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https://insideaiml.com/blog/Python-3---Numbers-470	[ "", null, "", null, "#### Top Courses", null, "#### Machine Learning with Python & Statistics", null, "4 (4,001 Ratings)", null, "218 Learners\n\n#### Live Masterclass on \"Python for Artificial Intelligence\"", null, "Dec 4th (7:00 PM) 208 Registered\nMore webinars\n\n# Python 3 - Numbers", null, "Neha Kumawat\n\na year ago\n\nNumber data types store numeric values. They are immutable data types. This means, changing the value of a number data type results in a newly allocated object.\nNumber objects are created when you assign a value to them. For example −\n``````\nvar1 = 1\nvar2 = 10\n``````\nYou can also delete the reference to a number object by using the del statement. The syntax of the del statement is −\n``````\ndel var1[,var2[,var3[....,varN]]]]\n``````\nYou can delete a single object or multiple objects by using the del statement. For example −\n``````\ndel var\ndel var_a, var_b\n``````\nPython supports different numerical types −\n• int (signed integers) − They are often called just integers or ints. They are positive or negative whole numbers with no decimal point. Integers in Python 3 are of unlimited size. Python 2 has two integer types - int and long. There is no 'long integer' in Python 3 anymore.\nint (signed integers) − They are often called just integers or ints. They are positive or negative whole numbers with no decimal point. Integers in Python 3 are of unlimited size. Python 2 has two integer types - int and long. There is no 'long integer' in Python 3 anymore.\n• float (floating point real values) − Also called floats, they represent real numbers and are written with a decimal point dividing the integer and the fractional parts. Floats may also be in scientific notation, with E or e indicating the power of 10 (2.5e2 = 2.5 x 102 = 250).\nfloat (floating point real values) − Also called floats, they represent real numbers and are written with a decimal point dividing the integer and the fractional parts. Floats may also be in scientific notation, with E or e indicating the power of 10 (2.5e2 = 2.5 x 102 = 250).\n• complex (complex numbers) − are of the form a + bJ, where a and b are floats and J (or j) represents the square root of -1 (which is an imaginary number). The real part of the number is a, and the imaginary part is b. Complex numbers are not used much in Python programming.\ncomplex (complex numbers) − are of the form a + bJ, where a and b are floats and J (or j) represents the square root of -1 (which is an imaginary number). The real part of the number is a, and the imaginary part is b. Complex numbers are not used much in Python programming.\nIt is possible to represent an integer in hexa-decimal or octal form\n``````\n>>> number = 0xA0F #Hexa-decimal\n>>> number\n2575\n\n>>> number = 0o37 #Octal\n>>> number\n31\n``````\n\n### Examples\n\nHere are some examples of numbers.\nA complex number consists of an ordered pair of real floating-point numbers denoted by a &plus bj, where a is the real part and b is the imaginary part of the complex number.\n\n## Number Type Conversion\n\nPython converts numbers internally in an expression containing mixed types to a common type for evaluation. Sometimes, you need to coerce a number explicitly from one type to another to satisfy the requirements of an operator or function parameter.\n• Type int(x) to convert x to a plain integer.\nType int(x) to convert x to a plain integer.\n• Type long(x) to convert x to a long integer.\nType long(x) to convert x to a long integer.\n• Type float(x) to convert x to a floating-point number.\nType float(x) to convert x to a floating-point number.\n• Type complex(x) to convert x to a complex number with real part x and imaginary part zero.\nType complex(x) to convert x to a complex number with real part x and imaginary part zero.\n• Type complex(x, y) to convert x and y to a complex number with real part x and imaginary part y. x and y are numeric expressions\nType complex(x, y) to convert x and y to a complex number with real part x and imaginary part y. x and y are numeric expressions\n\n## Mathematical Functions\n\nPython includes the following functions that perform mathematical calculations.\nThe absolute value of x: the (positive) distance between x and zero.\nThe ceiling of x: the smallest integer not less than x.\ncmp(x, y)\n-1 if x < y, 0 if x == y, or 1 if x > y. Deprecated in Python 3. Instead use return (x>y)-(x\nThe exponential of x: ex\nThe absolute value of x.\nThe floor of x: the largest integer not greater than x.\nThe natural logarithm of x, for x > 0.\nThe base-10 logarithm of x for x > 0.\nThe largest of its arguments: the value closest to positive infinity\nThe smallest of its arguments: the value closest to negative infinity.\nThe fractional and integer parts of x in a two-item tuple. Both parts have the same sign as x. The integer part is returned as a float.\nThe value of x*y.\nx rounded to n digits from the decimal point. Python rounds away from zero as a tie-breaker: round(0.5) is 1.0 and round(-0.5) is -1.0.\nThe square root of x for x > 0.\n\n## Random Number Functions\n\nRandom numbers are used for games, simulations, testing, security, and privacy applications. Python includes the following functions that are commonly used.\nA random item from a list, tuple, or string.\nA randomly selected element from range(start, stop, step).\nA random float r, such that 0 is less than or equal to r and r is less than 1\nSets the integer starting value used in generating random numbers. Call this function before calling any other random module function. Returns None.\nRandomizes the items of a list in place. Returns None.\nA random float r, such that x is less than or equal to r and r is less than y.\n\n## Trigonometric Functions\n\nPython includes the following functions that perform trigonometric calculations.\nReturn the arc cosine of x, in radians.\nReturn the arc sine of x, in radians.\nReturn the arc tangent of x, in radians.\nReturn atan(y / x), in radians.\nReturn the cosine of x radians.\nReturn the Euclidean norm, sqrt(xx + y*y).\nReturn the sine of x radians.\nReturn the tangent of x radians.\nConverts angle x from radians to degrees.\nConverts angle x from degrees to radians.\n\n## Mathematical Constants\n\nThe module also defines two mathematical constants −\npi\nThe mathematical constant pi.\ne\nThe mathematical constant e." ]	[ null, "https://www.facebook.com/tr", null, "https://insideaiml.com/assets/images/footer-logo.png", null, "https://ambrapaliaidata.blob.core.windows.net/ai-storage/landingPage/exploreCourses/machine%20learning%20with%20python.webp", null, "https://insideaiml.com/assets/images/staar.png", null, "https://insideaiml.com/assets/images/v.svg", null, "https://ambrapaliaidata.blob.core.windows.net/ai-storage/webinar%20images/4th%20dec%20without%20register.jpg", null, "https://insideaiml.com/assets/images/01img.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5870766,"math_prob":0.9946702,"size":394,"snap":"2021-43-2021-49","text_gpt3_token_len":130,"char_repetition_ratio":0.14358975,"word_repetition_ratio":0.0,"special_character_ratio":0.33756346,"punctuation_ratio":0.15277778,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9985768,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-03T13:24:08Z\",\"WARC-Record-ID\":\"<urn:uuid:9b832d0a-4f14-4561-a0fc-c517852b3be6>\",\"Content-Length\":\"256544\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3bc94448-8038-4335-bde3-d7187ca88d31>\",\"WARC-Concurrent-To\":\"<urn:uuid:1c1ce2f2-d117-41fa-b74e-889c6d7ec74b>\",\"WARC-IP-Address\":\"143.110.253.149\",\"WARC-Target-URI\":\"https://insideaiml.com/blog/Python-3---Numbers-470\",\"WARC-Payload-Digest\":\"sha1:HUXEZ33K23EVOQKYTZULWAG2U7XL4RLK\",\"WARC-Block-Digest\":\"sha1:2CPNBERUDOD2HVIUWQZAMHHO7LLNWJSJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964362879.45_warc_CC-MAIN-20211203121459-20211203151459-00259.warc.gz\"}"}
https://www.domestic-engineering.com/about/ignorance/rebellion.html	[ "# My rebellion against ignorance", null, "Photo credit to Torsten Dederichs\n\n## Math is for people who like understanding things\n\nThe world is full of literature about money for the mathematically ignorant and emotionally undisciplined. I am not writing to those people. If this blog is confusing or the equations unclear, the reader does not need me to hold their hand through an abbreviated explanation of calculus, they need a calculus and/or differential equations course which is available for free online in many places. No portion of this text requires more than a cursory understanding of calculus and the ability to follow step by step derivations. Most of the broad conclusions can be understood with algebra skills that all reasonably educated adults possess.\n\n## Ubiquitous and terrible ways to avoid writing equations\n\nPerhaps the most common ignorance inducing statement made in the financial math world is the attempt to describe the interest on a loan with a couple of examples.\n\nFrom Nerdwallet.com retrieved 2017-05-22\n\na 30-year, fixed-rate $300,000 mortgage with a 6% APR . . . for a total repayment amount of$647,515. If, however, you took out the same mortgage and paid $40,000 in one-time fees upfront, you would have a 4% APR and end up paying . . . a total cost of$555,607.\n\nFrom DaveRamsey.com retrieved 2017-05-22\n\nA $175,000, 30-year mortgage with a 4% interest rate will cost you$68,000 more over the life of the loan than a 15-year mortgage will.\n\nThat is to say, describe the following equation\n\n$$\\text{Interest} = f(B_0,r,t_\\text{term})$$\n\nusing only a few example points. $$B_0 =$$ loan amount, $$r =$$ interest rate, and $$t_\\text{term} =$$ the loan term. This is fatuously incomplete because if we know nothing else about that equation, we have no idea what kinds of functions we might be dealing with. Is interest a linear or exponential function of $$B_0$$? of $$r$$? logarithmic? sinusoidal? The author imparts almost no information with their examples except illustrations of their \"Debt is really costly\" or \"Low-interest rates are preferable to high ones\" thesis.\n\nThe real equation is not as complicated as the black box version suggests it might be because it is not an independent function of $$r$$ and $$t_\\text{term}$$ it is a function of their product $$rt_\\text{term}$$. It also scales linearly with $$B_0$$ That information takes the mystery from a function of 3 variables to a scaling relation of 1.\n\n$$\\frac{\\text{Interest}}{B_0} = f(rt_\\text{term})= \\frac{rt_\\text{term}}{1-e^{-rt_\\text{term}}}-1$$\n\nThis equation is simple to understand, easy to apply, and relevant to every loan with a constant repayment rate. For reasons I cannot understand, it also seems to be a giant secret in the finance guru business. This problem is widespread in the world of financial literature, not limited to describing the interest on a loan (though that is the most common case).\n\n## Business Insider thinks $$e$$ is too confusing to publish\n\nConsider this pile of garbage from businessinsider.com in an article \"11 Personal Finance Equations You Need To Know\" which is introduced with the overselling line \"We've rounded up 11 math equations that can be used every single day. Write them down, whip out your pencil, and prepare to budget like a genius\" despite the fact that only one of their 11 equations has anything to do with budgeting and it's so tautological, it must be a joke. (Variables changed for consistency with this blog.)\n\n$$P = \\frac{B_0 \\frac{r}{f}}{1-\\left(1+\\frac{r}{f}\\right)^{-t_\\text{term} f}}$$\n\nThe equation describes the payments on a loan as a function of the loan size, the payment frequency, the loan term, and the interest rate. It is misleading or insulting for three reasons: it appears to be derived from a finite difference analysis of the differential equation of a loan balance which it is not, no work is shown, and the approximations made are not stated.\n\nIf we look at the real equation, the one which is the result of a 1st order ODE, we see that the authors at business insider decided the readers would not know what $$e$$ is so they gave it a cumbersome approximation. They assume a sizable portion of their readership would be able to properly multiply and convert units in frequency and interest rate, raise quantities to fractional powers, and apply order of operations to a complex fraction but $$e$$ was too foreign. Replacing $$e$$ with 2.71 was also not an option for some indefensible, unknown reason.\n\n$$P = \\frac{B_0 \\frac{r}{f}}{1-e^{-rt_\\text{term}}}$$\n\nFrom the same article is this ridiculous attempt to compute the net present value of an annuity. $$F$$ is the cash flow rate (\\$ per year), $$r$$ is the rate of return, and $$t$$ is the length of the annuity.\n\n$$\\text{NPV} = F\\left( \\frac{1}{r}-\\frac{1}{r(1+r)^t}\\right)$$\n\nThe real equation that any 2nd-year college student in a STEM curriculum should be able to derive is both more correct and concise.\n\n$$\\text{NPV} = \\frac{F}{r} \\left( 1 - e^{-rt} \\right)$$\n\nAgain, Business Insider believes its readership is too uneducated to understand $$e$$? Do they think the same thing about $$\\pi$$? What is wrong with their editors? Why is their instinct to publish cumbersome, inaccurate formulas instead of combating ignorance with explanation?" ]	[ null, "https://www.domestic-engineering.com/about/ignorance/tagimage_mini.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9610002,"math_prob":0.9940554,"size":5619,"snap":"2023-14-2023-23","text_gpt3_token_len":1210,"char_repetition_ratio":0.10828139,"word_repetition_ratio":0.0021367522,"special_character_ratio":0.22192562,"punctuation_ratio":0.0900474,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9969987,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-04-01T03:52:16Z\",\"WARC-Record-ID\":\"<urn:uuid:62dc098a-7314-465f-8696-db9fac15edf9>\",\"Content-Length\":\"12185\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2468a929-bbb6-4901-b34b-83ad5690c208>\",\"WARC-Concurrent-To\":\"<urn:uuid:c7c11166-c7e0-4162-bb36-d59c5b1fb8df>\",\"WARC-IP-Address\":\"172.102.240.42\",\"WARC-Target-URI\":\"https://www.domestic-engineering.com/about/ignorance/rebellion.html\",\"WARC-Payload-Digest\":\"sha1:GA5QOKKEAUA2ICHOG7YAMQ4PDZQTLZEN\",\"WARC-Block-Digest\":\"sha1:3WVYCH5VLJ6UCGSAM5FIYS33WWJLI2VA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296949701.0_warc_CC-MAIN-20230401032604-20230401062604-00679.warc.gz\"}"}
https://tech-story.net/basic-electronics-diodes/	[ "", null, "Basic Electronics - Diodes", null, "# Basic Electronics – Diodes\n\nAfter having known about various components, let us focus on another important component in the field of electronics, known as a Diode. A semiconductor diode is a two terminal electronic component with a PN junction. This is also called as a Rectifier.", null, "The anode which is the positive terminal of a diode is represented with A and the cathode, which is the negative terminal is represented with K. To know the anode and cathode of a practical diode, a fine line is drawn on the diode which means cathode, while the other end represents anode.", null, "As we had already discussed about the P-type and N-type semiconductors, and the behavior of their carriers, let us now try to join these materials together to see what happens.\n\n## Formation of a Diode\n\nIf a P-type and an N-type material are brought close to each other, both of them join to form a junction, as shown in the figure below.", null, "A P-type material has holes as the majority carriers and an N-type material has electrons as the majority carriers. As opposite charges attract, few holes in P-type tend to go to n-side, whereas few electrons in N-type tend to go to P-side.\n\nAs both of them travel towards the junction, holes and electrons recombine with each other to neutralize and forms ions. Now, in this junction, there exists a region where the positive and negative ions are formed, called as PN junction or junction barrier as shown in the figure.", null, "The formation of negative ions on P-side and positive ions on N-side results in the formation of a narrow charged region on either side of the PN junction. This region is now free from movable charge carriers. The ions present here have been stationary and maintain a region of space between them without any charge carriers.\n\nAs this region acts as a barrier between P and N type materials, this is also called as Barrier junction. This has another name called as Depletion region meaning it depletes both the regions. There occurs a potential difference VD due to the formation of ions, across the junction called as Potential Barrier as it prevents further movement of holes and electrons through the junction.\n\n## Biasing of a Diode\n\nWhen a diode or any two-terminal component is connected in a circuit, it has two biased conditions with the given supply. They are Forward biased condition and Reverse biased condition. Let us know them in detail.\n\n### Forward Biased Condition\n\nWhen a diode is connected in a circuit, with its anode to the positive terminal and cathode to the negative terminal of the supply, then such a connection is said to be forward biased condition. This kind of connection makes the circuit more and more forward biased and helps in more conduction. A diode conducts well in forward biased condition.\n\n### Reverse Biased Condition\n\nWhen a diode is connected in a circuit, with its anode to the negative terminal and cathode to the positive terminal of the supply, then such a connection is said to be Reverse biased condition. This kind of connection makes the circuit more and more reverse biased and helps in minimizing and preventing the conduction. A diode cannot conduct in reverse biased condition.", null, "Let us now try to know what happens if a diode is connected in forward biased and in reverse biased conditions.\n\n## Working under Forward Biased\n\nWhen an external voltage is applied to a diode such that it cancels the potential barrier and permits the flow of current is called as forward bias. When anode and cathode are connected to positive and negative terminals respectively, the holes in P-type and electrons in N-type tend to move across the junction, breaking the barrier. There exists a free flow of current with this, almost eliminating the barrier.", null, "With the repulsive force provided by positive terminal to holes and by negative terminal to electrons, the recombination takes place in the junction. The supply voltage should be such high that it forces the movement of electrons and holes through the barrier and to cross it to provide forward current.\n\nForward Current is the current produced by the diode when operating in forward biased condition and it is indicated by If.\n\n## Working under Reverse Biased\n\nWhen an external voltage is applied to a diode such that it increases the potential barrier and restricts the flow of current is called as Reverse bias. When anode and cathode are connected to negative and positive terminals respectively, the electrons are attracted towards the positive terminal and holes are attracted towards the negative terminal. Hence both will be away from the potential barrier increasing the junction resistance and preventing any electron to cross the junction.\n\nThe following figure explains this. The graph of conduction when no field is applied and when some external field is applied are also drawn.", null, "With the increasing reverse bias, the junction has few minority carriers to cross the junction. This current is normally negligible. This reverse current is almost constant when the temperature is constant. But when this reverse voltage increases further, then a point called reverse breakdown occurs, where an avalanche of current flows through the junction. This high reverse current damages the device.\n\nReverse current is the current produced by the diode when operating in reverse biased condition and it is indicated by Ir. Hence a diode provides high resistance path in reverse biased condition and doesn’t conduct, where it provides a low resistance path in forward biased condition and conducts. Thus we can conclude that a diode is a one-way device which conducts in forward bias and acts as an insulator in reverse bias. This behavior makes it work as a rectifier, which converts AC to DC.\n\n### Peak Inverse Voltage\n\nPeak Inverse Voltage is shortly called as PIV. It states the maximum voltage applied in reverse bias. The Peak Inverse Voltage can be defined as “The maximum reverse voltage that a diode can withstand without being destroyed”. Hence, this voltage is considered during reverse biased condition. It denotes how a diode can be safely operated in reverse bias.\n\n## Purpose of a Diode\n\nA diode is used to block the electric current flow in one direction, i.e. in forward direction and to block in reverse direction. This principle of diode makes it work as a Rectifier.\n\nFor a circuit to allow the current flow in one direction but to stop in the other direction, the rectifier diode is the best choice. Thus the output will be DC removing the AC components. The circuits such as half wave and full wave rectifiers are made using diodes, which can be studied in Electronic Circuits tutorials.\n\nA diode is also used as a Switch. It helps a faster ON and OFF for the output that should occur in a quick rate.\n\n## V – I Characteristics of a Diode\n\nA Practical circuit arrangement for a PN junction diode is as shown in the following figure. An ammeter is connected in series and voltmeter in parallel, while the supply is controlled through a variable resistor.", null, "During the operation, when the diode is in forward biased condition, at some particular voltage, the potential barrier gets eliminated. Such a voltage is called as Cut-off Voltage or Knee Voltage. If the forward voltage exceeds beyond the limit, the forward current rises up exponentially and if this is done further, the device is damaged due to overheating.\n\nThe following graph shows the state of diode conduction in forward and reverse biased conditions.", null, "During the reverse bias, current produced through minority carriers exist known as “Reverse current”. As the reverse voltage increases, this reverse current increases and it suddenly breaks down at a point, resulting in the permanent destruction of the junction.", null, "" ]	[ null, "https://mc.yandex.ru/watch/74485888", null, "https://tech-story.net/basic-electronics-diodes/", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://tech-story.net/basic-electronics-diodes/image/gif,GIF89a%01%00%01%00%80%00%00%00%00%00%FF%FF%FF%21%F9%04%01%00%00%00%00%2C%00%00%00%00%01%00%01%00%00%02%01D%00%3B", null, "https://i1.wp.com/www.paypal.com/en_FR/i/scr/pixel.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9531726,"math_prob":0.8550034,"size":7657,"snap":"2021-43-2021-49","text_gpt3_token_len":1514,"char_repetition_ratio":0.15248922,"word_repetition_ratio":0.09034268,"special_character_ratio":0.18597361,"punctuation_ratio":0.08028169,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96114784,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],"im_url_duplicate_count":[null,null,null,1,null,9,null,9,null,9,null,9,null,9,null,9,null,9,null,9,null,9,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-09T07:51:31Z\",\"WARC-Record-ID\":\"<urn:uuid:447a1055-eaee-47b9-a135-cd555c34b755>\",\"Content-Length\":\"187320\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:524a9fb6-7562-4290-b41b-ec13cadfd9d9>\",\"WARC-Concurrent-To\":\"<urn:uuid:7badd1ed-d6f8-49ab-9aff-3a6bc01de63c>\",\"WARC-IP-Address\":\"172.67.182.101\",\"WARC-Target-URI\":\"https://tech-story.net/basic-electronics-diodes/\",\"WARC-Payload-Digest\":\"sha1:QG4E63ZIKHMZM4TTE5QKXUWHG5AUDTE2\",\"WARC-Block-Digest\":\"sha1:SS7UR6DGLD372HN27FMGK2MY77U3732E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363689.56_warc_CC-MAIN-20211209061259-20211209091259-00480.warc.gz\"}"}
https://mb3is.megx.net/gustame/constrained-analyses/cca/partial-canonical-correspondence-analysis	[ "### Partial Canonical Correspondence Analysis\n\n#### b\n\nFigure 1: An illustration of \"partialling out\" the influence of a set of variables (W) from a model. a) Both the explanatory variable(s) in matrices X and Y explain a portion of the variation in the response data (Y). b) After the partialling out the effect of W (which may be a single variable or a set of variables), only the variance in the response variables (Y) which can be exclusively explained by the variance in one set of explanatory variables (X) is retained.\n\nPartial canonical correspondence analysis (pCCA) is an extension of CCA wherein the influence of a set of variables stored in an additional matrix can be controlled for. The concept is related to partial correlation.\n\nThis is particularly useful when one wishes to control for a set of variables whose influence is known or at least anticipated and which are not of immediate interest. Examples include geographic distance, latitudinal temperature gradients, or depth-dependent photogradients.\n\nControlling for the effect of different sampling or measurement times or locations between samples is also possible. It must be determined whether time and/or space are best represented by dummy variables or appropriately transformed quantitative variables. As CCA is a method tuned to represent centroids (multivariate means) of data sets, if the influence of time and/or space is restricted to shifting these centroids then its effect can be well controlled for. However, if time/space effects show more complex influence or interact with other explanatory or control variables, these factors cannot be controlled for by pCCA.\n\nThe method can also be applied to examine the effect of a single variable in a matrix of explanatory variables using pCCA, while controlling for the other variables. This is done by placing all other explanatory variables in a matrix of control variables. Their effects may then be partialled out. A single canonical axis and eigenvalue will be generated which express the variation that the variable of interest is responsible for.\n\nMASAME pCCA app\n\nReferences" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9122127,"math_prob":0.93291944,"size":1565,"snap":"2020-45-2020-50","text_gpt3_token_len":280,"char_repetition_ratio":0.12684177,"word_repetition_ratio":0.0,"special_character_ratio":0.16932908,"punctuation_ratio":0.06870229,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9518992,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-28T13:03:44Z\",\"WARC-Record-ID\":\"<urn:uuid:a242512f-c888-4152-9166-0e59481077f5>\",\"Content-Length\":\"47235\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:90eb8767-9083-469a-89e0-cfea0449cfc9>\",\"WARC-Concurrent-To\":\"<urn:uuid:65f56656-18f1-4ba5-9896-b6b5799b0cb6>\",\"WARC-IP-Address\":\"194.95.6.8\",\"WARC-Target-URI\":\"https://mb3is.megx.net/gustame/constrained-analyses/cca/partial-canonical-correspondence-analysis\",\"WARC-Payload-Digest\":\"sha1:HLWONC5D2DG7OUDD6WV57KU33C5D6A6N\",\"WARC-Block-Digest\":\"sha1:65FMPIYHB2OOT5SN32YZZBRVKDXYVKGP\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141195656.78_warc_CC-MAIN-20201128125557-20201128155557-00170.warc.gz\"}"}
https://www.colorhexa.com/180216	[ "# #180216 Color Information\n\nIn a RGB color space, hex #180216 is composed of 9.4% red, 0.8% green and 8.6% blue. Whereas in a CMYK color space, it is composed of 0% cyan, 91.7% magenta, 8.3% yellow and 90.6% black. It has a hue angle of 305.5 degrees, a saturation of 84.6% and a lightness of 5.1%. #180216 color hex could be obtained by blending #30042c with #000000. Closest websafe color is: #000000.\n\n• R 9\n• G 1\n• B 9\nRGB color chart\n• C 0\n• M 92\n• Y 8\n• K 91\nCMYK color chart\n\n#180216 color description : Very dark (mostly black) magenta.\n\n# #180216 Color Conversion\n\nThe hexadecimal color #180216 has RGB values of R:24, G:2, B:22 and CMYK values of C:0, M:0.92, Y:0.08, K:0.91. Its decimal value is 1573398.\n\nHex triplet RGB Decimal 180216 `#180216` 24, 2, 22 `rgb(24,2,22)` 9.4, 0.8, 8.6 `rgb(9.4%,0.8%,8.6%)` 0, 92, 8, 91 305.5°, 84.6, 5.1 `hsl(305.5,84.6%,5.1%)` 305.5°, 91.7, 9.4 000000 `#000000`\nCIE-LAB 2.67, 10.744, -6.66 0.543, 0.296, 0.787 0.334, 0.182, 0.296 2.67, 12.641, 328.205 2.67, 3.409, -3.675 5.437, 8.321, -4.782 00011000, 00000010, 00010110\n\n# Color Schemes with #180216\n\n• #180216\n``#180216` `rgb(24,2,22)``\n• #021804\n``#021804` `rgb(2,24,4)``\nComplementary Color\n• #0f0218\n``#0f0218` `rgb(15,2,24)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #18020b\n``#18020b` `rgb(24,2,11)``\nAnalogous Color\n• #02180f\n``#02180f` `rgb(2,24,15)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #0b1802\n``#0b1802` `rgb(11,24,2)``\nSplit Complementary Color\n• #021618\n``#021618` `rgb(2,22,24)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #161802\n``#161802` `rgb(22,24,2)``\n• #040218\n``#040218` `rgb(4,2,24)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #161802\n``#161802` `rgb(22,24,2)``\n• #021804\n``#021804` `rgb(2,24,4)``\n• #000000\n``#000000` `rgb(0,0,0)``\n• #000000\n``#000000` `rgb(0,0,0)``\n• #000000\n``#000000` `rgb(0,0,0)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #30042c\n``#30042c` `rgb(48,4,44)``\n• #470641\n``#470641` `rgb(71,6,65)``\n• #5f0857\n``#5f0857` `rgb(95,8,87)``\nMonochromatic Color\n\n# Alternatives to #180216\n\nBelow, you can see some colors close to #180216. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #150218\n``#150218` `rgb(21,2,24)``\n• #160218\n``#160218` `rgb(22,2,24)``\n• #180218\n``#180218` `rgb(24,2,24)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #180214\n``#180214` `rgb(24,2,20)``\n• #180212\n``#180212` `rgb(24,2,18)``\n• #180211\n``#180211` `rgb(24,2,17)``\nSimilar Colors\n\n# #180216 Preview\n\nThis text has a font color of #180216.\n\n``<span style=\"color:#180216;\">Text here</span>``\n#180216 background color\n\nThis paragraph has a background color of #180216.\n\n``<p style=\"background-color:#180216;\">Content here</p>``\n#180216 border color\n\nThis element has a border color of #180216.\n\n``<div style=\"border:1px solid #180216;\">Content here</div>``\nCSS codes\n``.text {color:#180216;}``\n``.background {background-color:#180216;}``\n``.border {border:1px solid #180216;}``\n\n# Shades and Tints of #180216\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #060005 is the darkest color, while #fef3fd is the lightest one.\n\n• #060005\n``#060005` `rgb(6,0,5)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #2a0427\n``#2a0427` `rgb(42,4,39)``\n• #3c0537\n``#3c0537` `rgb(60,5,55)``\n• #4e0748\n``#4e0748` `rgb(78,7,72)``\n• #600858\n``#600858` `rgb(96,8,88)``\n• #730a69\n``#730a69` `rgb(115,10,105)``\n• #850b7a\n``#850b7a` `rgb(133,11,122)``\n• #970d8a\n``#970d8a` `rgb(151,13,138)``\n• #a90e9b\n``#a90e9b` `rgb(169,14,155)``\n• #bb10ab\n``#bb10ab` `rgb(187,16,171)``\n• #cd11bc\n``#cd11bc` `rgb(205,17,188)``\n• #df13cd\n``#df13cd` `rgb(223,19,205)``\n``#ec1ad9` `rgb(236,26,217)``\n• #ed2cdc\n``#ed2cdc` `rgb(237,44,220)``\n• #ef3edf\n``#ef3edf` `rgb(239,62,223)``\n• #f050e2\n``#f050e2` `rgb(240,80,226)``\n• #f262e5\n``#f262e5` `rgb(242,98,229)``\n• #f374e8\n``#f374e8` `rgb(243,116,232)``\n• #f586eb\n``#f586eb` `rgb(245,134,235)``\n• #f698ee\n``#f698ee` `rgb(246,152,238)``\n• #f8aaf1\n``#f8aaf1` `rgb(248,170,241)``\n• #f9bcf4\n``#f9bcf4` `rgb(249,188,244)``\n• #fbcff7\n``#fbcff7` `rgb(251,207,247)``\n• #fce1fa\n``#fce1fa` `rgb(252,225,250)``\n• #fef3fd\n``#fef3fd` `rgb(254,243,253)``\nTint Color Variation\n\n# Tones of #180216\n\nA tone is produced by adding gray to any pure hue. In this case, #0e0c0e is the less saturated color, while #1a0018 is the most saturated one.\n\n• #0e0c0e\n``#0e0c0e` `rgb(14,12,14)``\n• #0f0b0f\n``#0f0b0f` `rgb(15,11,15)``\n• #100a0f\n``#100a0f` `rgb(16,10,15)``\n• #110910\n``#110910` `rgb(17,9,16)``\n• #120811\n``#120811` `rgb(18,8,17)``\n• #130712\n``#130712` `rgb(19,7,18)``\n• #140613\n``#140613` `rgb(20,6,19)``\n• #150514\n``#150514` `rgb(21,5,20)``\n• #160414\n``#160414` `rgb(22,4,20)``\n• #170315\n``#170315` `rgb(23,3,21)``\n• #180216\n``#180216` `rgb(24,2,22)``\n• #190117\n``#190117` `rgb(25,1,23)``\n• #1a0018\n``#1a0018` `rgb(26,0,24)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #180216 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5424887,"math_prob":0.7809197,"size":3651,"snap":"2020-34-2020-40","text_gpt3_token_len":1601,"char_repetition_ratio":0.13134083,"word_repetition_ratio":0.011049724,"special_character_ratio":0.56505066,"punctuation_ratio":0.23730685,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9935163,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-19T06:23:59Z\",\"WARC-Record-ID\":\"<urn:uuid:5fc3713e-9ce0-4313-a52c-7557ae5d831e>\",\"Content-Length\":\"36157\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:74cff6e8-9548-4892-8c8d-b918f704aed0>\",\"WARC-Concurrent-To\":\"<urn:uuid:82ea8d1e-42d1-413f-a2e8-a05c1b5e0c8e>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/180216\",\"WARC-Payload-Digest\":\"sha1:5VS2RZVUSK7QKNIDKWFEBKFAVLAKFFD6\",\"WARC-Block-Digest\":\"sha1:3G4YYRQ4QAIOOX27TA3FAL5UMIIWIIP2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400190270.10_warc_CC-MAIN-20200919044311-20200919074311-00200.warc.gz\"}"}
https://calculator.name/numberstowords/2	[ "# Write 2 in English Words\n\nHow to write 2 in english words? - The number 2 written in english words is \"two\". Spell, say, write number 2 in english by using our numbers to words calculator.\n\n 2 in words two 2 spelling two\nNumbers to Words Converter\n\n# 2 spelling\n\nHow do you spell number in currency spelling? Here we have made a list of the currency names you would need to write spellings in order to deposit money against your currency cheques, DD, loan payments or more.\nJust find the currency and get spelling for it.\n\n• Afghanistan → Afghani → two Afghani\n• Albania → Leke → two Leke\n• Algeria → Dinar → two Dinar\n• Andorra → Euro → two Euro\n• Angola → Kwanza → two Kwanza\n• Antigua and Barbuda → Dollar → two dollar\n• Argentina → Peso → two Peso\n• Armenia → Dram → two Dram\n• Australia → Dollar → two dollar\n• Austria → Euro → two Euro\n• Azerbaijan → Manat → two Manat\n• Bahamas → Dollar → two dollar\n• Bahrain → Dinar → two dinar\n• Bangladesh → Taka → two Taka\n• Barbados → Dollar → two dollar\n• Belarus → Ruble → two ruble\n• Belgium → Euro → two Euro\n• Belize → Dollar → two dollar\n• Benin → Franc → two Franc\n• Bhutan → Ngultrum → two Ngultrum\n• Bolivia → Boliviano → two Boliviano\n• Bosnia and Herzegovina → Marka → two Marka\n• Botswana → Pula → two Pula\n• Brazil → Real → two Real\n• Brunei → Dollar → two dollar\n• Bulgaria → Lev → two Lev\n• Burkina Faso → Franc → two Franc\n• Burundi → Franc → two franc\n• Cambodia → Riel → two Riel\n• Cameroon → Franc → two Franc\n• Canada → Dollar → two dollar\n• Cape Verde → Escudo → two escudo\n• Central African Republic → Franc → two Franc\n• Chad → Franc → two Franc\n• Chile → Peso → two Peso\n• China → Yuan → two Yuan\n• Colombia → Peso → two Peso\n• Costa Rica → Colon → two Colon\n• Croatia → Kuna → two Kuna\n• Cuba → Peso → two Peso\n• Cyprus → Pound → two pound\n• Czech Republic → Koruna → two Koruna\n• Denmark → Krone → two Krone\n• Djibouti → Franc → two franc\n• Dominica → Dollar → two dollar\n• Dominican Republic → Peso → two Peso\n• East Timor → Dollar → two dollar\n• Ecuador → Dollar → two dollar\n• Egypt → Pound → two pound\n• El Salvador → Dollar → two dollar\n• Equatorial Guinea → Franc → two Franc\n• Eritrea → Nakfa → two Nakfa\n• Estonia → Kroon → two Kroon\n• Ethiopia → Birr → two Birr\n• Fiji → Dollar → two dollar\n• Finland → Euro → two Euro\n• France → Euro → two Euro\n• Gabon → Franc → two Franc\n• Gambia → Dalasi → two Dalasi\n• Georgia → Lari → two Lari\n• Germany → Euro → two Euro\n• Ghana → Cedi → two Cedi\n• Greece → Euro → two Euro\n• Grenada → Dollar → two dollar\n• Guatemala → Quetzal → two Quetzal\n• Guinea → Franc → two franc\n• Guinea-Bissau → Franc → two Franc\n• Guyana → Dollar → two dollar\n• Haiti → Gourde → two Gourde\n• Honduras → Lempira → two Lempira\n• Hungary → Forint → two Forint\n• Iceland → Krona → two krona\n• India → Rupee → two Rupee\n• Indonesia → Rupiah → two Rupiah\n• Iraq → Dollar → two dollar\n• Ireland → Euro → two Euro\n• Israel → Shekel → two Shekel\n• Italy → Euro → two Euro\n• Jamaica → Dollar → two dollar\n• Japan → Yen → two Yen\n• Jordan → Dinar → two dinar\n• Kazakhstan → Tenge → two Tenge\n• Kenya → Shilling → two shilling\n• Kiribati → Dollar → two dollar\n• Korea, North → Won → two Won\n• Korea, South → Won → two Won\n• Kuwait → Dinar → two dinar\n• Kyrgyzstan → Som → two Som\n• Laos → Kip → two Kip\n• Latvia → Lats → two Lats\n• Lebanon → Pound → two pound\n• Lesotho → Maluti → two Maluti\n• Liberia → Dollar → two dollar\n• Libya → Dinar → two dinar\n• Liechtenstein → Franc → two franc\n• Lithuania → Litas → two Litas\n• Luxembourg → Euro → two Euro\n• Macedonia → Denar → two Denar\n• Madagascar → Franc → two franc\n• Malawi → Kwacha → two Kwacha\n• Malaysia → Ringgit → two Ringgit\n• Maldives → Rufiya → two Rufiya\n• Mali → Franc → two Franc\n• Malta → Euro → two Euro\n• Mauritania → Ouguiya → two Ouguiya\n• Mauritius → Rupee → two rupee\n• Mexico → Peso → two peso\n• Moldova → Leu → two Leu\n• Monaco → Euro → two Euro\n• Mongolia → Tugrik → two Tugrik\n• Montenegro → Euro → two Euro\n• Morocco → Dirham → two Dirham\n• Mozambique → Metical → two Metical\n• Myanmar → Kyat → two Kyat\n• Namibia → Dollar → two dollar\n• Nauru → Dollar → two dollar\n• Nepal → Rupee → two rupee\n• Netherlands → Euro → two Euro\n• New Zealand → Dollar → two dollar\n• Nicaragua → Cordoba → two cordoba\n• Niger → Franc → two Franc\n• Nigeria → Naira → two Naira\n• Norway → Krone → two krone\n• Oman → Rial → two rial\n• Pakistan → Rupee → two rupee\n• Palau → Dollar → two dollar\n• Panama → Dollar → two dollar\n• Papua New Guinea → Kina → two Kina\n• Paraguay → Guaraní → two Guaraní\n• Philippines → Peso → two Peso\n• Poland → Zloty → two Zloty\n• Portugal → Escudo → two escudo\n• Qatar → Riyal → two riyal\n• Romania → Leu → two Leu\n• Russia → Ruble → two Ruble\n• Rwanda → Franc → two franc\n• St. Kitts and Nevis → Dollar → two dollar\n• St. Lucia → Dollar → two dollar\n• St. Vincent and the Grena → Dollar → two dollar\n• Samoa → Tala → two Tala\n• San Marino → Euro → two Euro\n• São Tomé and Príncipe → Dobra → two Dobra\n• Saudi Arabia → Riyal → two Riyal\n• Senegal → Franc → two Franc\n• Seychelles → Rupee → two rupee\n• Sierra Leone → Leone → two Leone\n• Singapore → Dollar → two dollar\n• Slovakia → Koruna → two Koruna\n• Slovenia → Euro → two euro\n• Solomon Islands → Dollar → two dollar\n• Somalia → Shilling → two shilling\n• South Africa → Rand → two Rand\n• Spain → Euro → two Euro\n• Sri Lanka → Rupee → two rupee\n• Sudan → Dinar → two Dinar\n• Suriname → Dollar → two dollar\n• Swaziland → Lilangeni → two Lilangeni\n• Sweden → Krona → two Krona\n• Switzerland → Franc → two franc\n• Syria → Pound → two pound\n• Taiwan → Dollar → two dollar\n• Tajikistan → Somoni → two somoni\n• Tanzania → Shilling → two shilling\n• Thailand → Baht → two baht\n• Togo → Franc → two Franc\n• Tonga → Pa'anga → two Pa'anga\n• Trinidad and Tobago → Dollar → two dollar\n• Tunisia → Dinar → two dinar\n• Turkey → Lira → two lira\n• Turkmenistan → Manat → two Manat\n• Tuvalu → Dollar → two dollar\n• Uganda → Shilling → two shilling\n• Ukraine → Hryvna → two Hryvna\n• United Arab Emirates → Dirham → two dirham\n• United Kingdom → Pound → two Pound\n• United States → Dollar → two dollar\n• Uruguay → Peso → two peso\n• Uzbekistan → Sum → two sum\n• Vatican City → Euro → two Euro\n• Venezuela → Bolivar → two Bolivar\n• Vietnam → Dong → two Dong\n• Western Sahara → Tala → two Tala\n• Yemen → Rial → two Rial\n• Zambia → Kwacha → two Kwacha\n• Zimbabwe → Dollar → two dollar\n\nHere are some more examples of numbers to words converter" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.90297514,"math_prob":0.52304196,"size":513,"snap":"2022-40-2023-06","text_gpt3_token_len":119,"char_repetition_ratio":0.1611002,"word_repetition_ratio":0.0,"special_character_ratio":0.22612086,"punctuation_ratio":0.09615385,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9831118,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-03T04:13:03Z\",\"WARC-Record-ID\":\"<urn:uuid:aea767b6-144e-491a-ad0d-fc1bfe82c1b2>\",\"Content-Length\":\"16051\",\"Content-Type\":\"application/http; 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https://www.rdocumentation.org/packages/spatstat/versions/1.2-1/topics/quadscheme	[ "0th\n\nPercentile\n\n##### Generate a Quadrature Scheme from a Point Pattern\n\nGenerates a quadrature scheme (an object of class \"quad\") from point patterns of data and dummy points.\n\nKeywords\nspatial\n##### Usage\nquadscheme(data)\nquadscheme(data, dummy=default.dummy(data), method=\"grid\", ...)\n##### Arguments\ndata\nThe observed data point pattern. An object of class \"ppp\" or in a format recognised by as.ppp()\ndummy\nThe pattern of dummy points for the quadrature. An object of class \"ppp\" or in a format recognised by as.ppp()\nmethod\nThe name of the method for calculating quadrature weights: either \"grid\" or \"dirichlet\".\n...\nParameters of the weighting method (see below)\n##### Details\n\nThis is the primary method for producing a quadrature schemes for use by mpl. The function mpl fits a point process model to an observed point pattern using the Berman-Turner quadrature approximation (Berman and Turner, 1992; Baddeley and Turner, 2000) to the pseudolikelihood of the model. It requires a quadrature scheme consisting of the original data point pattern, an additional pattern of dummy points, and a vector of quadrature weights for all these points. Such quadrature schemes are represented by objects of class \"quad\". See quad.object for a description of this class.\n\nQuadrature schemes are created by the function quadscheme. The arguments data and dummy specify the data and dummy points, respectively. There is a sensible default for the dummy points (provided by default.dummy). Alternatively the dummy points may be specified arbitrarily and given in any format recognised by as.ppp. There are also functions for creating dummy patterns including corners, gridcentres, stratrand and spokes. The quadrature region is the region over which we are integrating, and approximating integrals by finite sums. If dummy is a point pattern object (class \"ppp\") then the quadrature region is taken to be dummy$window. If dummy is just a list of$x, y$coordinates then the quadrature region defaults to the observation window of the data pattern, data$window.\n\nIf method = \"grid\" then the optional arguments (for ...) are (nx = default.ngrid(data), ny=nx). The quadrature region (see below) is divided into an nx by ny grid of rectangular tiles. The weight for each quadrature point is the area of a tile divided by the number of quadrature points in that tile. If method=\"dirichlet\" then the optional arguments are (exact=TRUE). The quadrature points (both data and dummy) are used to construct the Dirichlet tessellation. The quadrature weight of each point is the area of its Dirichlet tile inside the quadrature region.\n\n##### Value\n\n• An object of class \"quad\" describing the quadrature scheme (data points, dummy points, and quadrature weights) suitable as the argument Q of the function mpl() for fitting a point process model.\n\n##### References\n\nBaddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283--322. Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31--38.\n\nmpl, as.ppp, quad.object, gridweights, dirichlet.weights, corners, gridcentres, stratrand, spokes\n\n##### Examples\nlibrary(spatstat)\ndata(simdat)\nP <- simdat\nD <- default.dummy(P, 100)\nQ <- quadscheme(P, , \"grid\")" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.65202826,"math_prob":0.95702714,"size":3453,"snap":"2019-51-2020-05","text_gpt3_token_len":887,"char_repetition_ratio":0.157147,"word_repetition_ratio":0.03710575,"special_character_ratio":0.2319722,"punctuation_ratio":0.15566038,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9892463,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-29T16:51:49Z\",\"WARC-Record-ID\":\"<urn:uuid:9863ae4f-e422-461d-a3db-8d6f3c67839f>\",\"Content-Length\":\"20205\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b79c8f87-47c8-4041-9a53-36f30237ef7c>\",\"WARC-Concurrent-To\":\"<urn:uuid:7f7150e7-258b-45b6-b032-4eb342228326>\",\"WARC-IP-Address\":\"54.172.158.14\",\"WARC-Target-URI\":\"https://www.rdocumentation.org/packages/spatstat/versions/1.2-1/topics/quadscheme\",\"WARC-Payload-Digest\":\"sha1:2EO3YMM37UHUXLC4BE7SS3WH3POKAAGB\",\"WARC-Block-Digest\":\"sha1:EY4GOC7PQK63LNDLSZF6DRM6AA45KABT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579251801423.98_warc_CC-MAIN-20200129164403-20200129193403-00016.warc.gz\"}"}
