F_1}{F_2}) = 2\sqrt 6$$ and the required ratio is $${{5\sqrt 6 } \over {4\,.\,2\sqrt 6 }} = {5 \over 8}$$ 3 ### JEE Advanced 2014 Paper 2 Offline Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$ and $$S\,\,\left( {a{s^2},2as} \right)$$ be distinct points on the parabola $${y^2} = 4ax$$. Suppose that $$PQ$$ is the focal chord and lines $$QR$$ and $$PK$$ are parallel, where $$K$$ is the point $$(2a,0)$$ If $$st=1$$, then the tangent at $$P$$ and the normal at $$S$$ to the parabola meet at a point whose ordinate is A $${{{{\left( {{t^2} + 1} \right)}^2}} \over {2{t^3}}}$$ B $${{a{{\left( {{t^2} + 1} \right)}^2}} \over {2{t^3}}}$$ C $${{a{{\left( {{t^2} + 1} \right)}^2}} \over {{t^3}}}$$ D $${{a{{\left( {{t^2} + 2} \right)}^2}} \over {{t^3}}}$$ 4 ### JEE Advanced 2014 Paper 2 Offline Let $$a, r, s, t$$ be nonzero real numbers. Let $$P\,\,\left( {a{t^2},2at} \right),\,\,Q,\,\,\,R\,\,\left( {a{r^2},2ar} \right)$$ and $$S\,\,\left( {a{s^2},2as} \right)$$ be distinct points on the parabola $${y^2} = 4ax$$. Suppose that $$PQ$$ is the focal chord and lines $$QR$$ and $$PK$$ are parallel, where $$K$$ is the point $$(2a,0)$$ The value of $$r$$ is A $$- {1 \over t}$$ B $${{{t^2} + 1} \over t}$$ C $${1 \over t}$$ D $${{{t^2} - 1} \over t}$$ ### Joint Entrance Examination JEE Main JEE Advanced WB JEE ### Graduate Aptitude Test in Engineering GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN NEET Class 12	2022-05-29 03:11:49	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8735412955284119, "perplexity": 599.3266575396397}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663035797.93/warc/CC-MAIN-20220529011010-20220529041010-00645.warc.gz"}
http://jmre.dlut.edu.cn/cn/ch/reader/view_abstract.aspx?flag=1&file_no=20120412&journal_id=cn	Asymptotic Behavior of a Non-Local Hyperbolic Equation Modelling Ohmic Heating DOI：10.3770/j.issn:2095-2651.2012.04.012 作者单位汪先超国家数字交换系统工程技术研究中心, 河南郑州 450002 江成顺中南财经政法大学武汉学院, 湖北武汉 430079 本文研究了基于欧姆加热模型的一类非局部双曲问题解的渐近性态.研究发现该双曲问题的解只有三种情况:问题解整体有界且其唯一稳态解渐近稳定;问题解无穷远爆破;问题解有限时刻爆破.如果问题解在有限时刻爆破,该解在(0,1]的任意子区间上一致爆破,且爆破速度为$\lim_{t\rightarrow T^{}-}u(x,t)(T^{}-t)^{\frac{1}{\alpha+\beta p-1}}=(\frac{\alpha+\beta p-1}{1-\alpha})^{\frac{1}{1-\alpha-\beta p}}$,这里$T^$是爆破时间. In this paper, the asymptotic behavior of a non-local hyperbolic problem modelling Ohmic heating is studied. It is found that the behavior of the solution of the hyperbolic problem only has three cases: the solution is globally bounded and the unique steady state is globally asymptotically stable; the solution is infinite when $t\rightarrow\infty$; the solution blows up. If the solution blows up, the blow-up is uniform on any compact subsets of $(0,1]$ and the blow-up rate is $\lim_{t\rightarrow T^{}-}u(x,t)(T^{}-t)^{\frac{1}{\alpha+\beta p-1}}=(\frac{\alpha+\beta p-1} {1-\alpha})^{\frac{1}{1-\alpha-\beta p}}$, where $T^{}$ is the blow-up time.	2020-06-04 05:01:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7836147546768188, "perplexity": 682.9505223583642}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347439019.86/warc/CC-MAIN-20200604032435-20200604062435-00177.warc.gz"}
http://physics.stackexchange.com/questions/69443/fundamental-representation-of-su3-is-a-complex-representation	# Fundamental Representation of $SU(3)$ is a complex representation Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule: $\qquad \qquad \qquad [T^a_R, T^b_R]=if^{abc}T^c_R$. Now if $-(T^a_R)^* = T^a_R$, the representation $R$ is real. Again if we can find a unitary matrix, $V(\neq I)$ such that $\qquad \qquad \qquad -(T^a_R)^=V^{-1} T^a_R V \quad \forall a$ then the representation $R$ is pseudoreal. If a representation is neither real nor pseudoreal, the representation $R$ is complex. Claim: One way to show that a representation is complex is to show that at least one generator matrix $T^a_R$ has eigenvalues that do not come in plus-minus pairs. Now let us consider $SU(3)$ group. The generators in the fundamental representation are given by $T^a =\lambda^a/2; \quad a=1,...8$, where $\lambda^a$s are the Gell-Mann matrices. We see that $T^8$ has eigenvalues $(1/\sqrt{12}, 1/\sqrt{12}, -1/\sqrt{3} )$. My doubt is: According to the claim, is the fundamental representation of $SU(3)$ a complex representation? - An element of the fundamental representation of $SU(N)$ is a $nn$ complex matrix $M$ such that $y^\dagger M^\dagger Mx = y^\dagger x$, for every complex vectors $x, y$. – Trimok Jun 28 '13 at 16:39 This link might be useful: motls.blogspot.com/2013/04/complex-real-and-pseudoreal.html – Anne O'Nyme Jul 8 '14 at 9:48 First of all, we are dealing with unitary representations, so that the $T^a$s are always self-adjoint and the representations have the form $$U(v) = e^{i \sum_{a=1}^Nv^a T_a}$$ with $v \in \mathbb R^N$. When you say that $U$ is real you just mean that the representation is made of the very real, unitary, $n\times n$ matrices $U$. This way, the condition $(T_a)^* = -T_a$ is equivalent to the reality (in the proper sense) requirement. Let us come to your pair of questions. (1). You are right on your point: PROPOSITION. A unitary finite dimensional representation is complex (i.e. it is neither real nor pseudoreal) if and only if at least one self-adjoint generator $T_a$ has an eigenvalue $\lambda$ such that $-\lambda$ is not an eigenvalue. PROOF Suppose that $$(T_a)^* = -V T_a V^{-1}\tag{1}$$ for some unitary matrix $V$ and every $a=1,2,3,\ldots, N$. Since we also know that $T_a$ is self-adjoint, there is an orthogonal basis of eigenvectors $u_j^{(a)}\neq 0$, $j=1,\ldots, n$ and the eigenvalues $\lambda_j^{(a)}$ are real. Therefore: $$T_au_j^{(a)}= \lambda_j^{(a)} u_j^{(a)}\:.$$ Taking the complex conjugation and using (1) $$VT_aV^{-1}u_j^{(a)}= -\lambda_j^{(a)} u_j^{(a)}$$ so that $V^{-1}u_j^{(a)}\neq 0$ is an eigenvector of $T_a$ with eigenvalue $-\lambda_j$. We conclude that $\lambda$ is an eigenvalue if and only if $-\lambda$ is (consequently, if the dimension of the space is odd, $0$ must necessarily be an eigenvalue as well). Soppose,vice versa, that for the self-adjoint matrix $T^a$ its (real) eigenvalues satisfy the constraint that $\lambda$ is an eigenvalue if and only if $-\lambda$ is. As $T^a$ is self adjoint, there is a unitary matrix such that: $$T_a = U diag(\lambda_1, -\lambda_1, \lambda_2, -\lambda_2,..., \lambda_{n/2},-\lambda_{n/2}) U^{-1}$$ when $n$ is even, otherwise there is a further vanishing last element on the diagonal. Thus $$T^_a = U^* diag(\lambda_1, -\lambda_1, \lambda_2, -\lambda_2,..., \lambda_{n/2},-\lambda_{n/2}) U^{-1}$$ Notice that $U^$ is unitary if $U$ is such. Let us indicate by $e_1,e_2, \ldots, e_n$ the orthonormal basis of eigenvectors of $T^a$ where the matrix takes the above diagonal form. If $W$ is the (real) unitary matrix which swaps $e_1$ with $e_2$, $e_3$ with $e_4$ and so on (leaving $e_n$ fixed if $n$ is odd), we have that $$W diag(\lambda_1, -\lambda_1, \lambda_2, -\lambda_2,..., \lambda_{n/2},-\lambda_{n/2}) W^{-1} = - diag(\lambda_1, -\lambda_1, \lambda_2, -\lambda_2,..., \lambda_{n/2},-\lambda_{n/2})$$ and thus $$T^_a = U^W^{-1}(- UT_aU^{-1}) WU^{-1} = -S T_a S^{-1}$$ with $S= U^W^{-1}U$, which is unitary because composition of unitary matrices. We can conclude that, as you claim, a way to show that a representation is complex (i.e. it is not real) is to show that at least one generator matrix $T_a$ has (non-vanishing) eigenvalues that do not come in plus-minus pairs. QED (2). In view of (1), if the list of eigenvalues you presented is correct, the considered representation is obviously complex. - The N-dimensional fundamental representation of SU(N) for N greater than two is a complex representation whose complex conjugate is often called the antifundamental representation. Thus SU(3) fundamental representation is a complex representation. (see for example: Wiki) - – Angie38750 Feb 3 '14 at 4:16	2016-06-26 02:46:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.957257091999054, "perplexity": 120.14625679323736}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783394605.61/warc/CC-MAIN-20160624154954-00159-ip-10-164-35-72.ec2.internal.warc.gz"}
http://bootmath.com/what-is-the-intuitive-meaning-of-the-scalar-curvature-r.html	# What is the intuitive meaning of the scalar curvature R? Background: Let $M$ be a smooth, Riemannian manifold with metric $g$ and dimension $n$. Let $R^a_{bcd}$ be the Riemann tensor with respect to the Levi-Civita connection for $g$. Question: Is there any rigorous result that gives a good intuitive sense of the meaning of the scalar curvature $R = R^{ab} R_{ab}$? Discussion: What I have in mind is something like the following: For $n =2$, the volume of a geodesic ball of radius $\epsilon$ in $M$ is $\pi \epsilon^2 [1 – (R / 48) \epsilon ^2 + O(\epsilon^4) ]$. I may have the numerical factors wrong, but the point is this: $R$ tells you the difference between the volume of a geodesic ball and an ordinary Euclidean ball (for small radius). That’s the kind of result I’m looking for. My problem with this result is that it holds in normal geodesic coordinates but not in general coordinates. (Note that a choice of coordinates is necessary to define ‘a geodesic ball of radius $\epsilon$’). If you know how to generalize this result to arbitrary coordinates, or you know another result that gives some intuition for $R$, please let me know. Thanks for any help! #### Solutions Collecting From Web of "What is the intuitive meaning of the scalar curvature R?" What you have mentioned for $n=2$ is in fact true for higher dimension too: Quoted from here, we know that the ratio of the $n$-dimensional volume of a ball of radius $\epsilon$ in the manifold $M$ to that of a corresponding ball in Euclidean space $\mathbb{R}^n$ is given by, for small $\epsilon$, $$\frac{\mbox{Vol}(B_\epsilon(p)\subset M)}{\mbox{Vol}(B_\epsilon(p)\subset\mathbb{R}^n)}=1-\frac{R}{6(n+2)}\epsilon^2+O(\epsilon^4).$$ (Therefore, you are right with the constant for the case when $n=2$). Let us start with two familiar settings. On the sphere of radius 1, a spherical cap of geodesic radius $\rho$ has area $2 \pi (1 – \cos \rho).$ In the hyperbolic plane of curvature $-1,$ the area of a disk of radius $\rho$ has area $2 \pi ( \cosh \rho – 1).$ Writing out just a few terms of the power series for $\cos$ and $\cosh$ tells you the comparison to $\pi \rho^2.$ Actually, it is not necessary to specify a coordinate system of any sort to define distance on a Riemannian manifold, that is what geodesics are for. As a result, for small radii, the geodesic ball is also well-defined. Furthermore, in a pleasant feature, for small enough radii the geodesics do meet the sphere bounding the ball orthogonally, this is usually called the Gauss Lemma Other than that, see Paul’s answer…	2018-06-24 07:00:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9203613996505737, "perplexity": 125.49762515599501}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866888.20/warc/CC-MAIN-20180624063553-20180624083553-00257.warc.gz"}
https://www.commonlounge.com/python-regular-expressions-8237bf8f2b4641fca5bc4463d33a23bf/	# Python Regular Expressions March 08, 2018 Regular expressions are a powerful language for matching text patterns. This page gives a basic introduction to regular expressions and shows how regular expressions work in Python. The Python re module provides regular expression support. In Python a regular expression search is typically written as: match = re.search(pat, str) The re.search() method takes a regular expression pattern and a string and searches for that pattern within the string. If the search is successful, search() returns a match object or None otherwise. Therefore, the search is usually immediately followed by an if-statement to test if the search succeeded, as shown in the following example which searches for the pattern 'word:' followed by a 3 letter word (details below): str = 'an example word:cat!!' match = re.search(r'word:\w\w\w', str) # If-statement after search() tests if it succeeded if match: print 'found', match.group() ## 'found word:cat' else: print 'did not find' The code match = re.search(pat, str) stores the search result in a variable named match. Then the if-statement tests the match — if true the search succeeded and match.group() is the matching text (e.g. 'word:cat'). Otherwise if the match is false (None to be more specific), then the search did not succeed, and there is no matching text. The r at the start of the pattern string designates a python “raw” string which passes through backslashes without change which is very handy for regular expressions (Java needs this feature badly!). I recommend that you always write pattern strings with the r just as a habit. # Basic Patterns The power of regular expressions is that they can specify patterns, not just fixed characters. Here are the most basic patterns which match single chars: • a, X, 9, < — ordinary characters just match themselves exactly. The meta-characters which do not match themselves because they have special meanings are: . ^ $ * + ? { [ ] \ \| ( ) (details below) • . (a period) — matches any single character except newline '\n' • \w — (lowercase w) matches a “word” character: a letter or digit or underbar [a-zA-Z0-9_]. Note that although “word” is the mnemonic for this, it only matches a single word char, not a whole word. \W (upper case W) matches any non-word character. • \b — boundary between word and non-word • \s — (lowercase s) matches a single whitespace character — space, newline, return, tab, form [ \n\r\t\f]. \S (upper case S) matches any non-whitespace character. • \t, \n, \r — tab, newline, return • \d — decimal digit [0-9] (some older regex utilities do not support but \d, but they all support \w and \s) • ^ = start, $ = end — match the start or end of the string • \ — inhibit the “specialness” of a character. So, for example, use \. to match a period or \\ to match a slash. If you are unsure if a character has special meaning, such as '@', you can put a slash in front of it, \@, to make sure it is treated just as a character. # Basic Examples Joke: what do you call a pig with three eyes? piiig! The basic rules of regular expression search for a pattern within a string are: • The search proceeds through the string from start to end, stopping at the first match found • All of the pattern must be matched, but not all of the string • If match = re.search(pat, str) is successful, match is not None and in particular match.group() is the matching text ## Search for pattern 'iii' in string 'piiig'. ## All of the pattern must match, but it may appear anywhere. ## On success, match.group() is matched text. match = re.search(r'iii', 'piiig') ## found, match.group() == "iii" ## . = any char but \n match = re.search(r'..g', 'piiig') ## found, match.group() == "iig" ## \d = digit char, \w = word char match = re.search(r'\d\d\d', 'p123g') ## found, match.group() == "123" match = re.search(r'\w\w\w', '@@abcd!!') ## found, match.group() == "abc" # Repetition Things get more interesting when you use + and * to specify repetition in the pattern • + — 1 or more occurrences of the pattern to its left, e.g. 'i+' = one or more i’s • * — 0 or more occurrences of the pattern to its left • ? — match 0 or 1 occurrences of the pattern to its left # Leftmost & Largest First the search finds the leftmost match for the pattern, and second it tries to use up as much of the string as possible — i.e. + and * go as far as possible (the + and * are said to be “greedy”). # Repetition Examples ## i+ = one or more i's, as many as possible. match = re.search(r'pi+', 'piiig') ## found, match.group() == "piii" ## Finds the first/leftmost solution, and within it drives the + ## as far as possible (aka 'leftmost and largest'). ## In this example, note that it does not get to the second set of i's. match = re.search(r'i+', 'piigiiii') ## found, match.group() == "ii" ## \s* = zero or more whitespace chars ## Here look for 3 digits, possibly separated by whitespace. match = re.search(r'\d\s\d\s\d', 'xx1 2 3xx') ## found, match.group() == "1 2 3" match = re.search(r'\d\s\d\s\d', 'xx12 3xx') ## found, match.group() == "12 3" match = re.search(r'\d\s\d\s\d', 'xx123xx') ## found, match.group() == "123" ## ^ = matches the start of string, so this fails: ## but without the ^ it succeeds: match = re.search(r'b\w+', 'foobar') ## found, match.group() == "bar" # Emails Example Suppose you want to find the email address inside the string 'xyz [email protected] purple monkey'. We’ll use this as a running example to demonstrate more regular expression features. Here’s an attempt using the pattern r'\w+@\w+': str = 'purple [email protected] monkey dishwasher' match = re.search(r'\w+@\w+', str) if match: print match.group() ## 'b@google' The search does not get the whole email address in this case because the \w does not match the '-' or '.' in the address. We’ll fix this using the regular expression features below. # Square Brackets Square brackets can be used to indicate a set of chars, so [abc] matches 'a' or 'b' or 'c'. The codes \w, \s etc. work inside square brackets too with the one exception that dot (.) just means a literal dot. For the emails problem, the square brackets are an easy way to add '.' and '-' to the set of chars which can appear around the @ with the pattern r'[\w.-]+@[\w.-]+' to get the whole email address: match = re.search(r'[\w.-]+@[\w.-]+', str) if match: print match.group() ## '[email protected]' (More square-bracket features) You can also use a dash to indicate a range, so [a-z] matches all lowercase letters. To use a dash without indicating a range, put the dash last, e.g. [abc-]. An up-hat ^ at the start of a square-bracket set inverts it, so [^ab] means any char except 'a' or 'b'. # Group Extraction The group feature of a regular expression allows you to pick out parts of the matching text. Suppose for the emails problem that we want to extract the username and host separately. To do this, add parenthesis ( ) around the username and host in the pattern, like this: r'([\w.-]+)@([\w.-]+)'. In this case, the parenthesis do not change what the pattern will match, instead they establish logical groups inside of the match text. On a successful search, match.group(1) is the match text corresponding to the 1st left parenthesis, and match.group(2) is the text corresponding to the 2nd left parenthesis. The plain match.group() is still the whole match text as usual. str = 'purple [email protected] monkey dishwasher' match = re.search('([\w.-]+)@([\w.-]+)', str) if match: print match.group() ## '[email protected]' (the whole match) print match.group(1) ## 'alice-b' (the username, group 1) print match.group(2) ## 'google.com' (the host, group 2) A common workflow with regular expressions is that you write a pattern for the thing you are looking for, adding parenthesis groups to extract the parts you want. # findall findall() is probably the single most powerful function in the re module. Above we used re.search() to find the first match for a pattern. findall() finds all the matches and returns them as a list of strings, with each string representing one match. ## Suppose we have a text with many email addresses str = 'purple [email protected], blah monkey [email protected] blah dishwasher' ## Here re.findall() returns a list of all the found email strings emails = re.findall(r'[\w\.-]+@[\w\.-]+', str) ## ['[email protected]', '[email protected]'] for email in emails: # do something with each found email string print email # findall With Files For files, you may be in the habit of writing a loop to iterate over the lines of the file, and you could then call findall() on each line. Instead, let findall() do the iteration for you — much better! Just feed the whole file text into findall() and let it return a list of all the matches in a single step (recall that f.read() returns the whole text of a file in a single string): # Open file f = open('test.txt', 'r') # Feed the file text into findall(); it returns a list of all the found strings strings = re.findall(r'some pattern', f.read()) # findall and Groups The parenthesis ( ) group mechanism can be combined with findall(). If the pattern includes 2 or more parenthesis groups, then instead of returning a list of strings, findall() returns a list of tuples. Each tuple represents one match of the pattern, and inside the tuple is the group(1), group(2) .. data. So if 2 parenthesis groups are added to the email pattern, then findall() returns a list of tuples, each length 2 containing the username and host, e.g. ('alice', 'google.com'). str = 'purple [email protected], blah monkey [email protected] blah dishwasher' tuples = re.findall(r'([\w\.-]+)@([\w\.-]+)', str) print tuples ## [('alice', 'google.com'), ('bob', 'abc.com')] for tuple in tuples: print tuple[1] ## host Once you have the list of tuples, you can loop over it to do some computation for each tuple. If the pattern includes no parenthesis, then findall() returns a list of found strings as in earlier examples. If the pattern includes a single set of parenthesis, then findall() returns a list of strings corresponding to that single group. (Obscure optional feature: Sometimes you have paren ( ) groupings in the pattern, but which you do not want to extract. In that case, write the parens with a ?: at the start, e.g. (?: ) and that left paren will not count as a group result.) # RE Workflow and Debug Regular expression patterns pack a lot of meaning into just a few characters , but they are so dense, you can spend a lot of time debugging your patterns. Set up your runtime so you can run a pattern and print what it matches easily, for example by running it on a small test text and printing the result of findall(). If the pattern matches nothing, try weakening the pattern, removing parts of it so you get too many matches. When it’s matching nothing, you can’t make any progress since there’s nothing concrete to look at. Once it’s matching too much, then you can work on tightening it up incrementally to hit just what you want. # Options The re functions take options to modify the behavior of the pattern match. The option flag is added as an extra argument to the search() or findall() etc., e.g. re.search(pat, str, re.IGNORECASE). • IGNORECASE — ignore upper / lowercase differences for matching, so 'a' matches both 'a' and 'A'. • DOTALL — allow dot (.) to match newline — normally it matches anything but newline. This can trip you up — you think .* matches everything, but by default it does not go past the end of a line. Note that \s (whitespace) includes newlines, so if you want to match a run of whitespace that may include a newline, you can just use \s* • MULTILINE — Within a string made of many lines, allow ^ and $ to match the start and end of each line. Normally ^ / $ would just match the start and end of the whole string. # Greedy vs. Non-Greedy (optional) This is optional section which shows a more advanced regular expression technique. Suppose you have text with tags in it: <b>foo</b> and <i>so on</i> Suppose you are trying to match each tag with the pattern '(<.>)' — what does it match first? The result is a little surprising, but the greedy aspect of the . causes it to match the whole '<b>foo</b> and <i>so on</i>' as one big match. The problem is that the .* goes as far as is it can, instead of stopping at the first > (aka it is “greedy”). There is an extension to regular expression where you add a ? at the end, such as .? or .+?, changing them to be non-greedy. Now they stop as soon as they can. So the pattern '(<.?>)' will get just '<b>' as the first match, and '</b>' as the second match, and so on getting each <..> pair in turn. The style is typically that you use a .?, and then immediately its right look for some concrete marker (> in this case) that forces the end of the .? run. The ? extension originated in Perl, and regular expressions that include Perl’s extensions are known as Perl Compatible Regular Expressions — pcre. Python includes pcre support. Many command line utils etc. have a flag where they accept pcre patterns. An older but widely used technique to code this idea of “all of these chars except stopping at X” uses the square-bracket style. For the above you could write the pattern, but instead of . to get all the chars, use [^>]* which skips over all characters which are not > (the leading ^ “inverts” the square bracket set, so it matches any char not in the brackets). # Substitution (optional) The re.sub(pat, replacement, str) function searches for all the instances of pattern in the given string, and replaces them. The replacement string can include '\1', '\2' which refer to the text from group(1), group(2), and so on from the original matching text. Here’s an example which searches for all the email addresses, and changes them to keep the user (\1) but have yo-yo-dyne.com as the host. str = 'purple [email protected], blah monkey [email protected] blah dishwasher' ## re.sub(pat, replacement, str) -- returns new string with all replacements, ## \1 is group(1), \2 group(2) in the replacement print re.sub(r'([\w\.-]+)@([\w\.-]+)', r'\[email protected]', str) ## purple [email protected], blah monkey [email protected] blah dishwasher Content was originally based on https://developers.google.com/edu/python/regular-expressions, but has been modified since. Licensed under CC BY 3.0. Code samples licensed under the Apache 2.0 License.	2023-01-31 23:14:12	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3296523988246918, "perplexity": 3517.709629104175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499891.42/warc/CC-MAIN-20230131222253-20230201012253-00267.warc.gz"}
https://people.math.umass.edu/~schaffler/research.html	# Research ### In Preparation: • Geometric interpretation of toroidal compactifications of Deligne-Mostow ball quotients (with P. Gallardo, M. Kerr). • Compactifications of moduli of points and lines in $$\mathbb{P}^2$$ (with J. Tevelev). Available upon request. ### Master Thesis: • Distribution of rational points on algebraic curves. Master thesis, Roma Tre University (2012).	2020-04-04 21:09:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6754815578460693, "perplexity": 5345.236734421402}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370525223.55/warc/CC-MAIN-20200404200523-20200404230523-00454.warc.gz"}
https://docs.deepmodeling.com/projects/deepmd/en/v2.0.2/install/install-gromacs.html	# 1.5. Install GROMACS with DeepMD ## 1.5.1. Patch source code of GROMACS Download source code of a supported gromacs version (2020.2) from https://manual.gromacs.org/2020.2/download.html. Run the following command: export PATH=$PATH:$deepmd_kit_root/bin dp_gmx_patch -d $gromacs_root -v$version -p where deepmd_kit_root is the directory where the latest version of deepmd-kit is installed, and gromacs_root refers to source code directory of gromacs. And version represents the version of gromacs, only support 2020.2 now. You may patch another version of gromacs but still setting version to 2020.2. However, we cannot ensure that it works. ## 1.5.2. Compile GROMACS with deepmd-kit The C++ interface of deepmd-kit 2.x and tensorflow 2.x are required. And be aware that only deepmd-kit with high precision is supported now, since we cannot ensure single precision is enough for a GROMACS simulation. Here is a sample compile scipt: #!/bin/bash export CC=/usr/bin/gcc export CXX=/usr/bin/g++ export CMAKE_PREFIX_PATH="/path/to/fftw-3.3.9" # fftw libraries mkdir build cd build cmake3 .. -DCMAKE_CXX_STANDARD=14 \ # not required, but c++14 seems to be more compatible with higher version of tensorflow -DGMX_MPI=ON \ -DGMX_GPU=CUDA \ # Gromacs on ROCm has not been fully developed yet -DCUDA_TOOLKIT_ROOT_DIR=/path/to/cuda \ -DCMAKE_INSTALL_PREFIX=/path/to/gromacs-2020.2-deepmd make -j make install	2022-10-04 22:26:41	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4764224886894226, "perplexity": 14906.903382452461}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337529.69/warc/CC-MAIN-20221004215917-20221005005917-00521.warc.gz"}
http://crimson.chat/archives/5ec71558-fe0c-46ac-a58f-e5cd344bc81f	• 累计撰写 781 篇文章 • 累计创建 1 个标签 • 累计收到 1 条评论 # SSH 2022-01-09 / 0 评论 / 0 点赞 / 31 阅读 / 1,077 字 / 后端 / 部署 / SSH Specifying a different user name It is also possible to use a different username at the remote machine by entering the command as: ssh [email protected] The above can also be expressed with the syntax: ssh -l alternative-username sample.ssh.com Executing remote commands on the server The ssh command is often also used to remotely execute commands on the remote machine without logging in to a shell prompt. The syntax for this is: ssh hostname command For example, to execute the command: ls /tmp/doc on host sample.ssh.com, type the following command at a shell prompt: ssh sample.ssh.com ls /tmp/doc After authenticating to the remote server, the contents of the remote directory will be displayed, and you will return to your local shell prompt. -x Disables X11 forwarding. 0	2022-05-28 04:10:15	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3006358742713928, "perplexity": 13506.772453265285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663012542.85/warc/CC-MAIN-20220528031224-20220528061224-00268.warc.gz"}
https://brilliant.org/problems/just-fundamental/	Just Fundamental Number Theory Level 2 Find the remainder when $$2222^{5555} + 5555^{2222}$$ is divided by 7. ×	2016-10-26 05:48:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3176008462905884, "perplexity": 2192.4014544098013}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720737.84/warc/CC-MAIN-20161020183840-00184-ip-10-171-6-4.ec2.internal.warc.gz"}
http://mathoverflow.net/questions/135668/morphism-of-algebras	# Morphism of algebras Let $Q(i)$ be the extension of the rational numbers $Q$ obtained by adjoining a root i of the polynomial $X^2 + 1$. Consider the algebra B defined by the Hilbert symbol $(-2, -5)$ over $Q(i)$. So, by definition, $$B := Q(i)[[\alpha, \beta]]/(\alpha^2 = -2, \beta^2 = -5, \alpha\beta = -\beta\alpha)$$ here $[[\alpha, \beta]]$ means non-commutative polynomials in $\alpha, \beta$. The algebra $B$ is ramified at the primes of $Q(i)$ lying above $5$, and unramified at all other places. My question: Does there exist an injection of $Q$-algebras $$B \to M_4(Q),$$ where $M_4(Q)$ is the ring of $4 \times 4$ matrices over $Q$. - No. Consider the matrix corresponding to $i$. It is semisimple with eigenvalues $i,i,-i,-i$ . So it's centralizer is $8$-dimensional - clearly, it is $M_2(\mathbb Q(i))$. Since this algebra is ramified, it does not inject into the split algebra. I am not sure if I understand your answer, but are you aware of the following construction? Pick an isom of $Q$-vectorspaces $Q^4 \cong Q(i)^2$. Then $M_2(Q(i))$ acts by $Q(i)$-linear (hence also $Q$-linear) endomorphisms on $Q(i)^2$. Via $Q^4 \cong Q(i)^2$ we then get an injection $M_2(Q(i)) \rightarrow End_Q(Q(i)^2) = M_4(Q)$. – mnr Jul 4 '13 at 9:04 Yes. Moreover, that is the only quaterniom algebra of $\mathbb Q(i)$ with such an injection, because it is the centralizer of $i$, which is a representative of the unique conjugacy class of matrices satisfying $x^2+1=0$. – Will Sawin Jul 4 '13 at 14:35	2016-05-27 08:33:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9559980034828186, "perplexity": 187.15627182963505}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049276564.72/warc/CC-MAIN-20160524002116-00208-ip-10-185-217-139.ec2.internal.warc.gz"}