https://en.wikipedia.org/wiki/Woodbury_matrix_identity	[ "# Woodbury matrix identity\n\nIn mathematics (specifically linear algebra), the Woodbury matrix identity, named after Max A. Woodbury, says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report.\n\nThe Woodbury matrix identity is\n\n$\\left(A+UCV\\right)^{-1}=A^{-1}-A^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1},$", null, "where A, U, C and V are conformable matrices: A is n×n, C is k×k, U is n×k, and V is k×n. This can be derived using blockwise matrix inversion.\n\nWhile the identity is primarily used on matrices, it holds in a general ring or in an Ab-category.\n\nThe Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula. There are no published results concerning its error bounds. Anecdotal evidence suggests that it may diverge even for seemingly benign examples (when both the original and modified matrices are well-conditioned).\n\n## Discussion\n\nTo prove this result, we will start by proving a simpler one. Replacing A and C with the identity matrix I, we obtain another identity which is a bit simpler:\n\n$\\left(I+UV\\right)^{-1}=I-U\\left(I+VU\\right)^{-1}V.$", null, "To recover the original equation from this reduced identity, set $U=A^{-1}X$", null, "and $V=CY$", null, ".\n\nThis identity itself can be viewed as the combination of two simpler identities. We obtain the first identity from\n\n$I=(I+P)^{-1}(I+P)=(I+P)^{-1}+(I+P)^{-1}P$", null, ",\n\nthus,\n\n$(I+P)^{-1}=I-(I+P)^{-1}P$", null, ",\n\nand similarly\n\n$(I+P)^{-1}=I-P(I+P)^{-1}.$", null, "The second identity is the so-called push-through identity\n\n$(I+UV)^{-1}U=U(I+VU)^{-1}$", null, "that we obtain from\n\n$U(I+VU)=(I+UV)U$", null, "after multiplying by $(I+VU)^{-1}$", null, "on the right and by $(I+UV)^{-1}$", null, "on the left.\n\nPutting all together,\n\n$\\left(I+UV\\right)^{-1}=I-UV\\left(I+UV\\right)^{-1}=I-U\\left(I+VU\\right)^{-1}V.$", null, "where the first and second equality come from the first and second identity, respectively.\n\n### Special cases\n\nWhen $V,U$", null, "are vectors, the identity reduces to the Sherman–Morrison formula.\n\nIn the scalar case, the reduced version is simply\n\n${\\frac {1}{1+uv}}=1-{\\frac {uv}{1+uv}}.$", null, "#### Inverse of a sum\n\nIf n = k and U = V = In is the identity matrix, then\n\n{\\begin{aligned}\\left({A}+{B}\\right)^{-1}&=A^{-1}-A^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}\\\\&={A}^{-1}-{A}^{-1}\\left({A}{B}^{-1}+{I}\\right)^{-1}.\\end{aligned}}", null, "Continuing with the merging of the terms of the far right-hand side of the above equation results in Hua's identity\n\n$\\left({A}+{B}\\right)^{-1}={A}^{-1}-\\left({A}+{A}{B}^{-1}{A}\\right)^{-1}.$", null, "Another useful form of the same identity is\n\n$\\left({A}-{B}\\right)^{-1}={A}^{-1}+{A}^{-1}{B}\\left({A}-{B}\\right)^{-1},$", null, "which, unlike those above, is valid even if $B$", null, "is singular, and has a recursive structure that yields\n\n$\\left({A}-{B}\\right)^{-1}=\\sum _{k=0}^{\\infty }\\left({A}^{-1}{B}\\right)^{k}{A}^{-1}$", null, "if the spectral radius of $A^{-1}B$", null, "is less than one. That is, if the above sum converges then it is equal to $(A-B)^{-1}$", null, ".\n\nThis form can be used in perturbative expansions where B is a perturbation of A.\n\n### Variations\n\n#### Binomial inverse theorem\n\nIf A, B, U, V are matrices of sizes n×n, k×k, n×k, k×n, respectively, then\n\n$\\left(A+UBV\\right)^{-1}=A^{-1}-A^{-1}UB\\left(B+BVA^{-1}UB\\right)^{-1}BVA^{-1}$", null, "provided A and B + BVA−1UB are nonsingular. Nonsingularity of the latter requires that B−1 exist since it equals B(I + VA−1UB) and the rank of the latter cannot exceed the rank of B.\n\nSince B is invertible, the two B terms flanking the parenthetical quantity inverse in the right-hand side can be replaced with (B−1)−1, which results in the original Woodbury identity.\n\nA variation for when B is singular and possibly even non-square:\n\n$(A+UBV)^{-1}=A^{-1}-A^{-1}U(I+BVA^{-1}U)^{-1}BVA^{-1}.$", null, "Formulas also exist for certain cases in which A is singular.\n\n#### Pseudoinverse with positive semidefinite matrices\n\nIn general Woodbury's identity is not valid if one or more inverses are replaced by (Moore–Penrose) pseudoinverses. However, if $A$", null, "and $C$", null, "are positive semidefinite, and $V=U^{\\mathrm {H} }$", null, "(implying that $A+UCV$", null, "is itself positive semidefinite), then the following formula provides a generalization:\n\n{\\begin{aligned}(XX^{\\mathrm {H} }+YY^{\\mathrm {H} })^{+}&=(ZZ^{\\mathrm {H} })^{+}+(I-YZ^{+})^{\\mathrm {H} }X^{+\\mathrm {H} }EX^{+}(I-YZ^{+}),\\\\Z&=(I-XX^{+})Y,\\\\E&=I-X^{+}Y(I-Z^{+}Z)F^{-1}(X^{+}Y)^{\\mathrm {H} },\\\\F&=I+(I-Z^{+}Z)Y^{\\mathrm {H} }(XX^{\\mathrm {H} })^{+}Y(I-Z^{+}Z),\\end{aligned}}", null, "where $A+UCU^{\\mathrm {H} }$", null, "can be written as $XX^{\\mathrm {H} }+YY^{\\mathrm {H} }$", null, "because any positive semidefinite matrix is equal to $MM^{\\mathrm {H} }$", null, "for some $M$", null, ".\n\n## Derivations\n\n### Direct proof\n\nThe formula can be proven by checking that $(A+UCV)$", null, "times its alleged inverse on the right side of the Woodbury identity gives the identity matrix:\n\n{\\begin{aligned}&\\left(A+UCV\\right)\\left[A^{-1}-A^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}\\right]\\\\={}&\\left\\{I-U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}\\right\\}+\\left\\{UCVA^{-1}-UCVA^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}\\right\\}\\\\={}&\\left\\{I+UCVA^{-1}\\right\\}-\\left\\{U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}+UCVA^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}\\right\\}\\\\={}&I+UCVA^{-1}-\\left(U+UCVA^{-1}U\\right)\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}\\\\={}&I+UCVA^{-1}-UC\\left(C^{-1}+VA^{-1}U\\right)\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}\\\\={}&I+UCVA^{-1}-UCVA^{-1}\\\\={}&I.\\end{aligned}}", null, "### Alternative proofs\n\nAlgebraic proof\n\nFirst consider these useful identities,\n\n{\\begin{aligned}U+UCVA^{-1}U&=UC\\left(C^{-1}+VA^{-1}U\\right)=\\left(A+UCV\\right)A^{-1}U\\\\\\left(A+UCV\\right)^{-1}UC&=A^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}\\end{aligned}}", null, "Now,\n\n{\\begin{aligned}A^{-1}&=\\left(A+UCV\\right)^{-1}\\left(A+UCV\\right)A^{-1}\\\\&=\\left(A+UCV\\right)^{-1}\\left(I+UCVA^{-1}\\right)\\\\&=\\left(A+UCV\\right)^{-1}+\\left(A+UCV\\right)^{-1}UCVA^{-1}\\\\&=\\left(A+UCV\\right)^{-1}+A^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}.\\end{aligned}}", null, "Derivation via blockwise elimination\n\nDeriving the Woodbury matrix identity is easily done by solving the following block matrix inversion problem\n\n${\\begin{bmatrix}A&U\\\\V&-C^{-1}\\end{bmatrix}}{\\begin{bmatrix}X\\\\Y\\end{bmatrix}}={\\begin{bmatrix}I\\\\0\\end{bmatrix}}.$", null, "Expanding, we can see that the above reduces to\n\n${\\begin{cases}AX+UY=I\\\\VX-C^{-1}Y=0\\end{cases}}$", null, "which is equivalent to $(A+UCV)X=I$", null, ". Eliminating the first equation, we find that $X=A^{-1}(I-UY)$", null, ", which can be substituted into the second to find $VA^{-1}(I-UY)=C^{-1}Y$", null, ". Expanding and rearranging, we have $VA^{-1}=\\left(C^{-1}+VA^{-1}U\\right)Y$", null, ", or $\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}=Y$", null, ". Finally, we substitute into our $AX+UY=I$", null, ", and we have $AX+U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}=I$", null, ". Thus,\n\n$(A+UCV)^{-1}=X=A^{-1}-A^{-1}U\\left(C^{-1}+VA^{-1}U\\right)^{-1}VA^{-1}.$", null, "We have derived the Woodbury matrix identity.\n\nDerivation from LDU decomposition\n\nWe start by the matrix\n\n${\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}$", null, "By eliminating the entry under the A (given that A is invertible) we get\n\n${\\begin{bmatrix}I&0\\\\-VA^{-1}&I\\end{bmatrix}}{\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}={\\begin{bmatrix}A&U\\\\0&C-VA^{-1}U\\end{bmatrix}}$", null, "Likewise, eliminating the entry above C gives\n\n${\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}{\\begin{bmatrix}I&-A^{-1}U\\\\0&I\\end{bmatrix}}={\\begin{bmatrix}A&0\\\\V&C-VA^{-1}U\\end{bmatrix}}$", null, "Now combining the above two, we get\n\n${\\begin{bmatrix}I&0\\\\-VA^{-1}&I\\end{bmatrix}}{\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}{\\begin{bmatrix}I&-A^{-1}U\\\\0&I\\end{bmatrix}}={\\begin{bmatrix}A&0\\\\0&C-VA^{-1}U\\end{bmatrix}}$", null, "Moving to the right side gives\n\n${\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}={\\begin{bmatrix}I&0\\\\VA^{-1}&I\\end{bmatrix}}{\\begin{bmatrix}A&0\\\\0&C-VA^{-1}U\\end{bmatrix}}{\\begin{bmatrix}I&A^{-1}U\\\\0&I\\end{bmatrix}}$", null, "which is the LDU decomposition of the block matrix into an upper triangular, diagonal, and lower triangular matrices.\n\nNow inverting both sides gives\n\n{\\begin{aligned}{\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}^{-1}&={\\begin{bmatrix}I&A^{-1}U\\\\0&I\\end{bmatrix}}^{-1}{\\begin{bmatrix}A&0\\\\0&C-VA^{-1}U\\end{bmatrix}}^{-1}{\\begin{bmatrix}I&0\\\\VA^{-1}&I\\end{bmatrix}}^{-1}\\\\[8pt]&={\\begin{bmatrix}I&-A^{-1}U\\\\0&I\\end{bmatrix}}{\\begin{bmatrix}A^{-1}&0\\\\0&\\left(C-VA^{-1}U\\right)^{-1}\\end{bmatrix}}{\\begin{bmatrix}I&0\\\\-VA^{-1}&I\\end{bmatrix}}\\\\[8pt]&={\\begin{bmatrix}A^{-1}+A^{-1}U\\left(C-VA^{-1}U\\right)^{-1}VA^{-1}&-A^{-1}U\\left(C-VA^{-1}U\\right)^{-1}\\\\-\\left(C-VA^{-1}U\\right)^{-1}VA^{-1}&\\left(C-VA^{-1}U\\right)^{-1}\\end{bmatrix}}\\qquad \\mathrm {(1)} \\end{aligned}}", null, "We could equally well have done it the other way (provided that C is invertible) i.e.\n\n${\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}={\\begin{bmatrix}I&UC^{-1}\\\\0&I\\end{bmatrix}}{\\begin{bmatrix}A-UC^{-1}V&0\\\\0&C\\end{bmatrix}}{\\begin{bmatrix}I&0\\\\C^{-1}V&I\\end{bmatrix}}$", null, "Now again inverting both sides,\n\n{\\begin{aligned}{\\begin{bmatrix}A&U\\\\V&C\\end{bmatrix}}^{-1}&={\\begin{bmatrix}I&0\\\\C^{-1}V&I\\end{bmatrix}}^{-1}{\\begin{bmatrix}A-UC^{-1}V&0\\\\0&C\\end{bmatrix}}^{-1}{\\begin{bmatrix}I&UC^{-1}\\\\0&I\\end{bmatrix}}^{-1}\\\\[8pt]&={\\begin{bmatrix}I&0\\\\-C^{-1}V&I\\end{bmatrix}}{\\begin{bmatrix}\\left(A-UC^{-1}V\\right)^{-1}&0\\\\0&C^{-1}\\end{bmatrix}}{\\begin{bmatrix}I&-UC^{-1}\\\\0&I\\end{bmatrix}}\\\\[8pt]&={\\begin{bmatrix}\\left(A-UC^{-1}V\\right)^{-1}&-\\left(A-UC^{-1}V\\right)^{-1}UC^{-1}\\\\-C^{-1}V\\left(A-UC^{-1}V\\right)^{-1}&C^{-1}+C^{-1}V\\left(A-UC^{-1}V\\right)^{-1}UC^{-1}\\end{bmatrix}}\\qquad \\mathrm {(2)} \\end{aligned}}", null, "Now comparing elements (1, 1) of the RHS of (1) and (2) above gives the Woodbury formula\n\n$\\left(A-UC^{-1}V\\right)^{-1}=A^{-1}+A^{-1}U\\left(C-VA^{-1}U\\right)^{-1}VA^{-1}.$", null, "## Applications\n\nThis identity is useful in certain numerical computations where A−1 has already been computed and it is desired to compute (A + UCV)−1. With the inverse of A available, it is only necessary to find the inverse of C−1 + VA−1U in order to obtain the result using the right-hand side of the identity. If C has a much smaller dimension than A, this is more efficient than inverting A + UCV directly. A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.\n\nThis is applied, e.g., in the Kalman filter and recursive least squares methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new observation is processed at a time. This significantly speeds up the often real time calculations of the filter.\n\nIn the case when C is the identity matrix I, the matrix $I+VA^{-1}U$", null, "is known in numerical linear algebra and numerical partial differential equations as the capacitance matrix." ]	[ null, "https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffa2c14bb438728d93f2cdf7ea6657338ab8fb7", null, "https://wikimedia.org/api/rest_v1/media/math/render/svg/5eeb70301ac7e343fb8623767de8a0649d6e283e", null, "https://wikimedia.org/api/rest_v1/media/math/render/svg/e63b0d78508c049b7171f2c8012f60c0c077a9c5", null, "https://wikimedia.org/api/rest_v1/media/math/render/svg/d39971493c85bd36755ac1ef34a843f675cdc6a4", null, "https://wikimedia.org/api/rest_v1/media/math/render/svg/214a791b48df362d0ba51fbe3c6bc4c9ce5b207c", null, "https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9318fa3e0d6d84c24d203af60746322ff4e5e9", null, 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http://engineeronadisk.com/V3/engineeronadisk-164.html	[ "• Interfacing for Acquisition of Signals from Sensors and Generation of Signals for Actuators\n\n• Used by Computers, PLC’s, PID Controllers, etc.\n\n• Computers are designed to handle both input and output (I/O) data\n\n• Two main types of data I/O for computers\n\nAnalog\n\nDigital", null, "• A Continuous signal is sampled by the computer\n\n• The computer uses approximation techniques to estimate the analog value during the sampling window.\n\n• An example of an A/D, D/A control of a process is shown below", null, "• Multiplexers are used when a number of signals are to be input to a single A/D converter. This allows each of a number of channels to be sampled, one at a time\n\n• Signal conditioners are often to amplify, or filter signals coming from transducers, before they are read by the A/D converter.\n\n• Output drivers and amplifiers are often required to drive output devices when using D/A\n\n• Sampling problems occur with A/D conversion. Because readings are taken periodically (not continually), the Nyquist criterion specifies that sampling frequencies should be twice the frequency of the signal being measured, otherwise aliasing will occur.\n\n• Since the sampling window for a signal is short, noise will have added effect on the signal read. For example, a momentary voltage spike might result in a higher than normal reading.\n\n• When an analog value is converted to or from digital values, a quantization error is involved. The digital numbering scheme means that for an 8 bit A/D converter, there is a resolution of 256 values between maximum and minimum. This means that there is a round off error of approximately 0.4%.", null, "#### 26.1.1 Analog To Digital Conversions\n\n• When there are analog values outside a computer, and we plan to read these to digital values, there are a variety of factors to consider,\n\nwhen the sample is requested, a short period of time passes before the final sample value is obtained.\n\nthe sample value is ‘frozen’ after a sample interval.\n\nafter the sample is taken, the system may change\n\nsample values can be very sensitive to noise\n\nthe continuous values of the signal loose some accuracy when conversion to a digital number\n\n• Consider the conversion process pictured below,", null, "• Once this signal is processes through a typical A/D converter we get the following relations (these may vary slightly for different types of A/D converters).", null, "Problem 26.1 We are given a 12 bit analog input with a range of -10V to 10V. If we put in 2.735V, what will the integer value be after the A/D conversion? What is the error? What voltage can we calculate?\n\n• In most applications a sample is taken at regular intervals, with a period of ‘T’ seconds.\n\n• In practice the sample interval is kept as small as possible. (i.e., tau << T)\n\n• If we are sampling a periodic signal that changes near or faster that the sampling rate, there is a chance that we will get a signal that appears chaotic, or seems to be a lower frequency. This phenomenon is known as aliasing.\n\n• Quite often an A/D converter will multiplex between various inputs. As it switches the voltage will be sampled by a ‘sample and hold circuit’. This will then be converted to a digital value. The sample and hold circuits can be used before the multiplexer to collect data values at the same instant in time.\n\n• A simple type of A/D converter is shown below. It is known as a successive approximation type.", null, "#### 26.1.2 Analog Inputs With a PLC\n\n• To input analog values into a PLC we use the block transfer commands. These allow control information to the input card and retrieve results.\n\n• The example below shows ladder logic to do an analog input.", null, "• The block that needs to be written to an 1771-IFE analog input card is shown below. This is a 12 bit card, so the range will have up to 2**12 = 4096 values.", null, "• After the input card reads the values, the results are returned in a block. The structure of the block is shown below.", null, "26.2 Analog Outputs\n\n• After we have used a controller equation to estimate a value to put into our process, we must convert this from a digital value in the computers memory, to a physical voltage.\n\n• This voltage is typically limited to 20mA in most computer boards, and drawing near this current reduces accuracy and life of the board.\n\n• A simple circuit is shown below for a simple digital to analog converter.", null, "• The calculations for the A/D converter resolution and accuracy still apply.\n\nProblem 26.2 We need to select a digital to analog converter for an application. The output will vary from -5V to 10V DC, and we need to be able to specify the voltage to within 50mV. What resolution will be required? How many bits will this D/A converter need? What will the accuracy be?\n\n#### 26.2.1 Analog Outputs With A PLC\n\n• An example of an output card is 1771-OFE.\n\n• To output a value we only need to write a single value to the output card", null, "• The format for the block that is to be written to the card is shown below.", null, "26.3 Design Cases\n\n#### 26.3.1 Oven Temperature Control\n\n• Design an analog controller that will read an oven temperature, and when it passes 1200 degrees the oven will be turned off. The voltage from the thermocouple is passed through a signal conditioner that gives 1V at 500F and 3V at 1500F. The controller should have a start button and E-stop.\n\n#### 26.3.2 Statistical Process Control (SPC)\n\n• We can do SPC checking using analog inputs, and built in statistics functions.\n\n• Recall the basic equations for a control chart.", null, "• The general flow would be,\n\n1. Read sampled inputs.\n\n2. Randomly select values and calculate the average and store in memory. Calculate the standard deviation of the stored values.\n\n3. Compare the inputs to the standard deviation. If it is larger than 3 deviations from the mean, halt the process.\n\n4. If it is larger than 2 then increase a counter A, or if it is larger than 1 increase a second counter B. If it is less than 1 reset the counters.\n\n5. If counter A is =3 or B is =5 then shut down.\n\n6. Goto 1.\n\n26.4 Problems\n\nProblem 26.3 Write a program that will input an analog voltage, do the calculation below, and output an analog voltage.", null, "Problem 26.4 The following calculation will be made when input ‘A’ is true. If the result ‘x’ is between 1 and 10 then the output ‘B’ will be turned on. The value of ‘x’ will be output as an analog voltage. Create a ladder logic program to perform these tasks.", null, "", null, "Problem 26.5 You are developing a controller for a game that measures hand strength. To do this a ‘START’ button is pushed, 3 seconds later a ‘LIGHT’ is turned on for one second to let the user know when to start squeezing. The analog value is read at 0.3s after the light is on. The value is converted to a force ‘F’ with the equation below. The force is displayed by converting it to BCD and writing it to an output card (O:001). If the value exceeds 100 then a ‘BIG_LIGHT’ and ‘SIREN’ are turned on for 5sec. Use a structured design technique to develop ladder logic.", null, "", null, "" ]	[ null, "http://engineeronadisk.com/V3/engineeronadisk-3080.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3081.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3082.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3083.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3084.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3085.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3086.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3087.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3088.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3089.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3090.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3091.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3092.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3093.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3094.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3095.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3096.gif", null, "http://engineeronadisk.com/V3/engineeronadisk-3097.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8895471,"math_prob":0.975874,"size":7104,"snap":"2019-13-2019-22","text_gpt3_token_len":1622,"char_repetition_ratio":0.13239437,"word_repetition_ratio":0.009375,"special_character_ratio":0.2339527,"punctuation_ratio":0.09832636,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99089026,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],"im_url_duplicate_count":[null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-21T19:15:03Z\",\"WARC-Record-ID\":\"<urn:uuid:37fe5ed7-57e6-42a2-bbb7-9acdb6e917de>\",\"Content-Length\":\"24065\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:27041b91-bfce-4bb0-a902-dac22875afdc>\",\"WARC-Concurrent-To\":\"<urn:uuid:c4e1ba3e-7e66-4d02-8d37-710df50c8a60>\",\"WARC-IP-Address\":\"107.180.55.229\",\"WARC-Target-URI\":\"http://engineeronadisk.com/V3/engineeronadisk-164.html\",\"WARC-Payload-Digest\":\"sha1:MU7CYCGHAGWSBRJMXRX6JJZ3ZFRRSQNK\",\"WARC-Block-Digest\":\"sha1:MRTMJ75BXO5GOUUY6VIFLLJAN7WDJEHB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202530.49_warc_CC-MAIN-20190321172751-20190321194751-00236.warc.gz\"}"}
http://ixtrieve.fh-koeln.de/birds/litie/document/4566	[ "# Document (#4566)\n\nAuthor\nMiller, P.L.\nTitle\nPrototyping an institutional IAIMS/UMLS information environment for a academic medical center\nSource\nBulletin of the Medical Library Association. 80(1992) no.3, S.281-287\nYear\n1992\nAbstract\nA prototype design is discussed which shows the link between the US Matioanl Library of Medicine Integrated Academic Information Management System (IAIMS) and the Unified Medical Language System (UMLS)\nField\nMedizin\nObject\nIAIMS\nUMLS\n\n## Similar documents (author)\n\n1. Miller, G.A.: ¬The magical number, seven plus or minus two : some limits on our capacity for processing information (1956) 4.45\n```4.449747 = sum of:\n4.449747 = weight(author_txt:miller in 2752) [ClassicSimilarity], result of:\n4.449747 = fieldWeight in 2752, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.1195955 = idf(docFreq=93, maxDocs=42740)\n0.625 = fieldNorm(doc=2752)\n```\n2. Miller, M.L.: Automation and LCSH (1986) 4.45\n```4.449747 = sum of:\n4.449747 = weight(author_txt:miller in 2900) [ClassicSimilarity], result of:\n4.449747 = fieldWeight in 2900, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.1195955 = idf(docFreq=93, maxDocs=42740)\n0.625 = fieldNorm(doc=2900)\n```\n3. Miller, G.A.: Psychology and information (1968) 4.45\n```4.449747 = sum of:\n4.449747 = weight(author_txt:miller in 3245) [ClassicSimilarity], result of:\n4.449747 = fieldWeight in 3245, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.1195955 = idf(docFreq=93, maxDocs=42740)\n0.625 = fieldNorm(doc=3245)\n```\n4. Miller, D.C.: Evaluating CD-ROMs : to buy or what to buy (1987) 4.45\n```4.449747 = sum of:\n4.449747 = weight(author_txt:miller in 3305) [ClassicSimilarity], result of:\n4.449747 = fieldWeight in 3305, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.1195955 = idf(docFreq=93, maxDocs=42740)\n0.625 = fieldNorm(doc=3305)\n```\n5. Miller, D.J.: Advanced Freestyle searching with Lexis-Nexis (1997) 4.45\n```4.449747 = sum of:\n4.449747 = weight(author_txt:miller in 4225) [ClassicSimilarity], result of:\n4.449747 = fieldWeight in 4225, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n7.1195955 = idf(docFreq=93, maxDocs=42740)\n0.625 = fieldNorm(doc=4225)\n```\n\n## Similar documents (content)\n\n1. Squires, S.J.: Access to biomedical information : the Unified Medical Language System (1993) 0.79\n```0.78786916 = sum of:\n0.78786916 = product of:\n1.7333121 = sum of:\n0.016670894 = weight(abstract_txt:library in 6705) [ClassicSimilarity], result of:\n0.016670894 = score(doc=6705,freq=1.0), product of:\n0.047882635 = queryWeight, product of:\n1.0851197 = boost\n3.1831915 = idf(docFreq=4815, maxDocs=42740)\n0.013862374 = queryNorm\n0.34816158 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.1831915 = idf(docFreq=4815, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.038123764 = weight(abstract_txt:language in 6705) [ClassicSimilarity], result of:\n0.038123764 = score(doc=6705,freq=1.0), product of:\n0.08311307 = queryWeight, product of:\n1.4296288 = boost\n4.1938066 = idf(docFreq=1752, maxDocs=42740)\n0.013862374 = queryNorm\n0.45869762 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.1938066 = idf(docFreq=1752, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.039579257 = weight(abstract_txt:management in 6705) [ClassicSimilarity], result of:\n0.039579257 = score(doc=6705,freq=1.0), product of:\n0.08521523 = queryWeight, product of:\n1.4475956 = boost\n4.246512 = idf(docFreq=1662, maxDocs=42740)\n0.013862374 = queryNorm\n0.46446225 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.246512 = idf(docFreq=1662, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.05160368 = weight(abstract_txt:environment in 6705) [ClassicSimilarity], result of:\n0.05160368 = score(doc=6705,freq=1.0), product of:\n0.101701155 = queryWeight, product of:\n1.5814358 = boost\n4.6391315 = idf(docFreq=1122, maxDocs=42740)\n0.013862374 = queryNorm\n0.50740504 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.6391315 = idf(docFreq=1122, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.02569424 = weight(abstract_txt:information in 6705) [ClassicSimilarity], result of:\n0.02569424 = score(doc=6705,freq=3.0), product of:\n0.055812597 = queryWeight, product of:\n1.6567987 = boost\n2.430104 = idf(docFreq=10226, maxDocs=42740)\n0.013862374 = queryNorm\n0.4603663 = fieldWeight in 6705, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n2.430104 = idf(docFreq=10226, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.14776626 = weight(abstract_txt:medicine in 6705) [ClassicSimilarity], result of:\n0.14776626 = score(doc=6705,freq=1.0), product of:\n0.20507953 = queryWeight, product of:\n2.2456899 = boost\n6.5877166 = idf(docFreq=159, maxDocs=42740)\n0.013862374 = queryNorm\n0.7205315 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.5877166 = idf(docFreq=159, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.14991301 = weight(abstract_txt:unified in 6705) [ClassicSimilarity], result of:\n0.14991301 = score(doc=6705,freq=1.0), product of:\n0.20706102 = queryWeight, product of:\n2.2565129 = boost\n6.6194654 = idf(docFreq=154, maxDocs=42740)\n0.013862374 = queryNorm\n0.72400403 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.6194654 = idf(docFreq=154, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.039329555 = weight(abstract_txt:system in 6705) [ClassicSimilarity], result of:\n0.039329555 = score(doc=6705,freq=1.0), product of:\n0.10691241 = queryWeight, product of:\n2.293072 = boost\n3.3633559 = idf(docFreq=4021, maxDocs=42740)\n0.013862374 = queryNorm\n0.36786705 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.3633559 = idf(docFreq=4021, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.2025068 = weight(abstract_txt:medical in 6705) [ClassicSimilarity], result of:\n0.2025068 = score(doc=6705,freq=1.0), product of:\n0.31879282 = queryWeight, product of:\n3.95966 = boost\n5.8078184 = idf(docFreq=348, maxDocs=42740)\n0.013862374 = queryNorm\n0.6352301 = fieldWeight in 6705, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n5.8078184 = idf(docFreq=348, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n1.0221246 = weight(abstract_txt:umls in 6705) [ClassicSimilarity], result of:\n1.0221246 = score(doc=6705,freq=3.0), product of:\n0.65039563 = queryWeight, product of:\n5.655779 = boost\n8.295595 = idf(docFreq=28, maxDocs=42740)\n0.013862374 = queryNorm\n1.5715429 = fieldWeight in 6705, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n8.295595 = idf(docFreq=28, maxDocs=42740)\n0.109375 = fieldNorm(doc=6705)\n0.45454547 = coord(10/22)\n```\n2. Stuart, S.J.; Powell, T.; Humphreys, B.L.: ¬The Unified Medical Language System (UMLS) project (2002) 0.55\n```0.550092 = sum of:\n0.550092 = product of:\n1.100184 = sum of:\n0.007907954 = weight(abstract_txt:which in 263) [ClassicSimilarity], result of:\n0.007907954 = score(doc=263,freq=2.0), product of:\n0.040665183 = queryWeight, product of:\n2.9334934 = idf(docFreq=6181, maxDocs=42740)\n0.013862374 = queryNorm\n0.19446498 = fieldWeight in 263, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n2.9334934 = idf(docFreq=6181, maxDocs=42740)\n0.046875 = fieldNorm(doc=263)\n0.012374929 = weight(abstract_txt:library in 263) [ClassicSimilarity], result of:\n0.012374929 = score(doc=263,freq=3.0), product of:\n0.047882635 = queryWeight, product of:\n1.0851197 = boost\n3.1831915 = idf(docFreq=4815, maxDocs=42740)\n0.013862374 = queryNorm\n0.25844294 = fieldWeight in 263, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n3.1831915 = idf(docFreq=4815, maxDocs=42740)\n0.046875 = fieldNorm(doc=263)\n0.02310649 = weight(abstract_txt:language in 263) [ClassicSimilarity], result of:\n0.02310649 = score(doc=263,freq=2.0), product of:\n0.08311307 = queryWeight, product of:\n1.4296288 = boost\n4.1938066 = idf(docFreq=1752, maxDocs=42740)\n0.013862374 = queryNorm\n0.27801272 = fieldWeight in 263, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n4.1938066 = idf(docFreq=1752, maxDocs=42740)\n0.046875 = fieldNorm(doc=263)\n0.022023635 = weight(abstract_txt:information in 263) [ClassicSimilarity], result of:\n0.022023635 = score(doc=263,freq=12.0), product of:\n0.055812597 = queryWeight, product of:\n1.6567987 = boost\n2.430104 = idf(docFreq=10226, maxDocs=42740)\n0.013862374 = queryNorm\n0.3945997 = fieldWeight in 263, product of:\n3.4641016 = tf(freq=12.0), with freq of:\n12.0 = termFreq=12.0\n2.430104 = idf(docFreq=10226, maxDocs=42740)\n0.046875 = fieldNorm(doc=263)\n0.034907 = 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Nelson, S.J.; Powell, T.; Srinivasan, S.; Humphreys, B.L.: Unified Medical Language System® (UMLS®) Project (2009) 0.53\n```0.5345924 = sum of:\n0.5345924 = product of:\n1.470129 = sum of:\n0.014289338 = weight(abstract_txt:library in 1702) [ClassicSimilarity], result of:\n0.014289338 = score(doc=1702,freq=1.0), product of:\n0.047882635 = queryWeight, product of:\n1.0851197 = boost\n3.1831915 = idf(docFreq=4815, maxDocs=42740)\n0.013862374 = queryNorm\n0.2984242 = fieldWeight in 1702, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.1831915 = idf(docFreq=4815, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.032677513 = weight(abstract_txt:language in 1702) [ClassicSimilarity], result of:\n0.032677513 = score(doc=1702,freq=1.0), product of:\n0.08311307 = queryWeight, product of:\n1.4296288 = boost\n4.1938066 = idf(docFreq=1752, maxDocs=42740)\n0.013862374 = queryNorm\n0.39316937 = fieldWeight in 1702, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n4.1938066 = idf(docFreq=1752, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.012715351 = weight(abstract_txt:information in 1702) [ClassicSimilarity], result of:\n0.012715351 = score(doc=1702,freq=1.0), product of:\n0.055812597 = queryWeight, product of:\n1.6567987 = boost\n2.430104 = idf(docFreq=10226, maxDocs=42740)\n0.013862374 = queryNorm\n0.22782224 = fieldWeight in 1702, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n2.430104 = idf(docFreq=10226, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.12665679 = weight(abstract_txt:medicine in 1702) [ClassicSimilarity], result of:\n0.12665679 = score(doc=1702,freq=1.0), product of:\n0.20507953 = queryWeight, product of:\n2.2456899 = boost\n6.5877166 = idf(docFreq=159, maxDocs=42740)\n0.013862374 = queryNorm\n0.6175984 = fieldWeight in 1702, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.5877166 = idf(docFreq=159, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.12849687 = weight(abstract_txt:unified in 1702) [ClassicSimilarity], result of:\n0.12849687 = score(doc=1702,freq=1.0), product of:\n0.20706102 = queryWeight, product of:\n2.2565129 = boost\n6.6194654 = idf(docFreq=154, maxDocs=42740)\n0.013862374 = queryNorm\n0.6205749 = fieldWeight in 1702, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n6.6194654 = idf(docFreq=154, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.033711046 = weight(abstract_txt:system in 1702) [ClassicSimilarity], result of:\n0.033711046 = score(doc=1702,freq=1.0), product of:\n0.10691241 = queryWeight, product of:\n2.293072 = boost\n3.3633559 = idf(docFreq=4021, maxDocs=42740)\n0.013862374 = queryNorm\n0.31531462 = fieldWeight in 1702, product of:\n1.0 = tf(freq=1.0), with freq of:\n1.0 = termFreq=1.0\n3.3633559 = idf(docFreq=4021, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.2454753 = weight(abstract_txt:medical in 1702) [ClassicSimilarity], result of:\n0.2454753 = score(doc=1702,freq=2.0), product of:\n0.31879282 = queryWeight, product of:\n3.95966 = boost\n5.8078184 = idf(docFreq=348, maxDocs=42740)\n0.013862374 = queryNorm\n0.7700152 = fieldWeight in 1702, product of:\n1.4142135 = tf(freq=2.0), with freq of:\n2.0 = termFreq=2.0\n5.8078184 = idf(docFreq=348, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.8761068 = weight(abstract_txt:umls in 1702) [ClassicSimilarity], result of:\n0.8761068 = score(doc=1702,freq=3.0), product of:\n0.65039563 = queryWeight, product of:\n5.655779 = boost\n8.295595 = idf(docFreq=28, maxDocs=42740)\n0.013862374 = queryNorm\n1.3470367 = fieldWeight in 1702, product of:\n1.7320508 = tf(freq=3.0), with freq of:\n3.0 = termFreq=3.0\n8.295595 = idf(docFreq=28, maxDocs=42740)\n0.09375 = fieldNorm(doc=1702)\n0.36363637 = coord(8/22)\n```\n4. 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https://www.colorhexa.com/22c5b3	[ "# #22c5b3 Color Information\n\nIn a RGB color space, hex #22c5b3 is composed of 13.3% red, 77.3% green and 70.2% blue. Whereas in a CMYK color space, it is composed of 82.7% cyan, 0% magenta, 9.1% yellow and 22.7% black. It has a hue angle of 173.4 degrees, a saturation of 70.6% and a lightness of 45.3%. #22c5b3 color hex could be obtained by blending #44ffff with #008b67. Closest websafe color is: #33cccc.\n\n• R 13\n• G 77\n• B 70\nRGB color chart\n• C 83\n• M 0\n• Y 9\n• K 23\nCMYK color chart\n\n#22c5b3 color description : Strong cyan.\n\n# #22c5b3 Color Conversion\n\nThe hexadecimal color #22c5b3 has RGB values of R:34, G:197, B:179 and CMYK values of C:0.83, M:0, Y:0.09, K:0.23. Its decimal value is 2278835.\n\nHex triplet RGB Decimal 22c5b3 `#22c5b3` 34, 197, 179 `rgb(34,197,179)` 13.3, 77.3, 70.2 `rgb(13.3%,77.3%,70.2%)` 83, 0, 9, 23 173.4°, 70.6, 45.3 `hsl(173.4,70.6%,45.3%)` 173.4°, 82.7, 77.3 33cccc `#33cccc`\nCIE-LAB 71.909, -43.245, -2.248 28.76, 43.524, 49.531 0.236, 0.357, 43.524 71.909, 43.303, 182.976 71.909, -55.411, 3.264 65.973, -37.639, 1.668 00100010, 11000101, 10110011\n\n# Color Schemes with #22c5b3\n\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #c52234\n``#c52234` `rgb(197,34,52)``\nComplementary Color\n• #22c562\n``#22c562` `rgb(34,197,98)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #2286c5\n``#2286c5` `rgb(34,134,197)``\nAnalogous Color\n• #c56222\n``#c56222` `rgb(197,98,34)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #c52286\n``#c52286` `rgb(197,34,134)``\nSplit Complementary Color\n• #c5b322\n``#c5b322` `rgb(197,179,34)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #b322c5\n``#b322c5` `rgb(179,34,197)``\nTriadic Color\n• #34c522\n``#34c522` `rgb(52,197,34)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #b322c5\n``#b322c5` `rgb(179,34,197)``\n• #c52234\n``#c52234` `rgb(197,34,52)``\nTetradic Color\n• #178478\n``#178478` `rgb(23,132,120)``\n• #1a9a8b\n``#1a9a8b` `rgb(26,154,139)``\n• #1eaf9f\n``#1eaf9f` `rgb(30,175,159)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #27dac6\n``#27dac6` `rgb(39,218,198)``\n• #3dddcc\n``#3dddcc` `rgb(61,221,204)``\n• #52e1d1\n``#52e1d1` `rgb(82,225,209)``\nMonochromatic Color\n\n# Alternatives to #22c5b3\n\nBelow, you can see some colors close to #22c5b3. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #22c58a\n``#22c58a` `rgb(34,197,138)``\n• #22c598\n``#22c598` `rgb(34,197,152)``\n• #22c5a5\n``#22c5a5` `rgb(34,197,165)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #22c5c1\n``#22c5c1` `rgb(34,197,193)``\n• #22bcc5\n``#22bcc5` `rgb(34,188,197)``\n• #22aec5\n``#22aec5` `rgb(34,174,197)``\nSimilar Colors\n\n# #22c5b3 Preview\n\nText with hexadecimal color #22c5b3\n\nThis text has a font color of #22c5b3.\n\n``<span style=\"color:#22c5b3;\">Text here</span>``\n#22c5b3 background color\n\nThis paragraph has a background color of #22c5b3.\n\n``<p style=\"background-color:#22c5b3;\">Content here</p>``\n#22c5b3 border color\n\nThis element has a border color of #22c5b3.\n\n``<div style=\"border:1px solid #22c5b3;\">Content here</div>``\nCSS codes\n``.text {color:#22c5b3;}``\n``.background {background-color:#22c5b3;}``\n``.border {border:1px solid #22c5b3;}``\n\n# Shades and Tints of #22c5b3\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #020d0c is the darkest color, while #fbfefe is the lightest one.\n\n• #020d0c\n``#020d0c` `rgb(2,13,12)``\n• #051e1b\n``#051e1b` `rgb(5,30,27)``\n• #082e2a\n``#082e2a` `rgb(8,46,42)``\n• #0b3f39\n``#0b3f39` `rgb(11,63,57)``\n• #0e5049\n``#0e5049` `rgb(14,80,73)``\n• #116158\n``#116158` `rgb(17,97,88)``\n• #147167\n``#147167` `rgb(20,113,103)``\n• #168276\n``#168276` `rgb(22,130,118)``\n• #199385\n``#199385` `rgb(25,147,133)``\n• #1ca495\n``#1ca495` `rgb(28,164,149)``\n• #1fb4a4\n``#1fb4a4` `rgb(31,180,164)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #25d6c2\n``#25d6c2` `rgb(37,214,194)``\nShade Color Variation\n• #33dcc9\n``#33dcc9` `rgb(51,220,201)``\n• #43dfcd\n``#43dfcd` `rgb(67,223,205)``\n• #54e1d2\n``#54e1d2` `rgb(84,225,210)``\n• #65e4d6\n``#65e4d6` `rgb(101,228,214)``\n• #75e7db\n``#75e7db` `rgb(117,231,219)``\n• #86eadf\n``#86eadf` `rgb(134,234,223)``\n• #97ede4\n``#97ede4` `rgb(151,237,228)``\n• #a8f0e8\n``#a8f0e8` `rgb(168,240,232)``\n• #b8f3ec\n``#b8f3ec` `rgb(184,243,236)``\n• #c9f6f1\n``#c9f6f1` `rgb(201,246,241)``\n• #daf9f5\n``#daf9f5` `rgb(218,249,245)``\n• #ebfbfa\n``#ebfbfa` `rgb(235,251,250)``\n• #fbfefe\n``#fbfefe` `rgb(251,254,254)``\nTint Color Variation\n\n# Tones of #22c5b3\n\nA tone is produced by adding gray to any pure hue. In this case, #727575 is the less saturated color, while #07e0c8 is the most saturated one.\n\n• #727575\n``#727575` `rgb(114,117,117)``\n• #697e7c\n``#697e7c` `rgb(105,126,124)``\n• #608783\n``#608783` `rgb(96,135,131)``\n• #579089\n``#579089` `rgb(87,144,137)``\n• #4e9990\n``#4e9990` `rgb(78,153,144)``\n• #46a197\n``#46a197` `rgb(70,161,151)``\n• #3daa9e\n``#3daa9e` `rgb(61,170,158)``\n• #34b3a5\n``#34b3a5` `rgb(52,179,165)``\n• #2bbcac\n``#2bbcac` `rgb(43,188,172)``\n• #22c5b3\n``#22c5b3` `rgb(34,197,179)``\n• #19ceba\n``#19ceba` `rgb(25,206,186)``\n• #10d7c1\n``#10d7c1` `rgb(16,215,193)``\n• #07e0c8\n``#07e0c8` `rgb(7,224,200)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #22c5b3 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.53020036,"math_prob":0.7182757,"size":3713,"snap":"2019-13-2019-22","text_gpt3_token_len":1700,"char_repetition_ratio":0.12105689,"word_repetition_ratio":0.011111111,"special_character_ratio":0.5542688,"punctuation_ratio":0.23809524,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9796325,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-22T19:23:39Z\",\"WARC-Record-ID\":\"<urn:uuid:5542d4f2-8547-4901-874e-d7519c3acade>\",\"Content-Length\":\"36448\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:75c38525-5c45-4edf-9e2f-04a54c6ae098>\",\"WARC-Concurrent-To\":\"<urn:uuid:41777565-cc00-44fe-9db7-81677affce02>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/22c5b3\",\"WARC-Payload-Digest\":\"sha1:QQUSKYBXTLK2KWUAKDE3EAK6BZR3S7TO\",\"WARC-Block-Digest\":\"sha1:OEQH3KVJPHF3QT33INZ2C735ESRWS3O6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232256948.48_warc_CC-MAIN-20190522183240-20190522205240-00285.warc.gz\"}"}
https://www.scienceforums.com/topic/37923-minkowski-spacetime-diagrams-re-assigned/page/3/	[ "Science Forums\n\n# Minkowski SpaceTime diagrams re assigned\n\n## Recommended Posts\n\nand here's another paradox for you, (a paradox is an unsolvable problem, like the twin paradox, still unsolved despite many weak attempts)\n\nIn the pic, from the development of the lorentz equation by Sal of Khan academy, but also as presented by many other uni professors...\n\nNow because we also know that\n\nx = ct\n\nand also that\n\nx' = ct' Because we set this up to be exactly like this)\n\nTherefore these two equations end up proving that\n\n0 = 0.\n\nBut I could have told you that in primary school", null, "• Replies 157\n• Created\n\n#### Posted Images\n\n4 minutes ago, TheProdigalProdigy said:\n\nI don't see how y=1 here as you claim.\n\nI think your problem is in arithmetic. And even if it was 1 it's being multiplied by c so it's still equal\n\n\"y\" is the vertical axis correct? (just checking)\n\nit was Time, but that was replaced with the Distance \"ct\", which we have set on our \"y\" axis scales to be equal to one unit of distance of the horizontal axis. so one unit on x distance is exactly the same as one unit on the \"y\" axis distance. That's why Light is at 45 degrees.\n\nSo \"y\" is one unit, and so is X one identical unit. 1:1 ratio.\n\n##### Share on other sites\n\n3 minutes ago, TheProdigalProdigy said:\n\nBecause ct, or x, can't be zero. First of all because c is 299mill\n\nplease work out the algebra, simplify these equations. the end result is simply 0=0\n\n##### Share on other sites\n\n2 minutes ago, TheProdigalProdigy said:\n\nYou're welcome\n\nthat does not help with the problem does it?\n\nYou are ignoring the issues and talking in circles.\n\nDid you run those equations with real values? no.\n\nDid you do the algebra on the other equations that end up proving that 0 = 0?\n\nNo.\n\n##### Share on other sites\n\nJust now, TheProdigalProdigy said:\n\nYou do know beta is like 8,000 something, right?\n\nThe proton electron mass ratio?\n\nbeta is NOT 8000 in this example, beta here is the ratio of the observers velocity over the velocity of light.\n\nbut even if it was 8000, it matters not, as it cancels itself out perfectly.\n\n0 still = 0\n\nDO THE ALGEBRA\n\n##### Share on other sites\n\n3 minutes ago, TheProdigalProdigy said:\n\nWhat did you admit on the first post in this page? That where y=1 it actually equals c. Right? Which makes xy a 1:1 ratio in both pictures.\n\nSo again, you're welcome. The issue you had was a non issue.\n\nIts still the same issue. Even if you DRAW the graph so that a division of one second on the y axis is the same physical length as a division on the x axis, which is 30000000meters, the unit of one second is NOT identical to a distance of 3 million meters. One is SECONDS, the other is Distance.\n\nSo on casual appearance, because you have set the distances between units on both axies as identical, which allowed you to substitute for one second, the distance of 3 million... this is going to cause a problem when your original equations all worked in seconds and meters, not on meters and meters.\n\nYou cant deduct a velocity from a distance. You can divide a distance by a velocity, giving time. But there is no valid equation that allows deducting a velocity from a distance.\n\n##### Share on other sites\n\n8 minutes ago, TheProdigalProdigy said:\n\nRemember what I said about philosophizing about the axiomatic nature of velocity or distance?\n\nI said it has no place in math.\n\nyou simply CANNOT deduct 3 miles per hour from 6 miles. Period, no philosophy involved.\n\n##### Share on other sites\n\n7 minutes ago, TheProdigalProdigy said:\n\nAlso a rule there, velocity of observer < c. But that's already a rule because of time dilation. So that's why it's set up that way. Nice\n\nHere's why lorentz end up as zero according to the Khan academy derivation.", null, "##### Share on other sites\n\n7 hours ago, TheProdigalProdigy said:\n\nAlready came to the same result.\n\nok take the first equation, x' = gamma(x- betact)\n\nlet x =10\n\nrewrite x' = gamma ( 10 - beta 10)\n\nlet beta = 0.9 (can be anything less than 1)\n\nx' = gamma 9\n\nlet gamma = 0.9 (can be anything less than 1)\n\nx' = 8.1\n\nNOW carry on with the second equation.\n\nct' = 0.9( 10 - 0.9 10)\n\nx' = 10\n\nSo pick a result... is x' = 8.1 or does x'= 10?\n\n##### Share on other sites\n\n18 minutes ago, TheProdigalProdigy said:\n\nNeither, it's .9 & for ct'\n\ncrap i did it wrong again.\n\nUsing 90% light speed means that beta is 1.11111 and gamma is 2.294\n\nLet x =10\n\nNOW we do it with actual calculated values for gamma and beta and known distance for x\n\nwe get x' = minus 2.29\n\nBut x' was not going backwards! he was moving in the same direction as the light.\n\nanyway, its true that both equations balance . the results are minus 2.29 for both equations. and so all you have proven is that minus 2.29 is equal to minus 2.29\n\nwhich is not useful, as it could be any number, i.e, 42 = 42\n\nBut the number minus 2.29 is NOT the distance x' moved anyway! X' must be a positive distance.\n\nIf x' moved backwards, the input value for x should have been minus x.\n\nThe whole math was set up for both light and x' moving in the same direction.\n\nAlso, I'm no mathematician, im figuring it out myself here, so mistakes are possible.\n\nBut im right about Lorentz being crap and so too is Einstein.\n\nIf I get stuck with the math, (so far Im ok) I can just ask my x wife, who is a Maths Professor in a Chinese University, or my son who is following in her footsteps.\n\nThe upshot so far regarding any equation that just states that x=x is that its a useless equation.\n\nWhich is exactly what we have with x' = gamma(x - beta *ct)\n\na useless equation that just says that x=x.\n\n##### Share on other sites\n\n1 hour ago, TheProdigalProdigy said:\n\nNeither, it's .9 & for ct'\n\nThe error that Sal makes is when he substitutes ct for x, and ct' for x' in the first equations picture.\n\nHe did this because he said that x=ct and x'=ct'\n\nBUT this is ONLY true in a special case, that is referring to LIGHT, AND the graph has light at 45degrees.\n\nIf you draw a graph with light at any some other angle, by having different scale factors (which is more logical) then x can not = ct, and x' can not = ct'.\n\nPlugging ct into the equation replacing x is a massive error. Because the x in this case refers to the location of the observer who is NOT doing light speed, so his x can never equal ct.\n\n##### Share on other sites\n\nGreat, so you don't have an answer, so you resort to this personal attack.\n\nIs there some problem with my math this last time?\n\nWhat is incorrect with this statement, \"Plugging ct into the equation replacing x is a massive error. Because the x in this case refers to the location of the observer who is NOT doing light speed, so his x can never equal ct. \"\n\n##### Share on other sites\n\n5 hours ago, TheProdigalProdigy said:\n\nThe observer is (0,0)\n\nExcellent, but as the claim is ALSO made that x' +ct', so the the stationary observer and the MOVING observer are STILL at 0,0, and time is zero.\n\nSo nothing has occurred, but they use that particular time, when nothing has occurred in the experiment, to come up with the result that Time has shrunk for the guy how might move one day?\n\n##### Share on other sites\n\n2 minutes ago, TheProdigalProdigy said:\n\nc will still race ahead of an observer at ~300mill m/s no matter how close to the speed of light that observer gets to.\n\nand that statement has nothing whatever to do with what I just asked you.\n\nI said that t, t', x, x' must all be set to zero, meaning that the experiment has not commenced, so there is no data to run through you equation.\n\nAt least this is the condition that you gave to counter my argument. Your \"solution\" to negate my claim that its wrong to use x = ct, was to set t and x to zero. So lights not going anywhere either, not that that would change anything, the equations for Lorentz are still nonsense as i explained.\n\n##### Share on other sites\n\n1 hour ago, TheProdigalProdigy said:\n\nc will still race ahead of an observer at ~300mill m/s no matter how close to the speed of light that observer gets to.\n\nAnd that claim, that light always goes at c regardless of the speed of the one measuring it, is unsupported by either logic or experiment. That's the postulate of einstein's hypothesis, so its only a guess until his hypothesis is at least shown to make some sense! which it clearly does not.\n\nusing x = ct and x' = ct' is nonsense\n\n##### Share on other sites\n\n10 minutes ago, TheProdigalProdigy said:\n\nThe cancelling is just rise over run taking you to zero. There is still a rise and a run from zero, to negative x, ct. You have a lot to learn. I pity your comprehension powers.\n\nBut the Minkowsky spacetime diagram is probably the most useful invention since Edison invented electrify! Denial of Minkowski space spells imminent treason and dishonor\n\nWhat the hell is this?\n\nCan you at least TRY to respond to my criticism?\n\nHoping sideways is not getting around the problem i raised.\n\n##### Share on other sites\n\n2 hours ago, TheProdigalProdigy said:\n\nFinal warning\n\nYou're treading on thin ice whilst entering dangerous territory allegatiins online that 🛂You must CHOSE your next words carefully✍️\n\nShould you spend so much time playing video games? Its messing with your brain.\n\nMinkowski's diagram is a load of rubbish, Lorentz transform is also math nonsense, but together, Einstein has combined them both into the biggest load of steaming crap ever to come from a small hatted one. And they collectively have put ot some real zingers.\n\nAnyway, nothing you have said has solved the problem I raised about the failure of the development of the lorentz equation, so lets just agree that I'm correct, and pop pseudo science (mainstream science) is baseless nonsense.", null, "" ]	[ null, "https://www.scienceforums.com/uploads/monthly_2021_03/561808220_40-IntroductiontotheLorentztransform.thumb.jpg.5a57d1d78e6d20dc2db979650873f661.jpg", null, "https://www.scienceforums.com/uploads/monthly_2021_03/396676630_Untitled2.thumb.jpg.04c92ff80c6533df763820b670bb106b.jpg", null, "https://www.scienceforums.com/uploads/set_resources_1/84c1e40ea0e759e3f1505eb1788ddf3c_default_photo.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.974964,"math_prob":0.9505996,"size":641,"snap":"2023-40-2023-50","text_gpt3_token_len":147,"char_repetition_ratio":0.09576138,"word_repetition_ratio":0.0,"special_character_ratio":0.23712948,"punctuation_ratio":0.11023622,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9697469,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,3,null,3,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-03T01:24:05Z\",\"WARC-Record-ID\":\"<urn:uuid:37b0cb8d-94d2-4ff9-bf5c-46b32fdc476f>\",\"Content-Length\":\"298700\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e216d817-13a4-4053-b297-5169ad72c4b7>\",\"WARC-Concurrent-To\":\"<urn:uuid:85af0665-7c35-42fa-a7ad-b9b959044d2a>\",\"WARC-IP-Address\":\"104.21.51.222\",\"WARC-Target-URI\":\"https://www.scienceforums.com/topic/37923-minkowski-spacetime-diagrams-re-assigned/page/3/\",\"WARC-Payload-Digest\":\"sha1:B5RD6T3RLDMNTXPP4XU7A7VFQFOAIBYX\",\"WARC-Block-Digest\":\"sha1:KQ47P26SLEI7AUPOMVYNNADPNBQOZIWD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233511023.76_warc_CC-MAIN-20231002232712-20231003022712-00067.warc.gz\"}"}