https://relativitydigest.com/page/8/	## Analyzing Stephen Curry’s Play As a long-time Golden State Warriors fan (go Tim Hardaway and Chris Mullin!), I have been watching the Warriors this season with great interest. Stephen Curry has been getting a lot of attention. It is somewhat of a foregone conclusion that he will be the MVP this season, but, I am not completely convinced, in the sense that watching his play, he gets many open looks throughout the process of a game. I was therefore interested in analyzing his FG% has a function of his shot distance from the basket and the distance of the closest defender on the court. The NBA has made completing such an analysis somewhat easy with all of its new analytics tools like Shot Tracking but analyzing this question has proven difficult, because the trackers have not measured FG% as a function of two variables, rather, they have produced this statistic as function of each individual variable. One therefore ends up with a table of data as follows: FG% Distance from Basket (> 10 ft) Closest Defender Distance 1 56.5 10 NA 2 39.0 15 NA 3 46.9 20 NA 4 46.0 25 NA 5 60.0 30 NA 6 50.0 35 NA 7 36.4 40 NA 8 32.5 NA 0 9 42.4 NA 2 10 50.6 NA 4 11 47.8 NA 6 The “NA” values are the missing values as a result of not having the complete 3D set of data available. The only way I could see to alleviate this problem was to perform some type of interpolation . This way, I was able to perform the following surface regression: This regression to the interpolated data points had an R^2 value of: 0.99, so the fit actually was very good. The actual function for this surface was found to be: where $d$ denotes the closest defender distance, and $y$ denotes the distance from the basket for shots greater than 10 feet. Using this function and tools from multivariable calculus, we are able to conclude that: Min FG% = 38.164% at d = 1, y = 15 That is, Stephen Curry is expected to have his lowest field goal percentage with the closest defender within 1 foot of him while being within 15 feet of the basket. Certainly, looking at the plot above, we see that his FG% increases as defenders are further and further away. This can be also seen from the following contour plot obtained from computing the gradient of $FG(d,y)$ above: What about trends? Well, computing the gradient of $FG(d,y)$, we find that: $\nabla FG = (-6.813-0.6808d+1.0284d^2 + 0.9175 y - 0.3068 d y)\hat{d} + (-0.6783 + 0.9175d - 0.1534d^2)\hat{y}$ The charm of this is that we can now use methods of dynamical systems theory to obtain information about the trends! The vector field $\nabla FG$ is defined on the manifold $\mathbb{R}^2$ in the sense that it is a mapping: $\mathbb{R}^2 \to T\mathbb{R}^2$ that assigns to each point $m \in \mathbb{R}^2$ a vector in $T_{m} \mathbb{R}^2$. We can also interpret this vector field as the right-hand side of a system of first-order autonomous differential equations. Motivated by this, we see that the fixed points are thus found to be: $(d_1,y_1) = (0.864142, 10.1679)$ and $(d_2,y_2) = (5.11695,25.4915)$ Evaluating the Jacobian matrix in a neigbourhood of $(d_1,y_1)$ we find that the eigenvalues corresponding to this point are: $\lambda_1 = -2.21508, \lambda_{2} = 0.192138$. That is, the first point is a saddle point. Similarly, the eigenvalues of the second point are found to be: $\lambda_1 = 2.21509, \lambda_{2} = -0.192137$, which implies that this point is also another saddle point. So, in terms of trends, there certainly exist orbits where Stephen Curry tends to shoot away from defenders while also keeping a distance of more than 25 feet from the basket. There also exists orbits where he does the opposite. However, the following vector field plot is very illuminating in terms of displaying Steph Curry’s flow during the game: One sees that there is a tendency for his shots to converge where the defender is at least three feet away at a minimum distance of 25 feet away from the basket. The saddle point behaviour is very evident in the lower left and upper right corners of the vector field plot. ## 2016 Michigan Primary Predictions Using the Monte Carlo techniques I have described in earlier posts, I ran several simulations today to try to predict who will win the 2016 Michigan primaries. Here is what I found: For the Republican primaries, I predict: Trump: 89.64% chance of winning Cruz: 5.01% chance of winning Kasich: 3.29% chance of winning Rubio: 2.06% chance of winning The following plot is a histogram of the simulations: ## Stephen Curry and Mahmoud Abdul-Rauf? As usual, Phil Jackson made another interesting tweet today: And, as usual received many criticisms from “Experts”, who just looked at the raw numbers from each players, and saw that there is just no way such a statement is justified, but it is not that simple! When you compare two players (or two objects) who have very different data feature values, it is not that they can’t be compared, you must effectively normalize the data somehow to make the sets comparable. In this case, I used the data from Basketball-Reference.com to compare Chris Jackson’s 6 seasons in Denver to Stephen Curry’s last 6 seasons (including this one) and took into account 45 different statistical measures, and came up with the following correlation matrix/similarity matrix plot: Dark blue circles indicate a strong correlation, while dark red circles indicate a weak correlation between two sets of features. What would be of interest in an analysis like this is to examine the diagonal of this matrix, which offers a direct comparison between the two players: One can see that there are many features that have strong correlation coefficients. Therefore, it is true that Stephen Curry and Chris Jackson do in fact share many strong similarities! ## The Effect of Individual State Election Results on The National Election A short post by me today. I wanted to look at the which states are important in winning the national election. Looking at the last 14 presidential elections, I generated the following correlation plot: For those not familiar with how correlation plots work, the number bar on the right-hand-side of the graph indicates the correlation between a state on the left side with a state at the top, with the last row and column respectively indicating the national presidential election winner. Dark blue circles representing a correlation close to 1, indicate a strong relationship between the two variables, while orange-to-red circles representing a correlation close to -1 indicate a strong anti-correlation between the two variables, while almost white circles indicate no correlation between the two variables. For example, one can see there is a very strong correlation between who wins Nevada and the winner of the national election. Indeed, Nevada has picked the last 13 of 14 U.S. Presidents. Darker blue circles indicate a strong correlation, while lighter orange-red circles indicate a weak correlation. This also shows the correlation between winning states. For example, from the plot above, candidates who win Alabama have a good chance of winning Mississippi or Wyoming, but virtually no chance of winning California. This could serve as a potential guide in determining which states are extremely important to win during the election season! ## A Series of Lectures on Fine-Tuning in Biology A recent lecture and a series of interviews has been posted online where cosmologist George F.R. Ellis discusses the issue of fine-tuning in biology at considerable length and in considerable detail. Of course, the larger theme here is that to discuss and understand things like Darwinian evolution properly, one needs to have an understanding of the underlying physics, as it is laws of physics that allow life to emerge and for Darwinian evolution to occur in the first place. Here are the lectures: ## Notes on Dynamical Systems A big part of my research involves dynamical systems theory. A lot of people don’t know what this is, at least, they don’t have a very good idea. It has not helped that the vast majority of Canadian university physics programs have deemphasized classical mechanics and differential equations, but that is an another story! Anyways, here are some notes describing what they are and how they work.	2019-05-24 18:51:08	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 15, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7237313389778137, "perplexity": 662.0534192381625}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257731.70/warc/CC-MAIN-20190524184553-20190524210553-00004.warc.gz"}
http://mathhelpforum.com/discrete-math/182244-code-optimization-problem.html	# Math Help - A Code Optimization Problem 1. ## A Code Optimization Problem Let's say I have a matrix of cases. A row of the matrix corresponds to a sensor in a physical system, and a column of the matrix corresponds to a state that the system can be in. The entries in the matrix itself correspond to, say, a safe high limit for the sensor in that row. Here's an example: http://quicklatex.com/cache3/ql_9c51...d65d09a_l3.png The code used to produce this table is as follows: \begin{tabular}{c\|ccc}&\text{Off} &\text{Start} &\text{Run}\\ \hline \text{Temp} &90^{\circ}\text{C} &100^{\circ}\text{C} &150^{\circ}\text{C}\\ \text{Pressure} &10 \text{psig} &20 \text{psig} &30 \text{psig}\end{tabular} Now, then. I need to create nested case structures to handle outputting these limits to the checking routine. I want to minimize the total number of cases I have to code by altering the nesting of the cases. I can either check by state first, or by sensor first. For example, if the rows were "more uniform", then it would be better to have the state checking nested inside the sensor checking thus: Code: case sensor Temp case state Off Start Run end case Pressure case state Off Start Run end case end case That way, I could have fewer outer entries (I can combine cases if they're the same). On the other hand, if the columns were "more uniform", then it would be better to reverse the nesting order thus: Code: case state Off case sensor Temp Pressure end case Start case sensor Temp Pressure end Run case sensor Temp Pressure end case end case So I was thinking about a way to quantify whether the rows or columns are "more uniform". Here's my thought: to measure the uniformity of the columns, take each column, find its cardinality (the number of different items in that column), and sum over all the column cardinalities. To measure the uniformity of the rows, take each row, find its cardinality, and sum over all the row cardinalities. If the sum of the row cardinalities is smaller than the sum of the column cardinalities, then have the row cases on the outside. If the sum of the row cardinalities is greater than the sum of the column cardinalities, then have the row cases on the inside. Here's my question: will this procedure minimize the number of cases I have to code? Thank you! 2. First, if you change "tabular" into "array" in the LaTeX code, then it works in MHF. If the sum of the row cardinalities is smaller than the sum of the column cardinalities, then have the row cases on the outside. If the sum of the row cardinalities is greater than the sum of the column cardinalities, then have the row cases on the inside. Here's my question: will this procedure minimize the number of cases I have to code? I think your measure correctly determines the total number of cases if there is an outer case for each column or for each row (and the inner case may combine rows or columns when cells have the same values). This measure would not necessarily return the absolute minimum of the number of cases. For example, if the n x n matrix has the form $\begin{array}{cc}A & B\\B & A\end{array}$ where A and B are n/2 x n/2 submatrices where all elements are equal, then your measure will return 2n both for columns and for rows; however, it is enough to have 4 distinct cases. I think that your measure is good unless you have a lot of special matrices and the need for optimal solution is high.	2015-10-10 05:38:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7293481826782227, "perplexity": 603.1063837660702}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737940794.79/warc/CC-MAIN-20151001221900-00104-ip-10-137-6-227.ec2.internal.warc.gz"}
https://rodriguezanton.com/identifying-contact-states-for-2d-objects-using-pygad-and/	Way back in 2013 when I was a grad student at Marquette the majority of my research was in developing optimization routines to design passive flexible components to handle robotic assembly. I thought it would be fun to revisit some of the work I did and see if what it would look like with 10 years of improvement in opensource programs available for optimization. When I first worked on this most of the work was done with Matlab and its excellent optimization toolbox, with some additional programs written in C++ to improve technology. Some of the optimizations took so long and I was looking at so many variations that it ended up having to be modified to be distributed onto a Condor ( Now HTCondor) cluster at Marquette. The first step in the design task is to identify how two parts can contact and of those parts how many of them can actually occur within the bounds of our robot. ## Problem Setup The basic setup is to define a set of two objects A and B. Object A is fixed and has a bit of an incline to assist with assembly (basically this is because you need to be able to direct the force towards proper assembly). Object B is held by the manipulator of a robot. The robot assembly is setup within the bounds: INIT_MAX_THETA=np.pi/36 INIT_MIN_THETA=-np.pi/36 INIT_MAX_X=1.87 INIT_MAX_Y=24.5 INIT_MIN_X=-1.87 INIT_MIN_Y=0 Because we are going to be doing a lot of transformation and vector calculations we first need to define our objects in Python. We will also take advantage of the shapely to handle some of our geometric evaluations. VertA=np.array([ [-5.55, 4], [-5.55, -5], [5.55, -5], [5.55, 4 ], [4.55, 4], [0.55, 0], [0.55, -4], [-0.55, -4], [-0.55, 0], [-4.55, 4 ] ]) VertB=np.array([ [0.5, -4], [-0.5, -4], [-0.5, 0], [0.5, 0], ]) We are using numpy arrays because we are doing vector math and regular lists wont work with that. We need to prepare our objects for the evaluations so we define a function to do so. from shapely.geometry import Polygon, LineString, LinearRing def prepare_object(Vert: list, Verbose=False) -> list: ''' Function taks an list of vertices Returns and object dictionary with geometric properties. ''' #Check that object is defined as ccw LRObjB=LinearRing(Vert) if LRObjB.is_ccw is not True: Vert=[x for x in reversed(Vert)] Lines=[] Vectors=[] VectorsNorm=[] Edges=[] Verts2=[] # For each create an edge, a shapely line and also due to how we want iterate # rework our Vertices to mantain the shape. Normalize our Vectors for i in range(len(Vert)): Edge=[Vert[i-1],Vert[i]] Edges.append(Edge) Line=LineString([Vert[i-1],Vert[i]]) Lines.append(Line) Verts2.append(Vert[i-1]) Vector=np.array(Vert[i-1])-np.array(Vert[i]) Vectors.append(Vector) VectorNorm = Vector/np.linalg.norm(Vector) VectorsNorm.append(VectorNorm) # Get the vectors of each 2D object.Super easy for 2D Normals=[] for V in VectorsNorm: Norm=np.array([-V[1],V[0]]) Normals.append(Norm) if Verbose: print("Edges") print('\n'.join('{}: {}'.format(k) for k in enumerate(Edges))) print("Vertices") print('\n'.join('{}: {}'.format(k) for k in enumerate(Verts2))) print("Normals") print('\n'.join('{}: {}'.format(k) for k in enumerate(Normals))) print("Vectors") print('\n'.join('{}: {}'.format(k) for k in enumerate(Vectors))) print("Vectors Normalized") print('\n'.join('{}: {}'.format(k) for k in enumerate(VectorsNorm))) Obj={} Obj['Vertices']=np.asarray(Verts2) Obj['Edges']=Edges Obj['Lines']=Lines Obj['Vectors']=Vectors Obj['VectorsNorm']=VectorsNorm Obj['Normals']=Normals return Obj This returns an object dictionary (which really should be a class but I am a bit lazy). That has information regarding the Vertices, Edges, Lines, Vectors and Normals of the object. We also use the shapely class LinearRing to determine if our object is ccw or not. We want to keep this consistent so we can do our cross products the same way. We can check how our objects look. Since this is a simplification of a 3d problem our face normals are just the line normals with z=0 ## Concave Points If you spend any time looking at identifying contact between objects (or in general dealing with objects in space). You don't really want to deal with concave objects, and if you do you want to make sure you identify which vertices are causing it. For Obj A, it is clear it is concave. We identify this by taking a dot product and using the adjacent defining vectors. def get_concave(ObjData: list) -> list: """ Function takes an Object dictionary and returns a list of concave vertices for each one by looking at the attached vertices returns a lsit with the values """ ConcaveList=[] for i in range(len(ObjData['Vertices'])): x1=np.take(ObjData['Vertices'],i, axis=0, mode='wrap') be1=np.take(ObjData['Normals'],i-1, axis=0, mode='wrap') be2=np.take(ObjData['Normals'],i, axis=0, mode='wrap') be1=np.append(be1,0) be2=np.append(be2,0) v=np.cross(be1,be2) v0=np.append(x1,0) v2=np.append(x1,-1) res=np.dot(-v, (v2- v0)) ConcaveList.append(res) return ConcaveList If the value is negative then the vertex is concave. ## Identifying primitive Contact States Now we can get started with identifying the contact states. We want to reduce the time it takes so first we utilize our bounds and list of concave vertices to create a list of primitive contact states that we are going to look at feasibility for. This is a lengthy process so for this post I will only look at what we call Vertex-Edge contacts (V-E going forward). First we want to remove all of the contacts that involve the concave vertices, since for this problem they would be considered correct assembly. Then we want to look at how much we would have to rotate to actually have contact. So for example we would not be able to contact Obj A: V9 to Obj B: E0 since that would require fully fliping the part. def get_angle_vector(a: npt.DTypeLike, b: npt.DTypeLike) -> float: ''' Function to determine the angle bewteen two vectors ''' a1=np.arccos(a[0]/np.linalg.norm(a)) a2=np.arccos(b[0]/np.linalg.norm(b)) if a[1]<0: a1=-a1 if b[1]<0: a2=-a2 angle=a1-a2 if angle>np.pi: angle=angle-2np.pi elif angle<=-np.pi: angle=angle+2np.pi return angle def VE_feas(Angle_a: float, Angle_b: float) -> list[float, float, bool]: ''' Function to determine if a rotation angle is feasable returns the minimum, maxium and feasability of the operation ''' a_max=np.max([Angle_a, Angle_b]) a_min=np.min([Angle_a, Angle_b]) if a_max-a_min > np.pi: temp=a_max a_max=a_min+2np.pi a_min=temp if a_max<INIT_MIN_THETA or a_min>INIT_MAX_THETA: feas=False else: feas=True return a_min, a_max, feas # Initialize an empty contact state list ContactStates=[] for iA, (vA, cl) in enumerate(zip(ObjA['Vertices'], ObjA['Concave'])): # If contact doesnt include a concave vertex if cl>0: Ea=ObjA['Normals'][iA] Eb=ObjA['Normals'][iA-1] for iB, NB in enumerate(ObjB['Normals']): temp=-NB Angle_a=get_angle_vector(Ea, temp) Angle_b=get_angle_vector(Eb, temp) a_min, a_max, feas=VE_feas(Angle_a, Angle_b) if feas: print('V {}, E{}, Angle_Min {}, Angle_Max {}'.format(iA, iB, math.degrees(a_min), math.degrees(a_max))) cs={'Type':'V-E', 'ID':'V{}-E{}'.format(iA, iB), 'ElementA':vA, 'ElementB':ObjB['Edges'][iB], 'a_min':a_min, 'a_max':a_max} ContactStates.append(cs) This results in the following possible contact states V 0, E3, Angle_Min -45.0, Angle_Max 0.0 V 1, E0, Angle_Min -90.0, Angle_Max 0.0 V 1, E3, Angle_Min 0.0, Angle_Max 90.0 V 2, E0, Angle_Min 0.0, Angle_Max 90.0 V 2, E1, Angle_Min -90.0, Angle_Max 0.0 V 3, E1, Angle_Min 0.0, Angle_Max 90.0 V 3, E2, Angle_Min -90.0, Angle_Max 0.0 V 4, E2, Angle_Min 0.0, Angle_Max 90.0 V 4, E3, Angle_Min -90.0, Angle_Max 0.0 V 5, E3, Angle_Min 0.0, Angle_Max 45.0 V 6, E0, Angle_Min -45.0, Angle_Max 0.0 V 9, E2, Angle_Min 0.0, Angle_Max 45.0 ## Feasibility of Contact States Now that we have the possible contacts we need to actually determine if they are possible within the bounds we previously defined. In order to do so the approach is to generate an optimization to look at the possible configuration space and evaluate if that constitutes an valid assembly (for example V0-E4 which is outside the bounds of the robot). ### Getting the distance between Vertex and Edge There are two main values that determine if a contact state can occur. • The distance between the object features • The penetration between the objects The first value we look at is looking at the distance between the object features. For a Vertex Edge contact this is given to us by two values. The distance from the point to the line h1 and the distance along the line vector from a point projected onto it to the boundary points that define the segment h2. For h1 we just use the 3d vector formula. This means we take the cross product for the two vectors and divide by the normal length between the boundary points (A, B). $v_1=P-A$ $v_2=B-A$ $h1 = \frac{ v_2 \times v_1}{A-B}$ So we can use the following code to institute these measures def mu_f(x: float) -> float: """ Return the absolute value of x as lonf as it is les than 0 (-1E-3 to help with calculations) Returns the absolute value or zero """ if x < -1E-3: val = abs(x) else: val = 0 return val def get_h1(P: npt.DTypeLike,A: npt.DTypeLike ,B: npt.DTypeLike, verbose=False) -> list[float, np.array, float]: """ Return the distance from the point to the line by taking the cross product of the vector from the pt to a line segment boundary Return the distance and the two vectors """ v1=P-A v2=B-A # Make them be a three vector for sanity on cross products v1=np.append(v1,0) v2=np.append(v2,0) # Get the length of the vector from A to B normalLength=np.linalg.norm(A-B) # Get the crossproduct and divide by our normal. h1=abs(np.cross(v1,v2))/normalLength h1_2 = abs((P[0]-A[0])(B[1]-A[1])-(P[1]-A[1])(B[0]-A[0]))/normalLength if verbose: print('h1: {} h1_2: {}'.format(h1, h1_2)) # Return just the z component if h1[-1]<=thresh: res=0.0 else: res=h1[-1] return res, v1, v2 We also want to determine the distance from the projected point. If the perpendicular distance form the point lands inside the bounded segment that distance is zero, if not we get the measurement along the x component. $v_{1c} \frac{(v_1 \times v_w) \times (v_2)}{v_2 \dot v_2}$ $P_{p} = P + v_{1c}$ If $$A_x -B_x == 0)$$ then: $\alpha = \frac{P_{py} - A_y)\|A-B\|}{B_y-A_y}$ else $\alpha = \frac{P_{px} - A_x)\|A-B\|}{B_x-A_x}$ Then the value of h2 is going to be $h_2 = \mu(\alpha)+\mu(\|B-A\|-\alpha)$ Where $$\mu$$ is just a function that returns $$x \} if the \( x$$ negative else it returns $$0$$. The python code will look as follows: def get_h2(P: npt.DTypeLike, A: npt.DTypeLike, B: npt.DTypeLike) -> tuple[float, np.array, np.array]: """ Return the distance from a projected point on to the line defined by A,B to the boundary of the line defined by A,B. It returns 0 if the point is in the line. Return the mesure and the projection vector """ v1=P-A v2=B-A # Make them be a three vector for sanity on cross products v1=np.append(v1,0) v2=np.append(v2,0) # Get the length of the vector from A to B normalLength=np.linalg.norm(A-B) #Get a vector that projects the point perpedicular to our infinate line. temp=np.cross(v1,v2) v1c=np.cross(temp,v2)/np.dot(v2,v2) #Project the point Pp=P+v1c[:2] # Obtain alpha as the distance from projected point to line segment if A[0]-B[0]==0: alpha=((Pp[1]-A[1])normalLength)/(B[1]-A[1]) else: alpha=((Pp[0]-A[0])normalLength)/(B[0]-A[0]) # Set it to zero if the projected is on the line segment h2=mu_f(alpha)+mu_f(normalLength-alpha) if h2<thresh: h2=0 return h2, v1c, Pp So with the value h1+h2 we have a measure of the distance between the features (V-E) ## Getting the Penetration Ok so now we need to determine the amount of penetration that is occurring between the parts. This is a problem that is quite common in video game design, however most algorithms are more concerned with determining if contact occurs not so much the measure of the interference. The first step is to triangulate our objects into smaller triangles. This is done because as stated before concave objects don't play nice with most methods. To do so we use the sect package. from ground.base import get_context from sect.triangulation import Triangulation def constrained_triangulation(Obj: dict) -> list: """ Takes an object dictionary and triangulates Returns the suboject array """ context= get_context() Contour, Point = context.contour_cls, context.point_cls PolygonSect = context.polygon_cls objContour = Contour([Point(vct[0], vct[1]) for vct in Obj['Vertices']]) sect_objcontour = PolygonSect(objContour,[]) subObjs=Triangulation.constrained_delaunay(sect_objcontour, context=context).triangles() subObjsV = [[(vert.x, vert.y) for vert in subObj.vertices] for subObj in subObjs] return subObjsV This results in all our objects being broken up into smaller sub-objects. We now prepare the sub-objects as we did previously. # We prepare each of these subobjects as we initially did for the subObjsA=[prepare_object(subObjs) for subObjs in subObjsA] subObjsB=[prepare_object(subObjs) for subObjs in subObjsB] ### Growth Distance GJK-EPA like measure calculation In order to get a measure of the penetration we are going to use the growth distance as outline here. It operates similarly to the GJK-EPA algorithm. However in this case we are basically identifying the point at which the expansion of the the polygons by a given factor results in interaction. There a couple more complex point in the algorithm but I recommend reading the paper to understand it. It ends up such that values less that: • Values 0<x<1 are penetration • x=1 contact • x>1 expansion In order to do so we need to determine what a stationary point that will stay inside the object when in motion. Or in simpler terms for us, one that we can uniformly scale around. Thankfully for us, the centroid of a triangle is guaranteed to have that property. We also need to select an arbitrary point within the each face that forms the object, or for the 2D case the edges. We take a simple approach and just select the end point. What we want to do is solve the set of equations formed by the normals for the factor that results in contact $$x=1$$. This is a well established problem that we can solve using scipy.optimize.linprog , since its a linear objective function subject to inequality constraints of the form. $min_x c^Tx$ $\text{such that}$ $A_{ub} \le b_{ub}$ $l \le x \le u$ In our case $$A_{ub}$$ is defined by the normals $$N$$ and $$-(arb_{if} - seed_{if}) \times N_{if}$$ and $$b_{ub}$$ is $$seed_if \times N_{if}$$. Where $$if$$ refers to pt/vector $$i$$ defined in the frame $$f$$ of the unmoveble object (in our case A). We also define an inequality such that $$-x_4 < 0$$. The code to calculate this is as follows: # Now wrap the whole calculation within a function def growth_distance(subObjA, subObjB, conf): gdObjA = subObjA gdObjB = subObjB gdNormalsA = gdObjA['Normals'] gdNormalsB = gdObjB['Normals'] gdVertA = gdObjA['Vertices'] gdVertB = gdObjB['Vertices'] gdSeedA = gdObjA['Seed'] gdSeedB = gdObjB['Seed'] gdArbA = gdObjA['Arb'] gdArbB = gdObjB['Arb'] gdSeedBwrtA = transfer_pt(gdSeedB, conf) gdNormalsBwrtA = [transfer_vector(nrm, conf) for nrm in gdNormalsB] gdArbBwrtA = [transfer_pt(nrm, conf) for nrm in gdArbB] gdVertBwrtA = [transfer_pt(vert, conf) for vert in gdVertB] #Since we triangulated we know this is three. But for generalization we make sure FaceNumA = len(gdVertA) FaceNumB = len(gdVertB) #Initial Guesses for our GD and combine into the right shape GD = 1.0 # Number of constraints we need ConNum = FaceNumA+FaceNumB+1 # Our Object Coefficients we only need the growth function objCoeff = np.array([0, 0, 0, 1]) # Right side of the inequality constraint conB = [] conCoeff = [] for arb, Nrm in zip(gdArbA, gdNormalsA): arb_a = np.append(arb, 0) nrm_a = np.append(Nrm, 0) seed_a = np.append(gdSeedA, 0) temp = -np.dot(arb_a-seed_a, nrm_a) conCoeff.append([nrm_a[0], nrm_a[1], nrm_a[2], temp]) conB.append(np.dot(seed_a, nrm_a)) for arb, Nrm in zip(gdArbBwrtA, gdNormalsBwrtA): arb_b = np.append(arb, 0) nrm_b = np.append(Nrm, 0) seed_b = np.append(gdSeedBwrtA, 0) temp = -np.dot(arb_b-seed_b, nrm_b) conCoeff.append([nrm_b[0], nrm_b[1], nrm_b[2], temp]) conB.append(np.dot(seed_b, nrm_b)) conCoeff.append([0, 0, 0, -1]) conB.append(0) bounds = [(None, None) for x in range(1, 4+1)] res = linprog(np.array(objCoeff), A_ub=np.matrix(conCoeff), b_ub=np.array(conB), bounds=bounds) temp = np.dot(res.x, objCoeff) if temp <= 1-1e-4: gdDistance = 1-temp else: gdDistance = 0 fscale = temp res_lat = [] for lbl, mat in zip(['Obj', 'A_{{ub}}', 'B_{{ub}}'], [np.matrix(objCoeff).T, np.matrix(conCoeff), np.matrix(conB).T]): lat_str = '\\begin{{equation}} {}={} \end{{equation}}'.format( lbl, bmatrix(mat)) res_lat.append(lat_str) return gdDistance, res, res_lat This is probably a bit complex but we can check this plotting for the sub-objects and see that the result makes sense. The one thing to note here is that since we are going to be checking for no interference we are not worried about the expansion, so if the value of the result gdDistance is larger than 1 then we set the value to 0. fig, ax = plt.subplots(figsize=(10, 10)) shapeA = Polygon(gdVertA) shapeAs = scale(shapeA, fscale, fscale, origin=tuple(seed_a)) polyA = PolygonPatch(shapeA, fc="b", alpha=0.5) shapeB = Polygon(gdVertBwrtA) shapeBs = scale(shapeB, fscale, fscale, origin=tuple(seed_b)) polyBwrtA = PolygonPatch(shapeB, fc="g", alpha=0.5) ax.plot(gdSeedBwrtA[0], gdSeedBwrtA[1], 'ok', markersize=12, mfc='none') ax.plot(gdSeedA[0], gdSeedA[1], 'ok', markersize=12, mfc='none') ax.annotate('SeedA', tuple(gdSeedA), textcoords="offset points", ha='center', xytext=(0, 10), fontsize=12) ax.annotate('SeedBwrtA', tuple(gdSeedBwrtA), textcoords="offset points", ha='center', xytext=(0, 10), fontsize=12) ax.plot(res.x[0], res.x[1], 'xr', mfc='none', markersize=12) ax.set_aspect('equal', 'box') Since we want to determine a measure of how much penetration there is between all the sub objects then our total value is just the sum of all resulting growth distances. def poly_growth_fun(subObjsA, subObjsB,conf, verbose=False): gds=[] for An,subObjA in enumerate(subObjsA): for Bn, subObjB in enumerate(subObjsB): gd, res = growth_distance(subObjA, subObjB, conf)[:2] if verbose: print('A{}-B{} \| Status: {}, \ Success: {}, \ Value: {}'.format(An, Bn, res.status, res.success, res.fun)) if res.status==2: raise else: gds.append(gd) return np.array(gds).sum() ## Checking Feasibility Now that we can measure both the distance from feature to feature and the penetration we can crate a funtion that combines those two values. def opt_fun(cs, subObjsA, subObjsB, conf, verbose=False): Ve_value=valueVE(cs, conf) Gf_value=poly_growth_fun(subObjsA, subObjsB, conf) if verbose: print('V_E value {}, Growth_Value {}'.format(Ve_value, Gf_value)) return Ve_value+Gf_value ### Genetic Algorithm Since solving the equations to obtain a configuration that is has contact between the objects and also not penetrating is very complex (and not feasible once we get into more complex/ 3d objects). We instead will optimize the configuration in the possible space such that the result is equal to 0. If we can find a configuration then the contact state is possible. To perform this optimization we use a genetic algorithm that basically emulates the evolution by converting the attempted values $$x$$ into a chromosome and performing evolutionary changes to it based on a fitness function. Basically the xs that result in a higher function have a higher chance to reproduce. We keep some variety with mutations to help us avoid local minimums. For us each individual in our population is an attempted configuration in the space. Back in the day I used MATLAB's ga (and at some point GAlib). However that is not open source, there are a couple of options for python, most well know DEAP. However I decided to use Pygad to check it out. Pygad is different from MATLAB's ga in that it maximizes the fitness rather The first step in a genetic algorithm is to define a fitness function. For our case we just do the following. def fitness_func(conf, index): fval=opt_fun(current_cs, subObjsA, subObjsB, conf)+thresh fitness=1/np.abs(fval) return fitness Notice that we are minimizing fval and because of that we take 1/fval. Quick thing: In both our calculations of distances h1, h2 and gd we defined 0 as a value less than a threshold. The reason is that dividing by 0 in numpy can cause the genetic algo to not work properly Then we defined the bounds of our space of possible configurations gene_space=[] for l,u in zip(lb, ub): gene_space.append({'low':l, 'high':u}) Finally to show our work we create a callback after each generation. def on_generation(ga): print("Generation", ga.generations_completed) print(ga.best_solution()[1]) Finally we define our instance global current_cs # we cant pass aditional parameters to the function, so we pass the # contact state into a global variable, in this case we chose the last one current_cs = ContactStates[-1] sol_per_pop=30, num_genes=3, num_parents_mating=5, gene_type=np.float32, gene_space=gene_space, fitness_func=fitness_func, crossover_probability=0.4, mutation_type="random", mutation_probability=0.6, on_generation=on_generation, allow_duplicate_genes=False, stop_criteria=["reach_{}".format((1/thresh)-1), "saturate_50"]) We tweak each parameter depending on performance, the important ones are as follows. • num_generations number of generations, this can be played around with but the more generations the longer our optimization could take • sol_per_po the number of individuals per generation • num_genes how many chromosomes (values) exist in each individual, in our case this is 3 since we have configurations as $$x, y, \theta$$. • gene_type I defined this as a np.float32 since I don't have any interest in that many decimals and this reduces the calculation time. • gene_type the previous define bounds on created individuals • stop_criteria tells the optimization when to end. In our case we set it so that is ends if the value is (1/thresh) -1 which is basically 0 for our purposes and if there is no change after 50 generations (this can be played around). Because of the way we setup these functions there will be a siginificant jump once we actually get zero. Generation 1 25.727673594793036 Generation 2 25.727673594793036 Generation 3 59.70383773709654 Generation 4 59.70383773709654 Generation 5 59.70383773709654 Generation 6 59.70383773709654 Generation 7 59.70383773709654 Generation 8 59.70383773709654 Generation 9 59.70383773709654 Generation 10 1000.0 From that result we can see that the Contact State[-1] is actually possible. We can take a look at how the assembly looks for a given contact state. So we can determine the feasibility by doing checking the end fitness and seeing if its less than the thresh or not. fit_res=ga_res[1] if fit_res<=1/thresh: feasibility=True print('Contact State {} is feasable'.format(current_cs['ID'])) else: feasability=False print('Contact State {} is NOT feasable'.format(current_cs['ID'])) We can now iterate thru our list of primitive contact states and determine which ones are actually possible in the configuration space. ## Conclusion Well that was quite a lot of work and frankly gave me flashbacks to many sleepless nights. In all honesty Pygad is a great tool but I did miss the sleekness of Matlab's ga. The next step (and what I had to do for my thesis work) is to parallelize the code, and perhaps that is where a big improvement can be made, since there has been significant advancement in those areas in the last decade. The next steps require a lot more work that I might undertake at a different time. It requires determining the extremal positions of each contact state (including E-V, and the dual conbination of then E-V \| V-E, V-E \| E-V and and theoretically V-E \| V-E for shorter parts. Finally to actually design the flexible manipulator (or Admittance Matrix) we need to define a bunch of constraints in screw notation related to contact wrenches and optimizing on that. This part would be a stretch to do since frankly I get stressed out just thinking about remembering how to work in screw notation. Anyways a the jupyter notebook with all of this can be found in my github.	2022-06-28 21:40:22	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5136120915412903, "perplexity": 2775.5768940325356}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103617931.31/warc/CC-MAIN-20220628203615-20220628233615-00445.warc.gz"}