https://numbermatics.com/n/1520/	[ "# 1520\n\n## 1,520 is an even composite number composed of three prime numbers multiplied together.\n\nWhat does the number 1520 look like?\n\nThis visualization shows the relationship between its 3 prime factors (large circles) and 20 divisors.\n\n1520 is an even composite number. It is composed of three distinct prime numbers multiplied together. It has a total of twenty divisors.\n\n## Prime factorization of 1520:\n\n### 24 × 5 × 19\n\n(2 × 2 × 2 × 2 × 5 × 19)\n\nSee below for interesting mathematical facts about the number 1520 from the Numbermatics database.\n\n### Names of 1520\n\n• Cardinal: 1520 can be written as One thousand, five hundred twenty.\n\n### Scientific notation\n\n• Scientific notation: 1.52 × 103\n\n### Factors of 1520\n\n• Number of distinct prime factors ω(n): 3\n• Total number of prime factors Ω(n): 6\n• Sum of prime factors: 26\n\n### Divisors of 1520\n\n• Number of divisors d(n): 20\n• Complete list of divisors:\n• Sum of all divisors σ(n): 3720\n• Sum of proper divisors (its aliquot sum) s(n): 2200\n• 1520 is an abundant number, because the sum of its proper divisors (2200) is greater than itself. Its abundance is 680\n\n### Bases of 1520\n\n• Binary: 101111100002\n• Base-36: 168\n\n### Squares and roots of 1520\n\n• 1520 squared (15202) is 2310400\n• 1520 cubed (15203) is 3511808000\n• The square root of 1520 is 38.9871773795\n• The cube root of 1520 is 11.4977941579\n\n### Scales and comparisons\n\nHow big is 1520?\n• 1,520 seconds is equal to 25 minutes, 20 seconds.\n• To count from 1 to 1,520 would take you about twenty-five minutes.\n\nThis is a very rough estimate, based on a speaking rate of half a second every third order of magnitude. If you speak quickly, you could probably say any randomly-chosen number between one and a thousand in around half a second. Very big numbers obviously take longer to say, so we add half a second for every extra x1000. (We do not count involuntary pauses, bathroom breaks or the necessity of sleep in our calculation!)\n\n• A cube with a volume of 1520 cubic inches would be around 1 feet tall.\n\n### Recreational maths with 1520\n\n• 1520 backwards is 0251\n• 1520 is a Harshad number.\n• The number of decimal digits it has is: 4\n• The sum of 1520's digits is 8\n• More coming soon!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8567589,"math_prob":0.97160923,"size":2620,"snap":"2021-21-2021-25","text_gpt3_token_len":705,"char_repetition_ratio":0.112767585,"word_repetition_ratio":0.029953917,"special_character_ratio":0.30839694,"punctuation_ratio":0.1618497,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9935881,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-07T15:35:36Z\",\"WARC-Record-ID\":\"<urn:uuid:ca919d7c-8f8f-4ea4-ade9-7e9571a7085b>\",\"Content-Length\":\"18042\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a5993aec-19d5-44a1-a2cd-a301a8d4650c>\",\"WARC-Concurrent-To\":\"<urn:uuid:8c099eb1-42f9-45df-8209-5542e5361233>\",\"WARC-IP-Address\":\"72.44.94.106\",\"WARC-Target-URI\":\"https://numbermatics.com/n/1520/\",\"WARC-Payload-Digest\":\"sha1:FCAHJ6OPOOS3CMAHLO5NLAP5NJDHRX2I\",\"WARC-Block-Digest\":\"sha1:ICTUCUZHFIBZOMOUZDMJHHTZL5EY6326\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243988796.88_warc_CC-MAIN-20210507150814-20210507180814-00194.warc.gz\"}"}
https://gis.stackexchange.com/questions/317659/google-earth-engine-error-on-point-value-extraction	[ "# Google Earth Engine: Error on point value extraction\n\nI am trying to extract rainfall (CHIRPS) values for a set of locations but I got the following error:\n\nImage.reduceRegions: Unable to find a crs\n\nThis does not happens with other datasets (such as terraclimate)\n\nThe error arise when i start downloading the table from Tasks.\n\nHere is the link to the code I have run https://code.earthengine.google.com/c27f2156e81824b0990dcfe0b0a6f455\n\nThe error should be here:\n\n``````// do extraction\nvar ft = ee.FeatureCollection(ee.List([]));\n\n//Function to extract values from image collection based on point file and export as a table\nvar fill = function(img, ini) {\nvar inift = ee.FeatureCollection(ini);\nvar scale = ee.Image(MM.first()).projection().nominalScale().getInfo()\nvar ft2 = img.reduceRegions(pts, ee.Reducer.first(),scale);\nvar date = img.date().format(\"YYYYMM\");\nvar ft3 = ft2.map(function(f){return f.set(\"date\", date)});\nreturn inift.merge(ft3);\n};\n\n// Iterates over the ImageCollection\nvar profile = ee.FeatureCollection(MM.iterate(fill, ft));\n``````\n• May I ask what does the 'ini' mean in the function(img,ini),and why should I transfer this 'ini' into featurecollection? – Zhou XF Apr 15 at 2:40\n\n## 1 Answer\n\nSome of your images inside the collection do not contain bands. Therefore, the error is thrown. Filter out images which do not contain a band using (assuming the first images correctly contains a band):\n\n``````// Filter out empty images\nMM = MM.filter(ee.Filter.listContains('system:band_names', MM.first().bandNames().get(0)))\n``````\n\nFurthermore, I think you will need to get the scale using the following, based on this post:\n\n``````// get scale\nvar scale = MM.first().projection().nominalScale().multiply(0.05); print(scale)\n``````\n\nAs your feature collection was not shared, I drew a simple feature collection: Link script\n\n• it works perfectly.Thanks – Gianca Apr 5 at 10:02" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.55112666,"math_prob":0.80304265,"size":985,"snap":"2019-35-2019-39","text_gpt3_token_len":251,"char_repetition_ratio":0.117227316,"word_repetition_ratio":0.0,"special_character_ratio":0.26395938,"punctuation_ratio":0.20725389,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9535912,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-24T17:30:44Z\",\"WARC-Record-ID\":\"<urn:uuid:3f795a81-e107-4356-bb9c-91788c2a4a92>\",\"Content-Length\":\"135248\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:aced3456-0f0f-478e-89c9-713ea840864a>\",\"WARC-Concurrent-To\":\"<urn:uuid:802119dd-5fc9-4038-b37e-213b5f9611ce>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://gis.stackexchange.com/questions/317659/google-earth-engine-error-on-point-value-extraction\",\"WARC-Payload-Digest\":\"sha1:6CJ3PSTOEPAMZUFHI3XREAFECROECS7B\",\"WARC-Block-Digest\":\"sha1:HARH52T7JTYAVRHFVJC52K2W5BKQQXEM\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027321351.87_warc_CC-MAIN-20190824172818-20190824194818-00039.warc.gz\"}"}
https://www.geeksforgeeks.org/python-program-to-find-n-largest-elements-from-a-list/	[ "Related Articles\nPython program to find N largest elements from a list\n• Difficulty Level : Easy\n• Last Updated : 24 Apr, 2020\n\nGiven a list of integers, the task is to find N largest elements assuming size of list is greater than or equal o N.\n\nExamples :\n\n```Input : [4, 5, 1, 2, 9]\nN = 2\nOutput : [9, 5]\n\nInput : [81, 52, 45, 10, 3, 2, 96]\nN = 3\nOutput : [81, 96, 52]\n```\n\n## Recommended: Please try your approach on {IDE} first, before moving on to the solution.\n\nA simple solution traverse the given list N times. In every traversal, find the maximum, add it to result, and remove it from the list. Below is the implementation :\n\n `# Python program to find N largest ` `# element from given list of integers ` ` ` `# Function returns N largest elements ` `def` `Nmaxelements(list1, N): ` ` ``final_list ``=` `[] ` ` ` ` ``for` `i ``in` `range``(``0``, N): ` ` ``max1 ``=` `0` ` ` ` ``for` `j ``in` `range``(``len``(list1)): ` ` ``if` `list1[j] > max1: ` ` ``max1 ``=` `list1[j]; ` ` ` ` ``list1.remove(max1); ` ` ``final_list.append(max1) ` ` ` ` ``print``(final_list) ` ` ` `# Driver code ` `list1 ``=` `[``2``, ``6``, ``41``, ``85``, ``0``, ``3``, ``7``, ``6``, ``10``] ` `N ``=` `2` ` ` `# Calling the function ` `Nmaxelements(list1, N) `\n\nOutput :\n\n`[85, 41]`\n\nTime Complexity : O(N * size) where size is size of the given list.\nMethod 2:\n\n `# Python program to find N largest ` `# element from given list of integers ` ` ` `l ``=` `[``1000``,``298``,``3579``,``100``,``200``,``-``45``,``900``] ` `n ``=` `4` ` ` `l.sort() ` `print``(l[``-``n:]) `\n\nOutput:\n\n```[-45, 100, 200, 298, 900, 1000, 3579]\nFind the N largest element: 4\n[298, 900, 1000, 3579]\n```\n\nPlease refer k largest(or smallest) elements in an array for more efficient solutions of this problem.\n\nAttention geek! Strengthen your foundations with the Python Programming Foundation Course and learn the basics.\n\nTo begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course.\n\nMy Personal Notes arrow_drop_up\nRecommended Articles\nPage :" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5573988,"math_prob":0.9197699,"size":1515,"snap":"2021-04-2021-17","text_gpt3_token_len":484,"char_repetition_ratio":0.11647915,"word_repetition_ratio":0.06711409,"special_character_ratio":0.37623763,"punctuation_ratio":0.22089553,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97105396,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-23T22:06:27Z\",\"WARC-Record-ID\":\"<urn:uuid:ef9213e1-e2d6-47c5-93f8-c6360e0f43b0>\",\"Content-Length\":\"124828\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:77a2f218-5443-492a-accf-058f30a1ae7e>\",\"WARC-Concurrent-To\":\"<urn:uuid:67bc0af1-4e1a-4a59-a275-9097e03b2632>\",\"WARC-IP-Address\":\"23.12.144.19\",\"WARC-Target-URI\":\"https://www.geeksforgeeks.org/python-program-to-find-n-largest-elements-from-a-list/\",\"WARC-Payload-Digest\":\"sha1:2C5VCNZ4ZUIG6VX3ZYLOZHHCA3ACNPGX\",\"WARC-Block-Digest\":\"sha1:LOXQFTADA33EDWH447NBK2XCBSUYV6I2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703538431.77_warc_CC-MAIN-20210123191721-20210123221721-00535.warc.gz\"}"}
https://www.jianshu.com/p/6d16805537ef	[ "【译】RxJava变换操作符：.concatMap( )与.flatMap( )的比较\n\nObservable 转换\n\npublic class DataManager {\nprivate final List<Integer> numbers;\nprivate final Executor jobExecutor;\n\npublic DataManager() {\nthis.numbers = new ArrayList<>(Arrays.asList(2, 3, 4, 5, 6, 7, 8, 9, 10));\njobExecutor = JobExecutor.getInstance();\n}\n\npublic Observable<Integer> getNumbers() {\nreturn Observable.from(numbers);\n}\n\npublic List<Integer> getNumbersSync() {\nreturn this.numbers;\n}\n\npublic Observable<Integer> squareOf(int number) {\nreturn Observable.just(number * number).subscribeOn(Schedulers.from(this.jobExecutor));\n}\n}\n\nprivate final Func1<Integer, Observable<Integer>> SQUARE_OF_NUMBER =\nnew Func1<Integer, Observable<Integer>>() {\n@Override public Observable<Integer> call(Integer number) {\nreturn dataManager.squareOf(number);\n}\n};\n\npublic Observable<Integer> squareOf(int number) {\nreturn Observable.just(number * number).subscribeOn(Schedulers.from(this.jobExecutor));\n}\n\nlogcat 输出\n\nflatMap()与concatMap()的比较\n\nflatMap()操作符使用你提供的原本会被原始Observable发送的事件，来创建一个新的Observable。而且这个操作符，返回的是一个自身发送事件并合并结果的Observable。可以用于任何由原始Observable发送出的事件，发送合并后的结果。记住，flatMap()可能交错的发送事件，最终结果的顺序可能并是不原始Observable发送时的顺序。为了防止交错的发生，可以使用与之类似的concatMap()操作符。\n\nMerge operator\n\nConcat operator\n\nProblem solved\n\nconcatMap()的救赎。把flatMap()替换成concatMap()，问题迎刃而解。你可能会问：为什么不首先阅读文档（归功于RxJava的贡献者），有时候我们真的很懒，不到万不得已绝不会去查阅文档。这张图是经过测试后的最终结果（可以在最下面找到示例代码）：" ]	[ null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.8504084,"math_prob":0.7831907,"size":2854,"snap":"2019-43-2019-47","text_gpt3_token_len":1556,"char_repetition_ratio":0.17894737,"word_repetition_ratio":0.062176164,"special_character_ratio":0.19411352,"punctuation_ratio":0.12426036,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96408397,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-23T00:54:45Z\",\"WARC-Record-ID\":\"<urn:uuid:78c9389f-3514-4545-b620-db820539994c>\",\"Content-Length\":\"111749\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2d0250fb-dadb-4d25-9a29-d8d3ce79a4a2>\",\"WARC-Concurrent-To\":\"<urn:uuid:c2fa3dd6-9e5b-4d02-af47-765e5ebac442>\",\"WARC-IP-Address\":\"47.246.24.229\",\"WARC-Target-URI\":\"https://www.jianshu.com/p/6d16805537ef\",\"WARC-Payload-Digest\":\"sha1:FZCVKLCWG34SIIBVHGDTY2WRULY2YKC5\",\"WARC-Block-Digest\":\"sha1:JPCNB4KKTEXVOB7XWDXO6VW5YLWV56UB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987826436.88_warc_CC-MAIN-20191022232751-20191023020251-00206.warc.gz\"}"}
https://scirp.org/journal/paperinformation.aspx?paperid=58317	[ "On a System of Second-Order Nonlinear Difference Equations\n\nAbstract\n\nThis paper is concerned with dynamics of the solution to the system of two second-order nonlinear difference equations", null, ",", null, ",", null, ", where", null, ",", null, ",", null, ", i = 0, 1. Moreover, the rate of convergence of a solution that converges to the equilibrium of the system is discussed. Finally, some numerical examples are considered to show the results obtained.\n\nKeywords\n\nShare and Cite:\n\nBao, H. (2015) On a System of Second-Order Nonlinear Difference Equations. Journal of Applied Mathematics and Physics, 3, 903-910. doi: 10.4236/jamp.2015.37110.\n\nConflicts of Interest\n\nThe authors declare no conflicts of interest.\n\n Papaschinopoulos, G. and Schinas, C.J. (1998) On a System of Two Nonlinear Difference Equations. Journal of Mathematical Analysis and Applications, 219, 415-426. http://dx.doi.org/10.1006/jmaa.1997.5829 Clark, D., Kulenovic, M.R.S. and Selgrade, J.F. (2003) Global Asymptotic Behavior of a Two-Dimensional Difference Equation Modelling Competition. Nonlinear Analysis, 52, 1765-1776. http://dx.doi.org/10.1016/S0362-546X(02)00294-8 Clark, D. and Kulenovic, M.R.S. (2003) A Coupled System of Rational Difference Equations. Computers and Mathematics with Applications, 43, 849-867. http://dx.doi.org/10.1016/S0898-1221(01)00326-1 Yang, X. (2005) On the System of Rational Difference Equations . Journal of Mathematical Analysis and Applications, 307, 305-311. http://dx.doi.org/10.1016/j.jmaa.2004.10.045 Zhang, Q., Yang, L. and Liu, J. (2012) Dynamics of a System of Rational Third-Order Difference Equation. Advances in Difference Equations, 136, 1-8. http://dx.doi.org/10.1186/1687-1847-2012-136 Zhang, Q., Liu, J. and Luo, Z. (2015) Dynamical Behavior of a System of Third-Order Rational Difference Equation. Discrete Dynamics in Nature and Society, 2015, Article ID: 530453. http://dx.doi.org/10.1155/2015/530453 Ibrahim, T.F. (2012) Two-Dimensional Fractional System of Nonlinear Difference Equations in the Modeling Competitive Populations. International Journal of Basic & Applied Sciences, 12, 103-121. Din, Q., Qureshi, M.N. and Khan, A.Q. (2012) Dynamics of a Fourth-Order System of Rational Difference Equations. Advances in Difference Equations, 2012, 215. http://dx.doi.org/10.1186/1687-1847-2012-215 Kocic, V.L. and Ladas, G. (1993) Global Behavior of Nonlinear Difference Equations of Higher Order with Application. Kluwer Academic Publishers, Dordrecht. http://dx.doi.org/10.1007/978-94-017-1703-8 Liu, K., Zhao, Z., Li, X. and Li, P. (2011) More on Three-Dimensional Systems of Rational Difference Equations. Discrete Dynamics in Nature and Society, 2011, Article ID: 178483. Ibrahim, T.F. and Zhang, Q. (2013) Stability of an Anti-Competitive System of Rational Difference Equations. Archives Des Sciences, 66, 44-58. Zayed E.M.E. and El-Moneam, M.A. (2011) On the Global Attractivity of Two Nonlinear Difference Equations. Journal of Mathematical Sciences, 177, 487-499. http://dx.doi.org/10.1007/s10958-011-0474-8 Touafek, N. and Elsayed, E.M. (2012) On the Periodicity of Some Systems of Nonlinear Difference Equations. Bulletin Mathématiques de la Société des Sciences Mathématiques de Roumanie, 2, 217-224. Touafek, N. and Elsayed, E.M. (2012) On the Solutions of Systems of Rational Difference Equations. Mathematical and Computer Modelling, 55, 1987-1997. http://dx.doi.org/10.1016/j.mcm.2011.11.058 Kalabusic, S., Kulenovic, M.R.S. and Pilav, E. (2011) Dynamics of a Two-Dimensional System of Rational Difference Equations of Leslie-Gower Type. Advances in Difference Equations, 2011, 29. http://dx.doi.org/10.1186/1687-1847-2011-29 Ibrahim, T.F. (2012) Boundedness and Stability of a Rational Difference Equation with Delay. Revue Roumaine de Mathématiques Pures et Appliquées, 57, 215-224. Ibrahim, T.F. and Touafek, N. (2013) On a Third-Order Rational Difference Equation with Variable Coefficients. DCDIS Series B: Applications & Algorithms, 20, 251-264. Ibrahim, T.F. (2013) Oscillation, Non-Oscillation, and Asymptotic Behavior for Third Order Nonlinear Difference Equations. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 20, 523-532. Zhang, Q. and Zhang, W. (2014) On a System of Two High-Order Nonlinear Difference Equations? Advances in Mathematical Physics, 2014, Article ID: 729273. Pituk, M. (2002) More on Poincare’s and Peron’s Theorems for Difference Equations. Journal Difference Equations and Applications, 8, 201-216. http://dx.doi.org/10.1080/10236190211954", null, "", null, "", null, "", null, "", null, "[email protected]", null, "+86 18163351462(WhatsApp)", null, "1655362766", null, "", null, "Paper Publishing WeChat", null, "" ]	[ null, "https://file.scirp.org/image/Edit_b2aa5cd5-da42-4984-afac-c85ca32e5ca8.jpg", null, "https://file.scirp.org/image/Edit_746d9359-6c11-472d-afad-d30dec3491f2.jpg", null, "https://file.scirp.org/image/Edit_ab589f7b-f7ee-45a0-888a-353f0ac276d2.jpg", null, "https://file.scirp.org/image/Edit_6e0a337c-1315-456f-9b59-51d7b81a8987.jpg", null, "https://file.scirp.org/image/Edit_77ca3611-5a59-49bd-ad5c-64822edbea27.jpg", null, "https://file.scirp.org/image/Edit_b8614f00-b50a-409b-a2fb-595bb497c770.jpg", null, "https://scirp.org/images/Twitter.svg", null, "https://scirp.org/images/fb.svg", null, "https://scirp.org/images/in.svg", null, "https://scirp.org/images/weibo.svg", null, "https://scirp.org/images/emailsrp.png", null, "https://scirp.org/images/whatsapplogo.jpg", null, "https://scirp.org/Images/qq25.jpg", null, "https://scirp.org/images/weixinlogo.jpg", null, "https://scirp.org/images/weixinsrp120.jpg", null, "https://scirp.org/Images/ccby.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.59748477,"math_prob":0.41143405,"size":4015,"snap":"2021-43-2021-49","text_gpt3_token_len":1310,"char_repetition_ratio":0.19870357,"word_repetition_ratio":0.066287875,"special_character_ratio":0.37384808,"punctuation_ratio":0.28460687,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98655677,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-29T00:26:21Z\",\"WARC-Record-ID\":\"<urn:uuid:c0253407-e8b4-475c-b077-6cab169a0451>\",\"Content-Length\":\"90647\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7e076977-03f8-4ffa-a9e2-eb14d0e7d2ed>\",\"WARC-Concurrent-To\":\"<urn:uuid:9e63f355-54a3-409d-a732-4e7f1ab1f69f>\",\"WARC-IP-Address\":\"144.126.144.39\",\"WARC-Target-URI\":\"https://scirp.org/journal/paperinformation.aspx?paperid=58317\",\"WARC-Payload-Digest\":\"sha1:T2LXSMDARLO3XBG4NEIODSR73BNFF7SC\",\"WARC-Block-Digest\":\"sha1:D7YAQ6MD6DTICUY5RAGGVGD3O4VJVQXC\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358673.74_warc_CC-MAIN-20211128224316-20211129014316-00031.warc.gz\"}"}
https://eudml.org/search/page?q=sc.generalop.ANDl_0*c_0author_0eq%253A1.S.+Sengupta&qt=SEARCH	[ "## Currently displaying 1 – 9 of 9\n\nShowing per page\n\nOrder by Relevance \| Title \| Year of publication\n\nMetrika\n\nMetrika\n\nMetrika\n\n### Unbiased estimation of reliability for two-parameter exponential distribution under time censored sampling\n\nApplicationes Mathematicae\n\nThe problem considered is that of unbiased estimation of reliability for a two-parameter exponential distribution under time censored sampling. We give necessary and sufficient conditions for the existence of uniformly minimum variance unbiased estimator and also provide a characterization of a complete class of unbiased estimators in situations where unbiased estimators exist.\n\n### Estimation of the size of a closed population\n\nApplicationes Mathematicae\n\nThe problem considered is that of estimation of the size (N) of a closed population under three sampling schemes admitting unbiased estimation of N. It is proved that for each of these schemes, the uniformly minimum variance unbiased estimator (UMVUE) of N is inadmissible under square error loss function. For the first scheme, the UMVUE is also the maximum likelihood estimator (MLE) of N. For the second scheme and a special case of the third, it is shown respectively that an MLE and an estimator...\n\n### Unbiased estimation for two-parameter exponential distribution under time censored sampling\n\nApplicationes Mathematicae\n\nThe problem considered is that of unbiased estimation for a two-parameter exponential distribution under time censored sampling. We obtain a necessary form of an unbiasedly estimable parametric function and prove that there does not exist any unbiased estimator of the parameters and the mean of the distribution. For reliability estimation at a specified time point, we give a necessary and sufficient condition for the existence of an unbiased estimator and suggest an unbiased estimator based on a...\n\nMetrika\n\nMetrika\n\nPage 1" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8726098,"math_prob":0.9548276,"size":1441,"snap":"2021-21-2021-25","text_gpt3_token_len":263,"char_repetition_ratio":0.18162839,"word_repetition_ratio":0.110091746,"special_character_ratio":0.1721027,"punctuation_ratio":0.07438017,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9870169,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-24T18:39:49Z\",\"WARC-Record-ID\":\"<urn:uuid:369dbf02-e275-4794-a0db-cfa513f4a4ec>\",\"Content-Length\":\"72070\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:00330c06-bb17-492c-aab9-6b7d21f50a69>\",\"WARC-Concurrent-To\":\"<urn:uuid:1f19ccf8-abf3-4350-853c-56bf2a0b5a8b>\",\"WARC-IP-Address\":\"213.135.60.110\",\"WARC-Target-URI\":\"https://eudml.org/search/page?q=sc.generalop.ANDl_0*c_0author_0eq%253A1.S.+Sengupta&qt=SEARCH\",\"WARC-Payload-Digest\":\"sha1:D6PV3TUII46F6U3QK5TBLSI6LG6N2SG2\",\"WARC-Block-Digest\":\"sha1:E72EU47H4UHYCV5NCN5RKWQBCBLESAK3\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488556482.89_warc_CC-MAIN-20210624171713-20210624201713-00240.warc.gz\"}"}
https://linuxtut.com/en/b759028d3609a11072e0/	[ "# [JAVA] I want to perform Group By processing with Stream (group-by-count, group-by-sum, group-by-max)\n\nFor example, suppose a user and the amount paid by that user are given in the following form:\n\n``````public class Payment {\n\npublic static void main(String[] args) {\nvar payments = List.of(\nnew Payment(\"A\", 10),\nnew Payment(\"B\", 20),\nnew Payment(\"B\", 30),\nnew Payment(\"C\", 40),\nnew Payment(\"C\", 50),\nnew Payment(\"C\", 60)\n);\n}\n\nprivate String name;\nprivate int value;\n\npublic Payment(String name, int value) {\nthis.name = name;\nthis.value = value;\n}\npublic String getName() { return name; }\npublic int getValue() { return value; }\n}\n``````\n\nNow, for each user, we want to find the number of payments, the total amount paid, or the maximum amount. In SQL, it seems that you can easily find it by combining `GROUP BY` and window function, but in Java Stream, how should you write it?\n\n``````select name, count(*) from payment group by name;\nselect name, sum(value) from payment group by name;\nselect name, max(value) from payment group by name;\n``````\n\nThe overall policy is to use `Collectors.groupingBy`. First, the number of payments, that is, `group-by-count`.\n\n``````var counts = payments.stream().collect(Collectors.groupingBy(Payment::getName, Collectors.counting()));\ncounts.entrySet().stream().map(e -> e.getKey() + \"=\" + e.getValue()).forEach(System.out::println);\n// A=1\n// B=2\n// C=3\n``````\n\nThere is a well-known method called `Collectors.counting`, so it seems good to use it. The total amount paid next. The point is `group-by-sum`, but this also has a method with a descriptive name of` Collectors.summingInt`, so just use this.\n\n``````var sums = payments.stream().collect(Collectors.groupingBy(Payment::getName, Collectors.summingInt(Payment::getValue)));\nsums.entrySet().stream().map(e -> e.getKey() + \"=\" + e.getValue()).forEach(System.out::println);\n// A=10\n// B=50\n// C=150\n``````\n\nFinally, \"maximum amount paid\" = `group-by-max`, but I personally feel that it is the most controversial. As a basic policy, it seems to be quick to use `Collectors.maxBy`.\n\n``````var maxs = payments.stream().collect(Collectors.groupingBy(Payment::getName, Collectors.maxBy(Comparator.comparingInt(Payment::getValue))));\nmaxs.entrySet().stream().map(e -> e.getKey() + \"=\" + e.getValue().get().getValue()).forEach(System.out::println);\n// A=10\n// B=30\n// C=60\n``````\n\nAt this time, the type of the variable `maxs` is`Map <String, Optional <Payment >>`. ʻOptional` is like a marker to remind you that it may be null, but here in business logic the value of` maxs` cannot be null. In short, ʻOptional` here doesn't make much sense, so I want to get rid of it. In other words, I want to set the type of `maxs` to`Map <String, Payment>`, but in such a case, it seems to be quick to do as follows.\n\n``````var maxs = payments.stream().collect(Collectors.groupingBy(Payment::getName, Collectors.collectingAndThen(Collectors.maxBy(Comparator.comparing(Payment::getValue)), Optional::get)));\nmaxs.entrySet().stream().map(e -> e.getKey() + \"=\" + e.getValue().getValue()).forEach(System.out::println);\n// A=10\n// B=30\n// C=60\n``````\n\nHowever, at this point, the feeling of black magic begins to be good, so I want to moderate it (´ ・ ω ・｀)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7551405,"math_prob":0.791863,"size":3027,"snap":"2023-40-2023-50","text_gpt3_token_len":782,"char_repetition_ratio":0.14125042,"word_repetition_ratio":0.059808612,"special_character_ratio":0.2831186,"punctuation_ratio":0.24846625,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.967807,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-29T23:58:05Z\",\"WARC-Record-ID\":\"<urn:uuid:bbe848f2-48fc-4dd2-ae88-8ce32dda2aa9>\",\"Content-Length\":\"18817\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e79cf284-c9e8-4276-86c2-3c3fa645fe19>\",\"WARC-Concurrent-To\":\"<urn:uuid:4b812974-21db-4fd4-8ac9-18c1809e0c2f>\",\"WARC-IP-Address\":\"104.21.71.77\",\"WARC-Target-URI\":\"https://linuxtut.com/en/b759028d3609a11072e0/\",\"WARC-Payload-Digest\":\"sha1:IB3SBECC5DVLTNVWRLDG2ZFSIU2GH2P3\",\"WARC-Block-Digest\":\"sha1:FPWW62QQ2WR2XXAXCJUPAW53XDR6IEUQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510529.8_warc_CC-MAIN-20230929222230-20230930012230-00382.warc.gz\"}"}
https://www.psenterprise.com/concepts/model-targeted-experimentation	[ "# Model-targeted experimentation\n\n## Performing experiments to improve the model not the process\n\nThe guiding principle of model-targeted experimentation (also known as model-centric experimentation) is that experimentation used to improve the accuracy of the model rather than to improve the process itself. The model is then used to optimize the process design or operation.\n\nIf the empirical parameters – for example, reaction kinetic parameters, heat transfer coefficients – of a model are made as accurate and scale-invariant as possible, the model can be used to examine a wide design space much more rapidly and effectively than via experimentation alone.\n\nModel-targeted experimentation is also a key component of model-based innovation (MBI), a similar procedure applied at R&D level typically for early-stage process or product development.\n\n## What does model-targeted experimentation involve?\n\nIn a nutshell, model-targeted experimentation starts with a high-fidelity predictive model of the experimentation process and the key phenomena being studied.\n\nExperimental data is fitted to these models to calculate parameter values. Related model-based data analysis facilities also provide parameter confidence information, showing how accurately the proposed model represents what is observed.\n\nIf there is significant uncertainty in the parameters:\n\n• the model may not accurately reflect what is happening in real life. For example, a proposed reaction set may be missing one or two key reactions.\n• the experimental data may not contain sufficient or sufficiently-accurate data to calculate accurate and independent parameter values. In this case, the confidence information can be used as a guide in the design of subsequent experiments targeted at maximizing parameter accuracy.\n\n## Model-targeted experimentation – step-by-step\n\nThe key steps in model-targeted experimentation are:", null, "### Step 1. Construct first-principles models of the experiment", null, "This involves building a first principles model of the experimental setup used for gathering data, including representation of the key fundamental phenomena being studied (e.g. a detailed reaction set model).\n\nFor example, the model of a simple bench-scale stirred reactor experiment may involve:\n\n• a model of the stirred tank, complete with heat and material balances\n• a model of the cooling/heating jacket, with appropriate heat transfer equations\n• the reaction rate equations and species balance for the reactions occurring in the vessel.\n\nThe experiment model is constructed in a modular form allowing the components to be easily be implemented within the full equipment model in Step 3.\n\nInitial reaction kinetic parameter values, if not readily available, are typically found by literature search. The initial values are progressively refined in the steps below.\n\nModels may be written from scratch – for example, using equations found in research literature – but are typically taken from libraries such as the PSE Advanced Model Library for Fixed-Bed Catalytic Reactors (AML:FBCR).\n\nAn important note: when conducting model-targeted experiments, it is essential to use small-scale experimental apparatus where the phenomena being studied can be isolated as far as possible.\n\nFor example, experiments aimed at determining reaction kinetic parameter value should be as close to isothermal as possible to minimise temperature effects.\n\nLikewise, equipment should be small-scale to minimise the impact of mixing effects on results.\n\n### Step 2(a). Estimate the model parameters from data and analyse uncertainty", null, "The model constructed in Step 1 is used to estimate model parameters – typically kinetic constants or heat transfer coefficients – from initial experimental data using gPROMS’ parameter estimation techniques.\n\nIn addition to parameter values, model-based data analysis techniques built into the parameter estimation facility also yield estimates of the accuracy of these values in the form of confidence intervals, as well as estimates of the error behaviour of the measurement instruments.\n\nThis information is used to determine whether the parameters are sufficiently accurate for subsequent design and operational purposes. It is in fact possible to relate key performance indicators (KPIs) for the process back to uncertainty in parameter values, providing an indication of where further experimentation should be concentrated in order to minimise subsequent design risk.\n\n### Step 2(b). Design additional experiments, if necessary", null, "If the data analysis in Step 2 identifies areas of data uncertainty that are not within acceptable risk limits, additional experiments may need to be carried out.\n\nThese experiments can now be designed specifically to maximize information in the areas of interest, either informally or by using the model-based experiment design (MBED) techniques provided as a gPROMS option.\n\nMBED takes advantage of the significant amount of information that is already available in the form of the mathematical model to design the optimal next experiment that yields the maximum amount of parameter information (i.e. to minimise the uncertainty in the estimated parameters).\n\nMBED helps to achieve the required parameter accuracy with the minimum number – and hence time and cost – of experiments.\n\nModel-targeted experimentation may require a shift in thinking. Rather than aiming experiments at, for example, maximizing the yield of main product, more useful model information may be obtained by maximizing the yield of an impurity. The latter provides richer information for the model in characterising side reactions, which will be important in subsequent optimization calculations.\n\n### Step 2(c). Execute experiments and iterate if necessary", null, "Now carry out the experiment designed in Step 2(b). Repeat Steps 2(a)-2(b)-2(c)-2(a) until parameter values are within acceptable accuracy.\n\n## Repeating the cycle at different scales\n\nSteps 1 and 2 may be repeated to develop sub-models at different scales, covering different phenomena. Typically this proceeds in sequential fashion, keeping the parameters previously estimated constant at each subsequent stage.\n\nFor example, in a catalytic bed reactor, initial experiments may focus on small samples of catalyst in isolated conditions. The kinetic parameters estimated from these experiments are then fixed in subsequent experiments – for example to determine bed heat transfer characteristics.\n\n### Moving on to Step 3", null, "If experiments have been conducted under suitable conditions (e.g. small-scale, isothermal experiments for determining kinetic parameters), at the end of Step 2 you will have:\n\n• high-fidelity models of all key phenomena that closely represent the experimental data\n• scale-invariant parameters, meaning that the model can predict phenomena accurately over a range of scales and conditions\n• a good understanding of where the design risk lies and where further experimentation – if any – is required.\n\nThe model has effectively been validated against experimental data. Now, and only now, are you ready to build the full equipment model and proceed with design or operational analysis.", null, "" ]	[ null, "https://psewp.wpfuel.co.uk/wp-content/uploads/2021/01/mbe_cycle-1.png", null, "https://psewp.wpfuel.co.uk/wp-content/uploads/2021/01/mbe_step1.png", null, "https://psewp.wpfuel.co.uk/wp-content/uploads/2021/01/mbe_step2a.png", null, "https://psewp.wpfuel.co.uk/wp-content/uploads/2021/01/mbe_step2b.png", null, "https://psewp.wpfuel.co.uk/wp-content/uploads/2021/01/mbe_step2c.png", null, "https://psewp.wpfuel.co.uk/wp-content/uploads/2021/01/mbe_step3.png", null, "https://www.psenterprise.com/concepts/model-targeted-experimentation", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8690666,"math_prob":0.91909015,"size":7091,"snap":"2021-31-2021-39","text_gpt3_token_len":1263,"char_repetition_ratio":0.16410328,"word_repetition_ratio":0.0,"special_character_ratio":0.1745875,"punctuation_ratio":0.08089501,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96901584,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,4,null,4,null,4,null,4,null,4,null,4,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-29T02:25:49Z\",\"WARC-Record-ID\":\"<urn:uuid:6909cb2b-4b9a-41f1-a2d4-9b84092249ef>\",\"Content-Length\":\"57438\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6b6e2ff7-cd43-4885-be33-f05770e1d7f5>\",\"WARC-Concurrent-To\":\"<urn:uuid:95c597b4-8aee-458a-ba71-668f45599565>\",\"WARC-IP-Address\":\"104.21.27.172\",\"WARC-Target-URI\":\"https://www.psenterprise.com/concepts/model-targeted-experimentation\",\"WARC-Payload-Digest\":\"sha1:OM2JK723ZVPQGASLYCPA7QS7ERGGTV3M\",\"WARC-Block-Digest\":\"sha1:Q3S4AFUU4VY6UDRQPF5UEKVY6NSBBNU4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780061350.42_warc_CC-MAIN-20210929004757-20210929034757-00434.warc.gz\"}"}