http://www.chebfun.org/examples/approx2/Tucker.html	Warwick Tucker has shown me a beautiful example (unpublished). He considers the bivariate function $$f(x,y) = \sin(\cos(x^2)+10\sin(y^2)) - y\cos(x)$$ in the square $-5\le x, y \le 5$. What is the zero set of this function? In Chebfun2 we see that $f$ has rank 3: f = chebfun2(@(x,y) sin(cos(x.^2)+10sin(y.^2))-y.cos(x),[-5 5 -5 5]) f = chebfun2 object domain rank corner values [ -5, 5] x [ -5, 5] 3 [ 1.1 1.1 -1.7 -1.7] vertical scale = 6 The roots command finds the elegant zero set. tic c = roots(f); plot(c,'linewidth',1) axis([-5 5 -5 5]), axis square toc Elapsed time is 12.739323 seconds. Chebfun has found 79 components (the mathematically exact number would be even), size(c) ans = Inf 79 each of them parametrized by $s\in [-1,1]$, domain(c) ans = -1 1 and each component is represented by a polynomial of the same painfully high degree (i.e., c is an array-valued chebfun), even though some of them are very simple, length(c) ans = 3756 Though Chebfun2 roots can sometimes get outstanding accuracy, that has not happened in this case. To get an idea of the accuracy, suppose we find the 79 points corresponding to these curves at the arbitrary sample point $s=0.5$ and then evaluate $f$ at these 79 points. In principle the result should be a vector of 79 numbers close to machine epsilon, give or take a few powers of 10 since $f$ has large derivatives, but in fact, many of the numbers are much bigger than that: p = c(0.5,:); fp = f(p); semilogy(sort(abs(fp)),'.-') title('size of f at various pts on computed zero set') ylim([1e-16 1]), grid on A much faster way to see the zero set is with the Chebfun2 contour command: tic contour(f,[0 0],'linewidth',1) axis([-5 5 -5 5]), axis square toc Elapsed time is 0.380119 seconds.	2018-02-20 03:29:14	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7671085000038147, "perplexity": 1701.300186712978}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812873.22/warc/CC-MAIN-20180220030745-20180220050745-00682.warc.gz"}
https://www.nature.com/articles/s41598-019-55199-3?error=cookies_not_supported&code=b9ff872d-407d-4d41-b9c9-0843c4aaca40	## Introduction Sphingolipid metabolites have emerged as potent mediators of important signalling cascades responsible for regulating various cellular processes. Ceramide, sphingosine and sphingosine-1-phosphate (S1P) are the key players among sphingolipid metabolites1,2. Ceramide and sphingosine serve as ‘pro-apoptotic’ molecules and mediate cell cycle arrest and induce apoptosis. On the other hand, S1P functions as ‘pro-survival’ molecule and promotes cell proliferation and survival3,4. The balance between intracellular levels of these two interconvertible and oppositely acting sphingolipid metabolites forms a “sphingolipid-rheostat” which is critical in determining the cell fate5. A lean of this balance towards ceramide side makes the cell destined towards apoptotic or death pathways, whereas cell growth and survival is induced when S1P accumulates within the cell6. Interestingly, modulation of this rheostat so as to increase the levels of ceramide or sphingosine at the cost of S1P can be adopted as a therapeutic strategy to combat cancer7,8. Sphingosine kinase (SphK) is one of the key regulators of this rheostat as it generates S1P from sphingosine thereby decreasing the intracellular levels of both sphingosine and ceramide9. Two major isoforms of SphK has been found, SphK1 and SphK2, each having distinct intracellular location, tissue distribution and biological functions10. SphK1 is normally present in cytoplasm but translocated to cell membrane when activated, whereas SphK2 is found in nucleus. Notably, the two isoforms have been demonstrated to regulate diverse cellular processes. For example, SphK1 induces cell growth and proliferation, whereas SphK2 promotes cell growth arrest and apoptosis11. SphK1 is activated by several agonists, including mitogens, pro-inflammatory cytokines, and diverse growth factors, which is followed by its translocation to plasma membrane. The stimulated SphK1 produces S1P, a multifunctional lipid metabolite, from sphingosine leading to transient increase in its intracellular levels12,13. S1P plays key role in regulating diverse biological processes which are crucial for cancer progression and inflammation such as cell proliferation, differentiation, invasion and angiogenesis14,15,16. S1P acts both extracellularly by interacting with a family of five transmembrane G-protein-coupled receptors, termed as S1P receptor 1 to 5 (S1P1-5), and via intracellular targets such as various nuclear and cytoplasmic proteins which are involved in epigenetic regulation of specific genes, cell growth and calcium homeostasis17,18. Recently, SphK1 has received a great attention due to its involvement in a number of human pathologies, including cancer, rheumatoid arthritis, pulmonary fibrosis, diabetes, asthma and neurodegenerative disorders19,20. Over-expression of SphK1 has been observed in malignancies of various organs such as breast, lung, uterus, colon, pancreas, ovary, kidney, as well as in leukemia21,22,23,24,25,26. SphK1 up-regulation has also been linked to the poor prognosis of many human cancers24. Furthermore, SphK1 plays important role in processes like angiogenesis, tumorigenesis and chemotherapy resistance which are crucial for metastasis and cancer progression19. Li et al.27 has demonstrated that the over-expression of SphK1 induces NF-κB activation and shortened the cell cycle thereby promoting the proliferation of breast epithelial cells. In another study, researchers have found a strong correlation between elevated SphK1 expression levels and short survival times in patients with gastric cancer28. Owing to its remarkable role in cancer progression and metastasis, and other inflammatory diseases, SphK1 presents a novel therapeutic target to develop effective therapeutics to combat these diseases. The effects of some synthetic sphingosine analogues as SphK1 inhibitors have been investigated in animal models of human diseases29,30. But the synthetic therapeutics are often plagued with poor water solubility, limited bioavailability, and undesirable side effects31,32,33. Interestingly, compounds derived from natural sources impart no or minimal side effects and hence can be exploited to design highly specific and potent inhibitors against drug targets34,35. Additionally, the chemical structure of promising natural compound can be modified strategically so as to improve their aqueous solubility, absorption and metabolism36. Furthermore, the anticancer, anti-inflammatory, anti-diabetic and antioxidant effects of many plant derived compounds have been very well established that further makes their candidature strong as drug molecules37,38,39. The anti-inflammatory properties of some natural compounds in mouse model of SphK1-associated diseases have also been evaluated but the mechanism through which they exert their effects has not been assessed yet40,41. In the present study, we screened a series of plant-derived natural compounds to evaluate their inhibitory potential against SphK1. The interaction of compounds showing best inhibitory potential was evaluated at atomic level using molecular dynamics (MD) approach to get an insight into the binding site of compound with the SphK1, and the amino acid residues involved in the binding process. ## Materials and Methods ### Materials Luria broth and Luria agar were purchased from Himedia. Kanamycin and Tris were purchased MP Biomedicals, LLC (France). Ni-NTA resin was procured from Thermo Scientific (USA). BIOMOL® Green reagent was bought from Enzo (New York, USA). Isopropyl β-D-1-thiogalactopyranoside (IPTG), quercetin, vanillin and citral were purchased from Sisco Research Laboratories. Ursolic acid, capsaicin, DL-α tocopherol acetate and limonin were bought from Tokyo Chemical Industry Co., Ltd. (Tokyo, Japan). All reagents used were of analytical grade. ### Expression and purification of SphK1 SphK1 was successfully expressed in BL21-Gold (DE3) competent cells and subsequently purified by Ni-NTA affinity chromatography in a single step as described42. Briefly, the recombinant cells having plasmid with SphK1 gene insert were grown, and protein expression was induced by 1 mM at 37 °C for 3–4 hours. The cell pellet obtained was suspended in lysis buffer (50 mM Tris, 250 mM NaCl, 20 mM EDTA 0.1 mM PMSF and 1% of Triton 100, pH 8.0) and subjected to sonication to prepare inclusion bodies. Solubilisation of inclusion bodies was done by incubating them in buffer (50 mM Tris-HCl pH 8.0, 150 mM NaCl) containing 0.5% of N-Laurousyl sarcosine for 3–4 hours at room temperature followed by centrifugation at 10,000 rpm for 40 minutes. The supernatant obtained was loaded on Ni-NTA column for binding, followed by washing with 10 mM imidazole and elution with increasing concentrations of imidazole (20 mM to 400 mM). Purity of eluted fractions collected was assessed by SDS-PAGE. Fractions showing single band of protein were pooled and dialyzed extensively against 20 mM Tris-HCl buffer (pH 8.0) containing 100 mM NaCl. Protein concentration was determined using a molar absorption coefficient of 48275 M−1cm−1 at 280 nm on Jasco V-660 UV-visible spectrophotometer. ### Molecular docking Molecular docking and MD simulation studies were carried out on DELL® workstation with Intel® Xeon® CPU E5-2609 v3 @ 1.90 GHz processor with 64 GB RAM and two terabyte hard disk running on Ubuntu 18.04.2 LTS operating system. GROMACS 5.1.2 package was used to perform MD simulations. Computational tools such as PyMOL43, VMD (visual molecular dynamics)44 and QtGrace were used for visualization, evaluation and analysis of MD trajectories. Atomics coordinates of SphK1 structure were taken from the Protein Data Bank (PDB ID: 3VZB), and the structure of quercetin was downloaded from the PubChem database and processed in MGL tools45. AutoDock Vina46 was used for docking purpose. PyMOL and Discovery Studio Visualizer47 were employed to visualize the structures for the analysis of bound conformation and different interactions between quercetin and SphK1. ### MD simulations MD simulations were carried out for 100 ns on free SphK1 and SphK1-quercetin docked complex at 300 K of molecular mechanics level using GROMOS96 43a1 force-field in GROMACS 5.1.2. The structural coordinates of SphK1 were downloaded from the Protein Data Bank (PDB) with PDB ID: 3VZB and processed in SPDBV. The topology and force-field parameters for quercetin were generated from the PRODRG server and merged into the parent file of SphK1 to make complex. Both the systems were soaked in a 10 Å dimension sized cubic box for solvation in the SPC216 solvent model and were neutralized using counterions. Energy minimization was carried out using 1500 steps of steepest descent to remove bad contacts in the solvated systems. The temperature of both the systems was then raised up gradually from 0 K to 300 K during the equilibration time of 100 ps at constant volume, pressure (1 atm) and temperature (300 K) under periodic boundary conditions. The final MD run was set to 100,000 ps for both systems, and resulting trajectories were saved for further analysis using inbuilt utilities of GROMACS such as gmx energy, gmx rms, gmx rmsf, gmx gyrate, gmx sasa and gmx sham. A detailed description of MD simulations is reported elsewhere48,49,50,51,52. We also performed the principal component (PC) and free energy landscape (FEL) analysis of SphK1 before and after quercetin binding. These methods employ the calculation and diagonalization of the covariance matrix which can be calculated as: $$\begin{array}{ll}{C}_{ij}=\langle ({{\rm{x}}}_{i}-\langle {{\rm{x}}}_{i}\rangle )\,({{\rm{x}}}_{j}-\langle {{\rm{x}}}_{j}\rangle )\rangle & (i,\,j=1,\,2,\,3\,\ldots ,\,3{\rm{N}})\end{array}$$ here, xi = xj is the coordinate of the ith/jth atom of the system, whereas 〈−〉 represents an ensemble average. FELs were also constructed for both systems to understand the stability, folding and function behaviour of SphK1 before and after quercetin binding. The FEL can be constructed as: $${\Delta }G(X)=-\,{K}_{B}T\,\mathrm{ln}\,P(X)$$ where KB and T are the Boltzmann constant and absolute temperature, respectively. ΔG(X) is the probability distribution of the molecular system along the PCs. ### Fluorescence binding studies Fluorescence quenching experiments were carried out on Jasco spectroflourimeter (FP-6200) using 5 mm quartz cuvette. The temperature was maintained at 25 ± 0.1 °C by external thermostated peltier device. Stock solution of ligands was prepared in DMSO and diluted to a working concentration of 50 µM in Tris buffer just prior to taking measurements. Protein solution (4 µM) was titrated with increasing concentrations of ligand and the fluorescence emission spectrum was recorded in the range of 300–400 nm by keeping the excitation wavelength constant at 280 nm. The excitation and emission slit widths were kept at 10 nm. The final spectra were obtained by subtracting with the corresponding blank. For data analysis, fluorescence intensity at λmax was plotted against [ligand, μM] and then modified Stern-Volmer equation (Eq. 1) was used to derive binding parameters viz., binding constant (Ka) and number of binding sites (n) per molecule of protein for ligand-SphK1 system. $$\log ({{\rm{F}}}_{0}-{\rm{F}}/{\rm{F}})=\,\log \,{{K}}_{{\rm{a}}}+n\,\log \,[{\rm{Q}}]$$ (1) ### Isothermal titration calorimetry Binding affinity of SphK1 to quercetin was determined by using VP-ITC microcalorimeter from MicroCal, Inc (GE, MicroCal, USA). Protein sample was prepared by extensively dialyzing it against Tris buffer (20 mM Tris-HCl, pH 8.0 and 100 mM NaCl). Stock solution of compound was diluted in the last dialyzing buffer to avoid any baseline error. Equal amount of DMSO (1.0% v/v) was added to the protein solution to prevent signal stability problems during ITC measurements. 15 µM SphK1, which was kept in the sample cell with an effective volume of 2 ml, was titrated with 750 µM of compound filled in the titration calorimetry syringe. A programmed titration was performed with a first false injection of 2 µl followed by 24 successive injections of 10 µl ligand each at 300 seconds interval. The temperature of the sample and reference cell was isothermally maintained at 25 °C, and the syringe was stirred at 307 rpm during the experiment. Heat of dilution of compound into buffer solution (no protein) was also measured and subtracted from the protein’s titration data. The binding isotherm obtained was fitted with in-built origin ‘binding site’ model to derive the thermodynamic parameters viz., stoichiometry of binding (n), enthalpy change (ΔH) and association constant (Ka). ### Enzyme inhibition assay ATPase activity of purified SphK1 was checked by malachite green-based (BIOMOL® Green reagent, Enzo Life sciences) microtitre-plate assay. Fixed (2 µM) amount of protein was incubated with assay buffer (20 mM Tris-HCl, pH 8.0 and 100 mM NaCl) containing 10 mM MgCl2 and increasing concentrations of ATP for 30 minutes at 25 °C. The reaction was terminated by the addition of malachite green reagent followed by incubation for 20 minutes for colour development. The absorbance of each well was measured at 620 nm on a multiplate ELISA reader. Standard phosphate curve was prepared as described by manufacturer’s protocol and used to quantity the amount of free inorganic released from ATP by kinase activity. Thereafter, ATPase inhibition assay for SphK1 was performed in the presence of compounds. Firstly, Protein (2 µM) was pre-incubated with increasing concentrations of ligand at room temperature for 60 minutes in 96-well plate. 10 mM MgCl2 and 150 µM freshly prepared ATP was then added to the reaction mixture and incubated for 30 minutes at 25 °C. To terminate the reaction, BIOMOL® reagent was added and incubated for 20 minutes followed by absorbance measurement at 620 nm. The amount of free inorganic released was determined with the help of standard phosphate curve as described53. All the measurements were performed in triplicates. ## Results ### Expression and purification of recombinant SphK1 The recombinant his-tagged SphK1 was expressed in BL21 Gold (DE3) cells by IPTG induction. The protein was solubilised from inclusion bodies using N-Laurousyl sarcosine followed by its purification using Ni-NTA affinity chromatography in single step. The purity of SphK1 was evaluated by SDS-PAGE which showed a single protein band at 45 kDa (Fig. S1A). UV-absorption, far-UV circular dichroism and intrinsic fluorescence spectra of purified SphK1 reveal a proper folding of the recombinant protein without any aggregation (Fig. S1B–D). We analyzed the far-UV CD spectra of SphK1 to estimate the percentage of secondary structure elements in the native structure. SphK1 has 27% α-helix and 29% β-sheet which is close to the secondary structural content as reported in the crystal structure (PDB ID: 3VZB). The enzymatic activity of recombinant protein was further checked by ATPase assay suggesting the good quality of purified protein (Fig. S2). Overall, the results indicate that we have successfully expressed and purified recombinant SphK1 from E. coli in its biologically active native form. ### Molecular docking studies of natural compounds We have screened a series of natural compounds including quercetin, ursolic acid, capsaicin, DL-α tocopherol acetate, citral, limonin, vanillin and simvastatin for their possible interaction with SphK1 using molecular docking approach. Molecular docking helps us to analyze the binding pattern of each compound with SphK1 that further supports in identifying the interacting residues and calculating binding affinity. Binding energy estimated from the docking results of SphK1 with various ligands is shown in Table S1. The calculated binding affinities (ΔG) values were estimated as −8.2 and −8.1 kcal/mol for quercetin and capsaicin, respectively with SphK1. ### Fluorescence measurements To validate our docking results, the binding affinity of selected natural compounds with SphK1 was calculated using fluorescence measurements. A number of molecular interactions, including formation of an excited charge-transfer complex, intersystem crossing to the triplet state, molecular rearrangements, and ground-state complex formation between the fluorophore and quencher can lead fluorescence quenching54. SphK1 has 6 tryptophans that act as fluorophore and hence can be exploited for the ligand binding studies using fluorescence quenching approach. 4 µM of SphK1 was titrated by the successive addition of selected natural compounds from a 1.0 mM stock solution of ligands. The intrinsic fluorescence spectra were then collected in the range of 300–400 nm by keeping the excitation wavelength at 280 nm. The concentration of ligands was varied from 0 to 50 µM to obtain the saturation point. Figures 1 and S3 shows the fluorescence emission spectra of SphK1 with increasing concentrations of ligands. For the compounds having good binding interaction with SphK1, a remarkable decrease in Trp fluorescence was observed with every titration. Compounds exhibiting high binding affinity with the protein saturates at lower concentration during titration whereas others with no significant quenching effects do not bind to the protein appreciably even at higher concentrations. The quenching data was analyzed using modified Stern-Volmer equation to derive binding constant (Ka) and number of binding sites per SphK1 molecule (n). Table S1 shows the binding parameters of all studied natural compounds under investigation with SphK1. Among all the screened compounds, quercetin and capsaicin showed the best binding with SphK1 having Ka values of 4.38 × 105 M−1 and 1.53 × 104 M−1, respectively (Fig. 1 and Table 1). Other compounds do not show appreciable amount of fluorescence quenching and in-fact some of them induce large structural perturbations in SphK1 (Fig. S2). Hence these compounds were excluded from the studies done further to identify the potent inhibitors against SphK1. ### ITC measurements Fluorescence binding studies indicate that quercetin and capsaicin interact with high binding affinity to SphK1. To further validate the binding interaction, we performed ITC measurements and calculated the thermodynamic parameters viz., binding affinity, enthalpy change (ΔH), entropy change (ΔS), and stichiometry associated with SphK1-ligand binding reaction. Figure 2 shows the ITC isotherm generated from the titration of quercetin and capsaicin with SphK1 at 25 °C. The upper panel of binding isotherm with negative pulses of heat suggests that the interaction of SphK1 with quercetin and capsaicin is exothermic in nature yielding favourable values of binding enthalpy (Fig. 2 and Table 2). The lower panel shows the amount of heat released with each successive injection as a function of molar ratio of SphK1 and the compound. The blank titration measuring the heat of dilution of compound into Tris buffer was also performed. The heat associated with the addition of compound into buffer was subtracted from the heat changes by the titration of compound with protein. The corrected heat for the interaction of compound with SphK1 was plotted against the molar ratio of SphK1/Compound. The thermodynamic parameters presented in Table 2 for SphK1-ligand interaction were obtained by fitting the ITC isotherm with in-built binding model in origin software. The isotherm for SphK1-Quercetin interaction seems to be biphasic and would therefore be suggestive of the cooperative mode of interaction. The raw data was fitted to the sequential binding mode with 2, 3, 4, 5 and 6 binding sites. On comparing the chi square values for different binding modes, it was observed that the data fits best with 4 sequential binding sites having least chi square value. In contrast, SphK1-Capsaicin binding isotherm fit well with the single binding site model (Table 2). KD values estimated for both quercetin and capsaicin lays in micromolar range that correlate well with our fluorescence binding results. ### Enzyme inhibition assay SphK1 possess ATPase activity that can be exploited to check the inhibitory potential of natural compounds. ATPase inhibition assay was performed to evaluate the efficacy of quercetin and capsaicin (the best binding compounds) in diminishing the functional activity of the enzyme. Figure 3 shows the amount of inorganic phosphate released by SphK1 in the presence of increasing concentration of quercetin and capsaicin. It is quite clear that both quercetin and capsaicin are effective in inhibiting the enzymatic activity of SphK1 at micromolar concentrations (Fig. 3B,C). Quercetin and capsaicin showed an IC50 value (50% of ATPase inhibition) of ~2.8 µM and ~ 27.0 µM, respectively (Table 1). Overall, our activity results suggest that quercetin and capsaicin functions as potential inhibitors of SphK1 in micromolar range. ### Molecular docking of quercetin with SphK1 Quercetin was found to be most lethal for SphK1 activity and displayed the best binding affinity among all the natural compounds screened through a series of experiments, hence we performed docking study to get mechanistic details of SphK1-quercetin interaction at a molecular level. Molecular docking helps to predict a preferred orientation and binding prototype of a compound at the receptor’s binding pocket. In the analysis of SphK1-quercetin docked complex, we found that quercetin shows appreciable binding affinity (−8.2 kcal/mol) (Table S1), and preferentially occupy the substrate-binding pocket of SphK1. Quercetin is binding with several polar interactions and placed in the deep cavity of the SphK1 binding pocket (Fig. 4). During interaction analysis, we found that quercetin is present at the same place and mimicking the position where co-crystallized substrate D-sphingosine is present, which might presumably decreases the substrate accessibility of SphK1. Quercetin was found to occupied the distal end of the enclosed lipid pocket of SphK1 where the alkyl chain of sphingosine is found to bind, with the two phenolic rings A and C placed at the end of the pocket and the catechol ring B (3′,4′-dihydroxy group) directing toward the opening in the cleft between the N-terminal and the C-terminal domain (Fig. 4). The catechol ring forms three hydrogen bonds, two with Asp178 and one with Ile174, resulting in closed conformation for the lipid gate. At the other end, hydroxyl (−OH) and oxo (=O) groups of ring C are forming two hydrogen bonds with Thr196. The rest of the quercetin molecule forms several other non-covalent bonds such as π-Alkyl, π-Sigma and Van der Waals interactions with a similar set of non-polar residues (Phe173, Phe192, Leu259, Leu299, Val177, Leu268, Met272, Phe303 and Met306) that act to recognize the hydrophobic tail of the sphingosine (Fig. 5B and Table S2). Surface representation of SphK1 is indicating that the internal cavity of SphK1 is occupied by quercetin which is essentially binding with the substrate-binding pocket residues with an appreciable affinity (Fig. 5D). Hence, it could be speculated that quercetin bound at the catalytic site via interaction between its catechol group and the active site residue of SphK1 (Asp178) though hydrogen bond, and other stabilizing non-covalent interactions hinders the sphingosine accessibility, eventually leading to inhibit the activity of SphK1. Overall, the interaction analysis suggests that quercetin acts as a competitive inhibitor of sphingosine instead of being an ATP competitor like most of the kinase inhibitors. ### Structural changes in SphK1 upon quercetin binding MD simulations have been utilized to get insight into the structural dynamics and functional mechanism of a protein-ligand complex. It has been become a widely used approach to describe the dynamics of the binding prototype of a small compound with a protein in explicit solvent environment55. Here, we performed all-atom MD simulation of SphK1 in the free state, and quercetin-bound SphK1 complex to evaluate the conformational dynamics, stability and interaction mechanism of quercetin with SphK1. Analysis of MD trajectories provides a detailed mechanistic insight into how quercetin binding affects the structural and dynamic behaviour of SphK1. It reveals that the quercetin plays a key role in stabilizing the structural conformation of SphK1. We calculated the potential energy of SphK1 before and after quercetin binding to ascertain the equilibration and stability of the complexes prior to MD analysis. The calculated potential energy of SphK1 before and after quercetin binding was found to be −890,000 kJ/mol and −880,713 kJ/mol, respectively. Several important MD parameters calculated for both the systems are given in Table 3. A small molecule can cause large conformational changes in a protein structure after binding to its active pocket56,57,58,59. Calculating Root-mean-square deviation (RMSD) is one of the most widely used approaches to evaluate the structural deviation and conformational changes in a protein60. To validate the stabilization of SphK1 before and after quercetin binding, the RMSD of both systems were analyzed. Quercetin is found to stabilizes the global dynamics of SphK1 compared to free SphK1. During the analysis, an average of RMSD values for SphK1 before and after quercetin binding was found to be 0.37 nm and 0.38 nm, respectively (Table 3). Here we found no significant changes in RMSD values which suggests that the binding of quercetin stabilizes the structure and leads to less conformational changes from the native conformation of SphK1 (Fig. 6A). But, in RMSD