https://www.xtcsyb.com/wlbc/430.html	[ "# python魔法方法-属性转换和类的表示详解，python详\n\n``````D:Python27python.exe D:/HttpRunnerManager-master/HttpRunnerManager-master/test.py\nFalse\nFalse\n12\n234\nFalse\n``````\n\n## python魔法方法-属性转换和类的表示详解，python详解\n\n•__int__(self)\n\n•转换成整型，对应int函数。\n\n•__long__(self)\n\n•转换成长整型，对应long函数。\n\n•__float__(self)\n\n•转换成浮点型，对应float函数。\n\n•__complex__(self)\n\n•转换成复数型，对应complex函数。\n\n•__oct__(self)\n\n•转换成八进制，对应oct函数。\n\n•__hex__(self)\n\n•转换成十六进制，对应hex函数。\n\n•__index__(self)\n\n•首先，这个方法应该返回一个整数，可以是int或者long。这个方法在两个地方有效，首先是 operator 模块中的index函数得到的值就是这个方法的返回值，其次是用于切片操作，下面会专门进行代码演示。\n\n•__trunc__(self)\n\n•当 math.trunc(self) 使用时被调用.__trunc__返回自身类型的整型截取 (通常是一个长整型).\n\n•__coerce__(self, other)\n\n•实现了类型的强制转换，这个方法对应于 coerce 内建函数的结果（python3.0开始去掉了此函数，也就是该魔法方法也没意义了，至于后续的版本是否重新加入支持，要视官方而定。）\n\n•这个函数的作用是强制性地将两个不同的数字类型转换成为同一个类型，例如：", null, "``````class Foo(object):\ndef __init__(self, x):\nself.x = x\n\ndef __int__(self):\nreturn int(self.x) + 1\n\ndef __long__(self):\nreturn long(self.x) + 1\n\na = Foo(123)\nprint int(a)\nprint long(a)\nprint type(int(a))\nprint type(long(a))\n``````", null, "``````def __int__(self):\nreturn str(self.x)\n``````", null, "``````def __int__(self):\nreturn list(self.x)\n``````", null, "``````class Foo(object):\ndef __init__(self, x):\nself.x = x\n\ndef __int__(self):\nreturn long(self.x) + 1\n\ndef __long__(self):\nreturn int(self.x) + 1\n\na = Foo(123)\nprint int(a)\nprint long(a)\nprint type(int(a))\nprint type(long(a))\n``````", null, "__index__(self)：\n\n``````import operator\n\nclass Foo(object):\ndef __init__(self, x):\nself.x = x\n\ndef __index__(self):\nreturn self.x + 1\n\na = Foo(10)\nprint operator.index(a)\n``````", null, "``````class Foo(object):\ndef __init__(self, x):\nself.x = x\n\ndef __index__(self):\nreturn 3\n\na = Foo('scolia')\nb = [1, 2, 3, 4, 5]\nprint b[a]\nprint b\n``````", null, "``````class Foo(object):\ndef __init__(self, x):\nself.x = x\n\ndef __index__(self):\nreturn int(self.x)\n\na = Foo('1')\nb = Foo('3')\nc = [1, 2, 3, 4, 5]\nprint c[a:b]\n``````", null, "``````a = Foo('1')\nb = Foo('3')\nc = slice(a, b)\nprint c\nd = [1, 2, 3, 4, 5]\nprint d[c]\n``````\n\n__coerce__(self, other)：\n\n``````class Foo(object):\ndef __init__(self, x):\nself.x = x\n\ndef __coerce__(self, other):\nreturn self.x, str(other.x)\n\nclass Boo(object):\ndef __init__(self, x):\nself.x = x\n\ndef __coerce__(self, other):\nreturn self.x, int(other.x)\n\na = Foo('123')\nb = Boo(123)\nprint coerce(a, b)\nprint coerce(b, a)\n``````", null, "总结：是调用了第一个参数的魔法方法。\n\n•__str__(self)\n\n•定义当 str() 被你的一个类的实例调用时所要产生的行为。因为print默认调用的就是str()函数。\n\n•__repr__(self)\n\n•定义当 repr() 被你的一个类的实例调用时所要产生的行为。 str() 和 repr() 的主要区别是其目标群体。 repr() 返回的是机器可读的输出，而 str() 返回的是人类可读的。 repr() 函数是交换模式默认调用的\n\n•函数。\n\n•__unicode__(self)\n\n•定义当 unicode() 被你的一个类的实例调用时所要产生的行为。 unicode() 和 str() 很相似，但是返回的是unicode字符串。注意，如果对你的类调用 str() 然而你只定义了 __unicode__() ，那么其将不会\n\n•工作。你应该定义 __str__() 来确保调用时能返回正确的值，并不是每个人都有心情去使用unicode()。\n\n•__format__(self, formatstr)\n\n•定义当你的一个类的实例被用来用新式的格式化字符串方法进行格式化时所要产生的行为。例如， \"Hello, {0:abc}!\".format(a) 将会导致调用 a.__format__(\"abc\") 。这对定义你自己的数值或字符串类型\n\n•是十分有意义的，你可能会给出一些特殊的格式化选项。\n\n•__hash__(self)\n\n•定义当 hash()被你的一个类的实例调用时所要产生的行为。它必须返回一个整数，用来在字典中进行快速比较。\n\n•请注意，实现__hash__时通常也要实现__eq__。有下面这样的规则：a == b 暗示着 hash(a) == hash(b) 。也就是说两个魔法方法的返回值最好一致。\n\n•这里引入一个‘可哈希对象'的概念，首先一个可哈希对象的哈希值在其生命周期内应该是不变的，而要得到哈希值就意味要实现__hash__方法。而哈希对象之间是可以比较的，这意味着要实现__eq__或\n\n•者__cmp__方法，而哈希对象相等必须其哈希值相等，要实现这个特性就意味着__eq__的返回值必须和__hash__一样。\n\n•可哈希对象可以作为字典的键和集合的成员，因为这些数据结构内部使用的就是哈希值。python中所有内置的不变的对象都是可哈希的，例如元组、字符串、数字等；而可变对象则不能哈希，例如列表、\n\n•字典等。\n\n•用户定义的类的实例默认是可哈希的，且除了它们本身以外谁也不相等，因为其哈希值来自于 id 函数。但这并不代表 hash(a) == id(a)，要注意这个特性。\n\n•__nonzero__(self)\n\n•定义当 bool() 被你的一个类的实例调用时所要产生的行为。本方法应该返回True或者False，取决于你想让它返回的值。(python3.x中改为__bool__)\n\n•__dir__(self)\n\n•定义当 dir() 被你的一个类的实例调用时所要产生的行为。该方法应该返回一个属性的列表给用户。\n\n•__sizeof__(self)\n\n•定义当 sys.getsizeof() 被你的一个类的实例调用时所要产生的行为。该方法应该以字节为单位，返回你的对象的大小。这通常对于以C扩展的形式实现的Python类更加有意义，其有助于理解这些扩展。\n\n``````print to_int('str')\nprint to_int('str123')\nprint to_int('12.12')\nprint to_int('234')\nprint to_int('12#\\$%%')\n``````\n\npython学习3群：563227894\n\n``````def to_int(str):\ntry:\nint(str)\nreturn int(str)\nexcept ValueError: #报类型错误，说明不是整型的\ntry:\nfloat(str) #用这个来验证，是不是浮点字符串\nreturn int(float(str))\nexcept ValueError: #如果报错，说明即不是浮点，也不是int字符串。是一个真正的字符串\nreturn False\n``````\n\n``````s = \"12\"\ns = \"12.12\"\n``````" ]	[ null, "http://www.bkjia.com/uploads/allimg/160729/0305203413-0.png", null, "http://www.bkjia.com/uploads/allimg/160729/0305205500-1.png", null, "http://www.bkjia.com/uploads/allimg/160729/0305201956-2.png", null, "http://www.bkjia.com/uploads/allimg/160729/03052021U-3.png", null, "http://www.bkjia.com/uploads/allimg/160729/03052012L-4.png", null, "http://files.jb51.net/file_images/article/201607/201607220847308.png", null, 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https://www.nature.com/articles/s41598-018-37471-0?error=cookies_not_supported&code=801f3e6f-32db-40a9-8539-011c7a3cafdd	[ "## Introduction\n\nWater, an essential constituent of life1, remains an elusive target for modeling and simulation. Effective coarse-grained (CG) models of liquid water must balance computational savings, by handling fewer degrees of freedom, while at the same time capturing its essential physical properties2,3,4,5,6. CG water models have enabled simulations exceeding micro-meters/seconds that are relevant for processes in biophysical systems that are beyond the reach of conventional atomistic molecular dynamics (MD) simulations. CG modeling entails recasting the complex and detailed atomistic model into a simpler yet accurate representation. A CG model has the ability to model key quantities of interest (QoI) when it captures the effects of the eliminated degrees of freedom (DOFs)7,8,9. The CG process requires: (i) The identification of the system’s optimal resolution. Commonly, groups of atoms are described with pseudo-atoms/interaction sites and a “mapping” function is used to determine the relation between these sites and the atomistic coordinates. For a given system various coarse-graining levels can be employed. For example, existing CG lipid membrane models range from representations with a single anisotropic site10 to three sites per lipid thus differentiating between the head and the tail11,12, or by grouping three or four heavy atoms into beads thus capturing varying degrees of chemical properties13,14,15,16; (ii) The specification of the associated Hamiltonian. Here, DOFs can be reduced by simplifying the form or by neglecting specific terms in the Hamiltonian. For instance, one can neglect the bond/angle vibrations and resort to rigid models17.\n\nIn effective CG models, the removed DOFs are insignificant for the QoI. However, to what degree a specific DOF is negligible for a given observable is hardly ever known beforehand. Thus, the majority of the CG models are designed based on intuition or extrapolations from existing models. Additionally, the number of the removed DOFs must be large so that the diminished accuracy compared to the AT models is justified with the substantial computational gains. It is usually assumed that increasing the level of coarse-graining will decrease the model’s accuracy. However, and this is perhaps a key issue, the relation between the number of employed DOF in a model and its accuracy may not be a monotonic function7,18,19. Thus, one can end up (without realizing) in a worst case scenario, where the constructed model is redundant, i.e., better accuracy can be achieved with fewer DOFs (computationally less demanding model). For example, the two-site and four-site models of n-hexane molecule perform reasonably well, while a very similar three-site model fails19. A number of works have addressed the systematic selection of CG models in bio-molecular systems20,21,22,23,24,25.\n\nThe challenge of striking the optimal balance between accuracy and computational cost is crucial for CG models of water. At the same time, obtaining water-water interactions consumes the majority of the computational effort. Thus, many CG models of water were developed. These models differ in the coarse-graining resolution level, i.e., the mapping, which ranges from 1 to 11 water molecules per CG bead2,26. Models also differ in the employed Hamiltonian. For CG models where one bead represents one water molecule (1-to-1 mapping), the Hamiltonian is either derived from the atomistic simulations9,27 or parametrized based on analytic potentials ranging from a simple Lennard-Jones (LJ) to potentials incorporating tetrahedral ordering, dipole moment, and orientation-dependent hydrogen bonding interactions28,29,30,31. On a higher coarse-graining level, it was soon realized that chargeless models, such as the standard MARTINI model32,33, introduce unphysical features when applied to interfaces, such as an interface between water and a lipid membrane34,35,36. Thus, new CG models were developed which treat the electrostatics explicitly. In the PCGS model (3-to-1)37, the CG beads carry induced dipoles, in the polarizable MARTINI model (4-to-3)34 the electrostatic is modeled analog to the Drude oscillator, in the BMW model (4-to-3)35 the CG representation resembles a rigid water molecule with a fixed dipole and quadrupole moment, while the GROMOS CG model (5-to-2)38 introduces explicit charges with a fluctuating dipole. Note that in these models the extra interaction sites have no relation to the physical system making the intuitive construction of the model even more difficult.\n\nThus far, studies reporting the effects of the choices made in the coarse-graining level and model structure are relatively few. For water, the mapping was investigated by Hadley et al.39, where the investigated CG models were single-site models and the Hamiltonian was parameterized to reproduce the structural properties of water. The mapping 4-to-1 was found to give the optimal balance between efficiency and accuracy. However, by comparing the properties of the available water models it is hard to extract any physics as the models were developed to reproduce different properties. Furthermore, one should avoid artificially constructed scoring functions that could be biased but rather perform model selection based on rigorous mathematical foundation. In this respect, the Bayesian statistical framework can serve as a powerful tool which has become a popular technique to refine, guide and critically assess the MD models40,41,42,43,44,45.\n\nIn this work, we employ the Bayesian statistical framework to critically assess many CG water models (see Fig. 1). We investigate the biologically relevant CG resolution levels, i.e., mappings, where the number of grouped water molecules ranges from 1 to 6. At each resolution, multiple model structures are examined ranging from 1-site to 3-site models where we additionally investigate the rigid and flexible versions of the 2 and 3-site models for mapping M = 4. Our main objective is to determine the model evidence for all models and thus elucidate the impact of the mapping on the model’s performance and the relevant DOFs in CG modeling of water. Furthermore, we evaluate the speed-up for each developed model which allows us to assess efficiency-accuracy trade-off. Lastly, we investigate the transferability of the water models to different thermodynamics states, i.e., to different temperatures. To this end, we employ the hierarchical Bayesian framework46 that can accurately quantify the uncertainty in the parameter space for multiple QoI, i.e., different properties or the same property at different conditions.\n\n## Methods\n\nWe investigate a set of CG water models (partially shown in Fig. 1). For all models, we employ the interactions that are implemented in the standard MD packages. In the 1S model, a water cluster is modeled with a single chargeless spherical particle employing the LJ potential ULJ (rij) = 4ε[(σ/rij)12 − (σ/rij)6] between particles i and j. The model parameters are $${\\phi }_{1S}=(\\sigma ,\\epsilon )$$. The 2S model is a two-site model, where the sites are oppositely charged (±q) and constrained to a distance r0. The negatively charged (blue in Fig. 2) site interacts additionally with the LJ potential. The model parameters are $${\\phi }_{2S}=(\\sigma ,\\epsilon ,q,{r}_{0})$$. In order to satisfy the net neutrality of the water cluster, the three-site model can be constructed in two ways, which we denote as 3S and 3S* models. The 3S model resembles a big water molecule where all three particles are charged. The central (blue) site has a charge of −q, and the other two sites have a charge of +q/2. In the 3S* model, the central site is chargeless and the other two carry a ±q charge. Both three-site models have the parameters $${\\phi }_{3S\\mathrm{,3}S\\ast }=(\\sigma ,\\epsilon ,q,{r}_{0},{\\vartheta }_{0})$$. For all rigid model structures, we consider four levels of resolution with the number of grouped water molecules equal to 3, 4, 5, and 6. For the 1S model, we additionally consider the M = 1 mapping, while for the models with partial charges we investigate also M = 12. The level of resolution fixes the total mass of the CG representation. The mass ratio between the interaction sites in the two and three-sited models is fixed to 2 with the central particle carrying the larger mass. The electrostatic is in all cases modeled with the Coulomb’s interaction Ue (rij) = qiqj/(4πεε0rij), where we set the global dielectric screening to ε = 2.5. For M = 4, we consider also the flexible analogs of the models with charges. In the 2SF model, the two sites are interacting with a harmonic potential Ub (rij) = kb (rij − r0)2 with force constant kb. Therefore, the model parameters are $${\\phi }_{2SF}=(\\sigma ,\\epsilon ,q,{r}_{0},{k}_{b})$$. For the flexible three-site models 3SF and 3SF, the angle is unconstrained and modeled with the harmonic angle potential Ua (ϑij) = ka (ϑij − ϑ0)2 thus adding the force constant ka parameter to the parameter set, i.e., the model parameters are $${\\phi }_{3SF\\mathrm{,3}SF\\ast }=(\\sigma ,\\epsilon ,q,{r}_{0},{\\vartheta }_{0},{k}_{a})$$.\n\nWe remark that the data used as target QoI is part of the modeling choice. In this work, we use experimental data of density, dielectric constant, surface tension, isothermal compressibility, and shear viscosity, i.e., mostly thermodynamic properties. These are deemed of key importance for biophysical systems. The data used and the properties of the reference coarse-grained water models are reported in Table 1. The structural properties, e.g. radial distribution function or the dynamical properties, e.g. diffusion constant were not considered in this work because these properties cannot be measured experimentally for M > 1.\n\n### Uncertainty Quantification\n\n#### Bayesian Framework\n\nWe consider a computational model $${\\mathscr{C}}$$ that depends on a set of parameters $${\\phi }_{c}\\in {{\\mathbb{R}}}^{{N}_{\\phi }}$$ and a set of input variables or conditions $${\\boldsymbol{x}}\\in {{\\mathbb{R}}}^{{N}_{x}}$$. In the context of the current work, the computational model is the molecular dynamics solver, the model parameters correspond to the parameters of the potential and the input variables to the temperature of the system. Moreover, we consider an observable function $$F({\\boldsymbol{x}};\\,{\\phi }_{c})\\in {{\\mathbb{R}}}^{N}$$ that represents the output of the computational model. Here, the observable function is an equilibrium property of the system, e.g., the density. We are interested in inferring the parameters $${\\phi }_{c}$$ based on the a set of experimental data d = {di\| i = 1, …, N} that correspond to the fixed input parameters of the model x.\n\nIn the frequentist statistics framework, the parameters of the model are obtained by optimizing a distance of the model from the data, usually the likelihood function. In the Bayesian framework, the parameters follow a conditional distribution which is given by Bayes’ theorem,\n\n$$p(\\phi \|{\\boldsymbol{d}}, {\\mathcal M} )=\\frac{p({\\boldsymbol{d}}\|\\phi , {\\mathcal M} )\\,p(\\phi \| {\\mathcal M} )}{p({\\boldsymbol{d}}\| {\\mathcal M} )},$$\n(1)\n\nwhere $$p({\\boldsymbol{d}}\|\\phi , {\\mathcal M} )$$ is the likelihood function, $$p(\\phi \| {\\mathcal M} )$$ is the prior probability distribution and $$p({\\boldsymbol{d}}\| {\\mathcal M} )$$ is the model evidence. Here, $$\\phi$$ is the vector containing the computational model parameters $${\\phi }_{c}$$ and any other parameters needed for the definition of the likelihood or the prior density. $${\\mathcal M}$$ stands for the model under consideration and contains all the information that describes the computational and the statistical model.\n\nThe likelihood function is a measure of how likely is that the data d are produced by the computational model $${\\mathscr{C}}$$. Here, we make the assumption that the datum di is a sample from the generative model\n\n$${y}_{i}={F}_{i}({\\boldsymbol{x}};{\\phi }_{c})+{\\sigma }_{n}{d}_{i}\\varepsilon ,\\,\\varepsilon \\sim {\\mathscr{N}}\\mathrm{(0,}\\,\\mathrm{1).}$$\n(2)\n\nNamely, yi are random variables independent and normally distributed with mean equal to the observable of the model and standard deviation proportional to the data. The reason we choose this error model is because the set of experimental data d contains elements of different orders of magnitude, e.g., density is of order of 1 and surface tension of order 100. With this model the error allowed by the statistical model becomes proportional to the value of the data we want to fit47. The likelihood of the data $$p({\\boldsymbol{d}}\|\\phi )$$ has the form,\n\n$$p({\\boldsymbol{d}}\|\\phi )={\\mathscr{N}}({\\boldsymbol{d}}\|F({\\boldsymbol{x}},{\\phi }_{c}),{\\rm{\\Sigma }}),\\,{\\rm{\\Sigma }}={\\sigma }_{n}^{2}\\,{\\rm{diag}}\\,(\\,{{\\boldsymbol{d}}}^{2}),$$\n(3)\n\nwhere $$\\phi ={({\\phi }_{c}^{{\\rm{T}}},{\\sigma }_{n})}^{{\\rm{{\\rm T}}}}$$ is the parameter vector that contains the model and the error parameters.\n\nThe denominator of Eq. (1) is defined as the integral of the numerator and is called the model evidence. This quantity can be used for model selection48 as it is discussed in the next section. Finally, the prior probability encodes all the available information on the parameters prior to observing any data. If no prior information is known for the parameters, a non informative distribution can be used, e.g. a uniform distribution. In this work we use uniform priors, see SI for detailed information.\n\n#### Model Selection\n\nAssuming we have $${N}_{ {\\mathcal M} }$$ models $${ {\\mathcal M} }_{i},\\,\\,i=1,\\ldots ,{N}_{ {\\mathcal M} }$$ that describe different computational and statistical models, we wish to choose the model that best fits the data. In Bayesian statistics, this is translated into choosing the model with the highest posterior probability,\n\n$$p({ {\\mathcal M} }_{i}\|{\\boldsymbol{d}})=\\frac{p({\\boldsymbol{d}}\|{ {\\mathcal M} }_{i})p({ {\\mathcal M} }_{i})}{p({\\boldsymbol{d}})},$$\n(4)\n\nwhere $$p({ {\\mathcal M} }_{i})$$ encodes any prior preference to the model $${ {\\mathcal M} }_{i}$$. Assuming all models have equal prior probabilities, the posterior probability of the model depends only on the likelihood of the data. Taking the logarithm of the likelihood and using Eq. (1) we can write\n\n$$\\begin{array}{rcl}\\mathrm{ln}\\,p({\\boldsymbol{d}}\|{ {\\mathcal M} }_{i}) & = & \\int \\,\\mathrm{ln}\\,p({\\boldsymbol{d}}\|{ {\\mathcal M} }_{i})\\,p(\\phi \|{\\boldsymbol{d}},{ {\\mathcal M} }_{i}){\\rm{d}}\\phi \\\\ & = & \\int \\,\\mathrm{ln}\\,\\frac{p({\\boldsymbol{d}}\|\\phi ,{ {\\mathcal M} }_{i})\\,p(\\phi \|{ {\\mathcal M} }_{i})}{p(\\phi \|{\\boldsymbol{d}},{ {\\mathcal M} }_{i})}p(\\phi \|{\\boldsymbol{d}},{ {\\mathcal M} }_{i}){\\rm{d}}\\phi \\\\ & = & {\\mathbb{E}}[\\mathrm{ln}\\,p({\\boldsymbol{d}}\|\\phi ,{ {\\mathcal M} }_{i})]-{\\mathbb{E}}[\\mathrm{ln}\\,\\frac{p(\\phi \|{ {\\mathcal M} }_{i})}{p(\\phi \|{\\boldsymbol{d}},{ {\\mathcal M} }_{i})}],\\end{array}$$\n(5)\n\nwhere the expectation is taken with respect to posterior probability $$p(\\phi \|{\\boldsymbol{d}},{ {\\mathcal M} }_{i})$$. The first term is the expected fit of the data under the posterior probability of the parameters and is a measure of how well the model fits the data. The second term is the Kullback-Leibler (KL) divergence or relative entropy of the posterior from the prior distribution and is a measure of the information gain from data d under the model $${ {\\mathcal M} }_{j}$$. The KL divergence can be seen as a measure of the distance between two probability distributions.\n\nIf one would only consider the first term of Eq. (5) for the model selection, then the model that fits the data best would be selected. However, such an approach is prone to overfitting, i.e., choosing a too complex model, which reduces the predictive capabilities of the model. The second term serves as a penalization term. Models with posterior distributions that differ a lot from the prior, i.e., models that extract a lot of information from the data, are penalized more. Thus, model evidence can be seen as an implementation of the Ockham’s razor that states that simple models (in terms of the number of parameters) that reasonably fit the data should be preferred over more complex models that provide only slight improvements to the fit. For a detailed discussion on model selection and estimators of the model evidence, we refer to refs49,50.\n\n#### Hierarchical Bayesian Framework\n\nWe consider data structured as: $$\\overrightarrow{{\\boldsymbol{d}}}=\\{{{\\boldsymbol{d}}}_{1},\\ldots ,{{\\boldsymbol{d}}}_{{N}_{d}}\\}$$ where $${{\\boldsymbol{d}}}_{i}\\in {{\\mathbb{R}}}^{{N}_{i}}$$ corresponds to the conditions xi. For example, xi may correspond to different thermodynamic conditions under which the experimental data di are produced.\n\nThe classical Bayesian method for inferring the parameters of the computational model is to group all the data and estimate the probability $$p(\\phi \|\\overrightarrow{{\\boldsymbol{d}}})$$ (see Fig. 3 left). However, this approach may not be suitable when the uncertainty on $$\\phi$$ is large due to the fact that different parameters may be suitable for different data sets. On the opposite side, individual parameters $${\\phi }_{i}$$ can be inferred using only the data set di (see Fig. 3 middle). This approach preserves the individual information but any information that may be contained in other data sets is lost.\n\nFinally, a balance between retaining individual information and sharing information between different data sets can be achieved with the hierarchical Bayesian framework. In this approach, the independent models corresponding to different conditions are connected using a hyper-parameter vector ψ (see Fig. 3 right). The benefits of this approach is twofold: (i) better informed individual probabilities $$p({\\phi }_{i}\|\\overrightarrow{{\\boldsymbol{d}}})$$; and (ii) a data informed prior p(ψ\|d) is available in case new parameters $${\\phi }^{new}$$ that correspond to unobserved data need to be inferred. A detailed description of the sampling algorithm of this approach is given in Supporting information (SI).\n\n## Results\n\n### Impact of mapping\n\nFirst, we examine the impact of the level of resolution on the model accuracy using density, dielectric constant, surface tension, isothermal compressibility, and shear viscosity experimental data (see SI). In Fig. 4 the model accuracy, as measured by the model evidence, is shown as a function of mapping M, which denotes the number of water molecules represented by a given CG model. It is usually assumed the model’s performance is decreasing with the decreased resolution of the model. Indeed, for the 1S model, we observe precisely this trend. For the charged models, the evidence is still overall monotonically decreasing with M, however, compared to the 1S model the dependency of the evidence on M is much less drastic. To investigate this dependency further, we perform the UQ inference also for the charged models at M = 12. The observed evidences are comparable to the evidence of the 1S model at M = 4. Thus, with the models that incorporate partial charges, one can resort to models with higher mappings. According to the UQ, the best model for M = 1, 3 is the 1S model whereas for M > 3 the charged models are superior. However, one should keep in mind that the chargeless and charged models are not comparable as the chargeless models cannot provide the same amount of information, e.g., the dielectric constant is not defined. Comparing the evidences of the 2S, 3S, and 3S models, we see that the three models rank very closely with the 3S* model being somewhat better than the other two.\n\nWe emphasize that the model evidence encompasses much more than a mere evaluation of the model’s properties at the best parameters. Nonetheless, it is insightful to examine the target QoI and their dependency on the mapping. Figure 5 shows the density ρ, dielectric constant ε, surface tension γ, isothermal compressibility κ, and shear viscosity η for rigid models and mappings 1 to 6. The target QoI are obtained using the maximum a posteriori (MAP) parameters and evaluated as a mean of 5 independent simulation runs with different initial conditions. Note that ε is not defined for the 1S model and it is excluded from the target QoI in the second UQ inference of the charged models. We observe that the ρ and ε are within 10% of the experimental data for all mappings. On the contrary, the γ, κ, and η depend very strongly on the mapping. The general trend is similar for all models, i.e., as we increase the mapping the γ is decreasing, κ is increasing, and η is decreasing. This observation agrees with the general picture of coarse-graining. The more we increase the level of coarse-graining the softer are the interactions between the CG beads which correlates with increased κ and decreased γ and η. We observe that for some models there are no parameters σ and ε of the LJ potential that would fit well a certain target QoI (within the liquid state), in particular, the γ and η. A possible solution would be to replace the LJ non-bonded interaction with another interaction, e.g, the Born-Mayer-Huggins interaction that is used in the BMW model35. The 1S model with M = 4 can be directly compared with the MARTINI model as the models are equal but were developed with different target QoI. With our model, we observe very similar properties as reported for the MARTINI model. Additionally, the inferred parameters with the MAP estimates are also very close to those of the MARTINI (see SI).\n\n### Rigid vs. flexible models\n\nFor the mapping M = 4, we examine also the three flexible models 2SF, 3SF, and 3S* F. The resulting evidences for these models are listed in Table 2 along with the model evidences for the rigid counterparts. The physical motivation behind the flexible models is that they encompass the fluctuations in the dipole moment of the water cluster. In the two-site model, we incorporate them via bond vibrations, whereas in the three-site models with the angle fluctuations. Thus, the flexible models have 1 extra DOF compared to the rigid counterparts. However, as can be seen in Table 2, for the three-site models the flexible versions perform worse than the rigid ones. For the two-site model, the flexible model is only slightly better than the rigid model. Nonetheless, as flexible models usually demand smaller integration timesteps and consequently have a higher computational cost the model’s performance should be more substantial to justify the extra computational resources.\n\n### Accuracy vs. efficiency\n\nWe examine the accuracy vs. efficiency trade-off in Fig. 6 where we plot the evidence as a function of the speedup compared to the all-atom simulation. As a test simulation, we choose the NVE ensemble simulation at ambient conditions, a cubic domain with an edge of 5 nm and a simulation length of 10 ns. We also employ the maximal integration timestep still permitted by the model (see SI). The variation of the runtime varies extensively between the considered models. The computational cost depends on two factors: (i) on the number of particles, which in turn depends on the employed mapping and the number of interaction sites of the model; (ii) on the integration timestep that is increasing with increased coarse-graining since the interactions are softening. For a given mapping, we observe the smallest computational cost for the 1S model, followed by the 2S, and 3S* model while the 3S model has the highest computational cost. The difference in the 3S and 3S* models is due to the smaller timesteps required by the 3S model.\n\nThe trade-off between the accuracy and efficiency can be formally addressed as a decision problem, where the expected utility51 $${\\mathscr{U}}({ {\\mathcal M} }_{i};{\\boldsymbol{d}})$$ of an individual model $${ {\\mathcal M} }_{i}$$ given data d is given by:\n\n$${\\mathscr{U}}({ {\\mathcal M} }_{i};{\\boldsymbol{d}})=p({ {\\mathcal M} }_{i}\|{\\boldsymbol{d}})u({ {\\mathcal M} }_{i}\\mathrm{).}$$\n(6)\n\nWe define the utility function $$u({ {\\mathcal M} }_{i})$$ as the decimal logarithm of the computational speedup over the atomistic model. As shown in Fig. 6c the model with the maximal expected utility is found for the 1S model with M = 1. When we consider the models incorporating the partial charge: the 3S model is the most unfavorable, while the 2S and 3S* models are comparable in terms of their expected utility. In turn, the appropriate choice for the 2S and 3S* models are mappings M = 5 and 3, respectively.\n\n### Transferability to non-ambient TD conditions\n\nOne of the challenges of coarse-graining is the transferability of CG models. Typically, CG models are more sensitive to variations in the thermodynamic conditions than the atomistic models. Furthermore, the more we increase the level of coarse-graining, the more restricted the model is to the thermodynamics state at which it is parametrized. One way of making the model more robust to transferability is to parametrize it for different conditions. Within the Bayesian formalism, the hierarchical UQ allows us to merge multiple QoI. We test the transferability of three models 2SF, 3S, and 3S* for mapping M = 4. In Fig. 7, we plot the model evidences for the hierarchical UQ, where the temperatures T = 283, 298, 323 K are merged and the evidences for the classical UQ at each temperature. We observe that the 3S* model is the most transferable, having the highest hierarchical evidence. For the three-site models, we also observe that it is easier for the CG model to fit higher temperatures." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86410993,"math_prob":0.9957567,"size":39407,"snap":"2022-40-2023-06","text_gpt3_token_len":9468,"char_repetition_ratio":0.15229805,"word_repetition_ratio":0.014543051,"special_character_ratio":0.24411906,"punctuation_ratio":0.17490593,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99687475,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-28T01:15:36Z\",\"WARC-Record-ID\":\"<urn:uuid:ad67f79c-42dc-4438-a7ac-4c79d1774687>\",\"Content-Length\":\"356110\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:15417fac-f22c-4062-a8ce-4427a5387452>\",\"WARC-Concurrent-To\":\"<urn:uuid:4d26917a-2388-4636-8834-cff60ea7faf4>\",\"WARC-IP-Address\":\"146.75.32.95\",\"WARC-Target-URI\":\"https://www.nature.com/articles/s41598-018-37471-0?error=cookies_not_supported&code=801f3e6f-32db-40a9-8539-011c7a3cafdd\",\"WARC-Payload-Digest\":\"sha1:KVT7PCTGEBSLZPZCCWTBTMFAMSAJQHU3\",\"WARC-Block-Digest\":\"sha1:5LYRYFYNLSHKUZK5RO3XYEXP7BXKMHWI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499468.22_warc_CC-MAIN-20230127231443-20230128021443-00511.warc.gz\"}"}
https://www.r-bloggers.com/2021/07/double-descent-part-i-sample-wise-non-monotonicity/	[ "Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.\n\nOne of the most fascinating findings in the statistical/machine learning literature of the last 10 years is the phenomenon called double descent. For some reason, extremely over-parameterized models, where the number of parameters to estimate is 10-1000 times more than the number of observations, sometimes win competitions, prove to be the best models. How can this be in the presence of the bias-variance trade-off? How come the variance does not become enormous in these models? One explanation is that the U-shaped bias-variance trade-off curve takes a second downward turn at the end of the U, hence the name double descent. It’s almost magical. It’s almost like we pass through a portal, and in the new bizarro universe, the gravity pushes.\n\nToday, we will briefly talk about the idea behind the second descent and show its existence. In particular, we will see what happens when we know the functional form of the underlying data generating process, but we have small number of observations. This is one of the cases where double descent can occur, and it is called sample-wise double descent.\n\n### Bias-Variance Trade-off and Different Parameterization Regimes\n\nBias-variance trade-off is one of the most basic concepts that pretty much everybody doing prediction/forecasting should know. Briefly, it is saying that very simple models produce predictions with high bias and low variance, and as we construct more complex models to decrease the bias, we tend to increase the variance. At a certain point, the decline in bias is too small compared to the increase in the variance. Since the prediction error depends on both of them (for instance, $$MSPE = Variance + Bias^{2}$$), we need to find the sweet spot where neither the bias nor the variance is minimized, but both are small enough. In Figure 1, it is shown as the black point on the “Variance + Bias$$^2$$” curve. It is the point where the prediction error is minimized.1\n\nlibrary(tidyverse)\n#Prediction error\nf <- function(x){\n(-0.4 + 1/(x+0.5)) + (0.5exp(x))\n}\n#The point where the prediction error is minimized\noptimum <- optimize(f, interval=c(0, 1), maximum=FALSE, tol = 1e-8)\ntemp_data <- data.frame(x = optimum$minimum, y=optimum$objective)\n\nggplot(data = temp_data, aes(x=x, y=y)) +\nxlim(0,1) +\ngeom_function(fun = function(x) 0.5exp(x), color = \"red\", size = 2, alpha = 0.7) +\ngeom_function(fun = function(x) -0.4 + 1/(x+0.5), color = \"blue\", size = 2, alpha = 0.7) +\ngeom_function(fun = function(x) (-0.4 + 1/(x+0.5)) + (0.5exp(x)), color = \"forestgreen\", size = 2, alpha = 0.7) +\ngeom_point(size =3) +\ntheme_minimal() + ylab(\"Error\") + xlab(\"Model Complexity\") +\ntheme(axis.text=element_blank(),\naxis.ticks=element_blank()) +\nannotate(\"text\", x=0.2, y=0.11+1/(0.2+0.5)-0.35, label= expression(paste(\"B\", ias^2)), color = \"blue\", size =5) +\nannotate(\"text\", x=0.2, y=0.11+0.5exp(0.2), label= \"Variance\", color = \"red\", size =5) +\nannotate(\"text\", x=0.32, y=-0.35+ 0.11+(0.5exp(0.2) + 1/(0.2+0.5)), label= expression(paste(\"MSE = Variance + B\", ias^2)), color = \"forestgreen\", size =5)", null, "The double descent phenomenon, on the other hand, causes the U-shaped prediction error curve to decrease after a certain point. The variance starts to decrease as the model has more and more predictors (and parameters to estimate), and the bias is essentially zero.\n\nlibrary(tidyverse)\n\n#Prediction error function (it is piecewise so creating two of them).\nf1 <- function(x){\nifelse(x<=2, (-0.4 + 1/(x+0.5)) + (0.5exp(x)), NA)\n}\n\nf2 <- function(x){\nifelse(x>=2, (0 + 1/(1/(0.5exp(4/x)))), NA)\n}\n\n#Prediction variance function (it is piecewise so creating two of them).\nvar_f1 <- function(x){\nifelse(x<=2, (0.5exp(x)), NA)\n}\nvar_f2 <- function(x){\nifelse(x>=2, 1/(1/(0.5exp(4/x))), NA)\n}\n\n#Prediction bias function (it is piecewise so creating two of them).\nbias_f1 <- function(x){\nifelse(x<=2,-0.4 + 1/(x+0.5),NA )\n}\nbias_f2 <- function(x){\nifelse(x>=2,0,NA )\n}\n\nggplot(data = temp_data, aes(x=x, y=y)) +\nxlim(0,4) +\ngeom_function(fun = var_f1, color = \"red\", size = 2, alpha = 0.7) +\ngeom_function(fun = var_f2, color = \"red\", size = 2, alpha = 0.7) +\ngeom_function(fun = bias_f1, color = \"blue\", size = 2, alpha = 0.7) +\ngeom_function(fun = bias_f2, color = \"blue\", size = 2, alpha = 0.7) +\ngeom_function(fun = f1, color = \"forestgreen\", size = 2, alpha = 0.7) +\ngeom_function(fun = f2, color = \"forestgreen\", size = 2, alpha = 0.7) +\ngeom_vline(xintercept = 2, linetype = \"dashed\") +\ngeom_point() +\ntheme_minimal() + ylab(\"Error\") + xlab(\"Number of Predictors/Number of observations\") +\ntheme(axis.text=element_blank(),\naxis.ticks=element_blank()) +\nannotate(\"text\", x=0.32, y=-0.2+1/(0.2+0.5), label= expression(paste(\"B\", ias^2)), color = \"blue\") +\nannotate(\"text\", x=0.2, y=-0.2+0.5exp(0.2), label= \"Variance\", color = \"red\") +\nannotate(\"text\", x=0.26, y=0.21+(0.5exp(0.2) + 1/(0.2+0.5)), label= expression(paste(\"Variance + B\", ias^2)), color = \"forestgreen\") +\nannotate(\"text\", x=2.4, y=-0.2+1/(0.2+0.5), label= \"Interpolation limit\", color = \"black\") +\nggtitle(\"\")", null, "Figure 2: Double Descent\n\nWhat just happened here? We hit the interpolation limit and reach the over-parameterized regime, and suddenly the variance started to decrease. How did this happen? Let’s clarify something before we move further: We changed the x-axis label. It is no longer “model complexity”, but the ratio of the number of predictors to the number of observations. Does it matter? Before the interpolation threshold, not so much. After the interpolation threshold, it does. It depends on how we define the term “complexity”. We may say that once we hit the interpolation limit, the complexity no longer increases, because the model is as complex as the training data allows. Or we can define complexity in terms of the $$\\ell_{2}$$ norm of the coefficient estimates ($$(\\mathbf{\\beta^{T}\\beta})^{1/2}$$). Then, in fact, as the ratio increases, the norm may decrease.2\n\nBack to the main question: What just happened? We will examine this in more detail in later posts as well. Briefly, when the number of predictors is larger than the number of observations in the training sample, the estimator automatically regularizes itself. This happens even if we do not explicitly apply any of the regularization methods. The estimator becomes picky. Because there are more predictors (p) than the number of observations (n), it can focus on tasks other than minimizing the training error (which is usually 0 when $$n \\leq p$$). One of those tasks can be minimizing the norm, which allows the estimator to prefer the most powerful predictors to others.\n\n### Samplewise Non-monotonicity\n\nUsually, in the double descent literature, the focus is on the number of predictors. In particular, given the sample, whether the neural network should be wider or deeper is the main focus of attention. In this first part, however, we will hold the number of parameters constant and change the sample size. Since we are interested in the shape of the bias-variance trade-off curve and we have p/n in the x-axis, for our purposes, it matters little if n or p changes as long as p/n changes.\n\nIn this simple example, we consider the case where the underlying data generating process is as follows:\n\n$y = \\mathbf{X\\beta} + \\varepsilon$ with $$\\mathbf{X}$$ and $$y$$ are known and $$\\mathbf{\\beta}$$ and $$\\varepsilon$$ are unknown. Let’s say there are 100 known predictors ($$\\mathbf{X}$$ has 100 columns). The data is produced as follows:\n\nnum_predictor <- 100\ncreate_data <- function(sigma, training_sample_size){\nX <- matrix(rnorm(10000num_predictor), ncol = num_predictor)\n\n#Coefficients\nbeta <- runif(num_predictor)\nbeta <- beta/sqrt(sum(beta^2)) ## Normalizing so that the norm is 1\n\n#Error term\ne <- rnorm(10000, sd = sigma)\n#The outcome\ny <- X %% beta + e\n\nX_train <- X[1:training_sample_size,]\ny_train <- y[1:training_sample_size, ,drop = FALSE]\n\nX_test <- X[(training_sample_size+1):nrow(X),]\ny_test <- y[(training_sample_size+1):nrow(y), ,drop = FALSE]\n\nreturn(list(y_train=y_train, X_train=X_train,\ny_test=y_test, X_test=X_test,\ne=e, beta=beta))\n\n}\n\n#### Linear Models\n\nIt is surprisingly straightforward to obtain the sample-wise double descent with the linear model. The trick is that when the number of observations is smaller than the number of predictors (i.e. after the interpolation limit where there are multiple solutions), one can use the Moore-Penrose inverse to calculate $=$, as the $$\\mathbf{(X^{T}X)^{-1}}$$ part is undefined. Moore-Penrose inverse chooses the solution with the smallest Euclidean norm of the coefficients when there are multiple solutions. When the matrix is invertible and there is a unique solution, the Moore-Penrose inverse is numerically same as the regular inverse of the matrix. So we use it to obtain $$\\widehat{\\beta}$$ for any p/n.\n\nBased on Iyar Lin’s reproduction of the “More Data Can Hurt for Linear Regression: Sample-wise Double Descent” paper by Preetum Nakkiran, the following code produces two lines for two different data generating processes: one with no error term ($$\\mathbf{y} = \\mathbf{X} \\mathbf{\\beta}$$) and another with normally distributed error term with mean 0, standard deviation 0.1 ($y = +$).\n\nlibrary(furrr) # For parallel processing\nnum_workers <- parallel::detectCores() - 1L\n\n#Estimation\nLS_estimator <- function(sigma = 0.1, training_sample_size = num_predictor){\n# Putting the number generators inside to obtan the exact same data every time.\nset.seed(5)\nfull_data <- create_data(sigma=sigma, training_sample_size = training_sample_size)\n\n#Estimate coefficients with the training data and predict y_test\nbeta_hat <- MASS::ginv(full_data$X_train) %% full_data$y_train\ny_test_hat <- full_data$X_test %% beta_hat #Return the RMSE for given sigma and sample size. data.frame(rmse = sqrt(mean((full_data$y_test - y_test_hat)^2)),\nsigma = sigma,\ntraining_sample_size = training_sample_size)\n\n}\n\n# We are saving the file to save time.\nif (file.exists(\"all_results_lm.rds\")) {\n} else {\nplan(multisession, workers = num_workers)\nalternative_values <- expand.grid(\nsigma_values = c(0,0.1),\ntraining_sample_size_values = c(num_predictorc(0.5, 0.75, 0.9, 0.95, 1, 1.05, 1.1, 1.25, 1.5))\n)\nres <- future_map2_dfr(.x = alternative_values$sigma_values, .y = alternative_values$training_sample_size_values,\n.f = ~LS_estimator(sigma = .x,\ntraining_sample_size = .y),\n.options = furrr_options(scheduling = 25, seed = FALSE))\nplan(sequential)\nsaveRDS(res, \"all_results_lm.rds\")\n}\n\n#The plot\ndd_plot <- ggplot(data = res) +\ngeom_line(aes(x = training_sample_size, y= rmse, group = factor(sigma), color = factor(sigma)), size = 2) +\ngeom_point(aes(x = training_sample_size, y= rmse, group = sigma, color = factor(sigma)), size = 3) +\ntheme_bw() +\ncoord_cartesian(ylim = c(0, 1)) +ylab(\"RMSE\") + xlab(\"Training Sample Size\") +\ntheme(text=element_text(size=15))\n\ndd_plot$labels$colour <- \"Std. Dev. of Error\"\n\ndd_plot", null, "Figure 3: Samplewise Double Descent with Linear Estimator\n\nLet’s start with the unsurprising curve, the red one. What happens is that as the training sample size increases (p/n decreases), the RMSE decreases. When the number of observations in the training sample size is 100, we have 100 unknown variables and 100 equations, and the system has a unique solution. As a result, the RMSE hits 0, and remains 0 thereafter.\n\nNow, the surprising one curve, the turqoise colored line. In the beginning, everything is expected: more data brings more information and more information decreases the RMSE. But then, p/n is very close to 1, the RMSE suddenly shots up. It becomes so big that we have to cut the graph (the RMSE is more than 3 when p/n = 1). The error term completely confuses the estimator. However, the confusion only happens in the small neighborhood of p/n = 1, and the RMSE quickly decreases when the sample size increases.\n\n#### Neural Networks\n\nAnother question is whether we see the same pattern with the neural networks. Note that the spike in the RMSE occurs when there is a unique set of coefficients that solves the system and there is irreducible error; namely pure noise. So, these coefficients are correct and they are the only ones that solve the system when $$p/n \\leq 1$$. Neural networks tend to use gradient descent methods to obtain the coefficients. These coefficients will be approximately correct. Let’s see how neural networks perform in the same scenarios.\n\nlibrary(torch)\n\nNN_estimator <- function(sigma = 0.1, training_sample_size = num_predictor){\n# Putting the number generators inside to obtan the exact same data every time.\nset.seed(5)\ntorch_manual_seed(5)\n\nfull_data <- create_data(sigma=sigma, training_sample_size = training_sample_size)\n\nfull_data$x_train_tensor = torch_tensor(full_data$X_train, dtype = torch_float())\nfull_data$y_train_tensor = torch_tensor(full_data$y_train, dtype = torch_float())\nfull_data$x_test_tensor = torch_tensor(full_data$X_test, dtype = torch_float())\n\ntorch_dataset <- dataset(\n\nname = \"my_data\",\n\ninitialize = function() {\nself$x_train <- full_data$x_train_tensor\nself$y_train <- full_data$y_train_tensor\n},\n\n.getitem = function(index) {\n\nx <- self$x_train[index, ] y <- self$y_train[index, ]\n\nlist(x, y)\n},\n\n.length = function() {\nself$x_train$size()[]\n}\n)\n\ntrain_ds <- torch_dataset()\ntrain_dl <- train_ds %>%\ndataloader(batch_size = 25, shuffle = FALSE)\n\n# Pretty basic 1-layer neural network model (mathematically identical to the least squares estimator)\n#Naturally, this is extremely inefficient way to do this, but let's use nn_linear.\nnet = nn_module(\n\"class_net\",\n\ninitialize = function(){\n\nself$linear1 = nn_linear(num_predictor,1) }, forward = function(x){ x %>% self$linear1()\n\n}\n\n)\n\nmodel = net()\n\noptimizer <- optim_adam(model$parameters) n_epochs <- 2000 device <- \"cpu\" train_batch <- function(b) { optimizer$zero_grad()\noutput <- model(b[]$to(device = device)) target <- b[]$to(device = device)\n\nloss <- nnf_mse_loss(output, target)\nloss$backward() optimizer$step()\n\n}\n\nfor (epoch in 1:n_epochs) {\n\nmodel$train() coro::loop(for (b in train_dl) { loss <-train_batch(b) }) if (epoch %% 100 == 0 ){ cat(sprintf(\"\\nEpoch %d\", epoch)) } } model$eval()\n\ny_test_hat <- model(full_data$x_test_tensor) y_test_hat <- as_array(y_test_hat) #Return the RMSE for given sigma and sample size. data.frame(rmse = sqrt(mean((full_data$y_test - y_test_hat)^2)),\nsigma = sigma,\ntraining_sample_size = training_sample_size)\n\n}\n\nif (file.exists(\"all_results_nn.rds\")) {\n} else {\nalternative_values <- expand.grid(\nsigma_values = c(0,0.1),\ntraining_sample_size_values = c(num_predictorc(0.5, 0.75, 0.9, 0.95, 1, 1.05, 1.1, 1.25, 1.5))\n)\nres_nn <- data.frame()\nfor (zz in c(1:nrow(alternative_values))){\nres_nn <- bind_rows(res_nn , NN_estimator(\nsigma = alternative_values$sigma_values[zz], training_sample_size = alternative_values$training_sample_size_values[zz]\n))\n\n}\n\nsaveRDS(res_nn, \"all_results_nn.rds\")\n}\n\n#The plot\ndd_plot <- ggplot(data = res_nn) +\ngeom_line(aes(x = training_sample_size, y= rmse, group = factor(sigma), color = factor(sigma)), size = 2) +\ngeom_point(aes(x = training_sample_size, y= rmse, group = sigma, color = factor(sigma)), size = 3) +\ntheme_bw() +\ncoord_cartesian(ylim = c(0, 1)) +ylab(\"RMSE\") + xlab(\"Training Sample Size\") +\ntheme(text=element_text(size=15))\n\ndd_plot$labels$colour <- \"Std. Dev. of Error\"\n\ndd_plot", null, "Figure 4: Samplewise Double Descent with Neural Network\n\nWow!!! The sample-wise double descent did not appear as strongly with the neural networks. This is really interesting. The optimization algorithm has avoided the spike around p/n = 1 (or training sample size = 100).\n\nWhen the error term is non-existent (red curve), the neural networks did an OK job. The error is not 0 at p/n, which was the case for the linear model, but it was not too big either. But when the error term is non-negligible (turqoise curve), allowing some imperfections proved to be very helpful, especially if we are interested in preventing some catastrophic fails. There occurred some increase in RMSE at p/n = 1, but it is certainly not of the proportions that we have seen in the linear model graphs.\n\n#### Cost of Perfection\n\nWe know that the correct coefficient estimates, the set of coefficients that provide the unique solution, come from the linear model and we know that neural network coefficients are, in this sense, approximately correct. By decreasing the learning rate (the size of steps towards the correct solution that the neural network takes) and trying out alternative number of epochs, let’s see at what point do we observe the cost of perfection when p/n = 1 and standard deviation of the error term is not 0. The expectation is that in the beginning, when the number of epochs is too small, the RMSE will be quite high because we would be quite far from the correct solution. As we increase the number of epochs, the RMSE will go down initially. After a certain point, it will go up again.\n\nlibrary(torch)\n\nNN_estimator <- function(sigma = 0.1, training_sample_size = num_predictor, n_epochs){\n# Putting the number generators inside to obtain the exact same data every time.\nset.seed(5)\ntorch_manual_seed(5)\n\nfull_data <- create_data(sigma=sigma, training_sample_size = training_sample_size)\n\nbeta_hat <- MASS::ginv(full_data$X_train) %% full_data$y_train\ny_test_hat_lm <- full_data$X_test %% beta_hat y_train_hat_lm <- full_data$X_train %*% beta_hat\n\nfull_data$x_train_tensor = torch_tensor(full_data$X_train, dtype = torch_float())\nfull_data$y_train_tensor = torch_tensor(full_data$y_train, dtype = torch_float())\nfull_data$x_test_tensor = torch_tensor(full_data$X_test, dtype = torch_float())\n\ntorch_dataset <- dataset(\n\nname = \"my_data\",\n\ninitialize = function() {\nself$x_train <- full_data$x_train_tensor\nself$y_train <- full_data$y_train_tensor\n},\n\n.getitem = function(index) {\n\nx <- self$x_train[index, ] y <- self$y_train[index, ]\n\nlist(x, y)\n},\n\n.length = function() {\nself$x_train$size()[]\n}\n)\n\ntrain_ds <- torch_dataset()\ntrain_dl <- train_ds %>%\ndataloader(batch_size = 25, shuffle = FALSE)\n\n# Pretty basic 1-layer neural network model (mathematically identical to the least squares estimator)\n#Naturally, this is extremely inefficient way to do this, but let's use nn_linear.\nnet = nn_module(\n\"class_net\",\n\ninitialize = function(){\n\nself$linear1 = nn_linear(num_predictor,1) }, forward = function(x){ x %>% self$linear1()\n\n}\n\n)\n\nmodel = net()\n\noptimizer <- optim_adam(model$parameters, lr = 1e-5) device <- \"cpu\" train_batch <- function(b) { optimizer$zero_grad()\noutput <- model(b[]$to(device = device)) target <- b[]$to(device = device)\n\nloss <- nnf_mse_loss(output, target)\nloss$backward() optimizer$step()\nloss$item() } final_data <- data.frame() for (epoch in 1:n_epochs) { model$train()\ntrain_loss <- c()\ncoro::loop(for (b in train_dl) {\nloss <- train_batch(b)\ntrain_loss <- c(train_loss, loss)\n})\nmodel$eval() y_test_hat <- model(full_data$x_test_tensor)\ny_test_hat <- as_array(y_test_hat)\n\nfinal_data <- bind_rows(final_data, data.frame(rmse = sqrt(mean((full_data$y_test - y_test_hat)^2)), epoch = epoch, train_loss = mean(train_loss))) if (epoch %% 50 == 0 ){ cat(sprintf(\"\\nEpoch %d, training: loss: %3.5f \\n\", epoch, mean(train_loss))) cat(sprintf(\"\\nTot Epoch num %d\", n_epochs)) } } #Return the RMSE for given sigma and sample size. final_data <- final_data %>% mutate(rmse_lm = sqrt(mean((full_data$y_test - y_test_hat_lm)^2))) %>%\nmutate(rmse_lm_training = sqrt(mean((full_data\\$y_train - y_train_hat_lm)^2)))\n\nreturn(final_data)\n\n}\n\nif (file.exists(\"cost_of_perfection_nn.rds\")) {\n} else {\nres_nn_cp <- NN_estimator(\nn_epochs = 100000\n)\n\nsaveRDS(res_nn_cp, \"cost_of_perfection_nn.rds\")\n}\n\nggplot(data = res_nn_cp) +\ngeom_line(aes(x=epoch, y = rmse, color = \"Neural network\"), size = 2) +\ngeom_line(aes(x=epoch, y = rmse_lm, color = \"Linear model\"), size = 2) +\ntheme_bw() +\nscale_color_manual(name = \"Models\",\nvalues = c(\"Neural network\" = \"darkgreen\",\n\"Linear model\" = \"red\")) +\nylab(\"RMSE\") + xlab(\"Number of Epochs\") +\ntheme(text=element_text(size=15))", null, "Figure 5: Cost of Perfection\n\nWe do see the expected U-shaped RMSE curve, but the latter half of it never reaches the levels that the linear model reaches. The cost of perfection (though calling it cost of approximate perfection would be more apt) is quite small in the case of neural networks. Since the gradient descent algorithm never really reaches the exact set of coefficients, it never really pays the full price of the perfectionism. This occurs even when we select a relatively small learning rate ($$10^{-5}$$) and many epochs ($$10^6$$), and we achieve a training error that is smaller than $$10^{-13}$$.3 But the main point here is that the training error is not 0, so our solution is approximately correct for the training data. This has negligible effects for every p/n values other than 1. But when p/n = 1, the perfection vs. approximate perfection matters.\n\nThe general idea of allowing some imperfections to get better predictions should not be too novel, since the whole idea behind LASSO or Ridge regressions is to allow some bias to reduce variance. It is just nice to see how it can reveal itself under different scenarios. We will continue to see this pattern in the coming blog posts related to the double descent phenomenon as well.\n\n1. In this example, we used MSE as the prediction error. Other accuracy metric would yield qualitatively similar results in the sense that the sweet spot minimizes neither the variance nor the bias.↩︎\n\n2. This is shown in the paper titled “Triple descent and the Two Kinds of Overfitting: Where & Why do they Appear?” by d’Ascoli, Sagun, and Biroli.↩︎\n\n3. If the learning rate was infinitely small and the number of epochs was infinitely large, then we would reach the linear model solution.↩︎" ]	[ null, "https://i1.wp.com/dorukcengiz.netlify.app/post/double-descent-part-i-sample-wise-non-monotonicity/index.en_files/figure-html/unnamed-chunk-1-1.png", null, "https://i0.wp.com/dorukcengiz.netlify.app/post/double-descent-part-i-sample-wise-non-monotonicity/index.en_files/figure-html/unnamed-chunk-2-1.png", null, "https://i2.wp.com/dorukcengiz.netlify.app/post/double-descent-part-i-sample-wise-non-monotonicity/index.en_files/figure-html/unnamed-chunk-4-1.png", null, "https://i0.wp.com/dorukcengiz.netlify.app/post/double-descent-part-i-sample-wise-non-monotonicity/index.en_files/figure-html/unnamed-chunk-5-1.png", null, "https://i2.wp.com/dorukcengiz.netlify.app/post/double-descent-part-i-sample-wise-non-monotonicity/index.en_files/figure-html/unnamed-chunk-6-1.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7683508,"math_prob":0.9985239,"size":22043,"snap":"2021-31-2021-39","text_gpt3_token_len":5840,"char_repetition_ratio":0.12886247,"word_repetition_ratio":0.21067683,"special_character_ratio":0.28997868,"punctuation_ratio":0.14517766,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998302,"pos_list":[0,1,2,3,4,5,6,7,8,9,10],"im_url_duplicate_count":[null,3,null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-23T23:40:23Z\",\"WARC-Record-ID\":\"<urn:uuid:a182373e-9ae6-449a-9132-85143ff5a36e>\",\"Content-Length\":\"141517\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e8553e29-59b8-4054-ac81-3b6e3fd37125>\",\"WARC-Concurrent-To\":\"<urn:uuid:37919697-0f35-4e97-b6bd-1b33951def90>\",\"WARC-IP-Address\":\"172.64.135.34\",\"WARC-Target-URI\":\"https://www.r-bloggers.com/2021/07/double-descent-part-i-sample-wise-non-monotonicity/\",\"WARC-Payload-Digest\":\"sha1:7AOF524XWX4P3ARA4OZFGKQVBBSLQ4HK\",\"WARC-Block-Digest\":\"sha1:3C73NGM2TFCVI4PRKLNVKPIGXGDEZYWS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057479.26_warc_CC-MAIN-20210923225758-20210924015758-00471.warc.gz\"}"}