analysis of the SphK1 flap aa 170–180, it was observed that the quercetin-bound SphK1 exhibits significant structural deviation (Fig. S4A). Here, it was found that, as compared to the free SphK1, quercetin-bound SphK1 fluctuates considerably and has the least RMSD throughout the simulation (Fig. S4A). To see the residual movements and structural flexibility, the average fluctuation of each residue was evaluated and plotted as RMSFs in SphK1 before and after quercetin binding (Fig. 6B). We found several random fluctuations in the movement of each residue of SphK1 which were found to be minimized upon quercetin binding as depicted in the RMSF plot at region spanning from N- terminal to C- terminal. However, several increasing residual fluctuations were also seen upon quercetin binding, especially in the region spanning from amino acid residues 180 to 191 which overlaps with ATP-binding site residues (Arg185 and Arg191) (Figs. 4 and 6B). This suggests that structural flexibility induced by quercetin at ATP-binding pocket can also be presumed as a reason for reduced SphK1 activity along with the hindrance imparted by it for substrate binding as concluded from docking analysis. While in case of the flap of 170 to 180 residues that possess the substrate-binding site Asp178, RMSFs were found to be minimized reflecting significant conformational changes in SphK1 binding pocket upon interaction with quercetin (Fig. S4B). The fluctuation of substrate binding site Asp178 remains nearly constant and not changed significantly upon quercetin binding. However, increased fluctuation has been observed in Ile17, a functionally important residue of the substrate-binding lid of SphK1, upon binding to quercetin (Fig. S4B). The analysis shows that as compared to the free SphK1, quercetin-bound SphK1 fluctuates considerably throughout the simulation. Radius of gyration (Rg) is an important parameter of a protein accompanying with its overall conformational shape. It has been widely used to analyze the stability and conformational changes of a protein. The average Rg of free SphK1 and quercetin-bound SphK1 was calculated and found as 1.96 nm and 2.03 nm, respectively. Although the Rg plot shows little higher value in case of quercetin-bound SphK1, no significant structural switching was observed during the entire simulation (Fig. 6C). The protein attained a stable value of Rg equilibrated throughout the simulation. However, we observed significant changes in Rg of flap170–180 residues with decreased value, but no conformational shift was found suggesting least structural deviation in SphK1 upon quercetin binding (Fig. S4C). Solvent Accessible Surface Area (SASA) is the area of a protein that is directly accessible to the surrounding solvents61. An average of SASA values for free SphK1 and quercetin-bound SphK1 were calculated and found to be 145.61 nm2, and 153.09 nm2, respectively. We observed that the quercetin induces a little conformational change in SphK1 as depicted in increased SASA as compared to free SphK1 (Fig. 6D). In contrast, we observed a little decrement in SASA of the flap of residues 170 to 180 when in complexed with quercetin (Fig. S4D). In overall observation of flap 170 to 180, we found increased dynamics of SphK1 due to the binding of quercetin (Fig. S4). ### Interaction analysis of SphK1-quercetin complex Hydrogen bonding in a protein structure is a fundamental aspect of conformational stability62,63. The analysis of hydrogen bonds formed between a small molecule and a protein can be utilized to get insight into the stability of the complex which can provide directionality and specificity of protein-ligand interaction62. To assess the stability of SphK1 structure before and after quercetin binding, the intra-protein hydrogen bonds paired within 0.35 nm during the simulation were calculated as 251 and 254, respectively (Fig. 7A). An average of conventional hydrogen bonds between quercetin and SphK1 was also calculated with the stability of 3 bonds formed throughout the simulation. The quercetin was found to bind in SphK1 binding pocket with 4–5 conventional hydrogen bonds with higher fluctuation, and 3–4 hydrogen bonds with the least fluctuation (Fig. 7B). An average of bonds paired within 0.35 nm between quercetin and SphK1 was calculated 5 in number (Fig. 7C). ### Time evaluation of secondary structure changes in SphK1 The time evaluation of changes in the secondary structure content of SphK1 before and after quercetin binding was calculated and plotted after the completion of the 100 ns simulation. The analysis showed that the structural elements i.e. α-helix, β-sheet and turn in SphK1 remain constant and equilibrated throughout the simulation with least fluctuation (Fig. 8A). Here, the average number of residues participating in secondary structure content of SphK1-quercetin complex was found to be a little increased as a result of the formation of α-helix and β-bridges and a little decrease in coil and turn as compared with free SphK1 (Fig. 8B and Table S3). We found no significant changes in the secondary structure content of SphK1 before and after quercetin binding. ### Principal component and free energy landscape analysis of SphK1 Principal component (PC) analysis is an extensively used method to get insight into the global motion of a protein as they formed their specific function by collective motion of their atoms. It identifies the essential modes representing the major part of the collective motions of the protein. We performed PC analysis to capture SphK1 motions before and after quercetin binding to evaluate the complex stability. We captured the significant motions of SphK1 via the top two eigenvectors, PC1 and PC2 before and after quercetin binding. We plotted the two-dimensional projection of the ensembles these PCs to compare the conformational space sampling by free SphK1 and quercetin-bound SphK1 (Fig. 9A). We can see the regions explored both the systems overlap with a notable difference in conformational sampling between the free and quercetin-bound states of SphK1 (Fig. 9A). This analysis is also showing consistency with RMSD and RMSF results, where the structural dynamics was found to be slightly increased in case of SphK1-quercetin complex. Here, we can see a wider cluster of the stable states of SphK1 in presence of quercetin, but, no overall switching was observed in the motion of quercetin-bound SphK1 (Fig. 9). The principal motions of SphK1 residues were visualized and analyzed by representing their PCs in porcupine plots. The principal motion of free SphK1 and SphK1-quercetin complex, along the direction of PC1 and PC2, are showing in Fig. 9C. We observed that in the free SphK1, loop regions of the SphK1 binding pocket experienced more flexibility as compared to the remaining part of the protein. Particularly, only the arm loop of the SphK1 binding pocket shows significant fluctuations whereas the remaining secondary structures exhibit minimal motion. The analysis of porcupine plots is suggesting that the binding of quercetin to SphK1 has allosterically influenced the structural dynamics of SphK1 (Fig. 9C). Further, to analyse the conformational behaviour of SphK1 before and after quercetin binding, the Gibbs free energy landscapes (FELs) were plotted using the first two PCs, PC1 and PC2. The FELs of SphK1 and SphK1-quercetin complex are shown in Fig. 10. SphK1, before and after quercetin binding shows 3–4 stable global minima confined within 2–3 basins as depicted from FEL. However, we can see a noticeable change in the conformational behaviour of SphK1 in-presence of quercetin which progress to different energy minima as compared to free SphK1 (Fig. 10). ## Discussion The aberrant activation of signalling cascades is a frequent event in various types of human cancers64,65. Since protein and lipid kinases are key members of almost all signalling pathways, developing anticancer therapies targeting these central enzymes has always been a matter of choice among researchers66,67,68. To this end, SphK1 has emerged as a key enzyme as it regulates the sphingolipid rheostat responsible for determining cell fate. S1P generated by the catalytic action of SphK1 acts as pro-survival molecule and activates downstream targets involved in diverse pathological processes like cancer initiation, progression and inflammation. In fact, upregulation of SphK1 and its effector molecule S1P has been well established in various cancers and other human pathologies like pulmonary fibrosis, diabetes and Alzheimer’s disease. This makes SphK1 a potential drug target for the therapeutic intervention of diseases. The identification and development of therapeutic molecules, targeting human kinase, capable of selectively killing the cancerous cells without being cytotoxic is of immense importance, and has received the growing interest of scientists globally69,70,71. Since ages, many plant-derived compounds or phytochemicals have been known to be effective against several human diseases including cancer owing to their anti-inflammatory, antioxidant, anticancer, antiviral and pro-apoptotic effects36,37,72. Additionally, they offer many advantages over synthetic drug molecules such as no or minimal side effects and being inexpensive. The immense importance of dietary polyphenols and other natural compounds encouraged us to investigate them as potential and effective inhibitors of SphK1. Here, we studied the inhibitory potential of a series of plant-derived natural compounds. Among all, quercetin and capsaicin were found to possess best binding affinity towards SphK1 after the initial screening with molecular docking and fluorescence binding studies, and hence their role as SphK1 inhibitor was further evaluated. Quercetin is a polyphenolic flavonoid found in various fruits and vegetables whereas capsaicin, an active ingredient of chilli peppers, is a homovanillic acid derivative. The anti-carcinogenic, anti-angiogenic and anti-inflammatory effects of both of these dietary phytochemicals has been well established in the literature73,74,75,76,77,78. Recently, we have observed that quercetin and ellagic acid shows an excellent binding affinity to the pyruvate dehydrogenase kinase 3 and inhibits its enzyme activity79,80. Fluorescence binding studies reveal that both quercetin and capsaicin binds SphK1 with high affinity, being maximum for quercetin; whereas other studied compounds do not show significant fluorescence quenching (Figs. 1 and S1). The binding parameters calculated form ITC measurements also suggested a strong interaction of both quercetin and capsaicin with SphK1 (Fig. 2 and Table 2). We further examined the ATPase activity of SphK1 in the presence of increasing concentration of compounds which shows an IC50 value of quercetin and capsaicin as 2.8 and 27.0 µM, respectively (Fig. 3 and Table 1). Enzyme inhibition assay further validate our fluorescence and ITC results. Since quercetin was established as the most effective inhibitor of SphK1 among all screened natural compounds through a series of experiments, we tried to get mechanistic details of SphK1-quercetin interaction at molecular level by performing docking and MD simulation studies that could further facilitate our understanding of mechanism of inhibition by quercetin. Interestingly, a recent study by Zhang et al.40, illustrated that quercetin improves pulmonary fibrosis in vivo by inhibiting S1P/SphK1 signalling. This further encourages us to get mechanistic insights of the mode of inhibition of SphK1 by quercetin. Molecular docking of quercetin with SphK1 was done to examine the binding pattern. Analysis of docking results revealed a strong binding affinity (−8.2 kcal/mol) of SphK1-quercetin complex (Table 1). The interaction analysis suggests that quercetin occupies the same position where the natural substrate D-sphingosine binds (Fig. 4). This suggests that quercetin decreases the substrate accessibility of SphK1 by acting as a competitive inhibitor that ultimately leads to enzyme inhibition. The SphK1-quercetin complex was stabilized by 3 hydrogen bonds as well as π-π interactions with the residues of substrate binding pocket (Fig. 5B and Table S2). The stability and conformational changes occurring in SphK1 upon quercetin binding was further assessed by 100 ns MD simulation studies. The values of RMSD, RMSF, Rg and SASA suggest that quercetin binding stabilizes the SphK1 structure without any significant conformation shift (Table 3). However, several random fluctuations can be seen at initial time, but no conformational switching was observed during the entire simulations. This analysis is suggesting a strong stability of SphK1 upon quercetin binding. Notably, RMSF plot shows substantial conformational flexibility in the ATP binding pocket region upon quercetin binding which could probably disrupt the stabilizing interactions required for the ATPase activity of SphK1(Fig. 6B). This further explains the inhibitory potential of quercetin as assessed by our results of ATPase inhibition assay. We also investigated the anti-proliferative potential of quercetin on liver and lung cancer lines viz., HepG2 and A549. Quercetin was found to inhibit the growth of cancer cells in a dose dependent manner with an IC50 value of 49.10 µM and 55.90 µM for HepG2 and A549 cells, respectively (data not shown). At the same time, quercetin did not impart any toxicity towards normal counterparts (HEK293 cells). ## Conclusions In conclusion, this study indicates that quercetin and capsaicin act as potent inhibitors of SphK1; being quercetin as the best inhibitor by interacting directly with the substrate binding pocket. Hence, targeting SphK1 by these natural compounds can be a smart therapeutic approach to manage the clinical manifestations of cancer and other SphK1 associated human pathologies. Overall, our results encourage the use of dietary phytochemicals in the development of therapeutics against SphK1 and other disease related kinases.	2022-12-10 04:52:53	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4964176118373871, "perplexity": 6715.08866586137}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711712.26/warc/CC-MAIN-20221210042021-20221210072021-00294.warc.gz"}
https://opensourceconnections.com/glossary/hit/	# Hit « Back to Glossary Index A search result matching given criteria; sometimes used to denote the number of occurrences of a search term in a document.	2023-03-27 10:16:58	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9785658121109009, "perplexity": 1848.5817475674935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948620.60/warc/CC-MAIN-20230327092225-20230327122225-00497.warc.gz"}
https://www.physicsforums.com/threads/constant-angular-velocity-problem.310960/	Constant Angular Velocity Problem 1. Apr 30, 2009 math_girl 1. The problem statement, all variables and given/known data I encountered the following problem and I don't know where to begin or what formula to use... An object is moving in a circular path with a radius of 4.00 m. If the object moves through an arc length of 3.16m, then find the angular displacement? 2. Relevant equations I wish I knew. 3. The attempt at a solution Any help would be great thanks! 2. Apr 30, 2009 LowlyPion Welcome to PF. What is the circumference of a circle with radius 4? And what percentage of the circumference does 3.16 represent? That percentage of 360° then ... 3. Apr 30, 2009 PhanthomJay Just look up (then study) the relationship between arc length (s), radius (r) and angular displacement (theta). Note that theta is in radians. If you have no textbook, try Google. 4. May 2, 2009 math_girl so I got the following figured out, circumerence is 25.13 to find the percentage of the circumference that 3.16 represents should I take 3.16/25.13? If so that number is .126, than what should I do with figuring out the percentage of 360°? 5. May 2, 2009 LowlyPion What you determined is the fraction of the circumference. Since there are 360 degrees in a full circle or alternatively there are 2π radians in a circle ... If you want degrees, then multiply by 360. If you want it in radians, there are 2π (23.1415), then multiply .1266.283. 6. May 2, 2009 math_girl thanks thats what I needed to figure it out! Thanks	2017-08-23 11:22:30	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8325490355491638, "perplexity": 926.6470109859155}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886118195.43/warc/CC-MAIN-20170823094122-20170823114122-00195.warc.gz"}
https://indico.inp.nsk.su/event/8/contributions/1715/	# Instrumentation for Colliding Beam Physics (INSTR-17) 27 February 2017 to 3 March 2017 Budker Institute of Nuclear Physics Asia/Novosibirsk timezone ## Energy Scale Calibration of KEDR Detector Tagging System 28 Feb 2017, 17:00 1h Budker Institute of Nuclear Physics #### Budker Institute of Nuclear Physics 11, akademika Lavrentieva prospect, Novosibirsk, Russia Board: 1 Poster Colliders and detector integration ### Speaker Viacheslav Kaminskiy (Budker Institute of Nuclear Physics SB RAS) ### Description Tagging system of KEDR detector is a symmetrical focussing magnetic spectrometer for scattered at small angles electrons and positrons, which is embedded into the lattice of VEPP-4M collider. It is intended for two-photon processes study and measures scattered electron/positron energy with resolution $\Delta E/E_0 = 0.03\% \dots 0.6\%$ ($E_0$ is the beam energy). For precise energy scale calibration two methods are used: tagging of bremsstrahlung electron/positron by the photon energy measured by BGO calorimeter, and direct calibration using Compton backscattering spectrum edges. Also the energy scale is defined using the model of TS magnetic system with accuracy comparable to energy resolution. This report covers the design and current status of the calibration system. ### Primary author Viacheslav Kaminskiy (Budker Institute of Nuclear Physics SB RAS) ### Co-author Prof. Nikolai Muchnoi (Budker INP SB RAS) ### Presentation Materials There are no materials yet.	2020-10-24 01:01:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20360435545444489, "perplexity": 12062.593769812733}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107881551.11/warc/CC-MAIN-20201023234043-20201024024043-00083.warc.gz"}
https://homework.cpm.org/category/ACC/textbook/ccaa8/chapter/3%20Unit%204/lesson/CCA:%203.2.4/problem/3-70	Home > CCAA8 > Chapter 3 Unit 4 > Lesson CCA: 3.2.4 > Problem3-70 3-70. Use a generic rectangle to multiply the following expressions. Write each solution both as a sum and as a product. 1. $(2x+5)(x+6)$ • Draw a generic rectangle and label the length and width. $2x$ $+5$ $x$ $+6$ • Fill in the grid by multiplying the length and width of each box. $2x$ $+5$ $x$ $2x^2$ $5x$ $+6$ $12x$ $30$ $(2x+5)(x+6)=2x^2+17x+30$ Write an equation showing the product is the sum of the parts. Area as a sum: $2x^2+12x+5x+30=2x^2+17x+30$ Area as a product: $(2x+5)(x+6)$ 1. $(m−3)(3m+5)$ • See the help for part (a). $m$ $-3$ $3m$ $3m^2$ $-9m$ $+5$ $5m$ $-15$ Area as product: $(m−3)(3m+5)$ Area as sum: $3m^2−4m−15$ 1. $(12x+1)(x^2−5)$ • See the help for part (a). $\color{red}{12x}$ $\color{red}{+1}$ $\color{red}{x^2}$ $\color{red}{12x^3}$ $\color{red}{x^2}$ $\color{red}{-5}$ $\color{red}{-60x}$ $\color{red}{-5}$ 1. $(3−5y)(2+y)$ • See the help for part (a).	2020-05-25 14:46:47	{"extraction_info": {"found_math": true, "script_math_tex": 37, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8222638964653015, "perplexity": 967.2995646060439}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347388758.12/warc/CC-MAIN-20200525130036-20200525160036-00248.warc.gz"}
https://dials.tidymodels.org/reference/stop_iter.html	For some models, the effectiveness of the model can decrease as training iterations continue. stop_iter() can be used to tune how many iterations without an improvement in the objective function occur before training should be halted. ## Usage stop_iter(range = c(3L, 20L), trans = NULL) ## Arguments range A two-element vector holding the defaults for the smallest and largest possible values, respectively. trans A trans object from the scales package, such as scales::log10_trans() or scales::reciprocal_trans(). If not provided, the default is used which matches the units used in range. If no transformation, NULL. ## Examples stop_iter() #> # Iterations Before Stopping (quantitative) #> Range: [3, 20]	2022-05-29 10:58:05	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6134681701660156, "perplexity": 2650.7048131248175}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662644142.66/warc/CC-MAIN-20220529103854-20220529133854-00114.warc.gz"}
https://gateoverflow.in/310943/pgee-2019	224 views for(int i=0; i<=100;i++) { if (i % 3 == 0) printf("Great); if(i%5 == 0) printf("India"); } Count the number of times GreatIndia is printed. 1. 6 2. 20 3. 33 4. none of these edited \| 224 views 0 the answer should be $21$, please solve again. 0 Okay it means I have to take every possible combination of multiple of 3 and 5 here 0 For I = 0 nothing will be will be printed 0 $i=0$ "GreatIndia" will be printed. 0 For i=0 great india will be printed. 0 how 0 mod 3 wouldn't be equal to zero 0 0 mod 3 means remainder when we divide 0 by 3 so it is 0 0 Okay got confused it's zero is divided by three not the other way 0 if this question comes in gate then we will execute the loop step by step for 100 times ? 0 @ no, we can observe how condition satisfied and generalize the condition. 1 +1 vote 2	2019-07-17 07:01:24	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44909247756004333, "perplexity": 2162.2555248602803}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195525094.53/warc/CC-MAIN-20190717061451-20190717083451-00156.warc.gz"}
https://stanfordesp.org/learn/Splash/2013_Fall/catalog	Splash! Fall 2013 Course Catalog Filter Catalog by Grade: Biological and Medical Science Engineering Hobbies Life Skills Lunch Mathematics and Computer Science Physical Science Social Science Visual and Performing Arts Walk-in Activity Writing, Literature, and Language This is the catalog from a previous Splash program. Biological and Medical Science B3315: Why do we need flu shot? Difficulty: * Teachers: Mrinmoy Sanyal This class will give an overview of human immune system followed by an introduction on how our body recognizes different infectious agents like bacteria and virus. Then we will discuss how our immune system fights infection. Finally, benefits of immunization like flu shot will be discussed. Prerequisites None B3068: How to Distinguish Medical Knowledge from Hoaxes? Difficulty: Teachers: Kun-Hsing Yu There are lots of newspaper articles talking about the “recent advances in health sciences”. Some suggest that “beer helps to prevent cancer”, others assert “beer increases risks of getting cancer”. Which one should we believe? How do doctors decide what to do when coming across contrary scientific evidences? And what doctors don’t know about the surgery they perform or the drugs they prescribe? We will do some hands-on experiments on drawing conclusions in the world of uncertainty, and take a quick survey of current methodologies in medical sciences. Prerequisites Have flipped coins, or played any other games involving probability or uncertainty. B3078: A brief introduction to population genetics Difficulty: The human genome is comprised of three billion base pairs, of which, 99% are identical across the entire human population. Only a very small fraction of the genome harbors any variation. It is this small, variable fraction that plays an important role in natural selection and can inform us about events such as demography. With genomic sequencing technology becoming increasingly cheap and accessible to the public, we are now entering an exciting era of personalized genomics and medicine. In this course, we will learn about the signatures of genetic variation that can help us understand our susceptibility to diseases and our human demographic history. B3090: Mind and Body: How Your Mind Makes It Real Difficulty: ** Teachers: David Carreon Can a sugar-pill cause morphine release? Can hypnosis cure blindness? Can looks kill (literally)? Can getting shot not hurt? We'll talk about old history and new science developing around "mind-body" medicine, how your mind and brain affect your body in really interesting ways. We'll meet people with paralysis who can regain their movement, and blind people can regain their sight by the power of words. We'll see how the brain can produce pain completely independently of any "physical" cause. We'll discuss theories of how the brain might be involved in diseases like fibromyalgia and irritable bowel syndrome. If that's not enough, you'll learn about ritual executions that rely on the victim's expectation, and soldiers in WWII who get shot but don't seem to mind. In short, we'll explore the strange and perplexing frontier where Mind meets Body. B3242: DNA FUN Difficulty: * Teachers: Dena Leeman, Jana Lim In this class we will discuss DNA and the properties of DNA, and use this knowledge to figure out how to extract DNA from strawberries or other things (you can pick!)! Come ready to act like a scientist to solve the mystery of how to extract DNA - you will have the chance to experiment and see amazing things before your very eyes! B3197: White Blood Cells : aspect, function and...malfunction Difficulty: Teachers: Silviana Ilcus How do white blood cells look like? Why do we need them? What happens when things go awry with such a cell, and what causes common white blood cell related diseases? We will look at a couple of major such illnesses, and explore through the lens of the disease some of the mechanisms through which white blood cells perform their functions. Prerequisites knowing a bit about cells would be helpful, but not essential B3318: When DNA gets damaged Difficulty: Teachers: Erin Schwartz DNA contains the information for all of the parts of a cell. But what happens when DNA gets damaged? In this course, we discuss the different ways that cells have adapted to repair and tolerate changes in their DNA. We also discuss how defects in DNA repair pathways can lead to human disease, such as cancer. Prerequisites A general understanding of DNA and its structure would be helpful, but not essential. B3330: How do cells move? Difficulty: Teachers: Caleb Chan Cell migration is important for the development and maintenance of multicellular organisms. Biological processes such as embryonic development, wound healing, and immune responses all require the movement of cells in specific directions to specific locations. Come to this class to learn about the mechanisms that drive cell motility! B3072: Nutritional Food Science Difficulty: Teachers: Devan Diwanji We are what we eat, right? Whether you believe that statement or not you might as well learn why we need to eat. In this class, we’ll take a biologic perspective on eating by reviewing the basic food groups, conducting hands-on experiments, and applying such knowledge to medical case studies. B3110: Can you trust your senses? Difficulty: * Teachers: Jimmy Tobin How do we know what our eyes and ears are telling us are true? We'll be investigating our senses and finding places where they may break down. Some questions we'll ask: Why does orange juice taste bad after brushing my teeth? How do optical illusions work? How do ears and eyes work? We may not answer everything because some of these questions are still being reached, but we will be showing a bunch of fun illusions to learn more about perception and why we can't always trust it. Prerequisites None. B3149: Parasitic Worms: The Monsters Within Difficulty: * Teachers: Connie Fung, Katie Wu Parasitic worms have been living inside the human body since times before our earliest recorded history. Unlike the infectious microbes that cause diseases more familiar to us (i.e. the flu, the common cold, gastroenteritis, etc.), adult worms tend not to be microscopic in size—some can even reach 10-20 meters in length (imagine that living inside your gut)! Come and learn about how these disgusting yet fascinating creatures find their way into human hosts, where and how they survive within the human body, and the consequences that may result from worm infections, ranging from blindness to severe deformation of body parts to impairment of physical and mental development. Prerequisites Basic knowledge of biology would be great, and an interest in learning cool things about infectious organisms will be a plus! B3156: Dendrology Walk: An Introduction to Tree Biology and Identification Difficulty: * Teachers: Henrietta Bennett This class uses Stanford's collection of indigenous and imported plants to demonstrate interesting facts and features about trees. During a walking tour of campus, students will learn about tree anatomy, ecology, and the cultural relevance of certain tree species. Students will also learn the identifying characteristics of common tree families. B3324: Intro to Cancer Imaging Difficulty: Teachers: David Hsu Taking images of cancer is one of the most important areas of medical engineering. There are many methods to "take a picture" of a tumor, but each has its own pros and cons. Come learn about some of the most common methods of cancer imaging, such as CT, MRI, and PET. Prerequisites Students should have at least a very basic idea of what cancer tumors and x-rays are. B3335: “I’m HIV Positive”: A Brief Look into How HIV Infects and Affects Our Society Difficulty: Teachers: Renaud Vann This course will first quickly explore the basics of the Human Immunodeficiency Virus (HIV)–covering what HIV is, how HIV is transmitted, and how HIV affects humans physiologically in the form of AIDS. With that basis, we can then explore how HIV has become a considerable problem in many countries, how our views about sexuality lead to discrimination against HIV positive people, why there are so many misconceptions about HIV/AIDS, and what has been done to slow the spread of disease. Prerequisites A high-school-level Biology course is recommended, but absolutely not required. B3044: Intro to Epigenetics: From Development to Cancer Difficulty: ** Teachers: Melissa Ko Does DNA determine exactly how a living organism will look or behave? How do all the many cells in the human body function differently despite having the same DNA? Discover how non-genetic factors in our cells lead to a diversity of cell states or identities, allowing us to function as complex, multicellular organisms, but also causing serious problems such as cancer when these factors go awry. Prerequisites basic understanding of cancer and cell biology (if you know what oncogenes are, the structure of DNA, and what a cell is made of, you will be fine) B3282: Vaccines: Past, Present, and Preventing a Zombie Apocalypse Difficulty: * Primitive vaccines were developed in ancient China, India, and Africa and were the precursors for modern vaccine technology used today to prevent millions of people from suffering from diseases such as smallpox, whooping cough, and polio. How were vaccines developed? How do they work to prevent disease? What are the continuing challenges in vaccination? If a zombie apocalypse were eminent, could you be the vaccine developer to save the day? Prerequisites Enthusiasm for health or biology! B3050: Asthma Today Difficulty: Teachers: Derry Akin You've most likely heard about asthma before. Perhaps you know someone with it; a friend, a family member--maybe even yourself! In our class, we'll learn more about the condition, which is a growing problem in today's society. We'll be discussing a variety of asthma-related topics, such as what asthma really is, current treatment options, and where asthma treatment and research are headed today. B3073: HIV/AIDS: The History, The Virus, and The Present Difficulty: Teachers: Devan Diwanji Think you know what HIV is? Want to become an expert on HIV/AIDS? In our informal seminar we aim to investigate the history of the domestic and global HIV/AIDS epidemic, mechanisms of viral transmission and cellular entry, and current research breakthroughs. Prerequisites Basic High School Biology B3099: The Science of Optical Illusions Difficulty: * Teachers: Chris Baldassano Seeing the world around us feels effortless, but our visual system is actually incredibly complicated. We automatically use assumptions about the natural world to influence the way we see objects, and our brain has specialized areas to process different types of visual information. In this class, we’ll use optical illusions to investigate the surprisingly complex tricks that our brains use to help us understand the world. We’ll learn that all of us are partially blind, that colors are not what they seem, why faces are so important, and much more! B3211: Microbial Encounters in Daily Life Difficulty: Teachers: Kevin Meng, Katie Wu Have you ever wondered how many and what types of invisible bugs you encounter on a daily basis? In this class, we will explore the microbes that constantly surround us - everywhere from the kitchen sink to the inside of your mouth. The class will be a half-hour lecture followed by an interactive discussion. Hands-on modeling/drawing will demonstrate what different types of microbes look like and how they interact with their environment. Prerequisites Basic knowledge of biology B3306: The Heart: How it works and how we see it Difficulty: Teachers: Macy Zardeneta In this course, we will learn how the heart works and discuss cardiovascular imaging techniques. Prerequisites Basic biology recommended but not required and an interest in the heart. B3053: Human Evolution: from Wilt Chamberlain to Lance Armstrong Difficulty: * Teachers: Richard She Are human beings still evolving? Are performance enhancing drugs okay? How do we cope with a future in which genetic engineering will be possible? B3130: Embryology Difficulty: ** All animals start out as a single cell, the zygote. Embryology is the branch of science that studies the embryo: all of the stages of development between the zygote and animal's final form. This course will help answer questions such as: -How do embryos decide where to grow a head, where to grow a tail, where to grow arms and legs, where to grow eyes, etc.? -What do the embryos of different species of animal look like? What do they all have in common? What are the differences? -Why do human babies grow hands while bat babies grow wings? -Where does the embryo store all of the instructions it needs to grow and take the right shape? How does the embryo read those instructions? Can we change the instructions? What happens if we do? -How can we study embryos? What kind of experiments can scientists do with embryos? You will also be able to compare living and preserved embryos of different animals at different stages under the microscope. It'll be fun! :) B3313: Hepatitis B Difficulty: * Teachers: Jeffrey Kwong Learn about the Hepatitis B pandemic! Come hear about what the virus is, what the signs and symptoms are, and what YOU can do to protect your community from Hepatitis B. B3060: +1: Psychology Power-ups Difficulty: * Teachers: Zaviera Panlilio Wish you could use power ups like a real life Mario or Pokemon? This interactive course allows you to do things (in class) to replicate findings from researchers around the world. Learn how practicing gratitude, looking someone in the eye, and holding a friend's hand can make even an electrical shock feel less painful! Most of us know about prosocial behaviors--like sharing, caring and helping--but we may not realize the instant and long-term benefits they have for our physical and mental health! I will be exploring Positive Psychology and resiliency through my five years as a Physiological Psychology researcher and over 15 years volunteering with normative and clinical populations. Questions are encouraged and the aim for this course is to not only feel better--but also how to put research into practice at school, work, home and with friends to buffer you against everyday stressors. B3098: The Challenging Case of Robin Roberts Difficulty: ** Teachers: Suparna Dutt When standard treatments fail to cure blood cancers then bone marrow transplantation is the only curative option. Patients receive bone marrow cells from a healthy donor. This provides a new healthy immune system that helps keep the cancer cells from reappearing. Good Morning America host Robin Roberts received a bone marrow transplant last year for this reason. Stanford researcher Dr. Suparna Dutt will give an overview of Bone Marrow Transplantation- Blood Stem Cell Therapy that has saved thousands of lives over the past 30 years. B3207: Ethics of Scientific and Medical Research Difficulty: * Teachers: Paul Nuyujukian Learn about the core ethical ideas that govern all scientific and medical research. Discover the criteria that must be met for medical research and clinical trials. Explore the level of adherence of various forms of scientific research to these core principles and the means of oversight setup to ensure research is conducted in an ethical manner. We will also explore specific topics, examples, and cases; where the ethics of research are non-trivial to evaluate and often accompanied with social controversy. We will apply the core principles learned to actively debated areas of scientific and medical research. B3200: Topics in Structural Biology Difficulty: *** Teachers: Margreth Mpossi It's interesting how biological molecules work together in beautifully complex systems such as cellular processes and even whole organisms. Turns out structure and chemistry completely define behaviour of biomolecules! In this class we will go over how scientists figure out the composition, structure and specific chemical properties of biomolecules(e.g. proteins and RNA) while maintaining native and relevant states (i.e. physiological conditions). We will also talk about how function is affected by structure, and structure by chemical environment and composition. Also, why all of this is crucial to biology. Prerequisites AP bio would help. Curiosity and interest in science B3202: Nutrition 101: Fact Or Myth Difficulty: * Welcome any students and parents who would like to test their knowledge about nutrition. Can you tell what you know about nutrition is true or not? What are "made up" or are "assumptions"? What you eat, drink, and digest in your stomach is important, because this is about you and your well-being. Prerequisites Be interested about nutrition and would like to know what are factual information and myths. B3235: To Test or Not To Test? The Ethical Debates of Genetically Inherited Diseases Difficulty: ** If you could find out whether you will develop a disease with no known cure, would you want to know? This course will provide an introduction to the science behind Huntington’s Disease, which is a genetically inherited disease that affects both the mind and the body. After an overview of cool topics like DNA and genetics, we will talk about genetic testing – what that is, and how it applies to parents or children who may have Huntington’s Disease. You will then use this knowledge to debate the ethical concerns that arise when screening for inherited diseases. If your parents have Huntington’s Disease, would you get tested? If you were going to have children but did not know if you had the disease, would you get tested? Would you have kids if you tested positive? Prerequisites Some basic biology might be helpful, but we’ll provide a quick intro at the beginning of the course too! B3150: Sex 101: Basic Sexual Education Difficulty: * Curious about sex, sexual health/wellness, or about safer sex practices? Feel like your school sex ed hasn't given you all the information you want? If you said yes to either of those questions, this class is for you! In this class, we'll take a sex-positive approach to some of the basics of sexual education. Topics will likely include things like anatomy, sexually transmitted infections, and overall sexual wellness. B3232: Developmental Biology Difficulty: Ever wonder how an animal is formed? Ever wonder what animals look like before they are born? Did you know that all animals and even you started out as little more than a tiny ball of cells? How did these cells know to build you the way you are? In this class, we’ll examine how animals grow from such a simple form into a recognizable animal. We’ll talk about how cells in the embryo “decide” to form different tissues, and how they know where to form them. We'll also look at vertebrate embryos of different species as they develop to compare how similar/ different they are. You will be amazed by the similarities you can find between embryos of different species. We'll also crack open an egg and look at live chick embryos to explore different stages of development and talk about the different tools biologists use to study development. B3088: Practical Neuroscience (Updated!) Difficulty: Teachers: David Carreon What is the brain? How does it work? How can I make it stronger? You’ll learn about awesome experiments that show that the brain can be rewired, remolded and strengthened. You’ll meet someone operating with half a brain (literally), people who built physical strength just by thinking about it, and people who rewired their brains and cured mental illnesses with the power of thought. We’ll cover what you need to know about the brain. If you’ve got a brain, you should take this course! :) Updated with new materials for Fall 2013! Note: this course was extremely popular last Fall and Spring, so sign up soon! B3125: Your Microbes and You! Closed! Difficulty: ** Teachers: Julie Huang Did you know that from the day that you are born you become colonized with a multitude of microorganisms? Unlike germs that can make you sick, these microbes that live on and in your body provide many benefits that keep you healthy. Come learn about the microbes that you carry around every day and how they help protect you against disease! Prerequisites Some background in biology and an interest in microorganisms are a plus! B3139: Think Eat Save - Reduce Your FOODprint? Difficulty: * Teachers: Raji Lukkoor Have you ever heard the term FOODprint? Did you know that when you dump uneaten food into the trash can, it not only makes a kid go hungry in sub-Saharan Africa but also contributes to global warming? Here’s a statistic: Every year, consumers in rich countries waste almost as much food (222 million tons) as the entire net food production of sub-Saharan Africa (230 million tons)? Can you name some of the actions YOU can take to reduce your FOODprint? Come find out in this fascinating course, entitled “Think Eat Save – Reduce Your FOODprint.” Prerequisites none B3196: Sleep and Dreams 101 Difficulty: * Have you ever wondered what happens in the third of our lives that we spend sleeping? A lot more than you might think. Sleep stages, lucid dreaming, sleep disorders, and the dangers of sleep deprivation are all covered in this course. Drowsiness is red alert! B3209: Introduction to Neuroprosthetics Difficulty: * Teachers: Paul Nuyujukian Learn about the emerging field of neural prosthetics: electronic systems that interface with and connect to the brain and nervous system. We will start with a brief introduction to neuroscience as well as some of the guiding principles used in the field, transitioning to real-world examples. This class will cover descriptions of neural prosthetic systems that are commercially available or under active development. Systems discussed will include retinal prostheses, cochlear implants, and cortical communication and motor prosthetics. B3179: Genes and Genetic Engineering Difficulty: A basic introduction to the gene as the basic unit of biological function. We will also talk about the exciting prospect of using genetic engineering/synthetic biology to create a better world. Prerequisites Enthusiasm! B3217: Chocolate Food of the Gods Difficulty: Teachers: Howard Peters A fun Chocolate presentation on the history, biology, chemistry, biochemistry, manufacturing of chocolate. Some trivia, some saqmples and a free drawing for chocolate Prerequisites none B3275: When life gives you lemons, eat a miracle berry... Difficulty: * Learn about the science behind taste! How do we sense and distinguish different tastes? How do taste buds work? Find out all this and more! At the end of the class you will be able to try a "Miracle Berry Tablet" which will make sour foods taste sweet. B3091: Neuroscience of Pleasure, Pain and Purpose Difficulty: Teachers: David Carreon What motivates us? What systems in the brain drive us one way or another? Why did I just check Facebook again? In this talk, you’ll learn about sex (now I have your attention :P), addiction, pornography, drugs, altruism, self-control, and even the meaning of life. You’ll learn about how dopamine and oxytocin work and why they matter. You’ll also get some practical advice on ways to boost willpower, develop good habits and break bad ones. If you’ve got behaviors you’d like to change, you should take this course! :) B3100: Welcome to Your Brain Difficulty: Ever wonder how your brain makes you who you are? How does your brain help you see and move? Can we come up with a cure for brain diseases? This class is a hands-on introduction to the brain and its various functions. It's your chance to ask your burning questions about the brain to a bunch of people studying it! B3222: Top 5 and 1/2 Amazing Brain Facts Difficulty: Teachers: Keith Sudheimer Buried not so deeply inside the gooey fathoms of your brain are the secrets that have plagued poets and philosophers for millennia. That's right, they are just sitting there mocking us. But don't be fooled my friends. The brain can be a harsh mistress. Centuries of scientists laboring away their lives in windowless basement laboratories (which always smell oddly like dusty bologna) have discovered exactly 5 and 1/2 brain facts. Okay that's not exactly true. They have discovered a lot more than that. But only 5 and 1/2 of these facts were so face-meltingly awesome that upon learning them I fell out of my chair and wept uncontrollably on the floor for hours. I recommend you bring your own helmet and box of facial tissues to class. While your at it a change of pants isn't a terrible idea. (Okay just kidding you don't need any of that stuff) B3225: Microbes and Mud! Difficulty: Mud is not just for little children. There's a lot of great biology going on down in the muck. Did you know that there are microbes that breathe metal and ones that will die if they're exposed to oxygen? There's a whole world of bacteria that has nothing to do with the stuff we worry about being in your food. We'll be making and learning about Winogradsky columns (self sustaining microbial ecosystems that fit in a soda bottle) that show the amazing diversity of microbes in a clear, colorful way that you can take home and watch grow. Prerequisites A willingness to get your hands dirty B3229: How the brain responds to stress and trauma Difficulty: Teachers: Amy Garrett, Maisi Mayo Almost 60% of kids experience a traumatic event before age 16, such as getting hurt, seeing someone else get badly hurt, or other frightening events. These experiences can be difficult to forget, and can affect a person in a lot of ways. This class explains how teens might deal with trauma in different ways, and what brain imaging has taught us about the ways that the brain responds to trauma and stress, as well as how it can recover Prerequisites intro biology intro psychology B3251: Cancer Biology for Dummies Difficulty: Teachers: Stephanie Casey One in every three women and one in two men will contract it. We wear cute pink tee shirts and run 3 Days for it. You pay $20 of hard-earned cash to your friends who do Team in Training. But how much do you really know about cancer? In this class, we'll discuss causes, biology, types, and treatments for cancer. B3292: Biofilms: The Wonderful World of Bacterial Communities Difficulty: Teachers: Pascale Guiton Bacteria are everywhere and similar to humans they communicate with each other and reside within “small houses”. Enroll for this course to learn about the exciting features of a bacterial village and how it is produced during diseases. Prerequisites An interest in biology and micro-organisms will be a plus Engineering E3171: Wireless Music Transmission and Build Your Own Speaker Difficulty: Each student will build a simple speaker to take home. We will transmit music wirelessly and learn about sound, how speakers work, and basic circuits. Prerequisites Please bring a portable music player such as an MP3 player or smartphone if you have one. A few extras will be available to use if you do not have one. E3212: Engineering clean water- figuring out filtration Difficulty: * We will be teaching about water contamination and processes that are used to treat water to make it safe for drinking. The activity will allow students to design filters to clean water that is visibly dirty (i.e. turbid). We will split the students into groups of 3-4, and they will have the opportunity to make filters out of different materials to clean the water. E3269: Web App & Business Prototyping for Beginners Difficulty: We will teach you how to rapidly prototype and validate your ideas for starting your own business. By the end of the session, you will know how to create your own web application from scratch. We will also discuss how web businesses are formed. Instructors have backgrounds in engineering and product development at Stanford University, Microsoft, and several startups. Prerequisites Programming experience in Java, C/C++ E3175: Bending Light Difficulty: Teachers: Ilker Karakasoglu What is the fastest and most fascinating thing in the whole universe? LIGHT! In this class, we will delve into the magical world of light. We will talk about how ordinary sunlight is powerful enough to provide more energy in 2 hours than the world uses in one year. Then we will move on to much cooler topics such as how light can be harnessed to build Harry Potter's invisibility cloak, Anakin Skywalker's lightsaber or Iron Man's laser beams. Prerequisites Basic physics knowledge E3231: An Introduction to Earthquake Engineering Difficulty: * Teachers: Cristian Acevedo Earthquakes are one of Earth's most devastating phenomena. Come learn about earthquake mechanisms and design of structures in earthquake prone areas (like California) and experience shaking first hand! The class will cover the basic physics behind structural earthquake engineering design; the focus will be on explaining concepts through demonstrations. E3322: Engineering demos Difficulty: This class provides an overview of a few topics (including a demonstration) from some of the following technical fields: engineering, mathematics, computer science, and statistics. We hope this will be a great opportunity for students to learn about interesting projects and careers in engineering and other related fields. Questions and discussion are welcome! Students of all ages are encouraged, and the discussion will be tailored to the level of students enrolled. This event is partly sponsored by IEEE (Institute for electronics and electrical engineers). E3226: Digital Electronics Difficulty: Teachers: Sunil Pai Learn the basics of digital circuit design. I will teach students about switches, boolean operations and boolean gates, the basics of digital electronics. I will also teach students about how counters (like the decimal counter on a wristwatch) can be made using boolean gates and a clock pulse. I will then transition to more complex tools like multiplexers/demultiplexers and flip-flops, the latter of which is a crucial tool in computer hardware. Prerequisites Circuits at the level taught in high school physics classes. E3227: "How to live a crazy, amazing story: what I wish I knew in High School" Difficulty: Teachers: David Mora * DISCLAIMER: Taking this class will get you into Stanford. Guaranteed. * "Hi. I have been stalking you for the last 10 years of your life. I've compiled my observations into a 100% factual narrative of your life story. Want to read it?" Well... would you? Creepy hypothetical stalkers aside, we are all living out a story. This class is for those of us who want to make it a great one. WHAT WE WILL DO: 1. It's a secret. 2. (didn't I just say it was a secret?!) 3. Fine, some of it will involve airborne whip cream. ABOUT ME: My name is David, I love jazz piano, being crazy in public, performing slam poetry (is there a difference?), robotics, quantum mechanics, and listening to people's stories. E3262: Space Communications Difficulty: Teachers: Sawson Taheri This class will cover the basics of radio communication, with an emphasis on space based radio communication. Learn about: -Radio theory -Antennas -Time domain vs Frequency Domain -Digital communication -How to track and communicate with satellites Students will get a chance to make their own amateur radio satellite contact! Prerequisites -Completion of beginning Algebra -Motivation to learn E3172: Self-Healing Electronic Skin Difficulty: Teachers: Chao Wang The ability to spontaneously repair damage, which is termed as self-healing, is an important survival feature in nature because it increases the lifetime of most living creatures. Our human skin is a perfect example of self-healing capability: minor cut can be healed completely while severe cut can lead to scars. This class mainly focused on the examples of implanting this amazing self-healing feature onto electronics, like electronic skins. We can witness the miracles only in science fictions are now coming true! E3116: Polymers Difficulty: Teachers: Joseph Barakat Polymers are fascinating materials that can be found almost everywhere in our daily lives. These macromolecules have many repeating subunits that form long chains and networks. From the skin on our bones to the plastics we buy at the store, polymers come in all shapes and sizes. In this short course, we'll learn about polymers through fun hands-on activities that highlight some of their most important properties. E3240: Product Management 101 - How to Create Products Customers Love Difficulty: ** Teachers: Yisha Peng 1. PM role: What does PM do every day? What are required skills for a PM? What is the difference between top 1% PMs and 10% PMs? How to get prepared to become a great PM? Who are some great PMs in the US? … 2. Product Development Process: What is defined as a great product? How to create great products? How to measure the success of a product? How to work with different team to get things done? … E3157: Build Your Own Roller Coaster! Difficulty: * Come take part in a marble roller coaster design competition! We will discuss the basic physics behind designing a roller coaster and what components different engineers work on. Then we will group you into teams - you get to build your own roller coaster out of newspaper and tape! Prizes for the best design! Prerequisites None. Hobbies H3178: Cricket Difficulty: ** Teachers: Burjis Godrej Explaining the rules of cricket. Prerequisites - H3108: Learn to play bridge (the card game) Difficulty: * Question: What do Bill Gates and Warren Buffet have in common? Answer: They're both billionaires and they both love the card game bridge. If you too aspire to become a bridge-playing billionaire, then the first step is to learn how to play bridge! (Sadly, the second step is not covered in this class.) Bridge is a brainy card game somewhat like hearts. It's especially fun because it's played 2 on 2, so if you ever lose (hypothetically, of course) you've got someone other than yourself to blame! This class is for anyone and everyone who wants to learn bridge. NO EXPERIENCE NECESSARY! H3163: Games, Games, GAMES! Difficulty: * Teachers: Alison Buchsbaum Most people would agree that games are a lot of fun. Not as many people realize that they can have a purpose as well. There are lots of types, including ice breakers, get to know you games, and energizers. In this class, we will get to talk about why certain games exist and how to use them, but most importantly, have lots of fun playing games! H3317: Flexibility and Conditioning for Ballet Difficulty: ** Teachers: Laura Drohan This class will teach you some typical exercise routines that ballet dancers go through before each rehearsal. Inspired by the New York City Ballet workout, you'll learn how to increase your strength and flexibility to help in sports, dance, and health in general! It will include some fun music too! Prerequisites No prior dance experience required, just a little enthusiasm and some stretchy pants/socks or dance shoes! H3327: Vocaloid: The Fan Phenomenon Difficulty: * Teachers: Russell Chou Have you heard of supercell? Do you know who Hatsune Miku is? We’ll examine how Japanese MAD’s and the virtual singing idol software Vocaloid have impacted Japanese (and now the world’s) popular culture. H3328: What the heck is Touhou? Difficulty: * Teachers: Russell Chou You might have heard of Touhou Project, a series of indie shoot-em-up games that have taken the anime-comic-game community by storm, racking up hits on websites such as Pixiv, NicoNicoDouga, and conventions like Comiket and abroad. Come learn about why it is the #1 most popular fan franchise in internet history! H3338: Henna Your Hands! Creative, Temporary Body Art Difficulty: ** Teachers: Marcella Anthony Learn a brief history of Mehandi, known as Henna - a traditional body art. Explore traditional and modern uses and designs, learn the science of henna, then practice designing your own henna tattoo and adorn yourself! Participants will learn about the cultural history of henna and it’s common uses then and now from decorating hands and feet to coloring clothing and drums. After a brief lecture participants will practice drawing designs and apply a design to their hands. Participants will leave with a design, the instructors henna recipe, and care instructions. This session uses natural henna (www.hennaforhair.com) lemon juice, and hairspray as a fixative for the henna. The Henna will be pre-made in cones, but you will learn to make your own henna cone. Leave the paste on for a minimum of four hours up to overnight if you cover your design. Henna is a stain so don't wear your favorite outfit to this session in case you spill. Questions? Please email the instructor. H3065: Nutrition Label reading + Introduction to Chi Quong exercise Difficulty: * Teachers: May To Come and learn what you are eating!! We will explore label reading on packaged foods - fresh, frozen, canned, as a meal, desserts, cereals and more. There will be samples and hands on practice. At the end, let’s have some fun and strength a little. There will be a short but fun session on introduction to Chi Quong for health and everyday exercise. It is simple and easy to do. H3160: Magic: the Gathering! A Collectable Card Game Difficulty: Teachers: David Lam Magic: the Gathering is one of the oldest collectable card games and is still played today by thousands of people. In this class, we’ll go over the basic rules of the game, find out why Magic is such an awesome game, and then make our own decks and play them against other people! You’ll leave with a lot of free cards, an understanding of the game of Magic, and a great time! Prerequisites No experience required, but people that have some knowledge of Magic are advised to contact me @ [email protected] before signing up. All you have to bring is an open mind and excitement. H3326: Origami Roses w/ 4 petals Difficulty: Teachers: Russell Chou Learn how to fold a medium difficulty 4-petaled origami roses, based on the crane-base. Prerequisites Being able to fold a crane would be good H3102: Introduction to Photography Difficulty: * Teachers: Tony Jin Come find out what makes a great photo! You'll learn about topics like composition, exposure, lighting, etc. Don't know what those are? Great! We'll teach you all that. You'll come out of the class with a better understanding of how a camera works so you can use that knowledge to your advantage and show off your awesome photography to your friends! Prerequisites None. If you bring a camera, I can help you learn the controls, but there is no actual photography in this course. For that you should take my course "Photography Excursion" held later in the day. H3103: Technical Aspects of Photography Difficulty: *** Teachers: Tony Jin Advanced photographic knowledge and techniques, including exposure triangle, light value, depth of field, lens optics, filters, and image processing tools like Photoshop. Prerequisites Knowledge of basic photography, competence with math and physics. There is no actual photography in this course; for that you should take my course "Photography Excursion" held later in the day. Optional: Bring your best photos on a card, flash drive, computer, etc. to share with others. H3244: Paint with me! Difficulty: * Teachers: lisa sun, macy sun Relax and relieve your stress by exploring your artistic skills ! H3281: Learn Lacrosse! Difficulty: Teachers: Jacqueline Le Learn the lacrosse basics! In under 2 hours you'll play your first mini lacrosse game. No experience needed, and gear will be provided. H3045: Understanding Diplomacy Through Wargaming Difficulty: Much of historical European politics would have made more sense if you were there at the time. This class will give you a chance to recreate those politics. Take command of countries in a simple war game and learn about the balance of power by seeing it play out in action. H3104: Stanford Photowalk Difficulty: * Teachers: Tony Jin Join us in a walking tour of the magnificent Stanford campus. We will start in the Main Quad and visit many other locations, taking pictures as we go. At the end we can compare photos on a computer and even edit them if we have time. Prerequisites A camera is required for this course, preferably a camera with manual controls such as a DSLR. Photographic knowledge and skills will help you make better photos, so taking one or both of my previous courses ("Introduction to Photography" and "Technical Aspects of Photography") is highly recommended. H3169: Quidditch For Muggles Difficulty: * Stanford Quidditch, Silicon Valley Skrewts, and San Jose State Owls want you! We'll go over the basics of quidditch as it is played by high schools, colleges, and community teams throughout the world, and play some scrimmages. We'll provide the brooms! Wear shoes you can run in (e.g. sneakers). Bring a water bottle so you can keep hydrated. Sunscreen is recommended. H3325: Japanese (Riichi) Mahjong Difficulty: * Teachers: Russell Chou Learn how to play REAL mahjong, not mahjong solitaire. Riichi mahjong is the most common variant played in Japan, which is similar to but has a few key differences from Chinese mahjong. Prerequisites None, but knowing some basic Chinese characters such as numbers will help. Life Skills L3219: Invention & Inovation for Students using Patented Toys for Props Difficulty: Teachers: Howard Peters A fun presentation about invention and patenting using familiar toys (SLINKY, FRISBEE Yo-YO, etc) as props. A free drawing will be held at the end of the lectrue for a patented tooy and a copy of its US patent. Free web site pat2pdf.org #6469 is for A. Lincoln. Prerequisites none L3092: Justice - What’s the Right Thing to Do? Difficulty: Teachers: David Carreon Bank bailouts. Stealing to feed your hungry kid. Lying to save Jews in your basement. Waterboarding. What’s right and what’s wrong? And how do we know? This will be a crash course in Ethics, the rigorous discipline of determining what’s right. This lost science will be critical for anyone who will have to make decisions in their life. I’m modeling this course after the enormously popular class and book taught by Michael Sandel at Harvard (Google my