https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-15/issue-2/Normal-approximation-and-confidence-region-of-singular-subspaces/10.1214/21-EJS1876.full	[ "Translator Disclaimer\n2021 Normal approximation and confidence region of singular subspaces\nDong Xia\nElectron. J. Statist. 15(2): 3798-3851 (2021). DOI: 10.1214/21-EJS1876\n\n## Abstract\n\nThis paper is on the normal approximation of singular subspaces when the noise matrix has i.i.d. entries. Our contributions are three-fold. First, we derive an explicit representation formula of the empirical spectral projectors. The formula is neat and holds for deterministic matrix perturbations. Second, we calculate the expected projection distance between the empirical singular subspaces and true singular subspaces. Our method allows obtaining arbitrary k-th order approximation of the expected projection distance. Third, we prove the non-asymptotical normal approximation of the projection distance with different levels of bias corrections. By the $\\lceil \\log ({d_{1}}+{d_{2}})\\rceil$-th order bias corrections, the asymptotical normality holds under optimal signal-to-noise ratio (SNR) condition where ${d_{1}}$ and ${d_{2}}$ denote the matrix sizes. In addition, it shows that higher order approximations are unnecessary when $\|{d_{1}}-{d_{2}}\|=O({({d_{1}}+{d_{2}})^{1/ 2}})$. Finally, we provide comprehensive simulation results to merit our theoretic discoveries.\n\nUnlike the existing results, our approach is non-asymptotical and the convergence rates are established. Our method allows the rank r to diverge as fast as $o({({d_{1}}+{d_{2}})^{1/ 3}})$. Moreover, our method requires no eigen-gap condition (except the SNR) and no constraints between ${d_{1}}$ and ${d_{2}}$.\n\n## Funding Statement\n\nThis research is supported partially by Hong Kong RGC Grant ECS 26302019 and GRF 16303320.\n\n## Acknowledgments\n\nThe author would like to thank Yik-Man Chiang for the insightful recommendations on applying the Residue theorem, Jeff Yao for the encouragements on improving the former results, and an anonymous referee for pointing out the reference Kato (2013).\n\n## Citation\n\nDong Xia. \"Normal approximation and confidence region of singular subspaces.\" Electron. J. Statist. 15 (2) 3798 - 3851, 2021. https://doi.org/10.1214/21-EJS1876\n\n## Information\n\nReceived: 1 November 2020; Published: 2021\nFirst available in Project Euclid: 29 July 2021\n\nDigital Object Identifier: 10.1214/21-EJS1876\n\nSubjects:\nPrimary: 62H10 , 62H25\nSecondary: 62G20\n\nKeywords: Normal approximation , projection distance , Random matrix theory , Singular value decomposition , spectral perturbation\n\nJOURNAL ARTICLE\n54 PAGES", null, "SHARE\nVol.15 • No. 2 • 2021", null, "" ]	[ null, "https://projecteuclid.org/Content/themes/SPIEImages/Share_black_icon.png", null, "https://projecteuclid.org/images/journals/cover_ejs.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86739653,"math_prob":0.94562274,"size":1798,"snap":"2022-40-2023-06","text_gpt3_token_len":373,"char_repetition_ratio":0.11092531,"word_repetition_ratio":0.0,"special_character_ratio":0.20244716,"punctuation_ratio":0.124590166,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96901786,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-29T11:57:44Z\",\"WARC-Record-ID\":\"<urn:uuid:20a14847-6acf-4fbc-b920-2b6efd3ee76e>\",\"Content-Length\":\"150711\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c219f330-427f-4cd8-8c12-20282bf15985>\",\"WARC-Concurrent-To\":\"<urn:uuid:051dd497-a8b1-47fa-8400-230985e9efbe>\",\"WARC-IP-Address\":\"107.154.79.145\",\"WARC-Target-URI\":\"https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-15/issue-2/Normal-approximation-and-confidence-region-of-singular-subspaces/10.1214/21-EJS1876.full\",\"WARC-Payload-Digest\":\"sha1:EF47QTJYBBEDODDQSJNH4IBMRTE56P25\",\"WARC-Block-Digest\":\"sha1:3R6YN5AYTIBQCP44KMZRVE2DUZ5WNJIE\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335350.36_warc_CC-MAIN-20220929100506-20220929130506-00064.warc.gz\"}"}
https://crypto.stackexchange.com/questions/88607/ntru-euclidean-algorithm-the-inverse-of-f-modulo-p	[ "# NTRU Euclidean algorithm the inverse of f modulo p\n\nI am very new to the world of cryptography and have just began my research in the post quantum cryptography sector. I have been reading and trying to understand NTRU key generation and am struggling to understand how to practically do the inverse of f. I know that this is solved using the Euclidean algorithm, but I am unable to figure out what the steps would be. I was hoping someone would be able to show the steps to solve the example shown on the NTRUEncrypt wiki page shown below (fp or fq), or a smaller polynomial that gets the point across. I am more interested in the practical steps then the theory behind it, as I have found many resources for that.\n\nThank you!", null, "• It is found by the Extended-GCD. If you write \"find polynomial inverse modulo\" these words into your search engine you will see. Almost all books mentions this, too. See from handbook applied cryptography page 82 Mar 4, 2021 at 7:46\n\nI'll try to follow a similar notation to the example on wikipedia\n\nOur initial inputs are $$p(x) = x^{11}-1$$ which defines the ring and $$a(x)=-x^{10}+x^9+x^6-x^4+x^2-1$$ and working mod 3 rather than the mod 2 example on wikipedia.\n\nFor step 1 we have quotient $$q_1(x)=2x+2$$ and remainder $$r_1(x)=x^9+x^7+x^6+2x^5+2x^4+x^3+2x^2+1$$ so that $$t_1=x+1$$.\n\nFor step 2 we have quotient $$q_2=2x+1$$ and remainder $$r_2=x^8+2x^6+x^4+x^3+2x^2+2x+1$$ and $$t_2=x^2$$.\n\nFor step 3 we have quotient $$q_3=x$$ and remainder $$r_3=2x^7+x^6+x^7+x^4+2x^3+2x+1$$ and $$t_2=x^3+x+1$$.\n\nFor step 4 we have quotient $$q_4=2x+2$$ and remainder $$r_4=x^6+2x^5+x^4+x^2+2x+2$$ and $$t_4=2x^4+2x^3+2x^2+2x+1$$.\n\nFor step 5 we have quotient $$q_5=2x$$ and remainder $$r_5=2x^5+x^4+2x^2+x+1$$ and $$t_5=2x^5+2x^4+x^3+2x^2+2x+1$$.\n\nFor step 6 we have quotient $$q_6=2x$$ and remainder $$r_6=x^4+2x^3+2x^2+2$$ and $$t_6=x2x^6+2x^5+x^3+x^2+1$$.\n\nFor step 7 we have quotient $$q_7=2x$$ and remainder $$r_7=2x^3+2x^2+1$$ and $$t_7=x2x^7+2x^6+2x^5+2x^3+2x^2+1$$.\n\nFor step 8 we have quotient $$q_8=2x+2$$ and remainder $$r_8=x^2+x$$ and $$t_9=2x^9+x^8+2*x^7+x^5+2x^4+2x^3+2x+1$$.\n\nFor step 9 we have quotient $$q_9=2x$$ and remainder $$r_9=1$$ and $$t_9=2x^9+x^8+2x^7+x^5+2x^4+2x^3+x+2$$, which is the required inverse.\n\nI'll leave the mod 32 example as an exercise (sagemath is your friend).\n\n• Thank you, this is what I was hoping to see, I think I was using a invalid polynomial for my p(x). Thank you for your input! Mar 4, 2021 at 19:16\n• Thanks, this has been useful for me too. I have a follow-up question: so the Euclidean algorithm usually only runs in Euclidean domains (ED) and when working mod 3 you're running it in $\\mathbb{Z}_3[X]$ (right?) which I see is an ED because $\\mathbb{Z}_3$ is a field. But if you're working mod 32, then could there be problems as $\\mathbb{Z}_{32}[X]$ is no longer an ED?\n– wdc\nMay 29, 2021 at 2:15" ]	[ null, "https://i.stack.imgur.com/OCD9o.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9685527,"math_prob":0.9998528,"size":673,"snap":"2022-05-2022-21","text_gpt3_token_len":141,"char_repetition_ratio":0.10313901,"word_repetition_ratio":0.0,"special_character_ratio":0.20059435,"punctuation_ratio":0.06716418,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.000008,"pos_list":[0,1,2],"im_url_duplicate_count":[null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-19T12:37:43Z\",\"WARC-Record-ID\":\"<urn:uuid:cf0baeef-b21f-4959-a29c-2acde12eb3d9>\",\"Content-Length\":\"229153\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:913b8599-591f-48c5-9d66-cb389de9b535>\",\"WARC-Concurrent-To\":\"<urn:uuid:b98e744d-7bd8-4b9b-85de-91b06a193006>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://crypto.stackexchange.com/questions/88607/ntru-euclidean-algorithm-the-inverse-of-f-modulo-p\",\"WARC-Payload-Digest\":\"sha1:55OK2XI4NHG4IUP2AJRONYMMZC7GG7QA\",\"WARC-Block-Digest\":\"sha1:XVAXLKASHYAJOTWG5KZLJ4TRBSP5QCFB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662527626.15_warc_CC-MAIN-20220519105247-20220519135247-00238.warc.gz\"}"}
https://www.arxiv-vanity.com/papers/astro-ph/9711288/	[ "###### Abstract\n\nWe propose a simple model in which the cosmological dark matter consists of particles whose mass increases with the scale factor of the universe. The particle mass is generated by the expectation value of a scalar field which does not have a stable vacuum state, but which is effectively stabilized by the rest energy of the ambient particles. As the universe expands, the density of particles decreases, leading to an increase in the vacuum expectation value of the scalar (and hence the mass of the particle). The energy density of the coupled system of variable-mass particles (“vamps”) redshifts more slowly than that of ordinary matter. Consequently, the age of the universe is larger than in conventional scenarios.\n\nNSF-ITP/97-146\n\nNUHEP-TH-97-14\n\nastro-ph/9711288\n\nDark Matter with Time-Dependent Mass111Based on a talk by SMC at COSMO-97, International Workshop on Particle Physics and the Early Universe, 15-19 September 1997, Ambleside, Lake District, England.\n\nGreg W. Anderson Dept. of Physics and Astronomy, Northwestern University\n\n2145 Sheridan Rd., Evanston, IL 60208-3112, USA\n\nEmail:\n\nSean M. Carroll Institute for Theoretical Physics, University of California,\n\nSanta Barbara, California 93106, USA\n\nEmail:\n\n## 1 Introduction\n\nThe Big Bang model has proven extraordinarily successful as a framework for interpreting the structure and evolution of the universe on large scales. Within that framework, the cold dark matter scenario (featuring massive particles which bring the density of the universe to its critical value, and a scale-free spectrum of Gaussian density perturbations) has provided an elegant theory of structure formation, which unfortunately seems to fall short of perfect agreement with observation. Although the precise extent to which CDM disagrees with observation is arguable, there are two important areas in which the discrepancies are particularly troubling: predicting an age for the universe which is larger than the ages of the oldest globular clusters, and matching the COBE-normalized power spectrum of density fluctuations as measured by microwave background anisotropy experiments and direct studies of large-scale structure.\n\nOne way in which the simple CDM scenario may be modified, affecting the age of the universe as well as the evolution of density fluctuations, is to imagine that the closure density is provided by something different than (or in addition to) nonrelativistic particles. In a flat Robertson-Walker universe with metric\n\n ds2=−dt2+a2(t)(dx2+dy2+dz2) (1)\n\nand energy-momentum tensor\n\n Tμν=diag(−ρ,p,p,p) , (2)\n\nthe Friedmann equations imply that the time derivative of the scale factor satisfies\n\n ˙a2=8πG3a2ρ . (3)\n\nThe evolution of is therefore dependent on how the energy density scales with ; if , we have\n\n ¨aa=4πG3(2−n)ρ . (4)\n\nHence, the more slowly the energy density decreases as the universe expands, the more slowly the expansion will decelerate, implying a correspondingly older universe for any given value of the expansion rate today — for a flat universe dominated by such an energy density, the age is , where is the Hubble parameter and the subscript refers to the present time. (Eq. (4) can also be derived by positing an equation of state and using energy conservation; the two parameterizations are related by .) The energy density in a species of ordinary “matter” (a nonrelativistic particle species ) can be written , where is the mass of the particle and is its number density. The energy density of a matter-dominated universe is therefore proportional to , as the mass stays constant while the number density is proportional to the volume; the age of such a universe is .\n\nAlthough there is some controversy over the value of the Hubble constant, most recent determinations are consistent with a value , or . The upper limit on the age of a matter-dominated flat universe is therefore . Meanwhile, calculations of the ages of globular clusters imply an age , with a lower bound of . The apparent discrepancy between these values may be resolved by a revision in distance determinations to globular clusters, as suggested by recent measurements by the Hipparcos satellite ; while this would be the simplest solution, further work is necessary to accept it with confidence.\n\nAlternative resolutions are provided by models in which the density parameter , and some or all of the unseen energy density resides in a component which redshifts more slowly than nonrelativistic matter. The most popular such alternative is the introduction of a cosmological constant , for which . Such models have some attractive features, but are also plagued with both theoretical and observational disadvantages [4, 5]. A popular variation on this theme is to invoke a slowly-rolling scalar field, or equivalently a cosmological constant whose value varies with time, or simply an unspecified smooth component . More speculative possibilities include a network of cosmic strings or stable textures . We will not enumerate the good and bad qualities of each of these scenarios, noting only that none are sufficiently compelling to discourage the exploration of still further models.\n\nIn this paper we propose a simple model in which the dark matter consists of particles whose rest mass increases with time. This is achieved by having the rest mass derive from the expectation value of a scalar field ; the potential for depends on the number density of particles, and therefore increases naturally on cosmological timescales as the universe expands. As a result, the particle energy density decreases more slowly than , resulting in a larger age for the universe. (There is also a contribution from the potential energy of , which redshifts at the same rate.) We discuss some of the cosmological consequences of this proposal, including potential observational tests. The question of structure formation in the presence of such particles, as well as the construction of realistic particle physics models containing the necessary fields, is left for future work.\n\nAfter this paper was first submitted, we became aware of earlier an proposal for dark matter with time-dependent mass by Casas, Garcìa-Bellido, and Quirós . These authors considered models of scalar-tensor gravity, in which the scalar coupled differently to different species of particles.\n\n## 2 Scale factor and age of the universe\n\nThe model consists of a scalar and a particle species , which can be either bosonic or fermionic for the purposes of this work. The mass of is imagined to come from the vacuum expectation value of , with the constant of proportionality some dimensionless parameter :\n\n mψ=λ⟨ϕ⟩ . (5)\n\nMore elaborate dependences of on are certainly conceivable, but for the purposes of this paper we make this simple choice. The dynamics of are determined by a conventional kinetic term and a potential energy . The notable feature of the model is that we choose the potential to blow up at and roll monotonically to zero as . For simplicity we will write\n\n U(ϕ)=uoϕ−p , (6)\n\nalthough more complicated forms are again possible. While such a potential seems unusual, this form can arise for example due to nonperturbative effects lifting flat directions in supersymmetric gauge theories , as well as for moduli fields in string theory. (In fact this form of potential is not strictly necessary, as the phenomenon we will describe can occur with almost any potential; however, the effects are most dramatic with this choice.)\n\nThis model possesses no stable vacuum state; in empty space tends to roll to infinity. We consider instead the behavior of in a homogeneous background of ’s with number density . In that case, the dependence of the free energy on the value of comes both from the potential and the rest energy of the particles, which have a mass proportional to . The equilibrium value of a homogeneous configuration is therefore one which minimizes an effective potential of the form\n\n V(ϕ)=u0ϕ−p+λnψϕ . (7)\n\n(See Fig. 1.)", null, "Figure 1: Effective potential for ϕ. The light solid curve is the bare potential U(ϕ)∝ϕ−1. The effective potential at finite density, given by the solid curves, is obtained by adding a contribution linear in ϕ and proportional to the number density nψ. This is plotted for two different values of nψ, corresponding to two different stages in the evolution of the universe. As the universe expands, nψ decreases, and the equilibrium value of ϕ increases.\n\nThe additional contribution can be thought of as arising because increasing increases the energy density in ’s since it increases the mass of . The expectation value of is therefore\n\n ⟨ϕ⟩=(pu0λnψ)1/(1+p) . (8)\n\n(Such a configuration is not truly stable, as spatially inhomogeneous perturbations will tend to grow, but it can be stable enough for cosmological purposes.) Density-dependent potentials such as this have been discussed previously in other contexts; see e.g. .\n\nIn an expanding universe, the number density will change with time; in turn, the mass of both and will change, as will the vacuum energy. After the interactions of have frozen out, the number density can be written as , where is the density when , which we take to be the present epoch. Then evolves as\n\n ϕ=ϕ0a3/(1+p) , (9)\n\nwhere is the value of (8) at the present time. In terms of these variables the mass of the boson is given by\n\n m2ϕ=∂2V∂ϕ2=[p(p+1)u0ϕ−(p+2)0]a−3(2+p)/(1+p) , (10)\n\nand the mass of is\n\n mψ=λϕ0a3/(1+p) . (11)\n\nBoth and are therefore variable-mass particles, or “vamps”; a cosmological model in which vamps are the dominant component of the energy density at late times will be referred to as VDM.\n\nThere are a number of contributions to the energy density of the universe in this model. These include the energy in the scalar particles, in the particles, in the time derivative of the expectation value of , in the potential , and in ordinary components of matter and radiation. For reasonable values of the parameters, the energies in and in quanta are small; the former because is only changing on cosmological timescales, and the latter because the mass of is decreasing with time. (At early times, the bosons are very massive and rapidly decay.) The important new contribution is therefore simply , the sum of the fundamental potential and the rest energy in the ’s. (We assume for now that is is nonrelativistic. As we discuss later, it is most likely that the particles were relativistic when they decoupled, but at late times their momenta have redshifted sufficiently that they are slowly moving today.) Both of these components turn out to depend on the scale factor in the same way; the ratio of the energy density in particles to that in the potential for is simply\n\n ρψρU(ϕ)=1p . (12)\n\nIt is therefore convenient to deal with the sum of these two contributions,\n\n ρV=(1+p)u0ϕ−p=(1+pp)λϕnψ , (13)\n\nwhich evolves as\n\n ρV=ρV0a−3p/(1+p) . (14)\n\nThe parameter characterizing the effective equation of state of the coupled / system is therefore .\n\nThe energy density in ordinary massive particles (baryons plus a possible cold dark matter component) redshifts as , more slowly than , and will therefore be the dominant source of energy density in the universe for intermediate redshifts. The redshift at which is given by\n\n 1+zVM=(ρV0ρM0)(1+p)/3 . (15)\n\nThe age of the universe, meanwhile, will be larger than in conventional flat models. The age corresponding to a redshift is given by\n\n t=∫a0da′˙a′=H−10∫(1+z)−10[1−Ω0+ΩM0x−1+ΩV0x(2−p)/(1+p)]−1/2dx . (16)\n\nFor the limiting case , , we find that the age of the universe now is simply\n\n t0=23H−10(1+p−1) . (17)\n\nFig. 2 plots the age of universes with as a function of , for .\n\nAn interesting feature of this model, in comparison with alternative theories of rolling scalar fields and unusual equations of state, is that (because is massive and nonrelativistic at late times) it is at least conceivable that the energy density of the universe is dominated solely by baryons and vamps (without any significant cold dark matter component). For illustrative purposes, let us define the “minimal VDM model” as that with , , and km/sec/Mpc. This is a flat universe consisting solely of baryons and vamps, with the baryon density consistent with the prediction of Big Bang nucleosynthesis. In this minimal model, vamp-matter equality occurs at a redshift . The age of the universe turns out to be approximately years, in good accord with the (pre-Hipparcos) ages of the oldest globular clusters.", null, "Figure 2: Age of the universe in billions of years. The values in this plot are computed for flat universes consisting of only vamps and nonrelativistic matter, with p=1.\n\n## 3 Particle parameters and abundances\n\nThe properties we have deduced to this point depend on the present energy density , but not on any assumptions about the parameters of the particle physics model in which we imagine the necessary fields and interactions could arise. To understand the formation of large-scale structure in the model, however, it is necessary to know the mass and average velocity of the particles today, and to compute these requires some detailed knowledge of the interactions of our two fields. In the absence of a specific model, we will estimate these quantities under the minimal assumptions that was in thermal equilibrium at some high temperature and has evolved freely ever since.\n\nWe begin by considering the general problem of the motion of an otherwise free particle whose mass may vary throughout spacetime. The motion of such a particle extremizes the action\n\n S=∫√−pμpμdτ=λ∫ϕ(xμ)(−gμνdxμdτdxνdτ)1/2dτ , (18)\n\nwhere is the proper time along the particle’s trajectory and is the particle’s four-momentum. Variation of this action with respect to the path leads to an equation of motion\n\n Dpμdτ≡dpμdτ+Γμρσdxρdτpσ=−λ∇μϕ , (19)\n\nwhich can be written explicitly in terms of the path as\n\n d2xμdτ2+Γμρσdxρdτdxσdτ=−(gμν+dxμdτdxνdτ)∂ν(lnϕ) . (20)\n\nSince we are assuming that is constant along spacelike hypersurfaces of the metric (1), we can solve explicitly for the motion of a particle obeying (19). In terms of the magnitude of the spacelike 3-momentum,\n\n \|→p\|2=gijpipj=a2δijpipj , (21)\n\nwe find\n\n \|→p\|∝a−1 , (22)\n\njust as for conventional (constant-mass) particles. The distinction arises for the velocity; if the four-velocity satisfies , the magnitude of the three-velocity is proportional to . Thus, as the particles get more massive with time, they naturally slow down even more rapidly than ordinary test particles.\n\nAlthough we have not specified any explicit interactions between the vamps and visible matter, we may imagine that such reactions exist as long as they are sufficiently weak that they do not lead to consequences which would have already been observed. As a result of such interactions, we presume that the ’s were in thermal equilibrium at some high temperature. Their equilibrium phase-space distribution function is either a Fermi-Dirac or Bose-Einstein distribution,\n\n f(p)=gψh3P1eE/kTψ±1 , (23)\n\nwhere is the number of spin degrees of freedom, is Planck’s constant, is Boltzmann’s constant, is the temperature of the ’s, and is the energy, given by\n\n E=p0=(m2ψ+\|→p\|2)1/2 . (24)\n\nAs the temperature and density increase, goes up while goes down. At sufficiently early times, therefore, the particles were relativistic, and . Under these circumstances the ’s behave like ordinary relativistic particles; their temperature redshifts as , and their energy density as . When they become nonrelativistic, on the other hand, their kinetic temperature will scale as ; they cool off more rapidly than ordinary matter. (Strictly speaking it is incorrect to speak of a temperature after the particles become nonrelativistic, as the varying rates at which the particles slow down will distort the initially thermal distribution.)\n\nIt is reasonable to assume that was relativistic when it decoupled, and we may proceed under this assumption to show that it leads to a consistent picture. In that case the number density of today is given by the standard formula \n\n nψ0=825rψ cm−3 , (25)\n\nwhere is the ratio of , the effective number of degrees of freedom in , to , the total effective number of relativistic degrees of freedom at freeze-out. (In terms of the number of physical degrees of freedom , for bosons and for fermions.) For simplicity let us consider the case . Then we can directly determine the mass of in terms of the current density parameter and Hubble constant km/sec/Mpc:\n\n mψ=12.7Ωψ0h2r−1ψa3/2 eV . (26)\n\nIn terms of the Yukawa coupling , the other relevant parameters of the model are then\n\n u0=1.02×10−9Ω2ψ0h4λrψ (eV)5 (27)\n\nand\n\n mϕ=1.00×10−6λrψΩ1/2ψ0ha−9/4 eV . (28)\n\nThe temperature of the particles (while they are still relativistic) is diluted somewhat with respect to that of the photons, due to entropy production subsequent to the freeze-out of :\n\n Tψ=(g∗0g∗f)1/3Tγ0a−1=3.55×10−4a−1g−1/3∗f eV . (29)\n\nComparing (26) to (29), we find that the ’s first become non-relativistic at a redshift of\n\n zNR=66.3⎛⎜⎝Ωψ0h2g1/3∗frψ⎞⎟⎠2/5 . (30)\n\n## 4 Further consequences\n\nAlthough the VDM model helps to alleviate the age problem, there are a number of other cosmological tests that could conceivably rule it out. For example, nucleosynthesis places stringent limits on the number of degrees of freedom contributing to the energy density at . Particles which decouple at sufficiently high energies are not constrained by this test, as their number density is diluted by entropy production after decoupling. We do not know the temperature at which decouples, although there is no reason to believe that it isn’t sufficiently high to evade the nucleosynthesis bound. Meanwhile, the energy density in is much less than that in at high temperatures, and is therefore even less constrained.\n\nAny model which increases the age of the universe by changing the behavior of the scale factor with time will be subject to various cosmological tests which are sensitive to that relationship; these are conventionally used to place limits on the cosmological constant . Currently the most promising such tests are direct measurements of deviation from the linear Hubble law using high-redshift Type Ia supernovae , and volume/redshift tests provided by the frequency of gravitational lensing of distant quasars by intervening galaxies . These have recently been applied to a number of models with novel dependences of the scale factor on time, very similar to the scenario discussed in this paper. The results to date seem to indicate that these tests do not rule out the kind of models considered here, but may be able to do so in the near future when more data is available.\n\nOur investigation has been exclusively in the context of an unperturbed Robertson-Walker cosmology. The next step is to introduce perturbations and discuss CMB anisotropies and the formation of structure; work in this direction is in progress. However, it is worth noting some important features of the problem. There are two powerful effects which distinguish the growth of perturbations in a VDM cosmology from conventional cold dark matter, and they tend to affect the power spectrum in opposite ways. The first effect is the effectively negative pressure of the coupled system. At zero temperature, perturbations in vamps grow more rapidly than those in CDM; indeed, perturbations tend to grow even in the absence of gravity. The other effect, meanwhile, is the free streaming of the particles. The ’s decouple while relativistic, and in some respects act as hot dark matter. They will tend to flow out of overdense regions, damping the growth of perturbations until sufficiently late times. An accurate appraisal of the magnitude of this process requires numerical integration of the evolution equations, as the Boltzmann equation does not simplify as it would for massless or completely nonrelativistic particles . These two competing effects are not the entire story; for example, if the ’s are fermions they will be prevented from clustering on very small scales by the exclusion principle . The final perturbation spectrum is therefore the result of a number of processes, and cannot be reliably estimated analytically. In addition, of course, the simple model we have investigated here may be modified, either by altering the form of the potential (6) or by introducing other forms of energy in addition to baryons and vamps (e.g., ordinary hot or cold dark matter).\n\nAnother direction currently under investigation is the construction of particle physics models in which vamps may arise. A possible origin for the scalar is as one of the moduli of string theory; our understanding of the nonperturbative effects which give potentials to such fields is not sufficiently developed to attempt realistic model building at this time. In supersymmetric gauge theories, however, there are (perturbatively) flat directions whose dynamics are somewhat better understood, and in that context the search for a model may be more hopeful. In such a scenario there are a number of potentially dangerous effects which must be avoided; for example, if the expectation value of breaks supersymmetry, it may lead to gradual variations in the parameters of the standard model as the universe expands. Such variations are tightly constrained by a variety of data .\n\n## Acknowledgments\n\nWe would like to thank Edmund Bertschinger, Edward Farhi, and Mark Srednicki for helpful conversations. This work was supported in part by the National Science Foundation under grant PHY/94-07195." ]	[ null, "https://media.arxiv-vanity.com/render-output/7320551/x1.png", null, "https://media.arxiv-vanity.com/render-output/7320551/x2.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8952394,"math_prob":0.97261655,"size":24181,"snap":"2023-14-2023-23","text_gpt3_token_len":5818,"char_repetition_ratio":0.14191173,"word_repetition_ratio":0.012787724,"special_character_ratio":0.24990696,"punctuation_ratio":0.17565359,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9763525,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-29T18:28:49Z\",\"WARC-Record-ID\":\"<urn:uuid:c2c669fd-18cc-4227-9f0d-41abf99a4319>\",\"Content-Length\":\"398333\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:de64a056-3efd-4461-9040-7a639190e4fb>\",\"WARC-Concurrent-To\":\"<urn:uuid:28faa0ad-0c7b-426e-ac13-9745ad1b4234>\",\"WARC-IP-Address\":\"172.67.158.169\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/astro-ph/9711288/\",\"WARC-Payload-Digest\":\"sha1:JKTYF7VXTQQI64PDYOOZBE6YPRVT7RIM\",\"WARC-Block-Digest\":\"sha1:XYXLMRTER3VQXU5LVS7HIG4TDG7EBLZG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224644907.31_warc_CC-MAIN-20230529173312-20230529203312-00525.warc.gz\"}"}
https://russianpatents.com/patent/211/2115884.html	[ "# A method of measuring the displacement\n\n(57) Abstract:\n\nA method of measuring the movement relates to fiber-optic transmission systems in measurement technology and is designed to measure displacements of the object. Summarized in the measurement area, the modulated radiation at a frequency of1modulate and lower than1frequency 2and12. Emit the signal of the first harmonic of the modulation frequency 1. Then the signals of the first harmonic and second harmonic served on the block comparison, where they perform the comparison of the latter. The output signal of the block comparison serves to the input of the comparator where it is compared with a reference voltage. The output signal of the comparator is fed to the control input of the sampling device and storage, which includes the modulation signal with a frequency of2and the measured value of relative displacement of the end faces of fiber-optic channels is determined by the output signal of sample and hold. The invention eliminates the effect of multiplicative noise and increase the accuracy of the measurements. 5 Il.\n\nThe invention relates to fiber-optic transmission systems in measurement technology and can be used to measure premasticated the amount of movement of the object , at which measured value the effect on the mutual location of the receiving and transmitting ends of the fiber-optic channels or change the conditions of propagation of radiation between the fixed ends of the receiving and transmitting optical fiber channels. For this monochromatic radiation through the transmitting fibre channel fail in the measurement zone, where they form the flux enclosed in a cone of aperture of the light guide. Part of the radiation flux illuminating the input end of the receiving fiber-optic channel, derived them from the zone dimensions, and is applied to the photodetector, where the radiation is converted into a proportional electrical signal, which is used to determine the measured physical quantities. The physical basis of this method of measurement is the change in the intensity of radiation under the action of a measured parameter, which takes place from the end face of the transmitting optical fiber channel on the receiving end of a fiber link in accordance with the directivity, the light transmission fiber-optic channels, the influence of the measured values and different noise.\n\nHowever, this method of measuring displacements have the military reduces the measurement accuracy.\n\nFamous work , which presents a way to measure, for example, the moving object by using a Fabry-Perot interferometer, namely, that form a monochromatic radiation, by using the transmitting fibre channel fail him in the measurement zone, then through a host fibre channel down radiation to the photodetector, where transform it into a proportional electrical signal. It uses homodyne methods of measurement of various physical quantities, changing according to the harmonic law, which laid the basis for the study of harmonic components of the signal at the output of the homodyne system with further decoding and analysis of its envelope. Thus, to implement one of the methods described use the decomposition of the signal taken from the output of the measuring system in the spectrum. Set the value of the phase difference so that sin = 1. Then from the state of rest smoothly excite oscillations and find the first maximum amplitude of the harmonic component at the fundamental frequency of the investigated object 1. Then measure an unknown amplitude: to re-establish the value of = /2 +is hronicheskoi component at a frequency of1. Further in the formula to find the unknown value.\n\nThe main disadvantages are described in method are necessary to calculate the arguments of the Bessel functions and units values of the phase difference in the measuring system, the limitation of the range of measurements associated with the scope of uniqueness of Bessel functions, as well as the assumption that when the required two units of the magnitude of the phase difference remains constant characteristics of laser radiation (frequency stability, the laser intensity noise) and environmental parameters. To implement these conditions in practice is extremely difficult.\n\nClosest to the invention in its technical essence is a way of measuring displacements . This way, selected as a prototype, is that form of monochromatic radiation, modulate its intensity and wavelength at a frequency of1according to the harmonic law and light through the transmitting fibre channel surface of the object to the measured distance that is experiencing interference phenomena, the result of which are non-linear distortion occurring in the optical system. On the which selects the signal of the second harmonic of the modulation frequency1and the magnitude of its amplitude is determined by the desired distance. In this case, the implementation of the method is based on the following physical phenomenon: the power and the wavelength of a semiconductor laser depends on its current pumping .\n\nThe disadvantages of this method are the relatively low accuracy of the movement, the noise immunity and complicated implementation. This is because, firstly, there is no accounting for multiplicative noise, secondly, although the second harmonic and is a periodic function of the phase difference, its amplitude nonlinear changes over the period. Therefore, the determination of the unknown quantity on the basis of the amplitude of the second harmonic is inaccurate due to the nonlinearity of the latter. Let us consider the interference, which, as is well known, are divided into multiplicative and additive. To reflect the radiation power in the optical channel P can be expressed as follows :\n\nP = f(t,z)P0+ A(t,z), (1)\n\nwhere\n\nf(t,z) is the expression for the multiplicative noise;\n\nP0- source optical power;\n\nA(t,z) is the expression for the additive noise;\n\nt - time;\n\nz - external influence.\n\nAdditive interference voznikali on the photodetector. Their suppression is relatively easy: to conduct a more thorough protection to sensitive components from external radiation. Multiplicative noise due to the following factors: the instability of radiation sources; heterogeneity transparent environment fiber optic tract associated with aging fiber, its microengine, temperature. To compensate for multiplicative noise requires a fundamental change in the design of the device, method for determining the desired value.\n\nThe disadvantages include the limitation on the range of measurements associated with a scope ambiguity function on the period.\n\nThe task of the invention is to eliminate these disadvantages, i.e., improving the reliability and accuracy of measurement of moving object.\n\nThis object is achieved by a method for motion measurement, namely, that form a monochromatic radiation, modulate its intensity and wavelength at a frequency of1according to the harmonic law, by the transmitting fibre channel modulated radiation down to the zone of measurement light input end face of the receiving fiber optic canal, located on rosstanoutinmo channel radiation is led to the photodetector and the device, emit the signal of the second harmonic of the modulation frequency1that is different from the known fact that the emission modulate and also at a lower frequency2and1>>2emit the signal of the first harmonic of the modulation frequency1then the signals of the first and second harmonics served on the block comparison, where they perform the comparison of the latter, the output signal of the block comparison serves to the input of the comparator where it is compared with a reference voltage, the output signal of the comparator is fed to the control input of the sampling device and storage (water economy Department), at the entrance of the water economy Department serves the modulation signal with a frequency of2and the measured value of relative displacement of the end faces of fiber-optic channels is determined by the output signal UHV.\n\nThe main features that distinguish the proposed method from the known, are an additional modulation of the radiation at the lower frequency2further allocation of the first harmonic of the modulation frequency1with the subsequent comparison of the signals of the first and second harmonics, which defines the novelty. From the above it follows that the proposed method meets the criterion of \"inventive step\".\n\nThis gives the advantage of WPI is th measurements by improved signal processing.\n\nIn Fig. 1 presents a functional diagram of a device that implements the proposed method, Fig.2 is a plot of the ratio between the amplitudes of the second harmonic to the first signal frequency modulation1from the measured distance between the receiving and transmitting ends of the fiber-optical channels; Fig.3 is a diagram of signal processing for the case of comparison by division; in Fig.4 is a diagram of signal processing for the case of comparison by subtracting Fig.5 is a graph of the depth of modulation signal at a frequency of 2depending on the base of the Fabry - Perot interferometer (IFP).