course title). L3263: Teaching 101 Difficulty: Teachers: Kimberly Taylor When we think of teaching, we tend to envision a classroom setting with lots of desks and a teacher at the front. This is a form of teaching, but there are many other things that also use its skillset - running a summer camp, tutoring, and docenting are all examples of the many places you can apply teaching skills besides the classroom. The purpose of this class is to go over basic skills and knowledge to aid in learning how to teach effectively. It is a very interaction and discussion-based class, NOT lecture-based, so come prepared to participate and actively contribute to the class! L3194: Anyone Can Make It - Success Stories from Underrepresented Students at Stanford Difficulty: * Teachers: Andrew Guo In this class, you’ll talk to a panel of students from underrepresented and low-income backgrounds who made it to Stanford and are thriving as leaders today. Many of these students come from low-income rural and inner-city public and charter schools. Many are also first generation college students whose parents did not attend college. They’ll tell their stories and provide their thoughts on the key elements to success as a student from an underserved or underrepresented area. They will also discuss how to make the most out of your education and programs like Splash! This is NOT a class to ask about how to get into Stanford. This is a class geared towards the needs of the low-income, underrepresented students that come to Splash through financial aid. L3066: Interview Skills for Internship, College, and Job Applications Difficulty: ** Teachers: Oriana Li Halevy Are you anxious about internship, college, or job interviews? Come to this interactive course to receive solid tips from a Class of 1992 Harvard College pre-med turned United Nations intern turned US Department of State diplomatic interpreter turned multinational law firm corporate attorney turned venture investor, cross-border business consultant and strategist, and bilingual communications specialist and published translator/editor who has been on both sides of these interviews since high school. This course is for anyone wishing to develop and fine-tune interviewing skills that can be applied in a variety of settings. Topics will include: Preparation Presentation Common interview questions Common pitfalls Closing the interview Thank you notes L3295: Social Media Ettiquette: How Twitter can Make or Break your Life Full! Difficulty: * Teachers: Diana Marquez You might think Facebook and Twitter are just an outlet for your thoughts and a platform to connect with friends, but you're wrong! Your online persona is increasingly more important for future professors, landlords, and employers to size you up before they ever meet you. Learn the Do's and Don'ts of online media. L3101: Living in the moment: mindfulness and meditation Difficulty: * Teachers: Adam Perelman What does it mean to live in the moment? And why would we want to? Mindfulness and meditation are all about waking up and appreciating each moment of our lives. They can help us rewire our brains for the better, helping us to deal with stress, to feel happier and healthier, and to act with kindness and compassion. And we can apply them anywhere, anytime--whether we're eating a chocolate chip cookie, or talking to a friend, or doing homework, or just breathing! In this class, we'll talk about what mindfulness and meditation are all about, and we'll learn some simple ways to apply them right now in your own life. “In the end, just three things matter: How well we have lived How well we have loved How well we have learned to let go” ― Jack Kornfield L3174: The Power of Networking Difficulty: Teachers: Matthew Clark Do you wonder why certain people seem to have all the connections and you don't? Want to get ahead and make the right connections to help you in college and in the work force? Come on in and we will explore what to do and why to do it. L3182: Let's Design a Sustainable Satisfying Life Difficulty: Teachers: Tom Kabat Let’s brainstorm to design a satisfying and sustainable life. We'll have group discussion of values, choices and results. We'll explore the intersection of satisfaction, sustainability, consumption and community. People increasingly are viewed as "consumers". Let's explore the balance of many possible roles in our lives and how they can add satisfaction and promote sustainability L3301: Visioning and Executing for Life, Love and Everything in between Difficulty: * Teachers: Shiva Arunachalam Every wondered how to get the most out of life? With so many options out there, how do you decide what you like and what you can succeed in? What this class will offer is a framework for anyone interested in exploring the infinite possibilities life has to offer. L3329: Outdoor Leadership 101 Difficulty: * In this introduction to outdoor leadership we'll talk about group dynamics, community-building, risk management and a quick intro to some outdoor living skills. This class is a good fit for anyone with an interest in the outdoors -- all backgrounds welcome! Prerequisites None! L3134: Money Smart Kids Difficulty: * Teachers: Jitesh Shah Introduction to personal finance basics to help lay the foundation for lifelong learning about money management and its impact on your life. What will be covered - Basics of money, difference between cash and credit, banking and its purpose, checking and savings account, inflation, compound interest and elementary basics of stocks and bonds and its overall influence in our economy. Prerequisites None L3254: Applied Creativity Difficulty: * Teachers: Matthew S Ideas are the force that drives science and technology. They are the spark that begins every new invention, and every new solution to every problem – even the little ones that we have to deal with every day. But creativity is often thought of as restricted to arts and crafts, and its importance in problem solving is underestimated. I will teach you how I systematically approach new situations, with an emphasis on how to come up with new ideas. Be prepared to think outside the box! L3283: College Consortiums - one big college vs. many small interconnected colleges Difficulty: * Teachers: Lenore Byers, Ana Villa What is a consortium? What are the advantages and disadvantages when considering college? We will discuss examples (Claremont Colleges, Five Colleges, etc.) and their differences/similarities between higher education in general and each other. L3155: Leadership/Managerial Skills Difficulty: * Teachers: Melisa Rillera Discuss different leadership and managerial skills you will need not only in your professional career but in many aspects of your life. Talk about how to bring these aspects out of you. We'll go over the skills, what it will take to develop and refine them, and how to apply them to your daily life. L3305: Happiness 101 Difficulty: ** Teachers: Macy Zardeneta Scientists in positive psychology and neuroscience have been focusing on what we are all looking for, happiness. Come learn about their findings and some practical ways in which we can all find happiness right now, exactly where we are. We will discuss daily practices and mindsets in order to create a happier life. Prerequisites An open mind L3154: Why and How to Volunteer Locally and Abroad? Difficulty: * Why should you volunteer local or abroad? How and Where? During this class we will answer these questions and also touch on how to use volunteer experience on college applications and job resumes. We'll provide personal examples of volunteering as co-organizers of our own volunteer group and unique experiences such as volunteering in prisons and various countries abroad. L3123: The Art of Happiness Difficulty: * Teachers: Alfred Delena Ever wonder what it means to be happy? Ever wonder if happiness is something you can truly achieve? Well, fear not, in The Art of Happiness, we will go on a journey together through some of the research that has been done on Happiness and positive psychology. From this, you will learn different practices to express happiness, coping mechanisms to deal with stress and overall, how to create your own authentic source of pure happiness! Note: Presentation will be very interactive and engaging, student responses will be asked. Prerequisites Come with an open mind, a readiness to learn, to engage and to be happier! L3188: Bicycle Maintenance Difficulty: * Teachers: Tom Kabat Let's adjust gears, brakes, and the way a bike fits so your ride improves. We even patch tires, and fix klunky, squeky things. Bring your bike if you can. L3190: How Girls Can Change the World Difficulty: * Teachers: Priyanka Jain An interactive discussion about how the next generation can get involved in solving world problems. We will specifically work with the United Nation's Foundation Girl Up campaign. Girl Up provides a platform for girls like you to channel your energy and compassion to help the world's hardest to reach girls. Learn how to become a leader in your community and get free swag through our interactive workshop! Join us to learn more about how you can make a difference in our world. L3159: Friendship 101 Difficulty: * Has a friend ever come to you in a time of need? Have you ever felt like you didn’t know what to say? Come practice with Stanford Peer Counselors who will help you become a better listener and a more supportive friend! Prerequisites None L3173: Miss CEO: Becoming an Effective Leader Difficulty: * The world needs great leaders to tackle its biggest problems… and that starts with YOU! Although women are underrepresented in today's leadership ranks, this class will inspire and teach you how to position yourself as a leader in high school, college, and beyond. Come learn about relevant leadership skills such as effective negotiation and clear communication that will help you excel in a variety of academic, personal, and professional situations. More importantly you will also learn how to put these skills into practice starting today --including securing dream mentors, finding internships, navigating the college application process, and getting on the right trajectory for career achievement early on. The instructors for this class feature women from the Stanford community who have extensive experience leading and making innovative contributions to their fields. They also have a passion for helping students achieve their leadership potential, which you can learn more about at www.missceo.org. Prerequisites A desire to learn, grow, and discover your inner leader. L3290: Decision Adventure Difficulty: * Teachers: Chris Spetzler Take part in Decision Simulation where student trek in Nepal and face a decision. Learn a framework for decision making. Prerequisites none L3319: Sports and Statistics Difficulty: If you want to learn about your favorite players, from LeBron to Kobe to Messi to Miguel Cabrera, we'll teach you how to compare them in each of their sports through statistics and other analysis. L3289: The Great Debaters: Mastering the Art of Public Speaking Difficulty: Teachers: Louise Lu Have you ever felt shy or nervous when speaking in front of a group? Want to learn how to win an argument (and do it in style)? This class will engage you in the thrill of public speaking and provide you with a taste of competitive debate! We will start off with fun drills to help you become fearless and confident in your public speaking skills, then progress to lively mini-debate rounds with your fellow classmates. It should be an interactive and fun time for all! [Taught by the team of volunteer student coaches from the Stanford Youth Debate Initiative] L3316: Thinking Styles Difficulty: ** Teachers: melinda walker Knowing how you think best can make you a better thinker! Discover the four ways of thinking needed to solve problems and be creative. Knowing your thinking style can not only help you think better, but it can also help you work more effectively in teams. L3071: Make Everyday Thanksgiving: The Science of Happiness Difficulty: * Teachers: Tim Huang Did you know that gratitude and compassion can rewire your brain and behavior for the better? Did you also know that they can help you deal with stress in school, improve your physical health, and transform your social life? In this class, you'll learn about the power of gratitude, mindfulness, and self-compassion to change your life and increase your happiness! We'll look at scientific research from positive psychology and neurobiology, discuss all things happiness, and engage in a few fun exercises that will be useful in your everyday life. By the end of the class, you'll have tangible tools to help improve your academic, personal, and social life. Prerequisites None, just an open mind! L3127: Preparing for College Difficulty: * Teachers: Julie Huang There's more to getting into college than making the grades. Come learn how to prepare for college while in high school. L3302: The 7 Habits of Highly Effective People Difficulty: * Teachers: Jenna Shapiro This class will expose students to an abridged version of Stephen Covey's 7 Habits of Highly Effective People. We will discuss each of the seven habits and take part in interactive activities about them. L3126: Tips and Advice to Making the Most of Your College Experience Difficulty: * Teachers: Julie Huang You're heading to college this fall. Congratulations! But now that you're in, how do you survive and thrive there? I was a first generation college student who learned many useful tips to make the most of my college experience. Come hear about all the things I wish I had known before going to college. First generation college students are especially encouraged to attend! Prerequisites The course is designed to help seniors who will be going to college in the near future. L3144: How to Bullshit Difficulty: Teachers: Benjamin Yang Facts. They are useful. But unfortunately, not always available. Luckily for you, this class is all about how to break your crippling dependency on facts. Come learn all about creating information without worrying about inconsequential things such as "reality", or "truth". Become an expert at being an expert. I know what I'm talking about, and so can you! L3184: Philosophy and Friendship Difficulty: Teachers: Joseph Topasna What is "friendship"? This class will explore historical, philosophical views of friendship from Plato and Aristotle to contemporary philosophers. While surveying historical views of friendship, we will aim at answering the first question and questions like: "Do we have 'true' friends?", "What does it mean to act justly or unjustly towards our friends?", and "Can we live without friends?". Most importantly, each student will be pushed to form a personal view of friendship. Prerequisites There are none! Students are encouraged to come in with their own idea of what friendship is, while being open to new ideas. L3221: The Art of Effective Communication: A Primer on Telling a Good Story Difficulty: * Do you like telling stories? Do you wish you were better at expressing your thoughts and ideas in front of an audience? Whether you're talking to a group of 2 or 200, whether you're a scientist, an engineer, or a politician, getting your audience to quickly understand your ideas and be convinced by your arguments is a very useful superpower to have. We'll cover a broad range of tips and tricks, from how to engage and hold your audience's attention, to how to organize your thoughts and express your ideas clearly. Our class philosophy is: this is a safe environment, with no grades or judgment. We're here to practice and learn something. So, whether you've never spoken in public before or you're looking to hone your skills, come by and let's talk. L3233: Cooking potstickers & a brief intro to The Pale King Difficulty: Teachers: Hilary Noad If you would like to learn how to make chicken-scallion potstickers, or if you would like to learn a bit about David Foster Wallace's unfinished work \emph{The Pale King}, this is the class for you! The literary portion will be suitable for vegetarians, but the food will not. We won't go into much depth in our study of TPK; my goal is simply to spark your interest and maybe even inspire you to read the whole book. L3234: Self-Care Techniques Difficulty: Teachers: Mango Martin Learn to take care of yourself, so that you can feel well. L3280: Get More Ideas! Difficulty: * Teachers: Laura Mappin Design your own conversations to get more ideas and go to new and unfamiliar places. We'll discuss different conversational styles and practice a couple short exercises in class. You'll leave with ideas about how you can design your own to fit your goals. Prerequisites None L3291: Decision Adventure Plus Difficulty: Teachers: Chris Spetzler Students participate in a group project where they are students trekking in Nepal and face a difficult decision. They learn a decision framework to handle the situation and future decisions they face in life. Compared with the shorter version of this course (45 minutes), students will have an additional fun activity that expands their understanding of decision making. Lunch L3336: Lunch Period Difficulty: Teachers: Lunch this year is included in the tuition fee and free for all Splash participants! You must wear your nametag to receive lunch. Prerequisites your Splash nametag L3337: Lunch Period Difficulty: Teachers: Lunch this year is included in the tuition fee and free for all Splash participants! You must wear your nametag to receive lunch. Prerequisites your Splash nametag Mathematics and Computer Science M3049: Rapid Web Prototyping Difficulty: Teachers: komal bhatia You need not know how to code to build a web application prototype quickly. This class teaches 'How to build web prototypes rapidly' for testing an idea with users, before developing the real application. Prerequisites None Recommended (not required) for the students to have a laptop plus internet connection M3082: How does the INTERNET work??? Difficulty: Teachers: Ruchi Bhindwale Have you ever imagined what happens behind the scenes when you go to a web page, say http://www.stanfordesp.org, on your web browser? How does the browser connect to the server (machine that serves the pages) and what protocols does it use to communicate? How does the server locate the page that it needs to serve and in what format does it send it back? Finally what is the process by which the web page is displayed? Join me in this class as we embark upon a fascinating journey through the inner workings of the INTERNET. M3067: Introduction to Python Difficulty: Teachers: Kevin Liang, Eric Xiao More modern programming languages are becoming more and more abstract these days. Python is one of them! In this class I will teach the Python language and also tell you guys why it is a great language to code in! M3093: Sampling, Genetic Algorithms, and Evolution Difficulty: *** Teachers: Gabor Angeli Sampling from a probability distribution is an essential component in tasks such as machine learning or simulation, and is often surprisingly difficult. For example: sampling animals according to how likely they are to survive (the analogy used in the class), sampling dynamic video game scenes according to how realistic they are, or helping a lost robot navigate by sampling over possible locations. The theoretical portion of the class introduces the basics of sampling, and algorithms up through genetic algorithms and Markov Chain Monte Carlo methods. The second half of the class will be a programming project implementing the methods from the first half to make your own learning program! Prerequisites Basic statistics (Bayes Rule, joint/conditional probability, etc.); Basic programming (Python) for the second half. M3095: Intro to SQL Databases Difficulty: * Teachers: Alvin Sng Ever wondered how to store massive amounts of data in an organized and efficient way? The answer is Databases! In this course we will learn the fundamentals of Databases and look at MySQL as an example. We will also learn to install and run a MySQL database. M3158: Signal Processing and Sound Design Difficulty: ** Teachers: Han Altae-Tran Ever wonder why knowledge of $$\hat{f}(s)=\int_{-\infty}^{\infty}dt\cdot e^{-2\pi ist}f(t)$$ tells you how your music sounds without even having to hear it? Find out how the Fourier transform can be used to create electronic music with a computer. We'll use a synthesizer and frequency filter to make some cool sounds. If you can, bring your laptop, and I'll show you how to get started. Prerequisites Calculus could be useful to understand the math, but ultimately, we'll look at the ideas from a conceptual point of view. M3204: I know that you know that... - Game of Common Knowledge Difficulty: * Teachers: Peng Hui How 'The extremely polite monks who lives on THIS farm likes to play tricks on each other. On a lightless night when brothers Alfred and Benjamin were both snoring soundly, brother Craig sneaks in and colored their shaven head blue. Both monks woke up noticing each other's blue head but too polite to tell, too polite to ask if they too, had their head colored blue. The silence broke when brother Dave broke in, blurted out the sentence, "At least one of you have a blue head". Two seconds later, the two monks blushed one after another. (Why? Hint: traverse down both monks' train of thought) Without Dave's innocent remark, neither train of thought would have been set on motion, yet, Daves tell them nothing - apparently - that they do not know already.' This is a classical prologue to a philosophical field, namely common knowledge. In layman account, knowing what one knows is insufficient, knowing what others know, what other knows that one know that ...... In this session, I will present several mathematical games that illustrate this concept. Join this session if you yearn for a mind blogging break after a week full of classes! Credit (Inspiration and exceprt source): Math Hysteria by Ian Stewart Prerequisites Basic Arithmetic Skill M3287: To Infinity and Beyond! Difficulty: Teachers: Jonathan Kang Ever wondered what is the biggest number? That's easy! There's no biggest number! But the notion of infinity is more than meets the eye. In this course, we will attempt to answer questions such as: What do we really mean when we say there are infinitely many natural numbers? How did we arrive at our present understanding of infinity? Are there different kinds of infinities? The infinite has preoccupied mathematicians and philosophers of centuries past. Come learn more about this fascinating topic! Prerequisites Familiarity with algebra, comfort with basic mathematical proofs. M3079: Mad Hatter Mathematics Difficulty: Teachers: Zandra Vinegar There is math. Like no math in school. And proofs full of wonder, mystery, and danger! Some say to survive them, you need to be as mad as a hatter! Prerequisites BYOH* and prepare yourself for logic that doesn't make any sense. Bring your own hat (something with a rim that can support an index card please) M3166: The Lambda Calculus Difficulty: * Teachers: Nelson Elhage This will be a rapid introduction to the Lambda Calculus, a theoretical model of computing originally proposed by Alonzo Church in the 1930s. We will explore how, starting from the barest primitive ideas, you can derive all the comutational power of a modern desktop computer! NOTE: This is not the calculus you've heard about in school! In fact, there are almost no numbers involved, and you don't need to know any math -- just have a curiosity for some slightly abstract reasoning. Prerequisites Some programming experience will be helpful but not at all required. M3216: How to Share Difficulty: * Teachers: Danny Bulmash Never fight over how to share again! Learn some simple (and some not-so-simple) techniques for fairly dividing things up, and learn the logic behind why they work! Then try them out for yourself. M3272: What Do Computer Scientists (Really) Do? Difficulty: * WHat is the difference between a computer scientist and a programmer? What is computer science mostly about? What do computer scientists do? What sort of topics do computer scientists specialize in? Learn about the hottest discipline of the 21st century (i.e. the one with the biggest demand in the job market according to projections of the bureau of labor statistics). Prerequisites None M3286: Math Golf Difficulty: Teachers: Nathan Benjamin Do you enjoy math, but don’t like getting bogged down in tedious algebra? Do you like beautiful results, but hate formal proofs that give no intuition? In this class, we'll go through bunch of seemingly intractable or messy problems and then give short, sweet solutions that you can understand in your head. It's like golf -- the fewest number of moves wins! This class will probably feature your favorite characters, like prisoners with hats, rational dinosaurs, and pirates doing pirate stuff. M3247: Jump into Coding Difficulty: Do you want to learn how to program? Have you heard about "coding" or "programming" before but don't know what it is? If you have no previous programming background, this class is for you. Prerequisites None M3333: Functions and Dimensions Difficulty: * Teachers: Anika Huhn We will build colorful visualizations of shapes and functions in different dimensions. You'll get to take home your very own model of a hypercube! M3151: Proofs About Infinity Difficulty: * Teachers: Elizabeth Yang "Some infinities are bigger than other infinities." - John Green There's something funny about working with infinity. How can some infinities be bigger than others? Concepts that are normally easy, like addition, are quite counterintuitive once we start working with infinity. In this class, we will go over some fundamentals about countability, denseness, and finding/proving bijections! We will also cover Cantor's diagonalization arguments, Hilbert's hotel, and briefly talk about the continuum hypothesis. Prerequisites Some basic exposure to proofs and an open mind! M3153: Crash Course in Graph Theory with an Eye towards Programming Difficulty: * Despite their simple definition (a set of points connected by some number of lines) the study of abstract graphs has far-reaching applications such as mapping, social networks, and routing resources in Computer Science. We'll have a look at some fundamental properties of graphs of interest to computer scientists and some algorithms that help us harness useful information from graphs. Along the way, we'll also have a look at some USACO problems that involve graph theory to help you develop some intuition for the subject. Prerequisites No formal prerequisites, but you'll appreciate the material a lot more if you have taken a basic class in programming. M3205: How Number Theory Secures our World: From Prime Numbers to RSA Difficulty: Teachers: Rishi Bedi It's really hard to factor a really big number into two primes. It seems like a simple idea, but it's the basis for the RSA cryptosystem which is at the core of our internet security infrastructure. Come learn how one of the most commonly-used cryptographic algorithms (and related friends!) work, and how to mathematically prove they're secure. What was once considered an esoteric realm of pure mathematics has profoundly practical implications in the domain of cryptography and encryption; learn how the study of prime numbers forms the basis for public-key encryption as we trace the evolution of number theory from Euclid's Fundamental Theorem of Arithmetic all the way to RSA and beyond. Prerequisites Basic algebra M3258: Coincidence? I think not! Difficulty: Teachers: Lavanya Jose Isn't it funny how after you learn a new word, it begins to show up everywhere? What's the chance that two people in your class were born on the same day? How about three people on the same day? We'll talk about how we can understand coincidences in everyday life using tools from probability (e.g., balls and bins model, poisson processes) and some ideas from psychology. Prerequisites Familiarity with limits, basic statistics, binomial expansion. M3081: How to Cut a CAKE Difficulty: Teachers: Zandra Vinegar If two people have to split a cake, is it "fair" if one cuts and the other gets to choose which piece they want? Does this still work for 3 people? What if 100 pirates need to split up their$1000 of loot? Or if you need to split a \$20 weekly allowance between you and your younger sibling? How should ESP decide who gets into what classes? And, if we publish the lottery algorithm, what makes a system easy or difficult to 'game'? This class will explore the concepts of "fairness" and of "game theory" - using the intersection to discuss practical cases where people care about the result... LIKE WHEN THERE'S CAKE INVOLVED!!! (yes, there will be cake, and it will not be a lie) M3117: Conjecture and Proof Difficulty: * Teachers: Jeremy Booher The number 41 is a sum of two squares (25+16). Can you write 37 as a sum of two squares? How about 43 or 47? To a mathematician, the next obvious step is to find the pattern and make a conjecture. Only once we know what is true is it possible to prove it. We will illustrate how mathematical research is done by finding an answer to the question of which numbers are a sum of two primes and then proving it using the arithmetic of the Gaussian integers. Prerequisites Familiarity with the complex numbers and the desire to talk about theoretical mathematics and proofs. M3135: Introduction to Artificial Intelligence Difficulty: *** Teachers: Peter Pham We will briefly cover the history of AI before discussing a variety of algorithms that are representative of major paradigms. The objective of our session will not be to understand the details of particular algorithms, but rather to get a feel for how one can approach various problems in AI. Prerequisites A good mathematical background and/or strong mathematical intuition would be helpful. M3143: The Music of the Spheres Difficulty: * Teachers: Daniel Litt Why does the sun orbit the earth in a perfect circle? How do we know that the universe is shaped like a luggage trunk, with the rectangular earth resting askew at its base? Why are there exactly six planets? And how can we hear the ethereal hum of the great crystal spheres on which the planets agglomerate? We'll discover the sublime and hilarious history of cosmology, from Pythagorus to Kepler (600BC-1600AD, approximately), and learn some of the beautiful mathematics underlying their insane theories of the universe. We'll also talk a bit about the personalities and aesthetics which motivated science before the scientific revolution. This will be a weird and dangerous mix of history, math, and philosophy. M3243: Building iPhone Apps Difficulty: * Teachers: Nick Troccoli Ever wondered how the apps on an iPhone work? Ever wanted to go behind the scenes and make your own? We’ll take a look at how an iPhone app is created, from designing the interface to writing the code. Get a taste of what real developers do on a daily basis, and how you can make your own apps! Prerequisites Core programming knowledge, including functions and variables (knowledge of object-oriented programming, including classes and methods, recommended). M3147: Into the fourth dimension Difficulty: Teachers: Grant Sanderson Every wondered what exactly people mean by the "fourth dimension"? Have you ever been skeptical of those who say it's impossible to visualize anything beyond three dimensions? Bollox! I say it absolutely is possible for humans to visualize higher dimensions, and while I believe it takes longer than one short class to do so, I can at least explain how you might start. We will first go over the basics of what the fourth dimensions is, then I will devote most of the class time to going through a series of visualizations for various four dimensional shapes and figures. If you enjoy any one of math, philosophy, pretty pictures, or