\n\nA device that implements the proposed method contains the emitter 1, made in the form of a semiconductor laser, which is a source of monochromatic radiation, the device modulation radiation 2 at a frequency of2the device modulation radiation 3 frequency1the forming device DC 4 connected to the emitter 1 sequentially transmitting fiber-optic channel 5, the input end of which is optically connected with the emitter 1 and the outlet end is located in the measurement area, the receiving fiber-optic channel 6, the input end of which is located in the measurement zone coaxially with the output end of the transmitting from the photodetector 7, and further devices emit the signal of the first harmonic 8 and the second harmonic 9 frequency modulation1, Comparer 10, where a comparison is made of the signals coming from devices 8 and 9, a further comparator 11 is connected with the block of the reference voltage 12, the output of comparator 11 is connected to the control input of the sampling device and the storage 13, an input connected with the device modulation radiation 2.\n\nThe forming device DC 4 outputs the emitter 1 to the operating point, the modulation device 2, and 3 change the pump current of the emitter 1, which in turn affects the intensity and spectral composition of the radiation of the latter . In this case, the device 3 modulates the radiation harmonic law on the high frequency1(1 - 10 MHz), the device 2 modulates the radiation at the lower frequency2(1 - 1000 kHz). For simplicity, we assume that the modulation at a frequency of2happens sawtooth law - slow linear increase of the signal to a certain level the latter, then quickly drops to the initial level. The frequency value1whichever is greater for a given hardware implementation of schemes 7 - 10, and the frequency value2- from the condition that the response time of the circuits is teaching at frequencies2and1. Modulation of the pump current device 3 causes the modulation of the radiation, namely in addition to the permanent component of wavelength0an additional value of1similarly, the device 2 provides a Supplement to the wavelength0the value of 2. The value of1is chosen from the condition that the modulation of the radiation at a frequency of1will not lead to the emergence of the next order of interference, i.e.\n\n< / BR>\nwhere\n\nm is the order of interference.\n\nModulation of the radiation at the frequency , in contrast, would lead to a shift of the order of the interference by one, that must match\n\n< / BR>\nTherefore, the amplitude modulation of the radiation by the sawtooth law i0is calculated based on the value of1which in turn directly determines for each specifically made IFP. The amplitude modulation of the radiation by the harmonic law, i should be much less than the value of i0, i.e., 1 < i0. Thus, the generated radiation is supplied by the transmitting fiber-optic channel 5 in the measurement zone. The output end of the transmitter 5 and the input receiving end 6 of the fiber-optic channels are mirrors And the rank difference of the rays . In addition, it depends on the parameters of the IFP and the supplied radiation. Under the influence of the measured physical parameter, in particular when moving the studied object, there is a change in the magnitude of the difference of the rays in the IFP, which linearly depends on the distance between the mirrors \n\n< / BR>\nwhere loptthe distance h between the mirrors IFP taking into account the refractive index n and the angle of incidence of rays, lopt= hn;\n\nThe pump current of the emitter 1 is modulated as follows on the same period \"saw\":\n\nI = I0+ i0t + icos(1t), (5)\n\nwhere\n\nI0- constant component of the pump current;\n\ni0- small value in comparison with I0representing the current modulation sawtooth law with frequency2;\n\ni - small value in comparison with I0representing the current modulation harmonic law with frequency1;\n\n1- frequency modulation;\n\nt - time.\n\nThen the power of the laser will be determined in the first approximation \n\nP0= aI + b,\n\nwhere\n\na is a constant of order 7,510-2W/a ;\n\nb is a constant of order - 2,510-3W .\n\nthe characteristic classical IFP has the following form :\n\n< / BR>\nwhere\n\nP0the radiation power at the input IFP;\n\n- reflectance mirrors IFP.\n\nBut as a result of modulation of the pump current, the wavelength is variable and can be expressed as follows :\n\n=0+ kI, (9)\n\nwhere\n\n0the wavelength at a constant pump current I0;\n\nk - d /dI 610-9m/A ,\n\nor (2) the wavelength is represented\n\n=0+ ki0t + kicos(1t). (10)\n\nThe power of the optical radiation transmitted IFP (i.e., falling into the input end of the transmitting fiber-optic channel 5 and then on the photodetector 6), taking into account (4) and (7) can be represented in the form\n\n< / BR>\nThe expression (11) describes the transfer characteristic of IPP as a function of time and frequency1modulation of the pump current. The graph of this function in the coordinates of the power (PIFP)) and time(1t) represents a curve that contains the number of extrema. The number and shape depend on the magnitude of the modulation of the pump current i, the parameters of the interferometer and emitter (a, b, k, ) and the main thing - from base IFP, i.e. the distance between the mirrors h. So, when changing h (all other parameters fixed) form of transfer is elsaelsa as the origin of the two following pulses from the edges (considered in the period), and then the process repeats.\n\nCurve (11) is periodic, it is lawful to judge its decomposition into a Fourier series of harmonics. In this case, interested in the amplitudes of the first and second harmonic frequency1. For this purpose, the signal from the photodetector 7 is fed to the input of the device emitting the signal of the first harmonic 8 and the second harmonic 9 frequency modulation1the output signals of the devices 8 and 9 is fed to the input of block comparison 10. The Comparer 10 operates as follows: electronically divides the amplitude of the second harmonic to the amplitude of the first harmonic or adjusts the signals of the first and second harmonics at one level and then subtracts one from the other. Both options give you the opportunity to get rid of the multiplicative noise and therefore are of technical interest. Here the first option signal processing. In this case, the signal at the output of block 10 is\n\nS = I2/I1, (12)\n\nwhere\n\nIithe amplitude of the i-th harmonic frequency1decomposition of a signal (11).\n\nThe graph of the function S on the distance between the mirrors IFP is periodic, with period there is a weak minimum and a pronounced maximum (Fig.2). In the prototype searched her. First, the multiplicative noise, which is superimposed on the radiation at the output of IPP, to the same extent affect the magnitude of the second harmonic. Therefore, the use of the latter as an informative parameter obviously leads to a distortion of the desired movement by the amount of interference. Secondly, the amplitude of the second harmonic nonlinear changes over the period, therefore, the definition of the desired amplitude of the second harmonic is inaccurate.\n\nIn this way it is proposed to use the peak of the second relationship to the first harmonic as a signal that controls the comparator 11. The latter compares the output signal of the block 10 with support Uop, which is provided by the block 12, i.e. e can be configured for a certain pre-selected mode of operation. This may be related, for example, a gauge scale, thus avoiding the limitations of the measurement range. In addition, electronic division leads to the suppression of the multiplicative noise contained in the signal and its harmonics.\n\nConsider, for example, a linear increase of size in IFP (measured move) (Fig. 3, a). The comparator 11 is activated according to the level of the incoming signal is in the front, is a control device for sampling and storage 13, which also receives the sawtooth signal frequency2from device 2 (Fig.3, g). The device 13 generates a signal in accordance with the measured parameter U () and stores it until the next control pulse from the comparator 11 (Fig.3, D.). The reference voltage Uopis selected so that the comparator 11 worked in the field, where the output signal Comparer 10 has the highest slope.\n\nIn the second case, the block comparison 10 subtracts the signals from one another, for example from the amplitude of the first signal depending on the amplitude of the second signal prior to their adjustment to the intersection. This is possible with the use of electronic amplifier, the gain of which kusis as follows: the signal amplitudes of the first and second harmonics become such that the graph of their difference crosses zero (the time axis) in its most steep section. The output signal of the block 10 is\n\nS = I2- I1,\n\nwith each period of the graph of the function S crosses zero twice (Fig.4, b and C). It also allows you to set the mode of operation of the comparator using a zero reference voltage U2, I1and is influenced by them, but their difference does not currently have the interference at the point where it vanishes. In addition, in the case of subtracting the hardware diagram of the device easier compared to the case of division. E-the division requires the use of digital technology, which complicates the implementation of the method. When the analog division division accuracy does not exceed 0.5 - 1%.\n\nSpecifically, the method can be implemented as follows.\n\nWill calculate the amplitudes of the currents pumping for GaAs semiconductor laser. We will use the numerical data given in .\n\nThe forming device DC 4 outputs the emitter 1 to the operating point, forming a constant current I0= 100 mA. For simplicity, we assume that the modulation current of the pumping device 2 at a frequency of 2happens sawtooth law, providing a Supplement to the wavelength0the value of2. This modulation of the radiation at a frequency of 2should lead to a shift of the order of the interference by one, which should correspond to (3) or\n\n< / BR>\nwhere\n\nm is the order of interference.\n\nWe define the amplitude modulation of the radiation by the sawtooth law i0. Ukazatel of refraction of the medium between the mirrors IFP;\n\nh is the distance between the mirrors IFP (base IFP);\n\nFormulas (14) and (15) lead to the following expression:\n\n< / BR>\nThe wavelength taking into account (2) can be represented\n\n< / BR>\nfrom here defined\n\n< / BR>\nFor the calculation we will take k = 610-9m/A,0= 810-7M, n = 1. The expression (18) leads to the following results: to measure changes h base IFP, part of the order of several millimeters, it is advisable to set the value of i0about 10 mA. A plot of the depth of modulation signal at a frequency of2the distance between the mirrors IFP h shown in Fig.5. When the modulation depth is about 10%.\n\nThe amplitude modulation of the first voltage pumping device 3 must be less than the value of i0much more, in order to ensure condition (2). In this case, we can take i = 1 mA.\n\nThus, unlike the prototype, the proposed method allows to eliminate the influence of multiplicative noise by suppressing them with electronic comparison signals. Furthermore, the method allows for more accurate measurements of the displacements due to the fact that an informative signal proportional measure the modulation her. - M.: Energoatomizdat, 1989, S. 5-8.\n\n2. Optical homodyne measurement methods. - The magazine \"Foreign Radioelectronics\", 1995, N6, S. 43 - 48.\n\n3. Author's certificate N 1516775 the USSR, CL G 01 B 11/14, 1989 (prototype).\n\n4. Butusov M M Fiber optics and instrumentation. M.: Mashinostroenie, 1987, 330 S.\n\nMethod for motion measurement, namely, that form a monochromatic radiation, modulate its intensity and wavelength at a frequency of 1according to the harmonic law, by the transmitting fibre channel modulated radiation down to the zone of measurement light input end face of the receiving fiber-optic channel at a distance from the output end of the transmitting fiber-optic channel, then using the receiving fibre channel radiation is led to the photodetector and the device emitting the signal of the second harmonic of the modulation frequency1, characterized in that the radiation modulate and also at a lower frequency2and 12emit the signal of the first harmonic of the modulation frequency1then the signals of the first and second harmonics served on the block comparison, where compare vyhodnoi signal of the comparator is fed to the control input of the sampling device and storage at the input of the sample and hold signal modulation with a frequency of2and the measured value of relative displacement of the end faces of fiber-optic channels is determined by the output signal of sample and hold.\n\nSame patents:", null, "The invention relates to the field of measurements, in particular for controlling the position of crane tracks in terms mainly of bridge cranes", null, "Vocabulary sensor // 2107258\nThe invention relates to the field of measurement technology and can be used in automatic process control systems of industrial enterprises", null, "Vocabulary sensor // 2107258\nThe invention relates to the field of measurement technology and can be used in automatic process control systems of industrial enterprises", null, "The invention relates to measuring technique and can be used in machine building, ferrous and nonferrous metallurgy in the production of rent, rubber and chemical industry in the manufacture of tubular products without stopping the process", null, "The invention relates to a device for measuring the size of the periodically moving object containing the optoelectronic measuring device that includes a transmitting-receiving elements located in at least one plane changes, perpendicular to the longitudinal axis of the object, and the processing unit, and the plane of the measuring portal is limited by at least two measuring beams arranged at an angle relative to each other", null, "Fiber optic sensor // 2082086\nThe invention relates to the field of measurement technology, in particular to the field of fiber-optical measuring instruments", null, "The invention relates to measurement devices, namely, devices for measuring the geometric parameters of the shells", null, "The invention relates to measurement techniques, in particular to a method of controlling parameters of objects, and in particular to methods of determining particle sizes, and can be used to determine the size of particles, their size composition and concentration in powders, suspensions, and aerosols", null, "The invention relates to measuring technique and can be used in fiber-optic technology in cable industry in the manufacture of optical fibers and cables, in measurement techniques in the creation and study of fiber-optic sensors, etc", null, "The invention relates to the field of measurements, in particular for controlling the position of crane tracks in terms mainly of bridge cranes", null, "The invention relates to the technical equipment and is intended to mark the boundaries of the active layer in the fuel rods in the process of their manufacture", null, "The invention relates to the control and measuring equipment, devices for measuring the temperature of the heated products in high-temperature processes", null, "The invention relates to the control and measurement technology, in particular to devices and devices to the measuring device for checking the alignment of parts, and can be used for the installation of steam turbines", null, "Vocabulary sensor // 2107258\nThe invention relates to the field of measurement technology and can be used in automatic process control systems of industrial enterprises", null, "Vocabulary sensor // 2107258\nThe invention relates to the field of measurement technology and can be used in automatic process control systems of industrial enterprises", null, "Vocabulary sensor // 2107258\nThe invention relates to the field of measurement technology and can be used in automatic process control systems of industrial enterprises", null, "The invention relates to measuring technique and can be used in machine building, ferrous and nonferrous metallurgy in the production of rent, rubber and chemical industry in the manufacture of tubular products without stopping the process", null, "" ]	[ null, "https://img.russianpatents.com/img_data/6/62775-s.jpg", null, "https://img.russianpatents.com/img_data/3/33449-s.jpg", null, "https://img.russianpatents.com/img_data/3/33449-s.jpg", null, "https://img.russianpatents.com/img_data/3/30202-s.jpg", null, "https://img.russianpatents.com/img_data/1/16404-s.jpg", null, "https://img.russianpatents.com/img_data/291/2912456-s.jpg", null, "https://img.russianpatents.com/img_data/319/3199972-s.jpg", null, "https://img.russianpatents.com/img_data/317/3179075-s.jpg", null, "https://img.russianpatents.com/img_data/313/3130679-s.jpg", null, "https://img.russianpatents.com/img_data/6/62775-s.jpg", null, "https://img.russianpatents.com/img_data/4/45088-s.jpg", null, "https://img.russianpatents.com/img_data/4/40896-s.jpg", null, "https://img.russianpatents.com/img_data/3/33450-s.jpg", null, "https://img.russianpatents.com/img_data/3/33449-s.jpg", null, "https://img.russianpatents.com/img_data/3/33449-s.jpg", null, "https://img.russianpatents.com/img_data/3/33449-s.jpg", null, "https://img.russianpatents.com/img_data/3/30202-s.jpg", null, "https://img.russianpatents.com/top.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91037685,"math_prob":0.9285597,"size":25199,"snap":"2019-51-2020-05","text_gpt3_token_len":5186,"char_repetition_ratio":0.19317324,"word_repetition_ratio":0.20805535,"special_character_ratio":0.20096035,"punctuation_ratio":0.08253187,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9618054,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36],"im_url_duplicate_count":[null,4,null,10,null,10,null,4,null,2,null,2,null,2,null,2,null,2,null,4,null,2,null,2,null,2,null,10,null,10,null,10,null,4,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-10T15:54:12Z\",\"WARC-Record-ID\":\"<urn:uuid:7fcae85b-25cf-452b-acef-29d6549b630f>\",\"Content-Length\":\"76081\",\"Content-Type\":\"application/http; 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https://forum.ansys.com/discussion/27275/import-source-in-var-fdtd	[ "# import source in var-fdtd\n\nMember Posts: 1\n\nI would like to get an E-field profile using 3D fdtd and then use var-fdtd to further propagate the light.\n\nFor example, light is coupled to a slab via a grating coupler (simulation using 3D fdtd) and then propagate it in the slab using var-fdtd.\n\nIs it possible to do it in Lumerical? I do not see an option for importing a source in var-fdtd.\n\nTagged:\n\n• An import source is not available in var-FDTD. If you think it can be very useful and has wide use cases, you might want to submit a feature request through the Idea Exchange (IX). As an alternative, you can consider the following workflow:\n\ni) In the 3D FDTD simulation, calculate how much light is coupled into each of the modes the waveguide supports. If you are considering modes with y-polarization, you only need to calculate the coupling coefficients for these modes only. Assuming propagation in the x-direction, the field can be expressed as a superposition of the individual y-polarized modes:\n\nEy(y,z) = a1Ey,1(y,z) + a2Ey,2(y,z)+ ....\n\nii) In varFDTD, the 2D field profile ,Ey(y,z), at a waveguide cross-section can be constructed by multiplying the slab mode, E(z), and the field profile, E(y). (See the \"Analyze Results\" slide of the varFDTD - Solver Physics - Algorithm course)\n\nEy(y,z) = Ey(y)M(z) = [a1Ey,1(y) + a2Ey,2(y)+ ....]M(z)\n\nwhere\n\n• M(z): slab mode in the varFDTD\n• Ey,1(y), Ey,2(y), ....: eigenmodes of the waveguide in the varFDTD (based on the effective index profile)\n\nSo, you can use the coupling coefficients from the FDTD to obtain the results, Ey(y), in the varFDTD:\n\nEy(y) = a1Ey,1(y) + a2Ey,2(y)+ ....\n\nIt should be noted that any fields that are not coupled into the waveguide modes with specific polarization are not accounted for in the varFDTD simulations." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87265646,"math_prob":0.9631796,"size":1632,"snap":"2021-21-2021-25","text_gpt3_token_len":436,"char_repetition_ratio":0.10687961,"word_repetition_ratio":0.0,"special_character_ratio":0.2542892,"punctuation_ratio":0.14206128,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98598194,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-07T04:35:09Z\",\"WARC-Record-ID\":\"<urn:uuid:eadae9f9-1489-4a16-8668-464fcba5a176>\",\"Content-Length\":\"57511\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a1fd77e5-f970-4322-8a1e-a32b07835277>\",\"WARC-Concurrent-To\":\"<urn:uuid:9b25bfff-2d66-4d12-bf16-eec9f7c2ccff>\",\"WARC-IP-Address\":\"23.51.165.207\",\"WARC-Target-URI\":\"https://forum.ansys.com/discussion/27275/import-source-in-var-fdtd\",\"WARC-Payload-Digest\":\"sha1:OE633R66IAFLFN3G6BPFMRQ433TFX536\",\"WARC-Block-Digest\":\"sha1:C5XANWQMCOMF5LLBO3C74OQQODXMVVVC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243988774.96_warc_CC-MAIN-20210507025943-20210507055943-00224.warc.gz\"}"}
http://arxiv-export-lb.library.cornell.edu/abs/2001.05781?context=math	[ "math\n\n# Title: Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations\n\nAbstract: The absolute value equations (AVE) $Ax - \|x\| - b = 0$ is of interest of the optimization community. Recently, the SOR-like iteration method has been developed (Ke and Ma [{\\em Appl. Math. Comput.}, 311:195--202, 2017]) and shown to be efficient for numerically solving the AVE with $\\nu=\\\|A^{-1}\\\|_2<1$ (Ke and Ma [{\\em Appl. Math. Comput.}, 311:195--202, 2017]; Guo, Wu and Li [{\\em Appl. Math. Lett.}, 97:107--113, 2019]). Since the SOR-like iteration method is one-parameter-dependent, it is an important problem to determine the optimal iteration parameter. In this paper, we revisit the convergence conditions of the SOR-like iteration method proposed by Ke and Ma ([{\\em Appl. Math. Comput.}, 311:195--202, 2017]). Furthermore, we explore the optimal parameter which minimizes $\\\|T(\\omega)\\\|_2$ and the approximate optimal parameter which minimizes $\\eta=\\max\\{\|1-\\omega\|,\\nu\\omega^2\\}$. The optimal and approximate optimal parameters are iteration-independent. Numerical results demonstrate that the SOR-like iteration method with the optimal parameter is superior to that with the approximate optimal parameter proposed by Guo, Wu and Li ([{\\em Appl. Math. Lett.}, 97:107--113, 2019]).\n Comments: 15 pages, 5 figures, 6 tables Subjects: Numerical Analysis (math.NA); Optimization and Control (math.OC) Cite as: arXiv:2001.05781 [math.NA] (or arXiv:2001.05781v1 [math.NA] for this version)\n\n## Submission history\n\nFrom: Cairong Chen [view email]\n[v1] Thu, 16 Jan 2020 13:13:36 GMT (1226kb)\n\nLink back to: arXiv, form interface, contact." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.68069315,"math_prob":0.953036,"size":1647,"snap":"2020-24-2020-29","text_gpt3_token_len":440,"char_repetition_ratio":0.13937919,"word_repetition_ratio":0.04385965,"special_character_ratio":0.30297512,"punctuation_ratio":0.1863354,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9922182,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-31T23:20:56Z\",\"WARC-Record-ID\":\"<urn:uuid:25a68de9-0ccf-44e7-9f4e-f34dc6d513a4>\",\"Content-Length\":\"15695\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:890f3ebe-fa26-4fb1-b3fb-04e77eee1ba2>\",\"WARC-Concurrent-To\":\"<urn:uuid:d9cfcc26-24d9-4142-ab3c-d701c152ba97>\",\"WARC-IP-Address\":\"128.84.21.203\",\"WARC-Target-URI\":\"http://arxiv-export-lb.library.cornell.edu/abs/2001.05781?context=math\",\"WARC-Payload-Digest\":\"sha1:KNJWBEDUNOFW477LB5WMJTE4KOP3L3TI\",\"WARC-Block-Digest\":\"sha1:TXXRRTNIJ6LWS2CW2JOVIF5U66Z5HKZH\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347413786.46_warc_CC-MAIN-20200531213917-20200601003917-00486.warc.gz\"}"}
https://ask.sagemath.org/question/10604/can-sage-be-used-to-solve-the-following-kind-of-problem/	[ "# Can sage be used to solve the following kind of problem?\n\n\"Given that f(1) = 9, f'(1) = 5, g(9) = 6 and g'(9) = 4, what is the approximate value of g(f(1.05))?\"\n\nedit retag close merge delete\n\nSort by » oldest newest most voted", null, "You can just use the following script to do it. You construct two functions f and g such that they have a slope of 5 and 4 respectively and add a constant such that f(1) = 9 and g(9)=6 then evaluate at 1.05 using g(f(1.05)).\n\nf(x)=5x+4\ng(x)=4x-30\ng(f(1.05))\n\nmore\n\nYou could even use the differential equation solvers to solve these as initial value problems and stick it in... that said, this sounds like a Hughes-Hallett homework problem." ]	[ null, "https://www.gravatar.com/avatar/e2127c859038194c0b0b9178ce60c719", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8214504,"math_prob":0.9972814,"size":917,"snap":"2019-51-2020-05","text_gpt3_token_len":307,"char_repetition_ratio":0.109529026,"word_repetition_ratio":0.2278481,"special_character_ratio":0.35986915,"punctuation_ratio":0.0990566,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9994174,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-21T08:42:40Z\",\"WARC-Record-ID\":\"<urn:uuid:fc3c00da-306d-40d6-9aed-b877314270c0>\",\"Content-Length\":\"54556\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3ca86e1a-390a-43f7-9300-d91f4c9bd67d>\",\"WARC-Concurrent-To\":\"<urn:uuid:1d5303c4-b55a-4556-a21b-1b63939d1fd1>\",\"WARC-IP-Address\":\"140.254.118.68\",\"WARC-Target-URI\":\"https://ask.sagemath.org/question/10604/can-sage-be-used-to-solve-the-following-kind-of-problem/\",\"WARC-Payload-Digest\":\"sha1:6B6MRML3TPGCN27TM36REYYW7AFLTAJK\",\"WARC-Block-Digest\":\"sha1:LZ3MEH25SBXJ5FZ6LFL3BONOPGBRNOHE\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250601628.36_warc_CC-MAIN-20200121074002-20200121103002-00504.warc.gz\"}"}
https://www.cses.fi/problemset/task/1073	[ "CSES - Towers\n• Time limit: 1.00 s\n• Memory limit: 512 MB\nYou are given $n$ cubes in a certain order, and your task is to build towers using them. Whenever two cubes are one on top of the other, the upper cube must be smaller than the lower cube.\n\nYou must process the cubes in the given order. You can always either place the cube on top of an existing tower, or begin a new tower. What is the minimum possible number of towers?\n\nInput\n\nThe first input line contains an integer $n$: the number of cubes.\n\nThe next line contains $n$ integers $k_1,k_2,\\ldots,k_n$: the sizes of the cubes.\n\nOutput\n\nPrint one integer: the minimum number of towers.\n\nConstraints\n• $1 \\le n \\le 2 \\cdot 10^5$\n• $1 \\le k_i \\le 10^9$\nExample\n\nInput:\n5 3 8 2 1 5\n\nOutput:\n2" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.58402807,"math_prob":0.9922737,"size":529,"snap":"2021-31-2021-39","text_gpt3_token_len":163,"char_repetition_ratio":0.13142857,"word_repetition_ratio":0.0,"special_character_ratio":0.32136106,"punctuation_ratio":0.14634146,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99538624,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-02T06:54:22Z\",\"WARC-Record-ID\":\"<urn:uuid:66d2cfa6-8d2f-4cf6-89d0-a2adab44b86e>\",\"Content-Length\":\"4396\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:48047f62-9ff2-4c97-b3d3-2e2b2a029d84>\",\"WARC-Concurrent-To\":\"<urn:uuid:ff6c51a1-29c5-4fef-bd89-4b024933250f>\",\"WARC-IP-Address\":\"188.166.104.231\",\"WARC-Target-URI\":\"https://www.cses.fi/problemset/task/1073\",\"WARC-Payload-Digest\":\"sha1:WYCHXWKO7RPNZFAXXH3JOWEW7XMHY4U2\",\"WARC-Block-Digest\":\"sha1:U2DVOTBDKH5K2TPEOYOTVIGOKWG73KBA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154304.34_warc_CC-MAIN-20210802043814-20210802073814-00252.warc.gz\"}"}
http://planning.cs.uiuc.edu/node800.html	[ "## 15.1.1 Stability\n\nThe subject of stability addresses properties of a vector field with respect to a given point. Let", null, "denote a smooth manifold on which the vector field is defined;", null, "may be a C-space or a phase space. The given point is denoted as", null, "and can be interpreted in motion planning applications as the goal state. Stability characterizes how", null, "is approached from other states in", null, "by integrating the vector field.\n\nThe given vector field", null, "is considered as a velocity field, which is represented as", null, "(15.1)\n\nThis looks like a state transition equation that is missing actions. If a system of the form", null, "is given, then", null, "can be fixed by designing a feedback plan", null, ". This yields", null, ", which is a vector field on", null, "without any further dependency on actions. The dynamic programming approach in Section 14.5 computed such a solution. The process of designing a stable feedback plan is referred to in control literature as feedback stabilization.\n\nSubsections\nSteven M LaValle 2012-04-20" ]	[ null, "http://planning.cs.uiuc.edu/img8.gif", null, "http://planning.cs.uiuc.edu/img8.gif", null, "http://planning.cs.uiuc.edu/img215.gif", null, "http://planning.cs.uiuc.edu/img215.gif", null, "http://planning.cs.uiuc.edu/img8.gif", null, "http://planning.cs.uiuc.edu/img14.gif", null, "http://planning.cs.uiuc.edu/img2984.gif", null, "http://planning.cs.uiuc.edu/img2985.gif", null, "http://planning.cs.uiuc.edu/img253.gif", null, "http://planning.cs.uiuc.edu/img2891.gif", null, "http://planning.cs.uiuc.edu/img6388.gif", null, "http://planning.cs.uiuc.edu/img8.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.96256256,"math_prob":0.93722016,"size":486,"snap":"2020-10-2020-16","text_gpt3_token_len":97,"char_repetition_ratio":0.13692947,"word_repetition_ratio":0.0,"special_character_ratio":0.19753087,"punctuation_ratio":0.07608695,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95664024,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,2,null,null,null,null,null,4,null,4,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-18T11:09:51Z\",\"WARC-Record-ID\":\"<urn:uuid:51a7c357-96c0-4bbb-bb87-1e0fd9c619c2>\",\"Content-Length\":\"7315\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e44f01b8-b9d4-41e1-93d6-200cf95e07b5>\",\"WARC-Concurrent-To\":\"<urn:uuid:6e217158-f456-499b-b973-2a4299d72e63>\",\"WARC-IP-Address\":\"192.17.58.220\",\"WARC-Target-URI\":\"http://planning.cs.uiuc.edu/node800.html\",\"WARC-Payload-Digest\":\"sha1:RP4UZMIY5Q3R6Q4XACIOMDMD2C7MTQ34\",\"WARC-Block-Digest\":\"sha1:7YE26PG4WKK7KMJX22SFT5AHLLQ5UXH7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875143646.38_warc_CC-MAIN-20200218085715-20200218115715-00458.warc.gz\"}"}
https://neuropsychology.github.io/NeuroKit/functions/signal.html	[ "# Signal Processing#\n\n## Preprocessing#\n\n### signal_simulate()#\n\nsignal_simulate(duration=10, sampling_rate=1000, frequency=1, amplitude=0.5, noise=0, silent=False, random_state=None)[source]#\n\nSimulate a continuous signal\n\nParameters:\n• duration (float) – Desired length of duration (s).\n\n• sampling_rate (int) – The desired sampling rate (in Hz, i.e., samples/second).\n\n• frequency (float or list) – Oscillatory frequency of the signal (in Hz, i.e., oscillations per second).\n\n• amplitude (float or list) – Amplitude of the oscillations.\n\n• noise (float) – Noise level (amplitude of the laplace noise).\n\n• silent (bool) – If `False` (default), might print warnings if impossible frequencies are queried.\n\n• random_state (None, int, numpy.random.RandomState or numpy.random.Generator) – Seed for the random number generator. See for `misc.check_random_state` for further information.\n\nReturns:\n\narray – The simulated signal.\n\nExamples\n\n```In : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : pd.DataFrame({\n...: \"1Hz\": nk.signal_simulate(duration=5, frequency=1),\n...: \"2Hz\": nk.signal_simulate(duration=5, frequency=2),\n...: \"Multi\": nk.signal_simulate(duration=5, frequency=[0.5, 3], amplitude=[0.5, 0.2])\n...: }).plot()\n...:\nOut: <Axes: >\n```", null, "### signal_filter()#\n\nsignal_filter(signal, sampling_rate=1000, lowcut=None, highcut=None, method='butterworth', order=2, window_size='default', powerline=50, show=False)[source]#\n\nSignal filtering\n\nFilter a signal using different methods such as “butterworth”, “fir”, “savgol” or “powerline” filters.\n\nApply a lowpass (if “highcut” frequency is provided), highpass (if “lowcut” frequency is provided) or bandpass (if both are provided) filter to the signal.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• sampling_rate (int) – The sampling frequency of the signal (in Hz, i.e., samples/second).\n\n• lowcut (float) – Lower cutoff frequency in Hz. The default is `None`.\n\n• highcut (float) – Upper cutoff frequency in Hz. The default is `None`.\n\n• method (str) – Can be one of `\"butterworth\"`, `\"fir\"`, `\"bessel\"` or `\"savgol\"`. Note that for Butterworth, the function uses the SOS method from `scipy.signal.sosfiltfilt()`, recommended for general purpose filtering. One can also specify `\"butterworth_ba\"` for a more traditional and legacy method (often implemented in other software).\n\n• order (int) – Only used if `method` is `\"butterworth\"` or `\"savgol\"`. Order of the filter (default is 2).\n\n• window_size (int) – Only used if `method` is `\"savgol\"`. The length of the filter window (i.e. the number of coefficients). Must be an odd integer. If default, will be set to the sampling rate divided by 10 (101 if the sampling rate is 1000 Hz).\n\n• powerline (int) – Only used if `method` is `\"powerline\"`. The powerline frequency (normally 50 Hz or 60Hz).\n\n• show (bool) – If `True`, plot the filtered signal as an overlay of the original.\n\nReturns:\n\narray – Vector containing the filtered signal.\n\nExamples\n\n```In : import numpy as np\n\nIn : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=10, frequency=0.5) # Low freq\n\nIn : signal += nk.signal_simulate(duration=10, frequency=5) # High freq\n\n# Visualize Lowpass Filtered Signal using Different Methods\nIn : fig1 = pd.DataFrame({\"Raw\": signal,\n...: \"Butter_2\": nk.signal_filter(signal, highcut=3, method=\"butterworth\",\n...: order=2),\n...: \"Butter_2_BA\": nk.signal_filter(signal, highcut=3,\n...: method=\"butterworth_ba\", order=2),\n...: \"Butter_5\": nk.signal_filter(signal, highcut=3, method=\"butterworth\",\n...: order=5),\n...: \"Butter_5_BA\": nk.signal_filter(signal, highcut=3,\n...: method=\"butterworth_ba\", order=5),\n...: \"Bessel_2\": nk.signal_filter(signal, highcut=3, method=\"bessel\", order=2),\n...: \"Bessel_5\": nk.signal_filter(signal, highcut=3, method=\"bessel\", order=5),\n...: \"FIR\": nk.signal_filter(signal, highcut=3, method=\"fir\")}).plot(subplots=True)\n...:\n```", null, "```# Visualize Highpass Filtered Signal using Different Methods\nIn : fig2 = pd.DataFrame({\"Raw\": signal,\n...: \"Butter_2\": nk.signal_filter(signal, lowcut=2, method=\"butterworth\",\n...: order=2),\n...: \"Butter_2_ba\": nk.signal_filter(signal, lowcut=2,\n...: method=\"butterworth_ba\", order=2),\n...: \"Butter_5\": nk.signal_filter(signal, lowcut=2, method=\"butterworth\",\n...: order=5),\n...: \"Butter_5_BA\": nk.signal_filter(signal, lowcut=2,\n...: method=\"butterworth_ba\", order=5),\n...: \"Bessel_2\": nk.signal_filter(signal, lowcut=2, method=\"bessel\", order=2),\n...: \"Bessel_5\": nk.signal_filter(signal, lowcut=2, method=\"bessel\", order=5),\n...: \"FIR\": nk.signal_filter(signal, lowcut=2, method=\"fir\")}).plot(subplots=True)\n...:\n```", null, "```# Using Bandpass Filtering in real-life scenarios\n# Simulate noisy respiratory signal\nIn : original = nk.rsp_simulate(duration=30, method=\"breathmetrics\", noise=0)\n\nIn : signal = nk.signal_distort(original, noise_frequency=[0.1, 2, 10, 100], noise_amplitude=1,\n...: powerline_amplitude=1)\n...:\n\n# Bandpass between 10 and 30 breaths per minute (respiratory rate range)\nIn : fig3 = pd.DataFrame({\"Raw\": signal,\n....: \"Butter_2\": nk.signal_filter(signal, lowcut=10/60, highcut=30/60,\n....: method=\"butterworth\", order=2),\n....: \"Butter_2_BA\": nk.signal_filter(signal, lowcut=10/60, highcut=30/60,\n....: method=\"butterworth_ba\", order=2),\n....: \"Butter_5\": nk.signal_filter(signal, lowcut=10/60, highcut=30/60,\n....: method=\"butterworth\", order=5),\n....: \"Butter_5_BA\": nk.signal_filter(signal, lowcut=10/60, highcut=30/60,\n....: method=\"butterworth_ba\", order=5),\n....: \"Bessel_2\": nk.signal_filter(signal, lowcut=10/60, highcut=30/60,\n....: method=\"bessel\", order=2),\n....: \"Bessel_5\": nk.signal_filter(signal, lowcut=10/60, highcut=30/60,\n....: method=\"bessel\", order=5),\n....: \"FIR\": nk.signal_filter(signal, lowcut=10/60, highcut=30/60,\n....: method=\"fir\"),\n....: \"Savgol\": nk.signal_filter(signal, method=\"savgol\")}).plot(subplots=True)\n....:\n```", null, "### signal_sanitize()#\n\nsignal_sanitize(signal)[source]#\n\nSignal input sanitization\n\nReset indexing for Pandas Series.\n\nParameters:\n\nsignal (Series) – The indexed input signal (`pandas Dataframe.set_index()`)\n\nReturns:\n\nSeries – The default indexed signal\n\nExamples\n\n```In : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=10, sampling_rate=1000, frequency=1)\n\nIn : df = pd.DataFrame({'signal': signal, 'id': [x2 for x in range(len(signal))]})\n\nIn : df = df.set_index('id')\n\nIn : default_index_signal = nk.signal_sanitize(df.signal)\n```\n\n### signal_resample()#\n\nsignal_resample(signal, desired_length=None, sampling_rate=None, desired_sampling_rate=None, method='interpolation')[source]#\n\nResample a continuous signal to a different length or sampling rate\n\nUp- or down-sample a signal. The user can specify either a desired length for the vector, or input the original sampling rate and the desired sampling rate.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• desired_length (int) – The desired length of the signal.\n\n• sampling_rate (int) – The original sampling frequency (in Hz, i.e., samples/second).\n\n• desired_sampling_rate (int) – The desired (output) sampling frequency (in Hz, i.e., samples/second).\n\n• method (str) – Can be `\"interpolation\"` (see `scipy.ndimage.zoom()`), `\"numpy\"` for numpy’s interpolation (see `np.interp()`),``”pandas”`` for Pandas’ time series resampling, `\"poly\"` (see `scipy.signal.resample_poly()`) or `\"FFT\"` (see `scipy.signal.resample()`) for the Fourier method. `\"FFT\"` is the most accurate (if the signal is periodic), but becomes exponentially slower as the signal length increases. In contrast, `\"interpolation\"` is the fastest, followed by `\"numpy\"`, `\"poly\"` and `\"pandas\"`.\n\nReturns:\n\narray – Vector containing resampled signal values.\n\nExamples\n\nExample 1: Downsampling\n\n```In : import numpy as np\n\nIn : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=1, sampling_rate=500, frequency=3)\n\n# Downsample\nIn : data = {}\n\nIn : for m in [\"interpolation\", \"FFT\", \"poly\", \"numpy\", \"pandas\"]:\n...: data[m] = nk.signal_resample(signal, sampling_rate=500, desired_sampling_rate=30, method=m)\n...:\n\nIn : nk.signal_plot([data[m] for m in data.keys()])\n```", null, "Example 2: Upsampling\n\n```In : signal = nk.signal_simulate(duration=1, sampling_rate=30, frequency=3)\n\n# Upsample\nIn : data = {}\n\nIn : for m in [\"interpolation\", \"FFT\", \"poly\", \"numpy\", \"pandas\"]:\n....: data[m] = nk.signal_resample(signal, sampling_rate=30, desired_sampling_rate=500, method=m)\n....:\n\nIn : nk.signal_plot([data[m] for m in data.keys()], labels=list(data.keys()))\n```", null, "Example 3: Benchmark\n\n```In : signal = nk.signal_simulate(duration=1, sampling_rate=1000, frequency=3)\n\n# Timing benchmarks\nIn : %timeit nk.signal_resample(signal, method=\"interpolation\",\n....: sampling_rate=1000, desired_sampling_rate=500)\n....: %timeit nk.signal_resample(signal, method=\"FFT\",\n....: sampling_rate=1000, desired_sampling_rate=500)\n....: %timeit nk.signal_resample(signal, method=\"poly\",\n....: sampling_rate=1000, desired_sampling_rate=500)\n....: %timeit nk.signal_resample(signal, method=\"numpy\",\n....: sampling_rate=1000, desired_sampling_rate=500)\n....: %timeit nk.signal_resample(signal, method=\"pandas\",\n....: sampling_rate=1000, desired_sampling_rate=500)\n....:\n```\n\n## Transformation#\n\n### signal_binarize()#\n\nsignal_binarize(signal, method='threshold', threshold='auto')[source]#\n\nBinarize a continuous signal\n\nConvert a continuous signal into zeros and ones depending on a given threshold.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• method (str) – The algorithm used to discriminate between the two states. Can be one of `\"mixture\"` (default) or `\"threshold\"`. If `\"mixture\"`, will use a Gaussian Mixture Model to categorize between the two states. If `\"threshold\"`, will consider as activated all points which value is superior to the threshold.\n\n• threshold (float) – If `method` is `\"mixture\"`, then it corresponds to the minimum probability required to be considered as activated (if `\"auto\"`, then 0.5). If `method` is `\"threshold\"`, then it corresponds to the minimum amplitude to detect as onset. If `\"auto\"`, takes the value between the max and the min.\n\nReturns:\n\nlist – A list or array depending on the type passed.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : import numpy as np\n\nIn : import pandas as pd\n\nIn : signal = np.cos(np.linspace(start=0, stop=20, num=1000))\n\nIn : binary = nk.signal_binarize(signal)\n\nIn : pd.DataFrame({\"Raw\": signal, \"Binary\": binary}).plot()\nOut: <Axes: >\n```", null, "### signal_decompose()#\n\nsignal_decompose(signal, method='emd', n_components=None, kwargs)[source]#\n\nDecompose a signal\n\nSignal decomposition into different sources using different methods, such as Empirical Mode Decomposition (EMD) or Singular spectrum analysis (SSA)-based signal separation method.\n\nThe extracted components can then be recombined into meaningful sources using `signal_recompose()`.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – Vector of values.\n\n• method (str) – The decomposition method. Can be one of `\"emd\"` or `\"ssa\"`.\n\n• n_components (int) – Number of components to extract. Only used for `\"ssa\"` method. If `None`, will default to 50.\n\n• kwargs – Other arguments passed to other functions.\n\nReturns:\n\nArray – Components of the decomposed signal.\n\nExamples\n\n```In : import neurokit2 as nk\n\n# Create complex signal\nIn : signal = nk.signal_simulate(duration=10, frequency=1, noise=0.01) # High freq\n\nIn : signal += 3 nk.signal_simulate(duration=10, frequency=3, noise=0.01) # Higher freq\n\nIn : signal += 3 * np.linspace(0, 2, len(signal)) # Add baseline and trend\n\nIn : signal += 2 * nk.signal_simulate(duration=10, frequency=0.1, noise=0)\n\nIn : nk.signal_plot(signal)\n```", null, "```# Example 1: Using the EMD method\nIn : components = nk.signal_decompose(signal, method=\"emd\")\n\n# Visualize Decomposed Signal Components\nIn : nk.signal_plot(components)\n```", null, "```# Example 2: USing the SSA method\nIn : components = nk.signal_decompose(signal, method=\"ssa\", n_components=5)\n\n# Visualize Decomposed Signal Components\nIn : nk.signal_plot(components) # Visualize components\n```", null, "### signal_recompose()#\n\nsignal_recompose(components, method='wcorr', threshold=0.5, keep_sd=None, kwargs)[source]#\n\nCombine signal sources after decomposition\n\nCombine and reconstruct meaningful signal sources after signal decomposition.\n\nParameters:\n• components (array) – Array of components obtained via `signal_decompose()`.\n\n• method (str) – The decomposition method. Can be one of `\"wcorr\"`.\n\n• threshold (float) – The threshold used to group components together.\n\n• keep_sd (float) – If a float is specified, will only keep the reconstructed components that are superior or equal to that percentage of the max standard deviaiton (SD) of the components. For instance, `keep_sd=0.01` will remove all components with SD lower than 1% of the max SD. This can be used to filter out noise.\n\n• kwargs – Other arguments used to override, for instance `metric=\"chebyshev\"`.\n\nReturns:\n\nArray – Components of the recomposed components.\n\nExamples\n\n```In : import neurokit2 as nk\n\n# Create complex signal\nIn : signal = nk.signal_simulate(duration=10, frequency=1, noise=0.01) # High freq\n\nIn : signal += 3 * nk.signal_simulate(duration=10, frequency=3, noise=0.01) # Higher freq\n\nIn : signal += 3 * np.linspace(0, 2, len(signal)) # Add baseline and trend\n\nIn : signal += 2 * nk.signal_simulate(duration=10, frequency=0.1, noise=0)\n\n# Decompose signal\nIn : components = nk.signal_decompose(signal, method='emd')\n\n# Recompose\nIn : recomposed = nk.signal_recompose(components, method='wcorr', threshold=0.90)\n\nIn : nk.signal_plot(components) # Visualize components\n```", null, "### signal_detrend()#\n\nsignal_detrend(signal, method='polynomial', order=1, regularization=500, alpha=0.75, window=1.5, stepsize=0.02, components=[-1], sampling_rate=1000)[source]#\n\nSignal Detrending Apply a baseline (order = 0), linear (order = 1), or polynomial (order > 1) detrending to the signal (i.e., removing a general trend). One can also use other methods, such as smoothness priors approach described by Tarvainen (2002) or LOESS regression, but these scale badly for long signals. :Parameters: * signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• method (str) – Can be one of `\"polynomial\"` (default; traditional detrending of a given order) or `\"tarvainen2002\"` to use the smoothness priors approach described by Tarvainen (2002) (mostly used in HRV analyses as a lowpass filter to remove complex trends), `\"loess\"` for LOESS smoothing trend removal or `\"locreg\"` for local linear regression (the ‘runline’ algorithm from chronux).\n\n• order (int) – Only used if `method` is `\"polynomial\"`. The order of the polynomial. 0, 1 or > 1 for a baseline (‘constant detrend’, i.e., remove only the mean), linear (remove the linear trend) or polynomial detrending, respectively. Can also be `\"auto\"`, in which case it will attempt to find the optimal order to minimize the RMSE.\n\n• regularization (int) – Only used if `method=\"tarvainen2002\"`. The regularization parameter (default to 500).\n\n• alpha (float) – Only used if `method` is “loess”. The parameter which controls the degree of smoothing.\n\n• window (float) – Only used if `method` is “locreg”. The detrending `window` should correspond to the 1 divided by the desired low-frequency band to remove (`window = 1 / detrend_frequency`) For instance, to remove frequencies below `0.67Hz` the window should be `1.5` (`1 / 0.67 = 1.5`).\n\n• stepsize (float) – Only used if `method` is `\"locreg\"`.\n\n• components (list) – Only used if `method` is `\"EMD\"`. What Intrinsic Mode Functions (IMFs) from EMD to remove. By default, the last one.\n\n• sampling_rate (int, optional) – Only used if `method` is “locreg”. Sampling rate (Hz) of the signal. If not None, the `stepsize` and `window` arguments will be multiplied by the sampling rate. By default 1000.\n\nReturns:\n\narray – Vector containing the detrended signal.\n\nExamples\n\n```In : import numpy as np\n\nIn : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : import matplotlib.pyplot as plt\n\n# Simulate signal with low and high frequency\nIn : signal = nk.signal_simulate(frequency=[0.1, 2], amplitude=[2, 0.5], sampling_rate=100)\n\nIn : signal = signal + (3 + np.linspace(0, 6, num=len(signal))) # Add baseline and linear trend\n\n# Apply detrending algorithms\n# ---------------------------\n# Method 1: Default Polynomial Detrending of a Given Order\n# Constant detrend (removes the mean)\nIn : baseline = nk.signal_detrend(signal, order=0)\n\n# Linear Detrend (removes the linear trend)\nIn : linear = nk.signal_detrend(signal, order=1)\n\n# Polynomial Detrend (removes the polynomial trend)\n\nIn : cubic = nk.signal_detrend(signal, order=3) # Cubic detrend\n\nIn : poly10 = nk.signal_detrend(signal, order=10) # Linear detrend (10th order)\n\n# Method 2: Tarvainen's smoothness priors approach (Tarvainen et al., 2002)\nIn : tarvainen = nk.signal_detrend(signal, method=\"tarvainen2002\")\n\n# Method 3: LOESS smoothing trend removal\nIn : loess = nk.signal_detrend(signal, method=\"loess\")\n\n# Method 4: Local linear regression (100Hz)\nIn : locreg = nk.signal_detrend(signal, method=\"locreg\",\n....: window=1.5, stepsize=0.02, sampling_rate=100)\n....:\n\n# Method 5: EMD\nIn : emd = nk.signal_detrend(signal, method=\"EMD\", components=[-2, -1])\n\n# Visualize different methods\nIn : axes = pd.DataFrame({\"Original signal\": signal,\n....: \"Baseline\": baseline,\n....: \"Linear\": linear,\n....: \"Cubic\": cubic,\n....: \"Polynomial (10th)\": poly10,\n....: \"Tarvainen\": tarvainen,\n....: \"LOESS\": loess,\n....: \"Local Regression\": locreg,\n....: \"EMD\": emd}).plot(subplots=True)\n....:\n\n# Plot horizontal lines to better visualize the detrending\nIn : for subplot in axes:\n....: subplot.axhline(y=0, color=\"k\", linestyle=\"--\")\n....:\n```", null, "References\n\n• Tarvainen, M. P., Ranta-Aho, P. O., & Karjalainen, P. A. (2002). An advanced detrending method with application to HRV analysis. IEEE Transactions on Biomedical Engineering, 49(2), 172-175\n\n### signal_distort()#\n\nsignal_distort(signal, sampling_rate=1000, noise_shape='laplace', noise_amplitude=0, noise_frequency=100, powerline_amplitude=0, powerline_frequency=50, artifacts_amplitude=0, artifacts_frequency=100, artifacts_number=5, linear_drift=False, random_state=None, silent=False)[source]#\n\nSignal distortion\n\nAdd noise of a given frequency, amplitude and shape to a signal.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• sampling_rate (int) – The sampling frequency of the signal (in Hz, i.e., samples/second).\n\n• noise_shape (str) – The shape of the noise. Can be one of `\"laplace\"` (default) or `\"gaussian\"`.\n\n• noise_amplitude (float) – The amplitude of the noise (the scale of the random function, relative to the standard deviation of the signal).\n\n• noise_frequency (float) – The frequency of the noise (in Hz, i.e., samples/second).\n\n• powerline_amplitude (float) – The amplitude of the powerline noise (relative to the standard deviation of the signal).\n\n• powerline_frequency (float) – The frequency of the powerline noise (in Hz, i.e., samples/second).\n\n• artifacts_amplitude (float) – The amplitude of the artifacts (relative to the standard deviation of the signal).\n\n• artifacts_frequency (int) – The frequency of the artifacts (in Hz, i.e., samples/second).\n\n• artifacts_number (int) – The number of artifact bursts. The bursts have a random duration between 1 and 10% of the signal duration.\n\n• linear_drift (bool) – Whether or not to add linear drift to the signal.\n\n• random_state (None, int, numpy.random.RandomState or numpy.random.Generator) – Seed for the random number generator. See for `misc.check_random_state` for further information.\n\n• silent (bool) – Whether or not to display warning messages.\n\nReturns:\n\narray – Vector containing the distorted signal.\n\nExamples\n\n```In : import numpy as np\n\nIn : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=10, frequency=0.5)\n\n# Noise\nIn : noise = pd.DataFrame({\"Freq100\": nk.signal_distort(signal, noise_frequency=200),\n...: \"Freq50\": nk.signal_distort(signal, noise_frequency=50),\n...: \"Freq10\": nk.signal_distort(signal, noise_frequency=10),\n...: \"Freq5\": nk.signal_distort(signal, noise_frequency=5),\n...: \"Raw\": signal}).plot()\n...:\n```", null, "```# Artifacts\nIn : artifacts = pd.DataFrame({\"1Hz\": nk.signal_distort(signal, noise_amplitude=0,\n...: artifacts_frequency=1,\n...: artifacts_amplitude=0.5),\n...: \"5Hz\": nk.signal_distort(signal, noise_amplitude=0,\n...: artifacts_frequency=5,\n...: artifacts_amplitude=0.2),\n...: \"Raw\": signal}).plot()\n...:\n```", null, "### signal_flatline()#\n\nsignal_flatline(signal, threshold=0.01)[source]#\n\nReturn the Flatline Percentage of the Signal\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• threshold (float, optional) – Flatline threshold relative to the biggest change in the signal. This is the percentage of the maximum value of absolute consecutive differences.\n\nReturns:\n\nfloat – Percentage of signal where the absolute value of the derivative is lower then the threshold.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=5)\n\nIn : nk.signal_flatline(signal)\nOut: 0.008\n```\n\n### signal_interpolate()#\n\nInterpolate a signal\n\nInterpolate a signal using different methods.\n\nParameters:\n• x_values (Union[list, np.array, pd.Series]) – The samples corresponding to the values to be interpolated.\n\n• y_values (Union[list, np.array, pd.Series]) – The values to be interpolated. If not provided, any NaNs in the x_values will be interpolated with `_signal_interpolate_nan()`, considering the x_values as equally spaced.\n\n• x_new (Union[list, np.array, pd.Series] or int) – The samples at which to interpolate the y_values. Samples before the first value in x_values or after the last value in x_values will be extrapolated. If an integer is passed, nex_x will be considered as the desired length of the interpolated signal between the first and the last values of x_values. No extrapolation will be done for values before or after the first and the last values of x_values.\n\n• method (str) – Method of interpolation. Can be `\"linear\"`, `\"nearest\"`, `\"zero\"`, `\"slinear\"`, `\"quadratic\"`, `\"cubic\"`, `\"previous\"`, `\"next\"` or `\"monotone_cubic\"`. The methods `\"zero\"`, `\"slinear\"`,``”quadratic”`` and `\"cubic\"` refer to a spline interpolation of zeroth, first, second or third order; whereas `\"previous\"` and `\"next\"` simply return the previous or next value of the point. An integer specifying the order of the spline interpolator to use. See here for details on the `\"monotone_cubic\"` method.\n\n• fill_value (float or tuple or str) – If a ndarray (or float), this value will be used to fill in for requested points outside of the data range. If a two-element tuple, then the first element is used as a fill value for x_new < x and the second element is used for x_new > x[-1]. If “extrapolate”, then points outside the data range will be extrapolated. If not provided, then the default is ([y_values], [y_values[-1]]).\n\nReturns:\n\narray – Vector of interpolated samples.\n\nExamples\n\n```In : import numpy as np\n\nIn : import neurokit2 as nk\n\nIn : import matplotlib.pyplot as plt\n\n# Generate Simulated Signal\nIn : signal = nk.signal_simulate(duration=2, sampling_rate=10)\n\n# We want to interpolate to 2000 samples\nIn : x_values = np.linspace(0, 2000, num=len(signal), endpoint=False)\n\nIn : x_new = np.linspace(0, 2000, num=2000, endpoint=False)\n\n# Visualize all interpolation methods\nIn : nk.signal_plot([\n...: nk.signal_interpolate(x_values, signal, x_new=x_new, method=\"zero\"),\n...: nk.signal_interpolate(x_values, signal, x_new=x_new, method=\"linear\"),\n...: nk.signal_interpolate(x_values, signal, x_new=x_new, method=\"cubic\"),\n...: nk.signal_interpolate(x_values, signal, x_new=x_new, method=\"previous\"),\n...: nk.signal_interpolate(x_values, signal, x_new=x_new, method=\"next\"),\n...: nk.signal_interpolate(x_values, signal, x_new=x_new, method=\"monotone_cubic\")\n...: ], labels = [\"Zero\", \"Linear\", \"Quadratic\", \"Cubic\", \"Previous\", \"Next\", \"Monotone Cubic\"])\n...:\n\nIn : plt.scatter(x_values, signal, label=\"original datapoints\", zorder=3)\n```", null, "### signal_merge()#\n\nsignal_merge(signal1, signal2, time1=[0, 10], time2=[0, 10])[source]#\n\nArbitrary addition of two signals with different time ranges\n\nParameters:\n• signal1 (Union[list, np.array, pd.Series]) – The first signal (i.e., a time series)s in the form of a vector of values.\n\n• signal2 (Union[list, np.array, pd.Series]) – The second signal (i.e., a time series)s in the form of a vector of values.\n\n• time1 (list) – Lists containing two numeric values corresponding to the beginning and end of `signal1`.\n\n• time2 (list) – Same as above, but for `signal2`.\n\nReturns:\n\narray – Vector containing the sum of the two signals.\n\nExamples\n\n```In : import numpy as np\n\nIn : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : signal1 = np.cos(np.linspace(start=0, stop=10, num=100))\n\nIn : signal2 = np.cos(np.linspace(start=0, stop=20, num=100))\n\nIn : signal = nk.signal_merge(signal1, signal2, time1=[0, 10], time2=[-5, 5])\n\nIn : nk.signal_plot(signal)\n```", null, "### signal_noise()#\n\nsignal_noise(duration=10, sampling_rate=1000, beta=1, random_state=None)[source]#\n\nSimulate noise\n\nThis function generates pure Gaussian `(1/f)beta` noise. The power-spectrum of the generated noise is proportional to `S(f) = (1 / f)beta`. The following categories of noise have been described:\n\n• violet noise: beta = -2\n\n• blue noise: beta = -1\n\n• white noise: beta = 0\n\n• flicker / pink noise: beta = 1\n\n• brown noise: beta = 2\n\nParameters:\n• duration (float) – Desired length of duration (s).\n\n• sampling_rate (int) – The desired sampling rate (in Hz, i.e., samples/second).\n\n• beta (float) – The noise exponent.\n\n• random_state (None, int, numpy.random.RandomState or numpy.random.Generator) – Seed for the random number generator. See for `misc.check_random_state` for further information.\n\nReturns:\n\nnoise (array) – The signal of pure noise.\n\nReferences\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : import matplotlib.pyplot as plt\n\n# Generate pure noise\nIn : violet = nk.signal_noise(beta=-2)\n\nIn : blue = nk.signal_noise(beta=-1)\n\nIn : white = nk.signal_noise(beta=0)\n\nIn : pink = nk.signal_noise(beta=1)\n\nIn : brown = nk.signal_noise(beta=2)\n\n# Visualize\nIn : nk.signal_plot([violet, blue, white, pink, brown],\n...: standardize=True,\n...: labels=[\"Violet\", \"Blue\", \"White\", \"Pink\", \"Brown\"])\n...:\n```", null, "```# Visualize spectrum\nIn : psd_violet = nk.signal_psd(violet, sampling_rate=200, method=\"fft\")\n\nIn : psd_blue = nk.signal_psd(blue, sampling_rate=200, method=\"fft\")\n\nIn : psd_white = nk.signal_psd(white, sampling_rate=200, method=\"fft\")\n\nIn : psd_pink = nk.signal_psd(pink, sampling_rate=200, method=\"fft\")\n\nIn : psd_brown = nk.signal_psd(brown, sampling_rate=200, method=\"fft\")\n\nIn : plt.loglog(psd_violet[\"Frequency\"], psd_violet[\"Power\"], c=\"violet\")\n\nIn : plt.loglog(psd_blue[\"Frequency\"], psd_blue[\"Power\"], c=\"blue\")\n\nIn : plt.loglog(psd_white[\"Frequency\"], psd_white[\"Power\"], c=\"grey\")\n\nIn : plt.loglog(psd_pink[\"Frequency\"], psd_pink[\"Power\"], c=\"pink\")\n\nIn : plt.loglog(psd_brown[\"Frequency\"], psd_brown[\"Power\"], c=\"brown\")\n```", null, "### signal_surrogate()#\n\nsignal_surrogate(signal, method='IAAFT', random_state=None, kwargs)[source]#\n\nCreate Signal Surrogates\n\nGenerate a surrogate version of a signal. Different methods are available, such as:\n\n• random: Performs a random permutation of the signal value. This way, the signal distribution is unaffected and the serial correlations are cancelled, yielding a whitened signal with an distribution identical to that of the original.\n\n• IAAFT: Returns an Iterative Amplitude Adjusted Fourier Transform (IAAFT) surrogate. It is a phase randomized, amplitude adjusted surrogates that have the same power spectrum (to a very high accuracy) and distribution as the original data, using an iterative scheme.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• method (str) – Can be `\"random\"` or `\"IAAFT\"`.\n\n• random_state (None, int, numpy.random.RandomState or numpy.random.Generator) – Seed for the random number generator. See for `misc.check_random_state` for further information.\n\n• kwargs – Other keywords arguments, such as `max_iter` (by default 1000).\n\nReturns:\n\nsurrogate (array) – Surrogate signal.\n\nExamples\n\nCreate surrogates using different methods.\n\n```In : import neurokit2 as nk\n\nIn : import matplotlib.pyplot as plt\n\nIn : signal = nk.signal_simulate(duration = 1, frequency = [3, 5], noise = 0.1)\n\nIn : surrogate_iaaft = nk.signal_surrogate(signal, method = \"IAAFT\")\n\nIn : surrogate_random = nk.signal_surrogate(signal, method = \"random\")\n\nIn : plt.plot(surrogate_random, label = \"Random Surrogate\")\n\nIn : plt.plot(surrogate_iaaft, label = \"IAAFT Surrogate\")\n\nIn : plt.plot(signal, label = \"Original\")\n\nIn : plt.legend()\n```", null, "As we can see, the signal pattern is destroyed by random surrogates, but not in the IAAFT one. And their distributions are identical:\n\n```In : plt.plot(nk.density(signal), label = \"Original\")\n\nIn : plt.plot(nk.density(surrogate_iaaft), label = \"IAAFT Surrogate\")\nOut: [<matplotlib.lines.Line2D at 0x20ac9ec2200>]\n\nIn : plt.plot(nk.density(surrogate_random), label = \"Random Surrogate\")\nOut: [<matplotlib.lines.Line2D at 0x20ac9ec3bb0>]\n\nIn : plt.legend()\n```", null, "However, the power spectrum of the IAAFT surrogate is preserved.\n\n```In : f = nk.signal_psd(signal, max_frequency=20)\n\nIn : f[\"IAAFT\"] = nk.signal_psd(surrogate_iaaft, max_frequency=20)[\"Power\"]\n\nIn : f[\"Random\"] = nk.signal_psd(surrogate_random, max_frequency=20)[\"Power\"]\n\nIn : f.plot(\"Frequency\", [\"Power\", \"IAAFT\", \"Random\"])\nOut: <Axes: xlabel='Frequency'>\n```", null, "References\n\n• Schreiber, T., & Schmitz, A. (1996). Improved surrogate data for nonlinearity tests. Physical review letters, 77(4), 635.\n\n## Peaks#\n\n### signal_findpeaks()#\n\nsignal_findpeaks(signal, height_min=None, height_max=None, relative_height_min=None, relative_height_max=None, relative_mean=True, relative_median=False, relative_max=False)[source]#\n\nFind peaks in a signal\n\nLocate peaks (local maxima) in a signal and their related characteristics, such as height (prominence), width and distance with other peaks.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• height_min (float) – The minimum height (i.e., amplitude in terms of absolute values). For example, `height_min=20` will remove all peaks which height is smaller or equal to 20 (in the provided signal’s values).\n\n• height_max (float) – The maximum height (i.e., amplitude in terms of absolute values).\n\n• relative_height_min (float) – The minimum height (i.e., amplitude) relative to the sample (see below). For example, `relative_height_min=-2.96` will remove all peaks which height lies below 2.96 standard deviations from the mean of the heights.\n\n• relative_height_max (float) – The maximum height (i.e., amplitude) relative to the sample (see below).\n\n• relative_mean (bool) – If a relative threshold is specified, how should it be computed (i.e., relative to what?). `relative_mean=True` will use Z-scores.\n\n• relative_median (bool) – If a relative threshold is specified, how should it be computed (i.e., relative to what?). Relative to median uses a more robust form of standardization (see `standardize()`).\n\n• relative_max (bool) – If a relative threshold is specified, how should it be computed (i.e., relative to what?). Relative to max will consider the maximum height as the reference.\n\nReturns:\n\ndict\n\nReturns a dict itself containing 5 arrays:\n\n• `\"Peaks\"`: contains the peaks indices (as relative to the given signal). For instance, the value 3 means that the third data point of the signal is a peak.\n\n• `\"Distance\"`: contains, for each peak, the closest distance with another peak. Note that these values will be recomputed after filtering to match the selected peaks.\n\n• `\"Height\"`: contains the prominence of each peak. See `scipy.signal.peak_prominences()`.\n\n• `\"Width\"`: contains the width of each peak. See `scipy.signal.peak_widths()`.\n\n• `\"Onset\"`: contains the onset, start (or left trough), of each peak.\n\n• `\"Offset\"`: contains the offset, end (or right trough), of each peak.\n\nExamples\n\n```In : import neurokit2 as nk\n\n# Simulate a Signal\nIn : signal = nk.signal_simulate(duration=5)\n\nIn : info = nk.signal_findpeaks(signal)\n\n# Visualize Onsets of Peaks and Peaks of Signal\nIn : nk.events_plot([info[\"Onsets\"], info[\"Peaks\"]], signal)\n```", null, "```In : import scipy.datasets\n\nIn : ecg = scipy.datasets.electrocardiogram()\n\nIn : signal = ecg[0:1000]\n\n# Find Unfiltered and Filtered Peaks\nIn : info1 = nk.signal_findpeaks(signal, relative_height_min=0)\n\nIn : info2 = nk.signal_findpeaks(signal, relative_height_min=1)\n\n# Visualize Peaks\nIn : nk.events_plot([info1[\"Peaks\"], info2[\"Peaks\"]], signal)\n```", null, "### signal_fixpeaks()#\n\nsignal_fixpeaks(peaks, sampling_rate=1000, iterative=True, show=False, interval_min=None, interval_max=None, relative_interval_min=None, relative_interval_max=None, robust=False, method='Kubios', kwargs)[source]#\n\nCorrect Erroneous Peak Placements\n\nIdentify and correct erroneous peak placements based on outliers in peak-to-peak differences (period).\n\nParameters:\n• peaks (list or array or DataFrame or Series or dict) – The samples at which the peaks occur. If an array is passed in, it is assumed that it was obtained with `signal_findpeaks()`. If a DataFrame is passed in, it is assumed to be obtained with `ecg_findpeaks()` or `ppg_findpeaks()` and to be of the same length as the input signal.\n\n• sampling_rate (int) – The sampling frequency of the signal that contains the peaks (in Hz, i.e., samples/second).\n\n• iterative (bool) – Whether or not to apply the artifact correction repeatedly (results in superior artifact correction).\n\n• show (bool) – Whether or not to visualize artifacts and artifact thresholds.\n\n• interval_min (float) – Only when `method = \"neurokit\"`. The minimum interval between the peaks.\n\n• interval_max (float) – Only when `method = \"neurokit\"`. The maximum interval between the peaks.\n\n• relative_interval_min (float) – Only when `method = \"neurokit\"`. The minimum interval between the peaks as relative to the sample (expressed in standard deviation from the mean).\n\n• relative_interval_max (float) – Only when `method = \"neurokit\"`. The maximum interval between the peaks as relative to the sample (expressed in standard deviation from the mean).\n\n• robust (bool) – Only when `method = \"neurokit\"`. Use a robust method of standardization (see `standardize()`) for the relative thresholds.\n\n• method (str) – Either `\"Kubios\"` or `\"neurokit\"`. `\"Kubios\"` uses the artifact detection and correction described in Lipponen, J. A., & Tarvainen, M. P. (2019). Note that `\"Kubios\"` is only meant for peaks in ECG or PPG. `\"neurokit\"` can be used with peaks in ECG, PPG, or respiratory data.\n\n• kwargs – Other keyword arguments.\n\nReturns:\n\n• peaks_clean (array) – The corrected peak locations.\n\n• artifacts (dict) – Only if `method=\"Kubios\"`. A dictionary containing the indices of artifacts, accessible with the keys `\"ectopic\"`, `\"missed\"`, `\"extra\"`, and `\"longshort\"`.\n\n`signal_findpeaks`, `ecg_findpeaks`, `ecg_peaks`, `ppg_findpeaks`, `ppg_peaks`\n\nExamples\n\n```In : import neurokit2 as nk\n\n# Simulate ECG data\nIn : ecg = nk.ecg_simulate(duration=240, noise=0.25, heart_rate=70, random_state=42)\n\n# Identify and Correct Peaks using \"Kubios\" Method\nIn : rpeaks_uncorrected = nk.ecg_findpeaks(ecg)\n\nIn : artifacts, rpeaks_corrected = nk.signal_fixpeaks(\n...: rpeaks_uncorrected, iterative=True, method=\"Kubios\", show=True\n...: )\n...:\n```", null, "```# Visualize Artifact Correction\nIn : rate_corrected = nk.signal_rate(rpeaks_corrected, desired_length=len(ecg))\n\nIn : rate_uncorrected = nk.signal_rate(rpeaks_uncorrected, desired_length=len(ecg))\n\nIn : nk.signal_plot(\n...: [rate_uncorrected, rate_corrected],\n...: labels=[\"Heart Rate Uncorrected\", \"Heart Rate Corrected\"]\n...: )\n...:\n```", null, "```In : import numpy as np\n\n# Simulate Abnormal Signals\nIn : signal = nk.signal_simulate(duration=4, sampling_rate=1000, frequency=1)\n\nIn : peaks_true = nk.signal_findpeaks(signal)[\"Peaks\"]\n\nIn : peaks = np.delete(peaks_true, ) # create gaps due to missing peaks\n\nIn : signal = nk.signal_simulate(duration=20, sampling_rate=1000, frequency=1)\n\nIn : peaks_true = nk.signal_findpeaks(signal)[\"Peaks\"]\n\nIn : peaks = np.delete(peaks_true, [5, 15]) # create gaps\n\nIn : peaks = np.sort(np.append(peaks, [1350, 11350, 18350])) # add artifacts\n\n# Identify and Correct Peaks using 'NeuroKit' Method\nIn : peaks_corrected = nk.signal_fixpeaks(\n....: peaks=peaks, interval_min=0.5, interval_max=1.5, method=\"neurokit\"\n....: )\n....:\n\n# Plot and shift original peaks to the right to see the difference.\nIn : nk.events_plot([peaks + 50, peaks_corrected], signal)\n```", null, "References\n\n• Lipponen, J. A., & Tarvainen, M. P. (2019). A robust algorithm for heart rate variability time series artefact correction using novel beat classification. Journal of medical engineering & technology, 43(3), 173-181. 10.1080/03091902.2019.1640306\n\n## Analysis#\n\n### signal_autocor()#\n\nsignal_autocor(signal, lag=None, demean=True, method='auto', show=False)[source]#\n\nAutocorrelation (ACF)\n\nCompute the autocorrelation of a signal.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – Vector of values.\n\n• lag (int) – Time lag. If specified, one value of autocorrelation between signal with its lag self will be returned.\n\n• demean (bool) – If `True`, the mean of the signal will be subtracted from the signal before ACF computation.\n\n• method (str) – Using `\"auto\"` runs `scipy.signal.correlate` to determine the faster algorithm. Other methods are kept for legacy reasons, but are not recommended. Other methods include `\"correlation\"` (using `np.correlate()`) or `\"fft\"` (Fast Fourier Transform).\n\n• show (bool) – If `True`, plot the autocorrelation at all values of lag.\n\nReturns:\n\n• r (float) – The cross-correlation of the signal with itself at different time lags. Minimum time lag is 0, maximum time lag is the length of the signal. Or a correlation value at a specific lag if lag is not `None`.\n\n• info (dict) – A dictionary containing additional information, such as the confidence interval.\n\nExamples\n\n```In : import neurokit2 as nk\n\n# Example 1: Using 'Correlation' Method\nIn : signal = [1, 2, 3, 4, 5]\n\nIn : r, info = nk.signal_autocor(signal, show=True, method='correlation')\n```", null, "```# Example 2: Using 'FFT' Method\nIn : signal = nk.signal_simulate(duration=5, sampling_rate=100, frequency=[5, 6], noise=0.5)\n\nIn : r, info = nk.signal_autocor(signal, lag=2, method='fft', show=True)\n```", null, "### signal_changepoints()#\n\nsignal_changepoints(signal, change='meanvar', penalty=None, show=False)[source]#\n\nChange Point Detection\n\nOnly the PELT method is implemented for now.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – Vector of values.\n\n• change (str) – Can be one of `\"meanvar\"` (default), `\"mean\"` or `\"var\"`.\n\n• penalty (float) – The algorithm penalty. Defaults to `np.log(len(signal))`.\n\n• show (bool) – Defaults to `False`.\n\nReturns:\n\n• Array – Values indicating the samples at which the changepoints occur.\n\n• Fig – Figure of plot of signal with markers of changepoints.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : signal = nk.emg_simulate(burst_number=3)\n\nIn : nk.signal_changepoints(signal, change=\"var\", show=True)\nOut: array([1751, 2750, 4500, 5500, 7250, 8250])\n```", null, "References\n\n• Killick, R., Fearnhead, P., & Eckley, I. A. (2012). Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association, 107(500), 1590-1598.\n\n### signal_period()#\n\nsignal_period(peaks, sampling_rate=1000, desired_length=None, interpolation_method='monotone_cubic')[source]#\n\nCalculate signal period from a series of peaks\n\nParameters:\n• peaks (Union[list, np.array, pd.DataFrame, pd.Series, dict]) – The samples at which the peaks occur. If an array is passed in, it is assumed that it was obtained with `signal_findpeaks()`. If a DataFrame is passed in, it is assumed it is of the same length as the input signal in which occurrences of R-peaks are marked as “1”, with such containers obtained with e.g., `ecg_findpeaks()` or `rsp_findpeaks()`.\n\n• sampling_rate (int) – The sampling frequency of the signal that contains peaks (in Hz, i.e., samples/second). Defaults to 1000.\n\n• desired_length (int) – If left at the default `None`, the returned period will have the same number of elements as `peaks`. If set to a value larger than the sample at which the last peak occurs in the signal (i.e., `peaks[-1]`), the returned period will be interpolated between peaks over `desired_length` samples. To interpolate the period over the entire duration of the signal, set `desired_length` to the number of samples in the signal. Cannot be smaller than or equal to the sample at which the last peak occurs in the signal. Defaults to `None`.\n\n• interpolation_method (str) – Method used to interpolate the rate between peaks. See `signal_interpolate()`. `\"monotone_cubic\"` is chosen as the default interpolation method since it ensures monotone interpolation between data points (i.e., it prevents physiologically implausible “overshoots” or “undershoots” in the y-direction). In contrast, the widely used cubic spline interpolation does not ensure monotonicity.\n\nReturns:\n\narray – A vector containing the period.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=10, sampling_rate=1000, frequency=1)\n\nIn : info = nk.signal_findpeaks(signal)\n\nIn : period = nk.signal_period(peaks=info[\"Peaks\"], desired_length=len(signal))\n\nIn : nk.signal_plot(period)\n```", null, "### signal_phase()#\n\nCompute the phase of the signal\n\nThe real phase has the property to rotate uniformly, leading to a uniform distribution density. The prophase typically doesn’t fulfill this property. The following functions applies a nonlinear transformation to the phase signal that makes its distribution exactly uniform. If a binary vector is provided (containing 2 unique values), the function will compute the phase of completion of each phase as denoted by each value.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• method (str) – The values in which the phase is expressed. Can be `\"radians\"` (default), `\"degrees\"` (for values between 0 and 360) or `\"percents\"` (for values between 0 and 1).\n\nReturns:\n\narray – A vector containing the phase of the signal, between 0 and 2pi.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=10)\n\nIn : phase = nk.signal_phase(signal)\n\nIn : nk.signal_plot([signal, phase])\n```", null, "..ipython:: python\n\nrsp = nk.rsp_simulate(duration=30) phase = nk.signal_phase(rsp, method=”degrees”) @savefig p_signal_phase2.png scale=100% nk.signal_plot([rsp, phase]) @suppress plt.close()\n\n```# Percentage of completion of two phases\nIn : signal = nk.signal_binarize(nk.signal_simulate(duration=10))\n\nIn : phase = nk.signal_phase(signal, method=\"percents\")\n\nIn : nk.signal_plot([signal, phase])\n```", null, "### signal_plot()#\n\nsignal_plot(signal, sampling_rate=None, subplots=False, standardize=False, labels=None, kwargs)[source]#\n\nPlot signal with events as vertical lines\n\nParameters:\n• signal (array or DataFrame) – Signal array (can be a dataframe with many signals).\n\n• sampling_rate (int) – The sampling frequency of the signal (in Hz, i.e., samples/second). Needs to be supplied if the data should be plotted over time in seconds. Otherwise the data is plotted over samples. Defaults to `None`.\n\n• subplots (bool) – If `True`, each signal is plotted in a subplot.\n\n• standardize (bool) – If `True`, all signals will have the same scale (useful for visualisation).\n\n• labels (str or list) – Defaults to `None`.\n\n• kwargs (optional) – Arguments passed to matplotlib plotting.\n\n`ecg_plot`, `rsp_plot`, `ppg_plot`, `emg_plot`, `eog_plot`\n\nReturns:\n\n• Though the function returns nothing, the figure can be retrieved and saved as follows\n\n• .. code-block:: console – # To be run after signal_plot() fig = plt.gcf() fig.savefig(“myfig.png”)\n\nExamples\n\n```In : import numpy as np\n\nIn : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=10, sampling_rate=1000)\n\nIn : nk.signal_plot(signal, sampling_rate=1000, color=\"red\")\n```", null, "``` # Simulate data\nIn : data = pd.DataFrame({\"Signal2\": np.cos(np.linspace(start=0, stop=20, num=1000)),\n...: \"Signal3\": np.sin(np.linspace(start=0, stop=20, num=1000)),\n...: \"Signal4\": nk.signal_binarize(np.cos(np.linspace(start=0, stop=40, num=1000)))})\n...:\n\n# Process signal\nIn : nk.signal_plot(data, labels=['signal_1', 'signal_2', 'signal_3'], subplots=True)\n\nIn : nk.signal_plot([signal, data], standardize=True)\n```", null, "### signal_power()#\n\nsignal_power(signal, frequency_band, sampling_rate=1000, continuous=False, show=False, normalize=True, kwargs)[source]#\n\nCompute the power of a signal in a given frequency band\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• frequency_band (tuple or list) – Tuple or list of tuples indicating the range of frequencies to compute the power in.\n\n• sampling_rate (int) – The sampling frequency of the signal (in Hz, i.e., samples/second).\n\n• continuous (bool) – Compute instant frequency, or continuous power.\n\n• show (bool) – If `True`, will return a Poincaré plot. Defaults to `False`.\n\n• normalize (bool) – Normalization of power by maximum PSD value. Default to `True`. Normalization allows comparison between different PSD methods.\n\n• kwargs – Keyword arguments to be passed to `signal_psd()`.\n\nReturns:\n\npd.DataFrame – A DataFrame containing the Power Spectrum values and a plot if `show` is `True`.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : import numpy as np\n\n# Instant power\nIn : signal = nk.signal_simulate(duration=60, frequency=[10, 15, 20],\n...: amplitude = [1, 2, 3], noise = 2)\n...:\n\nIn : power_plot = nk.signal_power(signal, frequency_band=[(8, 12), (18, 22)], method=\"welch\", show=True)\n```", null, "..ipython:: python\n\n# Continuous (simulated signal) signal = np.concatenate((nk.ecg_simulate(duration=30, heart_rate=75), nk.ecg_simulate(duration=30, heart_rate=85))) power = nk.signal_power(signal, frequency_band=[(72/60, 78/60), (82/60, 88/60)], continuous=True) processed, _ = nk.ecg_process(signal) power[“ECG_Rate”] = processed[“ECG_Rate”]\n\n@savefig p_signal_power2.png scale=100% nk.signal_plot(power, standardize=True) @suppress plt.close()\n\n```# Continuous (real signal)\nIn : signal = nk.data(\"bio_eventrelated_100hz\")[\"ECG\"]\n\nIn : power = nk.signal_power(signal, sampling_rate=100, frequency_band=[(0.12, 0.15), (0.15, 0.4)], continuous=True)\n\nIn : processed, _ = nk.ecg_process(signal, sampling_rate=100)\n\nIn : power[\"ECG_Rate\"] = processed[\"ECG_Rate\"]\n\nIn : nk.signal_plot(power, standardize=True)\n```", null, "### signal_psd()#\n\nsignal_psd(signal, sampling_rate=1000, method='welch', show=False, normalize=True, min_frequency='default', max_frequency=inf, window=None, window_type='hann', order=16, order_criteria='KIC', order_corrected=True, silent=True, t=None, kwargs)[source]#\n\nCompute the Power Spectral Density (PSD)\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• sampling_rate (int) – The sampling frequency of the signal (in Hz, i.e., samples/second).\n\n• method (str) – Either `\"welch\"` (default), `\"fft\"`, `\"multitapers\"` (requires the ‘mne’ package), `\"lombscargle\"` (requires the ‘astropy’ package) or `\"burg\"`.\n\n• show (bool) – If `True`, will return a plot. If `False`, will return the density values that can be plotted externally.\n\n• normalize (bool) – Normalization of power by maximum PSD value. Default to `True`. Normalization allows comparison between different PSD methods.\n\n• min_frequency (str, float) – The minimum frequency. If default, min_frequency is chosen based on the sampling rate and length of signal to optimize the frequency resolution.\n\n• max_frequency (float) – The maximum frequency.\n\n• window (int) – Length of each window in seconds (for “Welch” method). If `None` (default), window will be automatically calculated to capture at least 2 cycles of min_frequency. If the length of recording does not allow the formal, window will be default to half of the length of recording.\n\n• window_type (str) – Desired window to use. Defaults to `\"hann\"`. See `scipy.signal.get_window()` for list of windows.\n\n• order (int) – The order of autoregression (only used for autoregressive (AR) methods such as `\"burg\"`).\n\n• order_criteria (str) – The criteria to automatically select order in parametric PSD (only used for autoregressive (AR) methods such as `\"burg\"`).\n\n• order_corrected (bool) – Should the order criteria (AIC or KIC) be corrected? If unsure which method to use to choose the order, rely on the default (i.e., the corrected KIC).\n\n• silent (bool) – If `False`, warnings will be printed. Default to `True`.\n\n• t (array) – The timestamps corresponding to each sample in the signal, in seconds (for `\"lombscargle\"` method). Defaults to None.\n\n• kwargs (optional) – Keyword arguments to be passed to `scipy.signal.welch()`.\n\n`signal_filter`, `mne.time_frequency.psd_array_multitaper`, `scipy.signal.welch`\n\nReturns:\n\ndata (pd.DataFrame) – A DataFrame containing the Power Spectrum values and a plot if `show` is `True`.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=2, frequency=[5, 6, 50, 52, 80], noise=0.5)\n\n# FFT method (based on numpy)\nIn : psd_multitapers = nk.signal_psd(signal, method=\"fft\", show=True)\n```", null, "```# Welch method (based on scipy)\nIn : psd_welch = nk.signal_psd(signal, method=\"welch\", min_frequency=1, show=True)\n```", null, "```# Multitapers method (requires MNE)\nIn : psd_multitapers = nk.signal_psd(signal, method=\"multitapers\", show=True)\n```", null, "```# Burg method\nIn : psd_burg = nk.signal_psd(signal, method=\"burg\", min_frequency=1, show=True)\n```", null, "```# Lomb method (requires AstroPy)\nIn : psd_lomb = nk.signal_psd(signal, method=\"lomb\", min_frequency=1, show=True)\n```", null, "### signal_rate()#\n\nsignal_rate(peaks, sampling_rate=1000, desired_length=None, interpolation_method='monotone_cubic')[source]#\n\nCompute Signal Rate\n\nCalculate signal rate (per minute) from a series of peaks. It is a general function that works for any series of peaks (i.e., not specific to a particular type of signal). It is computed as `60 / period`, where the period is the time between the peaks (see func:.signal_period).\n\nNote\n\nThis function is implemented under `signal_rate()`, but it also re-exported under different names, such as `ecg_rate()`, or `rsp_rate()`. The aliases provided for consistency.\n\nParameters:\n• peaks (Union[list, np.array, pd.DataFrame, pd.Series, dict]) – The samples at which the peaks occur. If an array is passed in, it is assumed that it was obtained with `signal_findpeaks()`. If a DataFrame is passed in, it is assumed it is of the same length as the input signal in which occurrences of R-peaks are marked as “1”, with such containers obtained with e.g., :func:.`ecg_findpeaks` or `rsp_findpeaks()`.\n\n• sampling_rate (int) – The sampling frequency of the signal that contains peaks (in Hz, i.e., samples/second). Defaults to 1000.\n\n• desired_length (int) – If left at the default None, the returned rated will have the same number of elements as `peaks`. If set to a value larger than the sample at which the last peak occurs in the signal (i.e., `peaks[-1]`), the returned rate will be interpolated between peaks over `desired_length` samples. To interpolate the rate over the entire duration of the signal, set `desired_length` to the number of samples in the signal. Cannot be smaller than or equal to the sample at which the last peak occurs in the signal. Defaults to `None`.\n\n• interpolation_method (str) – Method used to interpolate the rate between peaks. See `signal_interpolate()`. `\"monotone_cubic\"` is chosen as the default interpolation method since it ensures monotone interpolation between data points (i.e., it prevents physiologically implausible “overshoots” or “undershoots” in the y-direction). In contrast, the widely used cubic spline interpolation does not ensure monotonicity.\n\nReturns:\n\narray – A vector containing the rate (peaks per minute).\n\nExamples\n\n```In : import neurokit2 as nk\n\n# Create signal of varying frequency\nIn : freq = nk.signal_simulate(1, frequency = 1)\n\nIn : signal = np.sin((freq).cumsum() * 0.5)\n\n# Find peaks\nIn : info = nk.signal_findpeaks(signal)\n\n# Compute rate using 2 methods\nIn : rate1 = nk.signal_rate(peaks=info[\"Peaks\"],\n....: desired_length=len(signal),\n....: interpolation_method=\"nearest\")\n....:\n\nIn : rate2 = nk.signal_rate(peaks=info[\"Peaks\"],\n....: desired_length=len(signal),\n....: interpolation_method=\"monotone_cubic\")\n....:\n\n# Visualize signal and rate on the same scale\nIn : nk.signal_plot([signal, rate1, rate2],\n....: labels = [\"Original signal\", \"Rate (nearest)\", \"Rate (monotone cubic)\"],\n....: standardize = True)\n....:\n```", null, "### signal_smooth()#\n\nsignal_smooth(signal, method='convolution', kernel='boxzen', size=10, alpha=0.1)[source]#\n\nSignal smoothing\n\nSignal smoothing can be achieved using either the convolution of a filter kernel with the input signal to compute the smoothed signal (Smith, 1997) or a LOESS regression.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• method (str) – Can be one of `\"convolution\"` (default) or `\"loess\"`.\n\n• kernel (Union[str, np.array]) – Only used if `method` is `\"convolution\"`. Type of kernel to use; if array, use directly as the kernel. Can be one of `\"median\"`, `\"boxzen\"`, `\"boxcar\"`, `\"triang\"`, `\"blackman\"`, `\"hamming\"`, `\"hann\"`, `\"bartlett\"`, `\"flattop\"`, `\"parzen\"`, `\"bohman\"`, `\"blackmanharris\"`, `\"nuttall\"`, `\"barthann\"`, `\"kaiser\"` (needs beta), `\"gaussian\"` (needs std), `\"general_gaussian\"` (needs power width), `\"slepian\"` (needs width) or `\"chebwin\"` (needs attenuation).\n\n• size (int) – Only used if `method` is `\"convolution\"`. Size of the kernel; ignored if kernel is an array.\n\n• alpha (float) – Only used if `method` is `\"loess\"`. The parameter which controls the degree of smoothing.\n\nReturns:\n\narray – Smoothed signal.\n\nExamples\n\n```In : import numpy as np\n\nIn : import pandas as pd\n\nIn : import neurokit2 as nk\n\nIn : signal = np.cos(np.linspace(start=0, stop=10, num=1000))\n\nIn : distorted = nk.signal_distort(signal,\n...: noise_amplitude=[0.3, 0.2, 0.1, 0.05],\n...: noise_frequency=[5, 10, 50, 100])\n...:\n\nIn : size = len(signal)/100\n\nIn : signals = pd.DataFrame({\"Raw\": distorted,\n...: \"Median\": nk.signal_smooth(distorted, kernel=\"median\", size=size-1),\n...: \"BoxZen\": nk.signal_smooth(distorted, kernel=\"boxzen\", size=size),\n...: \"Triang\": nk.signal_smooth(distorted, kernel=\"triang\", size=size),\n...: \"Blackman\": nk.signal_smooth(distorted, kernel=\"blackman\", size=size),\n...: \"Loess_01\": nk.signal_smooth(distorted, method=\"loess\", alpha=0.1),\n...: \"Loess_02\": nk.signal_smooth(distorted, method=\"loess\", alpha=0.2),\n...: \"Loess_05\": nk.signal_smooth(distorted, method=\"loess\", alpha=0.5)})\n...:\n\nIn : fig = signals.plot()\n```", null, "```# Magnify the plot\nIn : fig_magnify = signals[50:150].plot()\n```", null, "References\n\n• Smith, S. W. (1997). The scientist and engineer’s guide to digital signal processing.\n\n### signal_synchrony()#\n\nsignal_synchrony(signal1, signal2, method='hilbert', window_size=50)[source]#\n\nSynchrony (coupling) between two signals\n\nSignal coherence refers to the strength of the mutual relationship (i.e., the amount of shared information) between two signals. Synchrony is coherence “in phase” (two waveforms are “in phase” when the peaks and troughs occur at the same time). Synchrony will always be coherent, but coherence need not always be synchronous.\n\nThis function computes a continuous index of coupling between two signals either using the `\"hilbert\"` method to get the instantaneous phase synchrony, or using a rolling window correlation.\n\nThe instantaneous phase synchrony measures the phase similarities between signals at each timepoint. The phase refers to the angle of the signal, calculated through the hilbert transform, when it is resonating between -pi to pi degrees. When two signals line up in phase their angular difference becomes zero.\n\nFor less clean signals, windowed correlations are widely used because of their simplicity, and can be a good a robust approximation of synchrony between two signals. The limitation is the need to select a window size.\n\nParameters:\n• signal1 (Union[list, np.array, pd.Series]) – Time series in the form of a vector of values.\n\n• signal2 (Union[list, np.array, pd.Series]) – Time series in the form of a vector of values.\n\n• method (str) – The method to use. Can be one of `\"hilbert\"` or `\"correlation\"`.\n\n• window_size (int) – Only used if `method='correlation'`. The number of samples to use for rolling correlation.\n\n`scipy.signal.hilbert`, `mutual_information`\n\nReturns:\n\narray – A vector containing the phase of the signal, between 0 and 2pi.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : s1 = nk.signal_simulate(duration=10, frequency=1)\n\nIn : s2 = nk.signal_simulate(duration=10, frequency=1.5)\n\nIn : coupling1 = nk.signal_synchrony(s1, s2, method=\"hilbert\")\n\nIn : coupling2 = nk.signal_synchrony(s1, s2, method=\"correlation\", window_size=1000/2)\n\nIn : nk.signal_plot([s1, s2, coupling1, coupling2], labels=[\"s1\", \"s2\", \"hilbert\", \"correlation\"])\n```", null, "References\n\n### signal_timefrequency()#\n\nsignal_timefrequency(signal, sampling_rate=1000, min_frequency=0.04, max_frequency=None, method='stft', window=None, window_type='hann', mode='psd', nfreqbin=None, overlap=None, analytical_signal=True, show=True)[source]#\n\nQuantify changes of a nonstationary signal’s frequency over time The objective of time-frequency analysis is to offer a more informative description of the signal which reveals the temporal variation of its frequency contents.\n\nThere are many different Time-Frequency Representations (TFRs) available:\n\n• Linear TFRs: efficient but create tradeoff between time and frequency resolution\n\n• Short Time Fourier Transform (STFT): the time-domain signal is windowed into short segments and FT is applied to each segment, mapping the signal into the TF plane. This method assumes that the signal is quasi-stationary (stationary over the duration of the window). The width of the window is the trade-off between good time (requires short duration window) versus good frequency resolution (requires long duration windows)\n\n• Wavelet Transform (WT): similar to STFT but instead of a fixed duration window function, a varying window length by scaling the axis of the window is used. At low frequency, WT proves high spectral resolution but poor temporal resolution. On the other hand, for high frequencies, the WT provides high temporal resolution but poor spectral resolution.\n\n• Quadratic TFRs: better resolution but computationally expensive and suffers from having cross terms between multiple signal components\n\n• Wigner Ville Distribution (WVD): while providing very good resolution in time and frequency of the underlying signal structure, because of its bilinear nature, existence of negative values, the WVD has misleading TF results in the case of multi-component signals such as EEG due to the presence of cross terms and inference terms. Cross WVD terms can be reduced by using smoothing kernel functions as well as analyzing the analytic signal (instead of the original signal)\n\n• Smoothed Pseudo Wigner Ville Distribution (SPWVD): to address the problem of cross-terms suppression, SPWVD allows two independent analysis windows, one in time and the other in frequency domains.