having your mind twisted in a knot, this class is for you. Prerequisites None! M3080: Introduction to Programing a la Fractal Forgeries Difficulty: Teachers: Zandra Vinegar Want to learn how to program a cloud? or a rough, and unpredictable mountain? or an infinitely precisely shaded fern? Then sign up for this class and I will BOTH introduce you to JavaScript, a powerful visual programming tool, and show you around the psychedelic world of Fractals! Check out some of these images and see if you can tell which are real and which are mathematically-generated forgeries: http://tinyurl.com/8erkfxy Those which are forgeries are made using Fractals: mathematical objects which are produced by repeating very simple instructions over and over again. You'd never want to draw these images by hand, but with the aid of computers, we can plot hundreds of thousands of points in seconds. This ability enables us to decode natural objects which the "smooth" curves and platonic solids you learn about in high school can never emulate. M3152: Problem Solving with Computer Algorithms Difficulty: Teachers: Kapil Yedidi Computer algorithms are behind many of the things we do in our day to day life. Whether it’s finding mutual friends on Facebook, using the iPhone auto-correct, or searching for something on Google, computer algorithms enable us to do things that we otherwise could not. We’ll also cover the logic behind these algorithms and others in the products we use every day. We’ll discuss things that computer are really good at (like sorting a list of 1,000,000 numbers in less than a second), and things that computers are really bad at (like figuring out the optimal way to organize student’s schedules at a school). We’ll spend the first half of class understanding ways to break down problems like a Computer Scientist. In the second half of class, students will begin devise algorithms on paper to solve real-world problems. Much like the study of Computer Science in college, this course focuses on problem solving over writing code. No programming experience is needed for this course. M3199: Knot Theory: da Vinci & Math Difficulty: * Teachers: David Hyde Knot theory is an exciting branch of topology, the mathematical study of shape. With knot theory, we can analyze the very rich structures and patterns hidden within knots. We can also make fascinating discoveries - for example, did you know that knots appear in the Mona Lisa, The Last Supper, and many other of Leonardo da Vinci's works? This Splash class looks at knot theory from both mathematical and artistic perspectives and should be a lot of fun for anyone who likes math, puzzles, or art! Prerequisites Interest in puzzles, art, math, or some combination thereof! M3107: Uncool Computing: The SQL Difficulty: * Teachers: Ruth Byers ... where SQL ("Structured Query Language" if you must) is pronounced "sequel." Learn about SQL injection, a tactic for getting unauthorized data (read: "credit card numbers and passwords") through standard user interfaces (read: "by typing stuff into web forms"). We'll talk about the technical details of how the exploit works and how websites defend against it. Prerequisites You should have used some sort of computer programming language before, even briefly. M3145: Paradoxes of High Dimensional Statistics Difficulty: * We will discuss what has come to be known as "the curse of dimensionality" - - a major challenge of modern data analysis important for making predictions in health care, biology, economics, and even presidential elections. Paradoxically, sometimes more data can lead to illusions of structure, and erroneous not-so-scientific findings. We will discuss some of these paradoxes and how mathematicians and scientists solve high-dimensional problems. Prerequisites Calculus M3303: Polyhedra Flip-books Difficulty: Teachers: Andrea Hawksley Learn to make your own math-y animations by drawing polyhedra flip-books demonstrating the relationships between the platonic solids. M3058: Node.js Making websites live Difficulty: Teachers: Alvin Sng In this course, we learn to make interactive websites with Node.js, we will build a working chat website so if you have a laptop handy it can be used in demos. I will also show how you can easily upload your web app using Heroku. M3165: What is Infinity? Full! Difficulty: Teachers: Nelson Elhage What is infinity? Is "infinity" a number? Does \frac{1}{0} = \infty? What does it mean when a mathematician tells you that some infinities are bigger than others? We'll explore what mathematicians mean by "infinity", talk about different sizes of infinities and how to compare them, and more! Prerequisites Some familiarity with abstract mathematical reasoning M3192: Game Theory and Mechanism Design Difficulty: Teachers: Jeff Jacobs Come learn the fundamentals of Game Theory and Mechanism Design, the mathematics of strategy, by formally analyzing problems that occur in our day-to-day lives! Urban Dictionary defines the "Sidewalk Shuffle" as "the awkward dance that results when two people approach each other from opposite directions, each attempts to move out of the other's way, and both end up moving in the same direction. Often followed by further ungainly movement, apologies, and awkward laughter." Has this ever happened to you? Was it awful? Learn Game Theory and never have to experience it again! Now, imagine you have a giant golden hippopotamus statue taking up space in your room that you want to get rid of. You ask all your friends how much they'd pay you for it, then the highest bidder gets it in exchange for their bid, right? WRONG! Learn Mechanism Design and find out how to make even more money from your friends in this situation. Prerequisites High School Algebra 1 M3241: Practical Machine Learning Difficulty: ** Can machines learn? Will they ever achieve a level of sentience that rivals that of humans? These are great questions that we will not answer in this class. Instead, we'll layout some of the foundations for classic machine learning techniques. Starting with the maximum likelihood approach, we will cover topics such as binary classification, regression, or fitting to a mixture of gaussians, and will show you how to derive their update rules. We'll end with real world examples, potentially in biology. Prerequisites Calculus M3284: Hamming Codes, Error Correction, and Seven Questions with a Liar Difficulty: ** Teachers: Andrew Guo Quick, think of a number between 0-15! Done? Okay, now I’ll ask you few questions. To each question, you must answer "yes" or "no," but you are allowed to lie at most once in your answers. Ready? Here are the questions: 1) Is your number greater than or equal to eight? 2) Is your number one of 4,5,6,7,12,13,14,15? 3) Is your number one of 2,3,6,7,10,11,14,15? 4) Is your number odd? 5) Is your number one of 1,2,4,7,9,10,12,15? 6) Is your number one of 1,2,5,6,8,11,12,15? 7) Is your number one of 1,3,4,6,8,10,13,15? From your answers, I claim that I can determine what number you chose, and whether you lied or not. If you want to watch me try to read your mind - or find out my secret - come to this class! Our strategy will be derived from the mathematical field of "error correction"; we'll start out learning about the Hamming code, and how it helps people send messages through noisy channels without fear of information loss. Prerequisites Be able to understand this joke: "There are 10 types of people in this world: those who know binary, and those who don't." M3261: Tessellation: Making Cool Patterns Difficulty: * Teachers: Katie Dektar We will learn how to create tessellations, tiling the two-dimensional plane with a repeating pattern. Students will be introduced to basic tessellation techniques and a tiny bit of geometry, and get a chance to make their own. Think: a soccer ball or M.C. Escher, as in this photo: http://uploads7.wikipaintings.org/images/m-c-escher/lizard-1.jpg M3288: Permutations and Combinations Difficulty: Teachers: Vivek Kaul A brief introduction to the interesting topic of permutations and combinations. We will cover problems on fundamental principles of counting, permutations and combinations M3334: Introduction to Computer Programming Difficulty: Teachers: Anika Huhn This hands-on class is intended for people who want to start writing code (and learn how to use it) but have not been exposed to it before. I will start from the very beginning, and by the end of the class you will have seen enough to know how to start learning more on your own. I will also give you some direction as to what might be fun projects to take on. If you want to get yourself set up on a laptop, bring the laptop in. M3057: Intro to web design, HTML, CSS & JavaScript Difficulty: * Teachers: Alvin Sng In this class I will cover the basics of web design including learning HTML, CSS, & JavaScript. We will learn to make a website from scratch and also be uploading the site with the use of Heroku. M3133: The Wizardry Behind Modern Cryptography Difficulty: ** Teachers: David Wu Is it possible to perform computations on data without knowing what that data is? Is it possible to share a secret with 100 people in a way that any group of 10 or more people can learn the secret but any group with less than 10 people can not? Is it possible to vote in an election without revealing your vote to anyone? The answer to all of the above questions is YES. While on the surface, such protocols seem almost magical, they are all made possible by clever mathematical tricks, some of which are quite simple to grasp! In this course, we will pull back the curtains and examine some of the mathematical tricks that enable the wizardry behind modern cryptography. Prerequisites Comfort with basic mathematics (e.g. basic algebra) and abstract ideas. M3136: Cloud Computing Difficulty: * Teachers: Vaishali Deshpande Do you want to learn the hot topic in the computer industry? Join me to learn following: 1. Evolution of Computer Industry 2. Different technologies such as Virtualization in Cloud Computing 3. What is Cloud Computing? 4. Different types- Public, Private, Hybrid and Community 5. Cloud service models - Iaas, PaaS, Saas and more 6. Benefits of Cloud Computing 7. Challenges in the Cloud - Security, Multi Tenancy 8. Future of Cloud Prerequisites Knowledge of computer industry M3246: Introduction to Machine Learning: or, how I learned to stop worrying and let the machine teach itself Difficulty: * The field of Machine Learning is getting a lot of attention, and indeed, it's pretty cool. It's a field in which computers can actually teach themselves to do things that not even their programmers are capable of. This is your opportunity to learn about what Machine Learning is, where it came from, and form an introductory understanding of how it works. We'll go through several examples, run some demos, and, for those interested in programming, we'll discuss implementation. M3332: Google Search and Google Translate: How Humans (Accidentally) Teach Computers Difficulty: * Teachers: Aaron Kalb Suppose you have billions of books, and someone asks you for all the pages containing the word "eggplant", sorted by the quality of the books according to the world's eggplant experts. How long would it take you to deliver the results? If you answered longer than 300 milliseconds, then you have something to learn from Google. It takes a human longer than that to even read, let alone answer, such a question. In that time, Google actually fulfills thousands of such requests. Their process is brilliant: conceptually simple but technically complex, and in this fast paced tour, we'll zoom through the basics of both, covering: - PageRank - Inverted Indices - TFIDF - Text alignment - Language modeling and more... We'll end by discussing some limitations of the Google approach to Search and Translation and hinting at next steps in both fields... Difficulty: * Teachers: Alvin Sng Come and learn to make fun and awesome Facebook apps. In this class we will go over the various Facebook APIs and learn how you can make a web app that uses the large social network. Prerequisites Programing experience recommended. M3167: How to build a Grandmaster: An introduction to game AI Difficulty: Teachers: Nelson Elhage Have you ever wondered how computers are able to play board games like Chess? It turns out that the way humans play these games is very different from how we know how to make computers do so. We'll look at the minimax technique for building AIs for Chess or other games, and we'll discuss some of the tricks you need to do to make it perform on a truly grandmaster level. Physical Science P3089: The Birth of Science Difficulty: Teachers: David Carreon Where did science come from? Who were the heroes who invented it? Why wasn't it invented sooner? In this class, we'll talk about the exciting story of how we went from alchemy to chemistry, from astrology to astronomy. We'll talk about the epic debate about whether it was the earth or the sun in the center of the universe, discussing the discoveries and politics involved. You'll hear about popes and kings, monks and bishops, and even a few professors. If that's not enough, I'll tell you: Why practically nobody thought the earth was flat How Galileo accomplished the greatest troll of all time Why it's really hard to prove that the earth revolves around the sun (without satellites) [Warning: Not just a history class! We'll actually try to think scientifically] P3140: Identifying Rocks Difficulty: * Ever wonder what cooled lava looks like or how rocks from deep in the earth look like? Do you want to see the different rock types that come out of a volcano? In this class we will explore the different rock types, where they are found, and how to identify them in different places you see everyday. Hands on activities will show you how geologists look at rocks to interpret them to gain important information about our earth! Understanding what rocks are composed of and how they behaved in the past gives us insights to how they might behave in the future. Codevelopers: Elizabeth Morin, Meredith Townsend, Natalie Sievers P3351: CO2 and You Difficulty: ** Teachers: Antonio Baclig As an American, you are responsible for emitting 20 tons of CO2 every year (averaged over the population). Where do CO2 emissions come from and what can we do, as individuals and society, to reduce the greenhouse gas production that is warming our planet? This course will dive into everyday decisions (how does turning off the lights compare to driving an extra mile in your car?) as well as present a wider view of energy systems and infrastructure (why are coal plants still producing a significant portion of our electricity?). Connecting it all will be a focus on understanding physical mechanisms and quantifying energy use and CO2 emissions to give you the tools to weigh alternatives rationally. P3077: Sports / Dribble Like Pro Difficulty: * Teachers: Kenneth Huo We will be gathering in the courtyard and practice dribble, passing and some fun and basic basketball skills! Prerequisites n/a P3189: The Science of Jurassic Park Difficulty: Teachers: Samantha Zarate Why is a Brontosaurus simply wrong? What's so special about the La Brea Tar Pits? Are organisms that have been dead for millions of years really that interesting? The real question: Who hasn't always wanted to know more about dinosaurs and fossils? Come and fulfill your childhood dream of learning the basics of paleontology, mistakes Steven Spielberg made in Jurassic Park, and more! Why? Because dinosaurs are cool. Prerequisites None - you don't even need to have seen Jurassic Park; this is just a basic paleontology class! P3277: Lightbulbs & Luminol: What Lights Them Up? Difficulty: Have you ever wondered what the difference is between a fluorescent and incandescent light bulb? Want to learn about how luminol (used in CSI to find blood), glow sticks and a firefly’s glow work? If you have taken one year of chemistry and you’d like to know, this is the class for you. We will explain incadenscence (incadenscent light bulb), fluorescence (fluorescent light bulb) and chemiluminescence (firefly, glowstick, luminol) and do a hands on activity with glow sticks. Prerequisites High school chemistry (1 year) P3083: Magic Sand Difficulty: * Teachers: Austin Wardrip Hydrophobic “magic” sand is special sand that has been coated with a silicon compound, making it able to repel water. Because the silicon is only one nanometer thick, hydrophobic sand looks and feels like regular sand. However, it behaves very differently. P3106: Colorful Polymers Difficulty: Teachers: Junzhe Lou Have you heard about polymers? Do you know their applications in our society? And can you imagine how closely they are connected with our daily life? In this class, we'll talk about the secrets of polymers including the history, structures, properties, and especially applications in our daily life. We will also introduce the advanced polymer mateirals that contribute greatly to new technologies. You will understand how important and magic polymers are. Prerequisites A basic background of chemistry is helpful but not required. P3185: Exploration Geophysics: Treasure-Hunting with Physics Difficulty: Teachers: Emily Fay Learn how geophysics combines principles from physics and geology to explore and understand the Earth. This class will look at different geophysical techniques and how they can be used to find oil and gas, mineral resources, groundwater, caves, and even buried treasure! P3086: Real Science: What It Is and How to Spot BS Difficulty: ** Teachers: David Carreon Lots of people talk about what “Scientists say” and what is “experimentally proven”. We all want to be rational and make good decisions. But what is science, really? And how do you know when someone is just trying to BS you by throwing around the word “science”? We’ll talk about that, and, like real scientists, do some real experiments. P3181: Paint: A Complex Chemical Compound Difficulty: * Teachers: Michelle LaComb Cavemen! Egyptians! Romans! Forgeries! Learn about all of these and more by studying the science of paint. Paintings are more than pretty objects hanging from the wall; they are historically-linked results of the study of a complex chemical interaction between various pigments and binders. This class will give a brief introduction to the history and science behind making paints and pigments, and how you can use information about the historical use of pigments to catch forgers. P3094: Molecular Gastronomy: The Chemistry of Cooking Difficulty: We’re surrounded by chemistry each and every day but some instances are more obvious than others. Most people recognize that their medicine is the product of chemistry; far fewer would say the same about their lunch. But they should. The flavor of their grilled chicken is the result of complex browning reactions, their salad dressing an emulsion of immiscible liquids, and the texture of their ice cream governed by thermodynamic principles. In this class, you’ll learn how to view food and cooking from the standpoint of a chemist and hopefully you come to see why chemistry is useful (and pretty cool!) Prerequisites High school chemistry P3141: Identifying Minerals Difficulty: Have you seen fluorescent minerals before? What about a mineral reacting with acid? This hands-on class will show you those things and more! Learn how geologists identify minerals out in the field to understand a rocks history. Group challenges will teach you the different techniques used to tell minerals apart and why they are important when looking at different rocks. Fool's gold, also known as iron sulfide (pyrite), has led to many a "gold" rush. Learn about mineral properties and don't get fooled. P3177: History and Future of Manned Spaceflight Difficulty: * This course goes starts from the beginnings in the 60s, goes through the space race, the construction of the international space station, and talks about the future of spaceflight. Topics include propulsion systems, space habitation, and the terraforming (turning into an Earth-like planet) of Mars. P3239: Muon in a Jar Difficulty: ** Teachers: Spencer Gessner In this class we build a particle detector using stuff you can find around your house. Our particle detectors can see subatomic particles called muons coming from outer space! P3253: Baking Powder Submarines Difficulty: * Teachers: Gretchen Lam Have you ever wondered how submarines work? Learn about what buoyancy has to do with it and make your own submarine with baking powder and other common household materials. P3279: Coral reefs: oases of the ocean Difficulty: What are coral reefs? Why are they so important? How do they support such high biodiversity? How do the physics, chemistry and biology all interact to form a complex functioning ecosystem? What are the main threats to coral reefs worldwide and what can we do better protect these fragile, beautiful ecosystems? These are some of the questions we will explore using field examples from around the world and seek to learn how we can better protect them in the future. P3308: The Music and the Math Difficulty: Teachers: Timothy Anderson Exploring some of the mathematical and physical concepts behind musical pitch, harmony, and instruments. In the first part we'll look at some of the math and physics of waves and use these concepts to explain the overtone series and its applications in composition and instrument technique. In the second part will discuss musical harmony (dissonance and consonance) from the perspective of mathematics and explore its role in music history. Prerequisites Knowledge of math through precalculus (trigonometry and second year algebra) necessary for most of the material. Ability to read music and basic physics knowledge preferred. P3265: The Higgs Boson: Understanding the Nobel Prize Full! Difficulty: ** Teachers: Kiel Howe The discovery of the Higgs Boson is one of the most important advances in particle physics in the last fifty years! The physicists responsible for predicting this particle were just awarded this year's Nobel Prize in physics. We will discuss what it means to "discover" a "fundamental particle", the amazing machine that made the discovery of the Higgs Boson possible, and what this discovery means for the future of particle physics. Prerequisites The jumping off point for this class is that you've heard or read the words "atom", "nucleus", and "electron", and have a rough idea what they might mean. The focus of the class will not be mathematical. P3270: Introduction to Quantum Mechanics 1 Difficulty: * Teachers: shruti puri “Anyone who is not shocked by quantum theory does not understand it"-Neils Bohr Come let’s explore the fascinating world of quantum physics, where, truth is stranger than fiction. I will use very little math to explain important basic concepts in quantum mechanics. In the first part of the lecture, I will talk about the basic concepts like wave-particle duality, uncertainty principle, energy levels etc. Prerequisites Wave nature of light, Classical Mechanics (Newton's laws) P3132: The Hydrogen & Oxygen Games Difficulty: ** Teachers: Stefanie Morgan Think you know everything about hydrogen and oxygen? Come find out! We'll do lots of interactive, hands-on experiments to learn about how these two simple elements both influence our daily lives and interact with other basic elements! Prerequisites Some knowledge of basic chemistry (understand what chemical equations are and more or less how to read them). P3271: Introduction to Quantum Mechanics 2 Difficulty: * Teachers: shruti puri “Anyone who is not shocked by quantum theory does not understand it"-Neils Bohr Come let’s explore the fascinating world of quantum physics, where, truth is stranger than fiction. I will use very little math to explain important basic concepts in quantum mechanics. In the second section of this class I will cover entanglement, raising the Schrodinger cat states in the lab (what the Nobel prize was about), teleportation etc. Prerequisites Basic postulates of Quantum Mechanics. P3274: Why Clocks Run Faster on Mount Everest : An Introduction to General Relativity Difficulty: Teachers: Alfred Zong If you stay in the Bay area while your friend is at the top of Mount Everest, your friend's watch is going to run faster than yours! This is not due to manufacturing defect, nor due to special relativity (you and your friend are not moving relative to each other). The truth is the Earth's gravity "slows down your watch". If you are curious to find out why, come to this class! Don't be daunted by the word "General Relativity": no equation is involved to convince you of this weird fact! Prerequisites No advanced math or physics knowledge required. If you know F = ma and you're ready to embrace some really weird (but true) phenomenon, this class is for you! P3307: Fun Physics Brain Teasers Difficulty: Teachers: Joshua Yoon You ever wonder why scientists do physics for fun? You can find out in this class! We'll be going over some classic physics brainteasers that have been around for ages and then move on to problems which require a little bit more creativity. So come join me in learning about the joys of solving physics problems that don't require so much of the math, but more of your intuition. So, just bring your heads and get ready for an awesome time! Here's are two example problems that we'll be going over in class, just to give you an idea on what to expect from this class: (1) A dinghy, with a passenger and an anchor, floats in a small pool. The passenger throws the anchor into the water. Will the water level in the pool increase, decrease, or stay the same? (2) Are there more grains of salt in a salt mound than there are atoms in a single grain of salt? Prerequisites A good background in algebra and geometry should suffice. Some knowledge in physics would help, but is not necessary. This class aims to focus mostly on conceptual-based material. P3096: Elephant Toothpaste Full! Difficulty: ** This class will introduce students to the role of catalysts in chemical reactions. We will identify everyday examples of catalysts and discuss their significance. The class will include various demonstrations and hands-on activities. Prerequisites A basic introduction to chemical reactions. P3087: Stars and their Physical Properties Difficulty: * Teachers: Isabel Baransky Stars are more than pretty objects in the sky-- they are beautifully dynamic, evolving, gigantic chunks of mass that have created every element imaginable, including you! Together, we will explore the birth of stars, their life cycles, and their deaths. Come learn about the amazing phenomenon that is a black hole, and how to rip a hole in space time. Understand what exactly a "supernova" is, and why it's so important. Explore degeneracy and its immense power. And just have a good time! Prerequisites A basic background in Physics is helpful but not required. Fundamentals will be covered. P3105: Fun with Chemistry! Difficulty: * Chemistry is exciting and it happens all around us every day. In this class we will talk about the states of matter, a little about polymers (like plastics), and really anything else that gives us a chance to do cool demos for you all! You may get a chance to make a chemical souvenir to take home, too. You'll just have to come and find out! P3183: Black holes! And space, time, and all that Difficulty: *** Take a deep breath and start thinking about what our world must be built of. How to measure distances in a way that doesn't depend on the choice of a ruler? What's "distance," anyway? Why is time different from space? What kinds of universes can we think of, and which one of those do we live in? These questions will kick off and propel our discussion until we reach the lofty goal: understand what black holes are, how they behave, and why they are crucially important for our understanding of the world. Prerequisites Algebra, algebra, algebra. There will not be many equations, and those that do show up will not be significantly manipulated. However, understanding what they mean will be important. Having some notion about vectors and matrices will be especially useful. P3237: The real face of Superman - Superhero physics Difficulty: Teachers: Jean-Baptiste Boin Many of you are familiar with the adventures of the Man of Steel and his other superhero friends. You know how Superman can lift cars and Spiderman can climb up walls. But this is only the visible side of the story. In this class, we will lead an investigation to know how these superheroes can have such powers and what it tells us about them. And how could we do this better than by using the superpowers we all have access to? Welcome to the world of physics! P3320: The Lab-Manual of Sherlock Holmes Difficulty: Teachers: Victoria Chang Sherlock Holmes was a man of many talents. Aside from being a consummate detective, he was also a skilled chemist. In A Study in Scarlet, Dr. Watson calls his knowledge of chemistry “profound”, and in 2002, the Royal Society of Chemistry awarded him an honorary fellowship. Students will learn about some of the important discoveries made by chemists of the Holmes era (late 1800’s to early 1900’s), as well as some important basic chemical concepts. This class contains lecture and demonstration. Prerequisites Some chemistry background preferred, but not required. P3097: Quantum Information Theory Difficulty: *** Teachers: Alfred Zong Have you heard of quantum computers? Have you wondered how they work? In this class, you will be introduced to the theory of quantum information and its application to quantum computation! In case you’re wondering why quantum computers are so cool, think about this: if you have a set of $$N$$ numbers, a quantum computer can tell whether a given number $$x$$ is in the set or not without looking at all $$N$$ numbers, i.e. the complexity is less than $$O(N)$$! Come to the class if you want to know the magic behind the scene! Prerequisites Required: love physics and math, familiarity with mechanics (e.g. $$F=ma$$), vector, trigonometry, logarithm, calculus, matrix multiplication, linear operator/transformation, eigenvalue/eigenvector. Optional: PDE, information theory. Not required: quantum mechanics (we’ll learn the essentials during the class!) P3148: Make a hologram Difficulty: Have you ever wondered how 3D images can be captured on a flat film? In this class you will not only learn something about light and the basic underlying principles of holograms, but you will actually get to create a small hologram of an object of your choice. The only restrictions are: It should be about matchbox sized, not be green and have interesting 3D features. P3170: Coral Reefs: Exploring Real Experimental Problems in Marine Science Difficulty: Are you interested in the ocean, the environment or science in general? Staci is studying the oldest corals in the world and has a problem. She needs to develop a way to accurately measure the surface area of coral to determine how fast the colony grows. In teams, you will measure objects surface areas using conventional techniques and figure out for yourself why measuring coral is challenging. Finally, you will learn why this is an important technique and ways marine researchers currently determine surface area, including a demo. P3195: Dreaming in Color: The Science of Light and Matter Difficulty: Why is the sky blue? Why can’t you see your cell phone screen while wearing sunglasses? How can you tell the color of dinosaurs from fossils? How can a peacock disguise itself by taking a bath? How can you make something black just by making it rougher? When is gold green, blue, or purple? In this interactive class, you’ll discover the answers to questions like these with demonstrations and hands-on activities. You’ll also find out how the answers to these questions are helping scientists and engineers discover more about outer space, create more efficient solar cells, and treat cancer. Prerequisites Basic optics helpful but not necessary. Bring your own polarized sunglasses if you have them. P3339: Cosmic Distances: How far away are the farthest things? Difficulty: Teachers: Tony Li How far is the farthest thing you can imagine? The nearest star is over 24 trillion miles away. The Milky Way is over one hundred thousand light years across. The nearest galaxy like our own is 770 thousand parsecs away. Sure, these numbers are big, but what do all they really mean and how can we start to get a feel for them? More importantly, how do we even know them in the first place (without an impossibly long ruler)? And how far is a light year, anyway? In this class we'll learn the answers to these questions and more. We'll see just how astronomical distances are actually measured. And we'll start to get a feel for really, hugely, ridiculously big distances. A famous writer once said, "The universe is a big place, perhaps the biggest." Let's try and find out just how big it is. P3048: Talking to Aliens Difficulty: Teachers: Michael Busch The search for extraterrestrial intelligence (SETI) currently focuses on finding any intelligent aliens. But how could we communicate with any aliens that may exist? This class will review what we can currently say about where aliens may be. Then we'll move into investigating examples of the various schemes humans have invented for talking to aliens, and by using the students as substitutes for aliens we'll figure out which ones are the most likely to be understood. Prerequisites Interest in astronomy and/or planetary science, and liking to solve puzzles. P3070: Maxwell's Equations Difficulty: * Teachers: Zandra Vinegar $$\varepsilon \varoiint \mathbf E \cdot ds = \iiint \mathbf q_\mathbf v dv$$ $$\oint \mathbf B \cdot dl = \mathbf I + \varepsilon \frac{d}{dt} \iint \mathbf E \cdot ds$$ $$\oint \mathbf E \cdot dl = - \mu \frac{d}{dt} \iint \mathbf B \cdot ds$$ $$\mu \varoiint \mathbf B \cdot ds = 0$$ These four equations describe one of the most universal and elegant relations in physics. They are Maxwell’s equations, unifying all observations of relativity, electricity, and magnetism. Don’t let the notation scare you off – this class has no prerequisites (as in, just be able to graph a function), but we will rigorously derive Maxwell’s explanation of electromagnetic phenomena (including light, electricity, magnets, …). “Derive” with the catch that, as I don’t believe in writing long equations on the board, everything in this class will be presented as a series of intuitive /and/ rigorous deductions, preserving concepts rather than constants. We will begin with only two observations. First, the relativistic nature of light: you can’t catch up to a light beam – it will always move away from you at speed c. Second, our observations of the force between two charges. From these two observations, we will DERIVE the explanation of everything else. Aka, the world will unfold before you and it will be beautiful. Prerequisites All this said, and there being no “hidden prerequisites,” the world will need to unfold before you /very/ quickly. I basically just claimed that I would introduce all of single-variable calculus and about half of multivariable calculus in the first hour of class – which I believe is an attainable goal – but this class will be rigorous, will be extremely intense, and will require the full two hours. P3109: Introduction to Time Difficulty: Teachers: Yao-Yuan Mao What is time? Time seems to be a simple concept that we use every single second. (The concept of time even appears in my previous sentence.) Yet, what REALLY is time? What does it mean when people say time is a coordinate? What does it mean when Einstein said time is relative? There are still questions about time which are even more puzzling. Why does time flow? Or does it flow? Why can't we travel back in time? Or could we? So do you want to learn more about time? P3186: The Physics of Star Trek Difficulty: Teachers: Rachel Reddick Warp drive. Force fields. Time travel. Aliens. How do they work? (Or not work...) Not strictly limited to Star Trek, I'll be doing an interactive discussion with students about what science fiction what is and is not possible in reality. Exact subject matter covered will be guided by student interest. Prerequisites Some background in physics is helpful, but not required. P3276: The Geometry of Relativity Difficulty: ** Teachers: Robert Moffatt In this class, we will discuss the geometric view of the Special and General Theories of Relativity. We will discuss some famous results and paradoxes of these two theories. Concepts will be presented in an intuitive and visual format, and students will learn how to use geometry to visualize relativistic effects and to perform relativistic calculations. Prerequisites Knowledge of Algebra and Trigonometry will be helpful for understanding some of the concepts discussed in this class. P3142: CO2 and You Difficulty: * Teachers: Antonio Baclig As an American, you are responsible for emitting 20 tons of CO2 every year (averaged over the population). Where do CO2 emissions come from and what can we do, as individuals and society, to reduce the greenhouse gas production that is warming our planet? This course will dive into everyday decisions (how does turning off the lights compare to driving an extra mile in your car?) as well as present a wider view of energy systems and infrastructure (why are coal plants still producing a significant portion of our electricity?). Connecting it all will be a focus on understanding physical mechanisms and quantifying energy use and CO2 emissions to give you the tools to weigh alternatives rationally.	2023-02-08 00:14:09	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 4, "x-ck12": 0, "texerror": 0, "math_score": 0.21087583899497986, "perplexity": 2206.201418763144}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500664.85/warc/CC-MAIN-20230207233330-20230208023330-00855.warc.gz"}
http://dennis-scheiba.com/code/10_sc_grids.html	# SuperCollider Grids¶ Git repository is hosted at GitHub. A topographic drum sequencer for SuperCollider based on the original module by Mutable Instruments. ## Notes on the port¶ A quick remark on how the original source code was ported. When looking through the source code of the original module we will find two interesting files: pattern_generator.cc and resources.cc. ### The drum maps¶ In resources.cc we will find 25 maps which are used to generate the patterns so now it boils down to understand how those maps are used to generate patterns. ### Generating drum patterns logic¶ This is handled in pattern_generator.cc. At the beginning the 25 drum maps are arranged into a 5 by 5 array and stored in the variable drum_map. Keep in mind that these values are in a rather arbitrary order which we need to take into account later. Now lets start by taking a look at the ReadDrumMap method. uint8_t PatternGenerator::ReadDrumMap( uint8_t step, uint8_t instrument, uint8_t x, uint8_t y) { uint8_t i = x >> 6; uint8_t j = y >> 6; const prog_uint8_t* a_map = drum_map[i][j]; const prog_uint8_t* b_map = drum_map[i + 1][j]; const prog_uint8_t* c_map = drum_map[i][j + 1]; const prog_uint8_t* d_map = drum_map[i + 1][j + 1]; uint8_t offset = (instrument * kStepsPerPattern) + step; uint8_t a = pgm_read_byte(a_map + offset); uint8_t b = pgm_read_byte(b_map + offset); uint8_t c = pgm_read_byte(c_map + offset); uint8_t d = pgm_read_byte(d_map + offset); return U8Mix(U8Mix(a, b, x << 2), U8Mix(c, d, x << 2), y << 2); } Some remarks what happens here: x>>6 is a left bit shift of 6 on x. What exactly means this? Lets say we have the number 183 which has the unsigned byte representation of 1 0 1 1 0 1 1 1 Doing 6 bit shifts to the right yields 0 0 0 0 0 0 1 0 which is the binary representation of 2. Thinking a bit about this yields that >>6 is the equivalent of doing an integer division by 64 which stays in range 0 to 3 for the values of an unsigned 8 byte integer which ranges from 0 to 255. We do the same for the variable y and use these both coordinates of to look up 4 drum maps. Now it makes sense that we are limiting x and y between 0 and 3 as we also accessing the adjacent maps in drum_map which is a 5 by 5 array. After this an offset is calculated uint8_t offset = (instrument * kStepsPerPattern) + step; which tells us how the maps itself are structured. As each maps contains rows of 96 integers we can assume that they are stored as 32 beats in a row and after these 32 beats the beats for the next instrument starts. After this section the 4 integers are read from the map and we come to the line U8Mix(U8Mix(a, b, x << 2), U8Mix(c, d, x << 2), y << 2); which is a bit tricky to understand. #### U8Mix¶ By searching the repo we can not find this function so it must be included somewhere else. By taking a look at the used dependencies (the linked git repositories) we can indeed find the source code for this function which turns out to be in assembly static inline uint8_t U8Mix(uint8_t a, uint8_t b, uint8_t balance) { Word sum; asm( "mul %3, %2" "\n\t" // b * balance "movw %A0, r0" "\n\t" // to sum "com %2" "\n\t" // 255 - balance "mul %1, %2" "\n\t" // a * (255 - balance) "com %2" "\n\t" // reset balance to its previous value "eor r1, r1" "\n\t" // reset r1 after multiplication : "&=r" (sum) : "a" (a), "a" (balance), "a" (b) ); return sum.bytes[1]; } After some assembly research and thanks to the comments it turns out it is the same as $\text{balance} * b + (255 - \text{balance}) * a >> 8$ which mixes two values a and b according to a parameter balance which maps 0 to full a and 1 to full b and everything in between. The interesting bit that in mathematical terms we define the arithmetic mean between two values $$a$$ and $$b$$ by $\frac{(255-\text{balance})a + (\text{balance})b}{255}$ So the difference is that the source code is not dividing by $$255$$ but is instead using $$>>8$$ which is an equivalent of an integer division by 256 - this is a bit off because we do not observe the value 256 as a 8 bit integer only goes to 255 but it is probably more performant to use this bit shifting trick in trade for accuracy. We can implement both functions in Python to see how they compare def balance_a(a, b, balance): return int(((255-balance)a + (balance)b)/255) balance_a(100, 200, 100) --> 139 def balance_b(a, b, balance): return (balance * b) + (255-balance)a >> 8 balance_b(100, 200, 100) --> 138 We see both implementations differ in the result by 1 but this is probably acceptable. For our re-implementation in SuperCollider we will use the proper mix of two variables as we do not need to optimize on a bit shifting level. #### Mixing values with U8Mix¶ After this short excursion we can come back to understanding U8Mix(U8Mix(a, b, x << 2), U8Mix(c, d, x << 2), y << 2); First we need to understand what x << 2 is doing - lets again assume x is 183 so 1 0 1 1 0 1 1 1 with << 2 becomes 1 1 0 1 1 1 0 0 which is 220. This is another neat trick and is basically $\left( \frac{183}{64} \mod 1 \right) \cdot 256$ Having in mind that earlier we divided by 64 to get a position between 0 and 3 to select a map this gives us the remainder of the division by 64 and is used to mix the values of each map. So doing x>>6 (basically $$\frac{x}{64}$$) gives us a value between 0 and 3 to select a map and dropping the 2 highest bits via x<<2 (basically $$\left( \frac{x}{64} \mod 1 \right) \cdot 256$$) gives us the remainder of the division and therefore the position between the maps. We do this for both maps and therefore get a in between position of maps which yields enriching results. #### EvaluateDrums¶ Now we can take a look at the context when ReadDrumMap is called. This is the method EvaluateDrums which is described here. The interesting bits are the for loop for (uint8_t i = 0; i < kNumParts; ++i) { uint8_t level = ReadDrumMap(step_, i, x, y); // ... if (level > threshold) { if (level > 192) { } } } } which calculates the level of each instrument according to the map and if this level exceeds the threshold a bit will be raised in a bit mask which is the equivalent of hit in our step sequencer. ### Extracting data using Python¶ It turns out the repository has the drum maps stored as Python lists so we can simply load these maps into a Python interpreter. import numpy as np # ... # converting the nested python list to numpy array nodes = np.array(nodes) Now we need to remember how these values were mapped in the C++ source code. The first re-arrangement of the values was while mapping nodes to a 5 by 5 drum map. We can simply imitate this in Python using numpy indices = np.array([ [10, 8, 0, 9, 11], [15, 7, 13, 12, 6], [18, 14, 4, 5, 3], [23, 16, 21, 1, 2], [24, 19, 17, 20, 22], ]) drum_map = nodes[indices] drum_map.shape -> (5, 5, 96) Now we need to take care of the offset to separate the drum maps which was implemented by uint8_t offset = (instrument kStepsPerPattern) + step; So we want to separate our 5x5x96 matrix into 3 5x5x32 matrix. instrument_maps = {} for i, instrument in enumerate("kick", "snare", "hihat"]): instrument_maps[instrument] = drum_map[:, :, (i32):((i+1)32)] where the order of the instruments are taken from the bit map commentary. By showing us each instrument map in Jupyter Lab we now have them in a clean format to be used in SuperCollider.	2023-03-25 20:30:34	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.46996235847473145, "perplexity": 1861.4168673967938}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945372.38/warc/CC-MAIN-20230325191930-20230325221930-00171.warc.gz"}
https://hoomd-blue.readthedocs.io/en/stable/notation.html	# Notation¶ The HOOMD-blue documentation uses the following mathematical notation. General notation: $$x$$ Scalar. $$\vec{a}$$ Vector. $$a$$ Magnitude of the vector $$\vec{a}$$. $$\hat{a}$$ Unit vector in the direction of $$\vec{a}$$. $$\mathbf{A}$$ Matrix. $$\mathbf{b}$$ Quaternion. $$\vert \mathbf{b} \vert$$ Magnitude of a quaternion. $$i$$, $$j$$, $$k$$ Indices. $$i$$ $$\sqrt{-1}$$ in some contexts. $$\mathrm{function}$$ A mathematical function. a $$[\mathrm{length}]^2$$ The quantity “a” has dimensions of $$\mathrm{length}^2$$. Symbol definitions: $$\vec{v}_{ij}$$ $$\vec{v}_j - \vec{v}_j$$ $$\vec{r}$$ Position. $$\vec{v}$$ Velocity. $$\mathbf{q}$$ Orientation. $$m$$ Mass. $$\vec{p}$$ Momentum. $$I$$ Moment of inertia. $$q$$ Charge. $$N$$ Number. $$L$$ Length. $$V$$ Volume. $$P$$ Pressure. $$\rho$$ Number density. $$\vec{a}_1$$, $$\vec{a}_2$$, $$\vec{a}_3$$ Box unit cell vectors. $$\vec{F}$$ Force. $$\vec{\tau}$$ Torque. $$U$$ Potential energy. $$K$$ Kinetic energy. $$E$$ Total energy. $$T$$ Temperature. $$k$$ Boltzmann’s constant in some contexts. Typically appears multiplying temperature: $$kT$$. $$k$$ Spring constant in some contexts. $$\beta$$ $$\frac{1}{kT}$$ $$t$$ Time.	2023-01-28 03:18:46	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7402322292327881, "perplexity": 1707.0824232430807}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499470.19/warc/CC-MAIN-20230128023233-20230128053233-00686.warc.gz"}
https://johnmathews.eu/btc-fork-analysis.html	# Analysis of the mean and median value of transactions on 5 Blockchains Comparing the daily mean and median value of on-chain transactions gives an indication of the extent of organic blockchain use. Comparing the mean-median ratio (MMR) across different Bitcoin forks shows that Bitcoin (BTC) is the least influenced by “whales”. By John Mathews Category: Data Sep 02 2019 This analysis was prepared for Coin Metrics as part of their recruitment process. It is a short demonstration of my thought process. The additional steps required to develop this into a useful analysis are also discussed. # CoinMetrics Case Study¶ Objective - to evaluate skills and abilities in multiple ways: 1. Importing data 2. Wrangling data 3. Exploring data 4. Analysis 5. Modeling 6. Communicating results Provide: 1. A written explanation of how to approach the problem 2. Present the beginning phases of implementation using coin metrics data Of the four options made available in the case study, option 3 was chosen Produce quality research that is of value to potential clients (doesn’t have to be complete) with a particular focus on network data ### Initial ideas¶ My first rough ideas were: 1. Compare different Bitcoin based chains, (BTC, BCH, LTC, BSV) to test the influence of whales and compare this to their respective (evolving) claims to be a store of value (SoV) and/or alternative to cash. 2. Develop and expand some of the research by Willy Woo. I find his research to be outstanding. In particular I think the following metrics merit further investigation: 1. days destroyed 2. hodl waves 3. thermo cap 4. average cap 3. Tracking the number of twitter followers of various crypto-twitter thought leaders and celebrities to test the hypothesis that “an increase in follower numbers shows that new retail investors are entering crypto-markets, and an increase in price is expected soon” Thought leaders / crypto celebrities could be further grouped by what types of coins they speak about most - smart contracts, DeFi, privacy coins, etc. Weibo could be analysed as well as Twitter to understand Chinese markets, Korean twitter could be analysed for the Korean retail market,etc. 4. I have an existing side project which has the goal of using a recurrent neural net using an LSTM architecture to predict BTC price movements. The app (model, stored data, data pipeline, visualization of results) will run autonomously on Google Cloud Platform. Candle data is consumed from CoinAPI.io and stored in BigQuery. Technical indicators will be calculated and used as additional factors to the model. Sentiment analysis from news outlets (Bloomberg, FT) would be added later. The model would be written using TensorFlow, and the BigQuery tables names would use BQ’s date format capabilites. This would make the project faster and cheaper. Idea 1 seemed like a sensible option. Ideas 3 and 4 are interesting and worth investigating, but not possible within the scope of this exercise: ### Testing the influence of whales¶ and “normal users” on BTC and 4 BTC forks, and discussing results in the context of each chain’s claimed technical advantages and use cases as e.g. a store of value or alternative to cash This will be achieved by comparing daily mean USD transaction value to daily median USD transaction value. This is done by calculating the mean-median ratio of transaction value (MMR). Hypothesis: If a chain has a much smaller median transaction size than mean transaction size, then on chain activity is dominated not by regular users making normal daily transactions, but by whales moving large amounts of currency to artificially inflate usage metrics. This could contradict claims that a blockchain has an active user base that the blockchain is meeting user needs. We assume that: 1. If a blockchain is functioning as digital cash, then most of its transactions would be small. e.g less than 100 USD. It should be noted that 100 USD is not a particularly small amount even in western countries and due to a blockchains borderless nature, it is even futher above a noraml ‘day-to-day’ transaction amount in large parts of the world. 2. Conversely, if a blockchain has relatively little organic use by normal users then whales (users with large holdings) will make up a large proportion of on-chain activity and would have average transaction sizes much larger than a day-to-day transaction. An untested guess at a “whale threshold” could be 100,000USD. 3. Where the ratio of mean to median transaction value is relatively high, we have an environment where the mean value is much higher than the median value, which shows that daily total value transacted is dominated by a few relatively large transactions, rather than many small value transactions. This would imply that whales dominate the blockchain (and likely market behavior) rather than members of the general public or retail investors. ### Chains:¶ The chains that will be analysed here are all forks of BTC. They are: • BTC • BCH • BSV • LTC • DOGE ### Fields¶ using the coinmetrics api, the following metrics will be used: 1. TxTfrValMeanUSD The sum USD value of native units transferred divided by the count of transfers (i.e., the mean “size” in USD of a transfer) that interval. 1. TxTfrValMedUSD The median USD value transferred per transfer (i.e., the median “size” in USD of a transfer) that interval. In [1]: from IPython.display import HTML HTML('''</span> <span class="s1">code_show=false; </span> <span class="s1">function code_toggle() {</span> <span class="s1"> if (code_show){</span> <span class="s1"> $('div.input').show();</span> <span class="s1">$('div.output_prompt').show();</span> <span class="s1"> } else {</span> <span class="s1"> $('div.input').hide();</span> <span class="s1">$('div.output_prompt').hide();</span> <span class="s1"> }</span> <span class="s1"> code_show =! code_show</span> <span class="s1">} </span> <span class="s1"> This analysis was made using Python. You can toggle the code visibility by clicking here. ''') Out[1]: This analysis was made using Python. You can toggle the code visibility by clicking here. In [2]: # import and setup import requests import json import pandas as pd import plotly.graph_objs as go import chart_studio.plotly as py import plotly from plotly.offline import init_notebook_mode plotly.offline.init_notebook_mode() init_notebook_mode(connected=True) from IPython.core.interactiveshell import InteractiveShell InteractiveShell.ast_node_interactivity = "all" import warnings warnings.simplefilter('ignore')	2019-09-20 11:46:04	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19690808653831482, "perplexity": 4075.3185023142473}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514574018.53/warc/CC-MAIN-20190920113425-20190920135425-00397.warc.gz"}
https://ibn.idsi.md/ro/vizualizare_articol/9145/datacite	Mathematical Model of the Local Kinetics of a Nickel Oxide Electrode Conţinutul numărului revistei Articolul precedent Articolul urmator 432 0 SM ISO690:2012KOSHELI, Nicolai; KOSTYRYA, M.. Mathematical Model of the Local Kinetics of a Nickel Oxide Electrode . In: Surface Engineering and Applied Electrochemistry. 2011, nr. 5(47), pp. 408-412. ISSN 1068-3755. EXPORT metadate: Google Scholar Crossref CERIF BibTeXDataCiteDublin Core Surface Engineering and Applied Electrochemistry Numărul 5(47) / 2011 / ISSN 1068-3755 /ISSNe 1934-8002 Mathematical Model of the Local Kinetics of a Nickel Oxide Electrode Pag. 408-412 Kosheli Nicolai, Kostyrya M. Disponibil în IBN: 29 noiembrie 2013 Rezumat A new mathematical model of the local kinetics characteristic E = f(i, τ) for nickel oxide electrodes NiOOH/Ni(OH)2 is presented. A flat crystal of active substance with thickness h that reconstructs a particle with a size of 2h is the sample for the studies. A mathematical model of the process is based on the conception considering the potential and polarization as a function of the concentration of the protons at the surface layer of the crystal. The polarization is considered as a sum of the concentration, activation, and ohmic components. The concept considers the coefficients of the protons activity and the proton vacancies in the crystal lattice. The potential of the electrode in the process of the discharge is determined by the changes in the surface concentration of protons. The coefficient of diffusion, the rate of the electromigration of protons, and the resistivity of the solid phase change over the thickness of the crystal in accordance with the changes in the concentration of the protons and the activity factors. The distribution of activity concentration of activity protons was determined by solving a one dimension problem of diffusive and electromigration transfer. ### DataCite XML Export <?xml version='1.0' encoding='utf-8'?> <creators> <creator> <creatorName>Kosheli, N.D.</creatorName> </creator> <creator> <creatorName>Kostyrya, M.V.</creatorName> </creator> </creators> <titles> <title xml:lang='en'>Mathematical Model of the Local Kinetics of a Nickel Oxide Electrode </title> </titles> <publisher>Instrumentul Bibliometric National</publisher> <publicationYear>2011</publicationYear> <relatedIdentifier relatedIdentifierType='ISSN' relationType='IsPartOf'>1068-3755</relatedIdentifier> <dates> <date dateType='Issued'>2011-10-03</date> </dates> <resourceType resourceTypeGeneral='Text'>Journal article</resourceType> <descriptions> <description xml:lang='en' descriptionType='Abstract'>A new mathematical model of the local kinetics characteristic E = f(i, τ) for nickel oxide electrodes NiOOH/Ni(OH)2 is presented. A flat crystal of active substance with thickness h that reconstructs a particle with a size of 2h is the sample for the studies. A mathematical model of the process is based on the conception considering the potential and polarization as a function of the concentration of the protons at the surface layer of the crystal. The polarization is considered as a sum of the concentration, activation, and ohmic components. The concept considers the coefficients of the protons activity and the proton vacancies in the crystal lattice. The potential of the electrode in the process of the discharge is determined by the changes in the surface concentration of protons. The coefficient of diffusion, the rate of the electromigration of protons, and the resistivity of the solid phase change over the thickness of the crystal in accordance with the changes in the concentration of the protons and the activity factors. The distribution of activity concentration of activity protons was determined by solving a one dimension problem of diffusive and electromigration transfer. </description> </descriptions> <formats> <format>application/pdf</format> </formats> </resource>	2020-07-03 12:44:52	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4042256772518158, "perplexity": 1932.5864958137468}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655882051.19/warc/CC-MAIN-20200703122347-20200703152347-00599.warc.gz"}
https://mathoverflow.net/questions/260235/sum-d-leq-x-mud-d-log-x-d-elementary-estimates	# $\sum_{d\leq x} (\mu(d)/d) \log x/d$: elementary estimates? Let $$F(x) = \sum_{d\leq x} \frac{\mu(d)}{d} \log \frac{x}{d}.$$ s it possible/feasible to give an elementary proof of the fact that $F(x)= 1 + o(1)$ (and, ideally, $1+O(1/\log x)$, or better)? By "elementary" I mean "using the properties of $\zeta(s)$ only for $\Re(s)\geq 1$, and of preference only for $s$ real". (Call work with $s$ complex, $\Re(s)\geq 1$, "semi-elementary" if you wish.) I'd also need for it to be possible to make the bounds nicely explicit. [Note: I am well aware of Ramaré's and Balazard's work, which relies on estimates on $\sum_{m\leq x} \mu(m)$ (derived in turn from estimates on $\sum_{m\leq x} \Lambda(m)$). I am looking for (semi-)elementary estimates, in part because I would like something that can be easily adapted to analogous sums.] • In case you (ore other users) will find there something useful (in the answer or in the comments), I'll add a link to math.SE post Showing that $\sum_{n \leq x} \frac{\mu(n)}{n} \log \frac{x}{n} = O(1)$.. Found using Approach0. – Martin Sleziak Jan 22 '17 at 11:59 • Probably a very naive question of mine, but would it help in any way to write the considered sum as $\sum_{d\leqslant x}\frac{\mu(d)}{d}\log x-\sum_{d\leqslant x}\mu(d)(\log' d.\log d)$? – Sylvain JULIEN Jan 22 '17 at 12:55 • according to Sylvain julien , Try to check this Erdos bound , equation 3 and 4 :terrytao.wordpress.com/tag/divisor-function, in the sense of conditionally convergent series. Assuming one can justify this (which, ultimately, requires one to exclude zeroes of the Riemann zeta function on the line Re(s)=1, you can obtain your estimation – zeraoulia rafik Jan 22 '17 at 23:33 ## 1 Answer In chapter 4, p. 15-18 of the book H. Iwaniec, Lectures on the Riemann Zeta Function, American Mathematical Society, University Lecture Series nº 62, 2014 there is an elementary proof of the prime number theorem with a rest $\psi(x)=x +O(x(\log x)^{-A})$ with arbitrary A. In page 18 he propose as exercise to proof $$\sum_{m\le x} \frac{\mu(m)}{m}\ll (\log x)^{-A},\qquad \sum_{m\le x} \frac{\mu(m)}{m}\log m=-1+O((\log x)^{-A}.$$ From this your assertion follows. So it is definitely feasible. • Er - I'd like to see it done? – H A Helfgott Jan 25 '17 at 18:54 • He uses "semi-elementary methods" to prove $M(x)\ll x (\log x)^{-A}$. From this I think there is no much difficulty. Essentially this is done by Landau when he proves $\sum \mu(n) \log n/n=-1$. (Primzhalen p. 612-613) – juan Jan 25 '17 at 19:49 • Sure, I agree it's not difficult if you assume that. But what are "semi-elementary methods"? Do you have to work with $\zeta(s)$ to the left of the line $\Re(s)=1$? – H A Helfgott Jan 25 '17 at 19:51 • But Iwaniec do not use zeta at Re s=1 only for $\sigma>1$. Well it is true he uses the behavior near $\sigma=1$ but he appear to prove this in page 10 again without using $\sigma=1$. – juan Jan 25 '17 at 19:56 • Iwaniec treat the analytic continuation in Chapter 6 of his book. – juan Jan 25 '17 at 20:02	2019-10-23 22:11:57	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8065686821937561, "perplexity": 711.4248967582577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987836295.98/warc/CC-MAIN-20191023201520-20191023225020-00111.warc.gz"}
https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/171/3/88443/borsuk-8211-sieklucki-theorem-in-cohomological-dimension-theory	# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty ## Borsuk–Sieklucki theorem in cohomological dimension theory ### Tom 171 / 2002 Fundamenta Mathematicae 171 (2002), 213-222 MSC: 55M10, 54F45. DOI: 10.4064/fm171-3-2 #### Streszczenie The Borsuk–Sieklucki theorem says that for every uncountable family $\{X_{\alpha}\}_{\alpha \in A}$ of $n$-dimensional closed subsets of an $n$-dimensional ANR-compactum, there exist $\alpha \ne \beta$ such that $\mathop{\rm dim} (X_{\alpha} \cap X_{\beta}) = n$. In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum $X$ is ${\rm clc}^{n+1}_{{\Bbb Z}}$, where $n\geq 1$, and $G$ is an Abelian group. Let $\{X_{\alpha }\}_{\alpha \in J}$ be an uncountable family of closed subsets of $X$. If ${\rm dim} _GX={\rm dim} _GX_{\alpha }=n$ for all $\alpha \in J$, then $\mathop{\rm dim} _G(X_{\alpha }\cap X_{\beta })=n$ for some $\alpha \neq \beta$. For $G$ being a countable principal ideal domain the above result was proved by Choi and Kozlowski [C-K]. Independently, Dydak and Koyama [D-K] proved it for $G$ being an arbitrary principal ideal domain and posed the question of validity of the Theorem for quasicyclic groups (see Problem~1 in [D-K]). As applications of the Theorem we investigate equality of cohomological dimension and strong cohomological dimension, and give a characterization of cohomological dimension in terms of a special base. #### Autorzy • Margareta BoegeInstituto de Matemáticas UNAM Av. Universidad S//N, Col. Lomas de Chamilpa 62210 Cuernavaca, Morelos, México e-mail • Jerzy DydakDepartment of Mathematics University of Tennessee Knoxville, TN 37996, U.S.A. e-mail • Rolando JiménezInstituto de Matemáticas UNAM Av. Universidad S//N, Col. Lomas de Chamilpa 62210 Cuernavaca, Morelos, México e-mail • Akira KoyamaDivision of Mathematical Sciences Osaka Kyoiku University Kashiwara, Osaka 582-8582, Japan e-mail • Evgeny V. ShchepinSteklov Institute of Mathematics Gubkina 8 117966 Moscow GSP-1, Russia e-mail ## Przeszukaj wydawnictwa IMPAN Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki. Odśwież obrazek	2018-06-19 06:25:50	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.882521390914917, "perplexity": 5219.154680612215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267861980.33/warc/CC-MAIN-20180619060647-20180619080647-00568.warc.gz"}

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