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• sampling_rate (int) – The sampling frequency of the signal (in Hz, i.e., samples/second).\n\n• method (str) – Time-Frequency decomposition method.\n\n• min_frequency (float) – The minimum frequency.\n\n• max_frequency (float) – The maximum frequency.\n\n• window (int) – Length of each segment in seconds. If `None` (default), window will be automatically calculated. For `\"STFT\" method`.\n\n• window_type (str) – Type of window to create, defaults to `\"hann\"`. See `scipy.signal.get_window()` to see full options of windows. For `\"STFT\" method`.\n\n• mode (str) – Type of return values for `\"STFT\" method`. Can be `\"psd\"`, `\"complex\"` (default, equivalent to output of `\"STFT\"` with no padding or boundary extension), `\"magnitude\"`, `\"angle\"`, `\"phase\"`. Defaults to `\"psd\"`.\n\n• nfreqbin (int, float) – Number of frequency bins. If `None` (default), nfreqbin will be set to `0.5sampling_rate`.\n\n• overlap (int) – Number of points to overlap between segments. If `None`, `noverlap = nperseg // 8`. Defaults to `None`.\n\n• analytical_signal (bool) – If `True`, analytical signal instead of actual signal is used in Wigner Ville Distribution methods.\n\n• show (bool) – If `True`, will return two PSD plots.\n\nReturns:\n\n• frequency (np.array) – Frequency.\n\n• time (np.array) – Time array.\n\n• stft (np.array) – Short Term Fourier Transform. Time increases across its columns and frequency increases down the rows.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : sampling_rate = 100\n\nIn : signal = nk.signal_simulate(100, sampling_rate, frequency=[3, 10])\n\n# STFT Method\nIn : f, t, stft = nk.signal_timefrequency(signal,\n...: sampling_rate,\n...: max_frequency=20,\n...: method=\"stft\",\n...: show=True)\n...:\n```", null, "```# CWTM Method\nIn : f, t, cwtm = nk.signal_timefrequency(signal,\n...: sampling_rate,\n...: max_frequency=20,\n...: method=\"cwt\",\n...: show=True)\n...:\n```", null, "```# WVD Method\nIn : f, t, wvd = nk.signal_timefrequency(signal,\n...: sampling_rate,\n...: max_frequency=20,\n...: method=\"wvd\",\n...: show=True)\n...:\n```", null, "```# PWVD Method\nIn : f, t, pwvd = nk.signal_timefrequency(signal,\n...: sampling_rate,\n...: max_frequency=20,\n...: method=\"pwvd\",\n...: show=True)\n...:\n```", null, "### signal_zerocrossings()#\n\nsignal_zerocrossings(signal, direction='both')[source]#\n\nLocate the indices where the signal crosses zero\n\nNote that when the signal crosses zero between two points, the first index is returned.\n\nParameters:\n• signal (Union[list, np.array, pd.Series]) – The signal (i.e., a time series) in the form of a vector of values.\n\n• direction (str) – Direction in which the signal crosses zero, can be `\"positive\"`, `\"negative\"` or `\"both\"` (default).\n\nReturns:\n\narray – Vector containing the indices of zero crossings.\n\nExamples\n\n```In : import neurokit2 as nk\n\nIn : signal = nk.signal_simulate(duration=5)\n\nIn : zeros = nk.signal_zerocrossings(signal)\n\nIn : nk.events_plot(zeros, signal)\n```", null, "```# Only upward or downward zerocrossings\nIn : up = nk.signal_zerocrossings(signal, direction=\"up\")\n\nIn : down = nk.signal_zerocrossings(signal, direction=\"down\")\n\nIn : nk.events_plot([up, down], signal)\n```", null, "Any function appearing below this point is not explicitly part of the documentation and should be added. Please open an issue if there is one.\n\nSubmodule for NeuroKit.\n\nsignal_formatpeaks(info, desired_length, peak_indices=None, other_indices=None)[source]#\n\nFormat Peaks\n\nTransforms a peak-info dict to a signal of given length" ]	[ null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_simulate1.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_filter1.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_filter2.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_filter3.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_resample1.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_resample2.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_binarize.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_decompose1.png", null, "https://neuropsychology.github.io/NeuroKit/_images/p_signal_decompose2.png", null, 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https://solvedlib.com/n/0-10-mole-of-argon-gas-is-allowed-to-enter-an-empty-50-cm,21121669	[ "# 0.10 mole of argon gas is allowed to enter an empty 50-cm' container at 208 C. The gas is then\n\n###### Question:\n\n0.10 mole of argon gas is allowed to enter an empty 50-cm' container at 208 C. The gas is then heated in this sealed containcr to temperature of 300\" â‚¬. Draw this proccss on PV diagram; with proper scales On each axis_ Determine the final pressure of the gas after this heating: How much work was done to the gas during this process? Dctcrminc thc change in the gas $internal cnergy during- this process, The gas is then cooled back to 20? C in a constant in constant pressure proccss; where the lid of the containcr is allowed to movc. Draw this process on your PV diagram_ Determine the final volume after this process_ How much work was done to the gas during this process? Determine the change in the gas$ internal energy during- this process _", null, "", null, "#### Similar Solved Questions\n\n##### Can you explain why? Question 5 0/2 pts Which combination would make the MOST effective buffer...\nCan you explain why? Question 5 0/2 pts Which combination would make the MOST effective buffer solution? Correct Answer 0.1 M H2CO3 and 1.0 M NaHCO3 0.1 M H2CO3 and 2.0 M NaHCO3 0.15 MH2CO3 and 0.015 M NaHCO3 You Answered 0.1 MH2CO3 and 0.1 M NaHCO3...\n##### Heights were measured for a random sample of 10 plants grown while being treated with a...\nHeights were measured for a random sample of 10 plants grown while being treated with a particular nutrient. The sample mean and sample standard deviation of those height measurements were 31 centimeters and 12 centimeters, respectively. Assume that the population of heights of treated plants is nor...\n##### Utt Urz u(0,+) = 0, u(1,t) =0, t 20 u(z,0) (1-c) 0 < x <1 Ut (c,C x(1-x) 0 <t <1\nUtt Urz u(0,+) = 0, u(1,t) =0, t 20 u(z,0) (1-c) 0 < x <1 Ut (c,C x(1-x) 0 <t <1...\n##### 6-pirig-P)-toug;-Vuw junid ppWi / V 88 nD41 FB I]\n6-pirig-P)-toug;-Vuw junid ppWi / V 88 nD41 FB I]...\n##### Graph each function. $f(x)=\\log _{3}(x-1)$\nGraph each function. $f(x)=\\log _{3}(x-1)$...\n##### Nonsynonymous mutations in genes are nucleotide changes that alter the amino acid sequence of the translated...\nNonsynonymous mutations in genes are nucleotide changes that alter the amino acid sequence of the translated protein, potentially leading to a change in protein function. Using your knowledge of the genetic code, which ONE of the following amino acid replacements can be caused by a single-base chang...\n##### Evaljate the fclloxing indelinite integrel.(6' 5)-7 &\nEvaljate the fclloxing indelinite integrel. (6' 5)-7 &...\n##### Suppaoo tho poalllon ol nn oblcal moving I 0 atroght lino Qivan by D() \" 41? + 51+ 6 Find Iha Inalanlanooua vobclly en /-\"3 The intlanlanoous veloaty at ( =3 /\nSuppaoo tho poalllon ol nn oblcal moving I 0 atroght lino Qivan by D() \" 41? + 51+ 6 Find Iha Inalanlanooua vobclly en /-\"3 The intlanlanoous veloaty at ( =3 /...\n##### A skier starting from rest skis straight down a slope 50 meters long in 5 seconds\nA skier starting from rest skis straight down a slope 50 meters long in 5 seconds. What is the magnitude of the acceleration of the skier? How would you go about solving this type of question?...\n##### The sum of three consecutive even integers is 12 less than the middle integer. What's the answer?\nThe sum of three consecutive even integers is 12 less than the middle integer. What's the answer?...\n##### Explain the ways in which Native Americans viewed European arrivals and encroachments in the sixteenth and...\nExplain the ways in which Native Americans viewed European arrivals and encroachments in the sixteenth and seventeenth centuries p.s I apologize for my category choice. History was not an option!...\n##### 6.00 HC-2.00 HC 1.5 UC3.00 cm2.00 cm\n6.00 HC -2.00 HC 1.5 UC 3.00 cm 2.00 cm...\n##### Write an equation for the function graphed here:4 3fla:)\nWrite an equation for the function graphed here: 4 3 fla:)...\n##### Fnd Ihe exc} valuz66, Co 5 ton (11] 68, Cos Don '(+8)] Wile 4ie elpeso Iprfa â‚¬ elpress&. TF not neces%oryt Tahorolze Otn2 denomina if 74, tan(Sin-',) fr Ix) al\nFnd Ihe exc} valuz 66, Co 5 ton (11] 68, Cos Don '(+8)] Wile 4ie elpeso Iprfa â‚¬ elpress&. TF not neces%oryt Tahorolze Otn2 denomina if 74, tan(Sin-',) fr Ix) al...\n##### The figure shows a small positive charge some distance away from a larger positive charge. The...\nThe figure shows a small positive charge some distance away from a larger positive charge. The smaller charge is currently a distance rı away from the center of the larger charge, which is twice as far away as r2. Answer the three questions below, using three significant digits. Part A: What ha...\n##### Let T : Rz[r] R? be the linear transformation given by the formula T(a \| bx cr) = (a 6. â‚¬ 6) and let B {I + LI 1.22 \| 1} and {(1,1), (-1,1)} be bases for Rz and R? respectively: Find basis for KerT _ Find basis for ImT Determine [T]g' B: Verify that dim( Ker(T)) + dim(Im(T)) dim(Rz[z]).Compute the determinants of these matrices using cofactor formula:Date: May 16. 2021_LINEAR ALGEBRA HOMEWORKDEADLINE [8 MAY 2021. 23.59.2 =1 01 F1A =and B =and C .-1\nLet T : Rz[r] R? be the linear transformation given by the formula T(a \| bx cr) = (a 6. â‚¬ 6) and let B {I + LI 1.22 \| 1} and {(1,1), (-1,1)} be bases for Rz and R? respectively: Find basis for KerT _ Find basis for ImT Determine [T]g' B: Verify that dim( Ker(T)) + dim(Im(T)) dim(Rz[z]). C...\n##### (b) Eliminate the parameter and write the resulting rectangular equation whose graph represents the curve.Adjust the domain of the rectangular equation, if necessary:[0, [4, 0)not necessary\n(b) Eliminate the parameter and write the resulting rectangular equation whose graph represents the curve. Adjust the domain of the rectangular equation, if necessary: [0, [4, 0) not necessary...\n##### QUESTION 18 Match the following definition with their term: Table of Drugs and Chemicals (A-Z) 1]...\nQUESTION 18 Match the following definition with their term: Table of Drugs and Chemicals (A-Z) 1] Sarcoma 2] Pneumonia 3] Aspirin 4] Infection 1 2 3 4 4 points QUESTION 19 When coding the diagnostic statement, “wheezing and shortness of breath cau...\n##### 3. LetA =Determine the eigenvalues for A, and the subspaces corresponding to the eigenvalues_ Is A diagonalizable? What is the minimal polynomial for A?\n3. Let A = Determine the eigenvalues for A, and the subspaces corresponding to the eigenvalues_ Is A diagonalizable? What is the minimal polynomial for A?...\n##### 12. Prepare in journal form the entries necessary to record the following stock transactions of the...\n12. Prepare in journal form the entries necessary to record the following stock transactions of the Vallen Company during 2019: Oct. 1 Purchased 1,000 shares of its own common stock for $20 per share. Oct. 5 Sold 500 shares of treasury stock purchased on Oct 1. For$25 per share. Oct. 8 Sold 250 sha...\n##### Identify the oxidized substance, the reduced substance, the oxidizing agent, and the reducing agent in the...\nIdentify the oxidized substance, the reduced substance, the oxidizing agent, and the reducing agent in the redox reaction. Mg(s) + CI,() Mg?+(aq) + 2C1\" (aq) Which substance gets oxidized? OCH OCI Mg? * Mg Which substance gets reduced? OCH OM Ос, Mg? What is the oxidizingent OC OM Wh...\n##### For the reaction:Caco, (calcile)CacOs (aragonite) entropy and freo onergy change at 25\"C and atmosphere Calculate the enthalpy, Which phase Is stable under Ihese conditions? Calculate (AHlr)\" , at G00 C assuming (hat the system Is Isobaric Cp (calldeg-mole) = 24.98 5.24 X 10JT_ 6.20 X 10*T2 Given: calcite: aragonite: Cp (calldeg-mole) = 20.13 + 10.24 X 10JT_ 3.34 X 109T?\nFor the reaction: Caco, (calcile) CacOs (aragonite) entropy and freo onergy change at 25\"C and atmosphere Calculate the enthalpy, Which phase Is stable under Ihese conditions? Calculate (AHlr)\" , at G00 C assuming (hat the system Is Isobaric Cp (calldeg-mole) = 24.98 5.24 X 10JT_ 6.20 X 10...\n##### What is the algebraic expression for \"4 times the sum of q and p\"?\nWhat is the algebraic expression for \"4 times the sum of q and p\"?..." ]	[ null, "https://cdn.numerade.com/ask_images/f58d1e5d9728401baeb4075f239d1273.jpg ", null, "https://cdn.numerade.com/previews/ab784db9-3abe-43b3-a0bc-c0ebcebec73f_large.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95632184,"math_prob":0.95839393,"size":6495,"snap":"2023-14-2023-23","text_gpt3_token_len":1447,"char_repetition_ratio":0.18179017,"word_repetition_ratio":0.0097402595,"special_character_ratio":0.22278675,"punctuation_ratio":0.093226515,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99059796,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-29T17:15:43Z\",\"WARC-Record-ID\":\"<urn:uuid:8a8fe551-577b-47e3-a5e6-dcfe9783c883>\",\"Content-Length\":\"105457\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5a4f21f4-7f9b-41f8-a2f2-871e7a22aadc>\",\"WARC-Concurrent-To\":\"<urn:uuid:601855a1-905b-4d57-8d29-b7faaa66977d>\",\"WARC-IP-Address\":\"104.21.12.185\",\"WARC-Target-URI\":\"https://solvedlib.com/n/0-10-mole-of-argon-gas-is-allowed-to-enter-an-empty-50-cm,21121669\",\"WARC-Payload-Digest\":\"sha1:F7OQGVN4GRX7575YVXBNLOOPAT34BVCS\",\"WARC-Block-Digest\":\"sha1:ADCP5ZBXQOMNKZBVAJOCMTDDO6DMLIAD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296949009.11_warc_CC-MAIN-20230329151629-20230329181629-00165.warc.gz\"}"}
http://dimacs.rutgers.edu/archive/Events/2014/abstracts/shpilka.html	[ "### DIMACS Theoretical Computer Science Seminar\n\nTitle: Reed-Muller Codes with Respect to Random Errors and Erasures\n\nSpeaker: Amir Shpilka, Technion\n\nDate: Wednesday, October 8, 2014 11:00-12:00pm\n\nLocation: CoRE Bldg, Room 301A, Rutgers University, Busch Campus, Piscataway, NJ\n\nAbstract:\n\nIn TCS we usually study error correcting codes with respect to the Hamming metric, i.e. we study their behaviour with respect to worst case errors. However, in coding theory a more common model is that of random errors, where Shannon’s results show a much better tradeoff between rate and decoding radius.\n\nWe consider the behaviour of Reed-Muller codes in the Shannon model of random errors. In particular, we show that RM codes with either low- or high-degree (n^1/2 or n-n^1/2, respectively), with high probability, can decode from an 1-r fraction of random erasures (where r is the rate). In other words, for this range of parameters RM codes achieve capacity for the Binary-Erasure-Channel. This result matches experimental observations that RM codes can achieve capacity for the BEC, similarly to Polar codes. We also show that RM-codes can handle many more random errors than the minimum distance, i.e. roughly n^d/2 errors for codes of degree n-d (where the minimum distance is only 2^d).\n\nWe show that the questions regarding the behaviour of Reed-Muller codes wrt random errors are tightly connected to the following question. Given a random set of vectors in {0,1}^n, what is the probability the their d’th tensor products are linearly independent? We obtain our results by giving answer to this question for certain range of parameters.\n\nBased on a joint work with Emmanuel Abbe and Avi Wigderson." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8713826,"math_prob":0.84548396,"size":1705,"snap":"2019-35-2019-39","text_gpt3_token_len":411,"char_repetition_ratio":0.10875955,"word_repetition_ratio":0.0077220076,"special_character_ratio":0.22170088,"punctuation_ratio":0.13662791,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96543,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-22T16:36:52Z\",\"WARC-Record-ID\":\"<urn:uuid:c19abcb0-1b92-48b6-a303-786af029aa7e>\",\"Content-Length\":\"2416\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:df2652df-fff2-405e-8840-1232e060395c>\",\"WARC-Concurrent-To\":\"<urn:uuid:677edb62-9ff4-47e4-af16-d11118eb63ea>\",\"WARC-IP-Address\":\"128.6.75.106\",\"WARC-Target-URI\":\"http://dimacs.rutgers.edu/archive/Events/2014/abstracts/shpilka.html\",\"WARC-Payload-Digest\":\"sha1:KT2MYF4XM75FNYYJCPX3UZWJY6PJRGE4\",\"WARC-Block-Digest\":\"sha1:RFSK3IBGYJUT5CCQKRNKCGKL7SO3WICI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027317274.5_warc_CC-MAIN-20190822151657-20190822173657-00003.warc.gz\"}"}
https://moam.info/solution-of-the-poisson-equation-by-differential-wiley-online-library_5caa42d5097c473c488b45cf.html	[ "## Solution of the Poisson equation by differential ... - Wiley Online Library\n\nMay 6, 2019 - and Patankar' compared solutions of the two-dimensional Poisson equation ... This set has a unique solution for the weighting coefficients, uij, ...\n\nINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 19, 71 1-724 (1983)\n\nSOLUTION OF THE POISSON EQUATION BY DIFFERENTIAL QUADRATURE FARUK CIVAN AND C. M. SLIEPCEVICH\n\nUniversity of Oklahoma, Norman, OK, U.S.A.\n\nSUMMARY The method of differential quadrature is demonstrated by solving the two-dimensional Poisson equation. The results for three test problems are compared with the exact analytical solutions and the numerical solutions obtained by others for the Galerkin, the control-volume and the five-point finite difference methods. The method of differential quadrature leads to more accurate results for comparable levels of computational effort.\n\nINTRODUCTION Many applications of the Poisson equation to transport processes require numerical solutions which usually are based on some version of finite elements or finite differences. Ramadhyani and Patankar' compared solutions of the two-dimensional Poisson equation by the Galerkin, control-volume and five-point finite difference methods and concluded that the control-volume method produced the most accurate results. The purpose of this paper is to demonstrate the application of the method of differential quadrature to the identical test problems used by Ramadhyani and Patankar so that direct comparisons could be made with both their analytical and numerical (conventional methods) results. Bellman' introduced the method of diff erential quadrature and applied it to one-dimensional, initial value problems. Subsequently, Mingle6 applied the method to one-dimensional, initialboundary value problems. In this paper, the method is extended to the two-dimensional, boundary value problem. APPROXIMATION OF DERIVATIVES BY DIFFERENTIAL QUADRATURE Consider xi: i = 1 , 2 , . . . ,N are the sample points obtained by subdividing the x-variable into N discrete values and f(xi) are the function values at these points. If aij are the weights to attach to these function values at the sample points, the values of the function derivatives at these points are approximated by a weighted sum of the function values at these points as expressed by the following quadrature formula:'\n\nTo calculate the weighting coefficients the function f(x) is represented by an appropriate analytical function, such as a polynomial, f(x)=xk-*;\n\n0029-5981/83/050711-14\\$01.40 @ 1983 by John Wiley & Sons, Ltd.\n\nk = 1 , 2 , . .., N\n\n(2)\n\nReceived 6 July 1981 Revised 23 March 1982\n\n712\n\nF. CIVAN AND C. M. SLIEPCEVICH\n\nand its derivative,\n\naf(x)/ax\n\n= (k - l)xk-’\n\n(3)\n\nSubstituting equations (2) and (3)into equation (l),N linear algebraic equations are obtained: N\n\nc ui,.xjk-l = ( k - - l ) x f - ’ ;\n\nj=l\n\nk =1, 2 , . . . , N\n\nand i = 1 , 2 , . . . , N\n\n(4)\n\nThis set has a unique solution for the weighting coefficients, uij,because the matrix of elements is a Vandermonde matrix whose inverse can be obtained analytically as shown by Hamming.’ According to Bellman et d.,’the approximation formulae for the partial derivatives of second (and higher) order are obtained by iterating the quadrature approximation formula for the first-order partial derivative, given by equation (1).Hence, for example, the secondorder partial derivative approximation formula is derived from equation (1)by replacing f ( x ) by af(x)/ax,\n\nIn order to reduce the complexity of the derivative approximation formulae and thereby conserve on computational effort, it is advantageous to use quadrature approximation formulae for also the second6 and higher order derivative^.^ Of course, the weighting coefficients for each formula will be different from those for the first-order derivative. (This approach does not appear to be feasible for mixed partial derivatives, however.) Thus, for example, the second-order partial derivatives can be approximated by a linear weighted sum of function values at the sample points as\n\nin which bij are the weights attached to the function values at the sample points. As before, the weighting coefficients can be obtained by a procedure similar to that for equation (1). Again, the function given by equation (2) is used so that the second-order derivative is d z f ( x ) / d x 2 = (k - I)(& -2\n\n) ~ ~ - ~\n\n(7)\n\nSubstituting equation (2) and (7)into equation (6) results in a set of N linear algebraic equations for the weighting coefficients, bij, N\n\nC bijx;-’ =(k - l ) ( k - 2 ) ~ : ~ ~ k; = 1,2,. ..,N and i = 1,2,. .. ,N\n\n(8)\n\nj=l\n\nThis equation can be solved in the same manner as indicated for equation (4) above. The derivative approximation formulae derived for a function of one variable can be extended as follows for a function of two variables:\n\nSOLUTION OF POISSON EQUATION\n\n713\n\nwhere Nr and N Y are the number of sample points in the x- and y-directions, respectively. The second-order derivative approximation formulae are\n\nNY\n\nc bykf(Xi,yk)\n\n(12b)\n\nk=l\n\nSimilar formulae can be developed for all of the partial derivatives of any order for multidimensional systems,' but they will not be presented here since they are not needed for the examples which follow. APPLICATION O F APPROXIMATION FORMULAE TO POISSON EQUATION The following summary of the application of differential quadrature is general and, therefore, is not problem dependent. Although the technique illustrated below is specifically for the two-dimensional Poisson equation, it can be readily extended to the three-dimensional case. Consider the two-dimensional Poisson equation in normalized form:\n\nwhere p = L / H represents the aspect ratio and 0 s x , y s 1. Here the dimensionless quantities are given by x =X/L\n\nq5=\\$\n\nifS=O\n\nt 14)\n\n(16b)\n\nwhere 4, the dependent variable, is a function of the space co-ordinates, X and Y, and S represents a given source strength. L and H are the characteristic lengths of the physical domain in the X and Y directions. For numerical solution the two-dimensional rectangular grid system of Figure 1 is formed by subdividing the x and y variables into N\" and N Y discrete values, respectively. The spacings need not necessarily be equal; however, in this study they are assigned equally. The resulting discrete points in the x and y directions are indicated by subscripts i and j , respectively. Then the partial derivatives of the function, q5 (x, y), are replaced by the approximation formulae throughout the Poisson equation and the derivative boundary conditions to obtain a set of linear algebraic equations which can be solved using any appropriate methods. To utilize differential quadrature, two options are available. The first approach' is to replace the second-order partial derivatives with respect to x and y variables in equation (13)by the\n\n714\n\nF. CIVAN AND C. M. SLIEPCEVICH\n\n7\n\nh\n\n2\n\nI i - 4\n\ni\n\n----L\n\ni = f , Z , . . . ,N Figure 1. The unit square region and the grid for the normalized Poisson equation\n\napproximation formulae ( I l a ) and (12a) to obtain NY\n\nNY\n\nin which the following shorthand notation is used for convenience: dii=-4(xi, y j ) and Sij=S(xi, y i ) . The second approach6 is to replace the derivatives by the approximation formulae ( l l b ) and (12b), respectively:\n\nOf the two approaches, the latter saves appreciably on computational effort but requires ) additional storage for the bii's. Both the equations (17a) and (17b) contain ( N x ) ( N Yfunction values, some of which are the prescribed boundary values given by Dirichlet conditions and others by the boundary conditions involving the function derivatives, such as Neumann or mixed boundary conditions. The first step for the numerical model is to separate terms that are associated with the known boundary values and move these terms, as well as the source term, to the right of equation (17a) or (17b). The terms associated with the interior points, as well as the boundary points upon which boundary conditions involving function derivatives are imposed, must be collected on the left-hand side of equations (17a) or (17b). As a result, these equations can be represented by\n\ng . . = h.. 11\n\n(18)\n\n11\n\nwhich can be written as\n\nag-\n\nP\n\nC 3 d r n= h,; 4 a4rn\n\ni = 2 , 3 , . . . , (N\"-1)\n\nand j = 2 , 3 , .\n\n. . ,(Ivy-1)\n\n(19)\n\nHere, p and q are indices for those mesh points where the function values are to be calculated. In equation (19), agij/aq5, are the elements of the Jacobian matrix obtained from gii. For the first approach, they are given by\n\nSOLUTION OF POISSON EQUATION\n\n715\n\nand for the second approach,\n\nagij/a4,\n\n= ajqb;\n\n+si&'b;q\n\n(20b) Here, S,, are the Kroneker deltas whose values are equal to 1when rn = n but 0 when rn # n. The Jacobian matrix elements are constant numbers and have to be calculated only once for the Poisson equation. However, equation (20b) is preferred for relatively large mesh systems to conserve on computational effort. Since the elements of the Jacobian matrix are constant values, equation (19) represents (N\"- 2 ) ( N y- 2) linear algebraic equations. If these equations also contain the boundary values defined by conditions involving function derivatives, such as the Neumann or mixed type, in addition to the interior region function values, the function derivatives in these boundary conditions also need to be replaced by the quadrature approximation formulae to obtain an algebraic relation for the boundary values. For example, consider a boundary condition expressed by\n\nad(X, Y ) / a Y = f ( x > at Y = 1\n\n(21)\n\nwhere f ( x ) is a given function. Replacing the function derivative in equation (21) with the differential quadrature approximation given by equation (10) one obtains\n\nAfter collecting the term associated with the known boundary value (if there is any at all) on the right, equation (22) can also be expressed in a form similar to equation (19),\n\nwhere the Jacobian matrix elements are given by\n\nagiNy/a4, = &,a 'N,Y\n\n(24)\n\nOther types of boundary conditions involving the function derivatives can be treated similarly. The resulting set of algebraic equations are solved simultaneously using, for example, a Gaussian elimination technique for the unknown function values at points (p, 4). However, in the application problems presented next a slightly different approach will be used to reduce the size of the matrix equation that needs to be solved for the unknown function values. For this purpose, equation (22) is solved first for the unknown boundary values as\n\nUsing equation (25) the unknown boundary values will be eliminated throughout the discretized Poisson equation (17a) or (17b). In this manner the number of equations to be solved is reduced because equation (23) is not required any longer. The resulting equation can be expressed in the form of equation (19) according to\n\nag.. 1 -1L4pq=hij; q = 2 a4m\n\n( N x - 1 ) (Ny-1) p=2\n\ni = 2 , 3 , . . . , ( N x -1) and j = 2 , 3 , . . . , ( N y-1)\n\n7 16\n\nF. CIVAN AND C. M. SLIEPCEVICH\n\nwhich contains only the function values for the interior region. Upon simultaneous solution of equation (26) the unknown boundary values are obtained from their corresponding expressions, such as equation (25). Because the expressions for the Jacobian matrix elements contain the Kroneker deltas, the resulting Jacobian matrix contains many zero elements and therefore is not a 'full' matrix. One may, of course, take advantage of this situation to develop special matrix solvers to reduce the computing effort and to increase the accuracy of numerical solutions. However, for the sizes of the matrices encountered in the test problems which follow, a Gaussian elimination type of solver for a set of linear equations with a full coefficient matrix is acceptable. TEST PROBLEMS Three test problems involving the Poisson equation, which are identical to Ramadhyani and Patankar,' will be used to demonstrate the quality of the numerical solutions by the method of differential quadrature. Both of the aforementioned approaches were used for each test problem; as would be expected the results were identical. Therefore, in the interest of brevity, only the procedure for the first approach will be presented for the first problem and only the procedure for the second approach will be presented for the second and third problems.\n\nFirst problem Consider the following problem in normalized form:\n\n-+p a24 ax\n\n2a24 y=o;\n\nOsx,ysl\n\naY\n\nThe exact solution in normalized form is given by\n\n4 = sinh [wy/([email protected])]. sin (m/2)/sinh [7r/(2/3)]\n\n(32)\n\nReplacing the partial derivatives with the approximation formulae given by equations (9), ( l l a ) and (12a), the model equations (27)-(31) reduce to\n\n4il=O; i = 2 , 3 , . . . ,N\" 4N i Y = sin (7rxi/2); i = 2,3, . . . ,N\" 4lj=O; j = 1 , 2 , . . . , N Y\n\n(34)\n\nSOLUTION OF POISSON EQUATION\n\n717\n\nBy virtue of equation (36), equation (37) leads to the following expression for the value of the function on the symmetry boundary,\n\nUsing equations (34)-(38) and rearranging, equation (33) reduces to the following set of algebraic equations:\n\nfor i = 2 , 3 , . . . , (N\"- 1), and j = 2 , 3 , . . . , ( N y- l),in which the elements of the Jacobian matrix are given by\n\nand NY\n\nhij = -p2 sin (rxi/2)\n\n1 a&& 1=1\n\nSolution of equation (39) has been accomplished with subroutines DECOMP, SOLVE and SING given by Forsythe and Moler? which utilizes the method of Gaussian elimination, on the IBM 370/158 computer facility at the University of Oklahoma. Numerical calculations have been carried out using uniform grid systems, 5 x 5 and 7 x 7. A local error, defined by =- Idexact -4computedl\n\nE\n\n(42)\n\nwas calculated at every point of the uniform grid. , the maximum To facilitate comparisons, only the errors at the centre of the domain, E ~ and are shown in Figure 2 for various values of the aspect ratio L / H . error in the domain, E-, These results are compared directly with those reported by Ramadhyani and Patankar.7 It is evident that the method of differential quadrature using a 7 X 7 grid results in much lower errors than are obtained by the conventional finite element and finite difference techniques using a 7 x 7 grid.\n\nSecond problem The governing equations for the second problem in normalized form are\n\nC#l(x,O)=O;\n\nOcxsl\n\nl ) / a y = 0;\n\nOax s 1\n\n4(0,y)=O;\n\nOayal\n\na4 (I,y)/ax = 0;\n\no =S y a 1\n\na&(X,\n\n718\n\nF. CIVAN AND C. M. SLIEPCEVICH 10.0\n\n?\\$--.,\n\n1.0\n\n. . ... 0.\n\n'0\n\n0.1\n\n\\\n\n-\n\n\\\n\nl o 3 E,\n\n0.01\n\nx-.-\n\nX Galer\n\nin(7X7) Ref.7\n\n+-+Control 0\n\nvolume(7X7)\n\nRef.7\n\n5-Point f i n i t e d i f f . ( 7 X 7 ) Ref.7\n\n0---ODiff.\n\nq u a d . ( 7 ~ 7 ) p r e s e n t work\n\nI\n\nI\n\n0.001\n\n0.25\n\np r e s e n t work\n\nI\n\n0.75\n\n0.50\n\nI 1.0\n\nL/H\n\nFigure 2(a). The centre-point error for the first problem\n\nThe exact solution in normalized form is given by\n\n4\n\n1 =-(2P2 1-y2-4\n\na0\n\nC (-1)\" cosh (A,@x)cos(A,y)/[A:\n\ncosh ( A d ) ] }\n\nn=O\n\nwhere A, = (2n + 1)7~/2. For numericalsolutions, the partial derivativesare replaced with the approximationformulae given by the equations (9), (lo), (Ilb) and (12b). By rearrangingthe model, equations (43)-(47) are reduced to the following set of algebraic equations:\n\nfor i = 2 , 3 , . . . , (N' - 1) andj = 2 , 3 , . . . , (IVY- l),in which\n\nagij/a4,\n\n= 6jq(b;,,- b~N=U&x,/U&XNX)+6i,@2(biq- b & y U L Y q / U h y )\n\n(50)\n\nSOLUTION OF POISSON EQUATION\n\n719\n\nThe right of equation (49) is simply hij = -1. The function values along the symmetry boundaries are calculated by\n\n( z2 (Nx-1)\n\n4NY= -\n\nakxk#kj)/akxNx\n\n(51)\n\nand\n\nThe solution of equation (49) has been carried out using the same subroutines and grid systems as in the first problem. For various values of the aspect ratio L / H , the results in terms of the centre-point error c 0 and maximum error E,, obtained via the differential quadrature method are presented in Figure 3. Again the errors for differential quadrature with a 7 x 7\n\n720\n\nF. CIVAN AND C. M. SLIEPCEVICH\n\ngrid are lower than those obtained by the conventional finite element and finite difference technique using a 7 x 7 grid. In Figure 3(a) it should be noted that the centre-point errors for the finite element and the finite difference techniques increase with increasing aspect ratio; although this behaviour is anomalous, Ramadhyani and Patankar do not offer any comment. On the other hand, the maximum errors for the conventional techniques, shown in Figure 3(b), demonstrate normal behaviour. By contrast, both the centre-point and maximum errors produced via differential quadrature solutions exhibit the expected decreases with increasing aspect ratio. However, it is somewhat surprising to find that the 5 x 5 grid for differential quadrature gives smaller centre-point errors than the 7 x 7, as can be seen in Figure 3(a), whereas for the maximum error the 7 X 7 differential quadrature is more accurate than the 5 x 5 according to Figure 3(b).\n\nc 0 . -. 11\n\n.-N .\n\nx--\n\n\\\\\n\n+ \\ \\\n\n\\ \\\n\n\\\n\n\\\n\n'\\\n\n\\\n\n0\n\n\\\n\n\\\n\n0.01\n\nt-\n\nX-.-x\n\nG a l e r k i n ( 7 X 7 ) Ref.'\n\n+ -+ C o n t r o l\n\nvolume( 7x7) R e f . 7\n\n5-Point f i n i t e diff.(7X7)\n\n0.001\n\ni\n\nRef.7\n\nO---ODiff.\n\npresent work\n\no--aOiff.\n\np r e s e n t work\n\nI\n\nI\n\nI\n\nI\n\n1\n\nI\n\nI\n\nI\n\nFigure 3(a). The centre-point error for the second problem\n\nSOLUTION OF POISSON EQUATION\n\n100.0\n\n0.1\n\nn t\n\nx-.--ry\n\nGalerkin(7X7) Ref. 7\n\n+-+Control\n\nvolume(7X7)\n\nRef. 7\n\n5-Point f i n i t e diff.(7X:)\n\no---oOiff. o--nOiff. 0.01\n\n72 1\n\nI\n\nI\n\nRef.7\n\np r e s e n t work\n\np r e s e n t work\n\nI\n\nI\n\nI\n\nI\n\nI\n\nI\n\nThird problem This problem has been formed by considering a square region between two concentric circles of radii, rl and r2, and treating the one-dimensional problem in polar co-ordinates like a two-dimensional problem in Cartesian co-ordinates for a source strength, S,equal to zero. Moving the Cartesian co-ordinates to X1= Y1= r l / J 2 point and using equations (14) and (15) with L = H = ( r 2 - r l ) / J Z and equation (16b), the following normalized form of the Poisson equation was obtained:\n\na2d a24\n\n2+- 0. ax ay2- ’\n\n0S x , y S 1\n\n(53)\n\nfor which the Dirichlet boundary conditions along the boundaries and the exact solution are given by\n\n4 = In (r/rd/ln 2 1/2\n\nwhere r = (X2+Y )\n\n.\n\n(r2/r1)\n\n(54)\n\n722\n\nF. CIVAN AND C. M. SLIEPCEVICH\n\nAs before, the partial derivatives are replaced by the approximation formulae, equations (1lb) and (12b). Then, the terms involving the boundary values are separated and are collected on the right-hand side to obtain the following set of linear algebraic equations:\n\n10.0\n\n1 .o\n\n/ / /\n\n-\n\n0.1\n\n/ / /\n\n0\n\nl o 4 E,\n\n-\n\n0.01\n\nx-.-x\n\nGalerkin(llX11)\n\n+- + C o n t r o l 5-Point\n\n0\n\no---oOiff. 0.001 2\n\nvolume(llX11) Ref.7 finite diff.(llXll)Ref.7\n\nquad.(5x5) p r e s e n t work\n\no----oDiff. I\n\nRef. i\n\nq u a d . ( 7 X 7 ) p r e s e n t work 1\n\n4\n\nI\n\nI\n\n6\n\n1\n\nI\n\na\n\n1\n\nI 10\n\nr2'rl\n\nFigure 4(a). The centre-point error for the third problem\n\n723\n\nSOLUTION OF POISSON EQUATION 100.0\n\nI p’\n\n10.0\n\n- +/\n\n1.0\n\n/ /\n\n/\n\n/\n\nl o 4 Emax\n\n/ / / - 0\n\n0.1\n\nx-.-x\n\nG a l e r k i n ( 11x11) Ref. I\n\n+-+Control 0\n\no---ODiff. . f f iDl [ - - - -n\n\n0.01\n\n2\n\nvolume(llXl1)\n\n5-Point\n\nRef.7\n\nfinite diff.(IlXll)Ref.?\n\np r e s e n t work\n\np r e s e n t work\n\n1 a 4\n\n6\n\n10\n\nrz’rl\n\nFigure 4(b). The maximum error for the third problem\n\nUsing the same subroutines to solve equation (55) the resulting values of .I and .nax are compared with published values’ in Figure 4, It is clearly evident that the differential quadrature solution with a 7 X 7 grid is more accurate than the conventional finite element and finite difference methods with a 11x 11grid. DISCUSSION AND CONCLUSIONS Based on the foregoing comparison of results for these three test problems, it is evident that for comparable levels of computational effort, the method of differential quadrature generally produces smaller errors, although it is conceded that the conventional finite element and finite difference techniques probably give sufficiently accurate results for many applications of the Poisson equation. Thus, the principal advantage of differential quadrature is that it is basically easier to apply than conventional numerical techniques.\n\n724\n\nF. CIVAN A N D C. M. SLIEPCEVICH\n\nAlthough not evident from the test problems used herein, it has been demonstrated2 that the method of differential quadrature can be advantageously employed in solving accurately a variety of practical, multi-dimensional problems with a considerable savings in computational effort. The reason is that this technique is somewhat unique in that it appears to give optimum performance with grids as small as 5 x 5 or 7 x 7, and rarely beyond 15 x 15. In these cases, if solutions at intermediate points are required, they can be easily obtained by an appropriate interpolation, such as the Lagrange method. From the standpoint of the two approaches presented herein for applying the method of differential quadrature, the second approach, equation (17b), requires 25-75 per cent less computational effort than the first approach, equation (17a). ACKNOWLEDGEMENTS\n\nThe authors acknowledge the financial support of University Technologists, Inc. of Norman, Oklahoma, U.S.A. and the Merrick Computing Center of the University of Oklahoma.\n\nREFERENCES\n\n1. R.Bellman, B. G. Kashef and J. Casti, ‘Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations’, J. Comp. Phys. 10,40-52 (1972). 2. F. Civan, ‘Solution of transport phenomena type models by the method of differential quadratures’, Ph.D. dissert., Univ. of Oklahoma (1978). 3. F. Civan and C. M. Sliepcevich, ‘Application of differential quadrature to transport processes’, J. Math. Anal. Appl., to be published. 4. G . E. Forsythe and C. B. Moler, Computer Solution of Linear Algebraic Systems, sect. 17, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. 5. R. W. Hamming, Numerical Methods for Scientists and Engineers, 2nd edn, McGraw-Hill, New York, 1973,p. 124. 6. J. 0. Mingle, ‘Computational considerations in nonlinear diffusion’, Znt. J. num.Meth. Engng, 7 , 103-116 (1973). 7. S. Ramadhyani and S. V. Patankar, ‘Solution of the Poisson equation: comparison of the Galerkin and controlvolume methods’, Znt. J. num. Meth. Engng, 15,1395-1402 (1980)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8816661,"math_prob":0.99498016,"size":21786,"snap":"2021-04-2021-17","text_gpt3_token_len":5870,"char_repetition_ratio":0.14971077,"word_repetition_ratio":0.0816645,"special_character_ratio":0.270403,"punctuation_ratio":0.14079748,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9992724,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-20T10:24:48Z\",\"WARC-Record-ID\":\"<urn:uuid:a5213607-bef4-472f-9768-481820bf727a>\",\"Content-Length\":\"83802\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7d3cc2de-7e99-45b4-8b2b-378cbd387e06>\",\"WARC-Concurrent-To\":\"<urn:uuid:1f92da27-2ba3-4d36-9bb5-e64c97af8b87>\",\"WARC-IP-Address\":\"104.21.68.83\",\"WARC-Target-URI\":\"https://moam.info/solution-of-the-poisson-equation-by-differential-wiley-online-library_5caa42d5097c473c488b45cf.html\",\"WARC-Payload-Digest\":\"sha1:RSJM4BZCHHACOTPTF4RIP7DDT44UKGDU\",\"WARC-Block-Digest\":\"sha1:DYUOBJWVODM3LKW4WD64AOCBSLLSQC7F\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703519984.9_warc_CC-MAIN-20210120085204-20210120115204-00141.warc.gz\"}"}
https://electricmusicstore.com/products/dual-control-voltage-processor-model-257-rev2-0	[ "# DUAL CONTROL VOLTAGE PROCESSOR MODEL 257 REV2.0\n\n## \\$99.00\n\nA set of 1 PCB.\n\nConsists of two identical sections, each of which permits several applied control voltages to define a single output voltage according to the equation:\n\nV_a K + V_b (1 - M) + M * V_c + V_offset = V_out\n\nThe algebraic manipulations possible with this module include addition, subtraction, scaling, inversion, and multiplication. Also incorporated is the capability of using one control voltage (M) to transfer control from one applied voltage (V_b) to another (V_c)." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8348807,"math_prob":0.9305117,"size":534,"snap":"2021-31-2021-39","text_gpt3_token_len":129,"char_repetition_ratio":0.11320755,"word_repetition_ratio":0.0,"special_character_ratio":0.2434457,"punctuation_ratio":0.11702128,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95477444,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-22T12:23:45Z\",\"WARC-Record-ID\":\"<urn:uuid:7488a3c4-6ccc-4602-8c2a-97c09cf1bbad>\",\"Content-Length\":\"43862\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:64a82e16-bce5-4806-afa4-8f4ad5c08171>\",\"WARC-Concurrent-To\":\"<urn:uuid:e59dbc95-02d6-4281-9bb0-84cb31df1923>\",\"WARC-IP-Address\":\"23.227.38.32\",\"WARC-Target-URI\":\"https://electricmusicstore.com/products/dual-control-voltage-processor-model-257-rev2-0\",\"WARC-Payload-Digest\":\"sha1:BF66DDKJFJZKJXHVDADWUCF3QFX4PKWC\",\"WARC-Block-Digest\":\"sha1:C5CLW4K6UZUOEVQJYCF3S2763YFMLIJD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057347.80_warc_CC-MAIN-20210922102402-20210922132402-00578.warc.gz\"}"}
https://www.wolframalpha.com/input/?i=2(Gravitational+constant)%2F(45(speed+of+light)%5E5)(electron+mass)%5E2((hydrogen+atom+radius)%5E4)(compton+frequency)%5E6(6.022%C3%9710%5E23)	[ "", null, "2(Gravitational constant)/(45(speed of light)^5)(electron mass)^2((hydrogen atom radius)^4)(compton frequency)^6(6.022×10^23)" ]	[ null, "https://www.wolframalpha.com/_next/static/images/Logo_1tof6SJC.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6614153,"math_prob":0.9975755,"size":378,"snap":"2021-04-2021-17","text_gpt3_token_len":98,"char_repetition_ratio":0.09625668,"word_repetition_ratio":0.0,"special_character_ratio":0.26719576,"punctuation_ratio":0.11764706,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9952455,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-21T15:31:03Z\",\"WARC-Record-ID\":\"<urn:uuid:c1444cbe-5c58-4f99-ab77-911db0cc6492>\",\"Content-Length\":\"15711\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:09fbebf5-84f6-47fb-b740-fede7a6a1437>\",\"WARC-Concurrent-To\":\"<urn:uuid:581c8ed7-e09c-4999-8690-42a6ad0bf679>\",\"WARC-IP-Address\":\"140.177.16.37\",\"WARC-Target-URI\":\"https://www.wolframalpha.com/input/?i=2(Gravitational+constant)%2F(45(speed+of+light)%5E5)(electron+mass)%5E2((hydrogen+atom+radius)%5E4)(compton+frequency)%5E6(6.022%C3%9710%5E23)\",\"WARC-Payload-Digest\":\"sha1:FND75CDLK27NUVWVXUN2X6QSLYME75YV\",\"WARC-Block-Digest\":\"sha1:ZHZXFWPO2DR3Z7HLMYUODAXQVSZZO6WU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703524858.74_warc_CC-MAIN-20210121132407-20210121162407-00000.warc.gz\"}"}
https://discourse.julialang.org/t/replicating-rubys-dig-in-julia/20551	[ "# Replicating Ruby's dig() in Julia\n\nHello,\n\nin Ruby there’s a method useful for navigating nested dictionaries called dig()\n\n``````nested = Dict(\n:a => Dict(\n:b => \"b\",\n:c => Dict(\n:d => \"d\"\n)\n)\n)\n\ndig(nested, :a, :b) #=> \"b\"\ndig(nested, :a, :c, :d) #=> \"d\"\ndig(nested, :a, :e) #=> nothing\n``````\n\nI have written this function. The problem is that `args...` will produce a Tuple. When called recursively, i will have a Tuple of Tuples and I don’t know how to flatten it or to avoid this. I can fix it by making `args::Tuple` and calling like this: `dig(nested, (:a, :b))`, but I find this somewhat uglier.\n\n``````function dig(dict::Dict, args...)\naux = dict[args]\nif length(args[2:end]) > 0\ndig(access, args[2:end])\nelse\naux\nend\nend\nend\n\n``````\n\nHow can I “splat” args when maing the recursive call or flatten the Tuple?\n\n``````function dig(dict::Dict, args...)\nd = dict\nfor key in args\nd = d[key]\nelse\nreturn nothing\nend\nend\nd\nend\n``````\n1 Like\n``````dig(dict::AbstractDict, key, keys...) = dig(dict[key], keys...)\ndig(x) = x\n``````\n\nThis throws KeyError rather than returning `nothing` on missing key, but should be easy to modify if you really want to return `nothing` as in your example.\nEDIT: alternately\n\n``````dig2(dict::AbstractDict, keys...) = foldl(getindex, keys; init=dict)\n``````\n6 Likes\n``````dig(x) = x\ndig(d::AbstractDict, key, keys...) = dig(get(d, key, nothing), keys...)\n``````\n\n`get(dict, key, default)` is really nice for dictionaries, and also faster than first checking (`haskey`) and then retrieving.\n\nEdit: The `foldl` solution is much faster, but fails when the key isn’t found:\n\n``````julia> dig2(d, :a, :e)\n\n``````\n3 Likes\n\n@DNF I didn’t find `foldl` to be much faster, at least after doing the necessary modifications so it doesn’t fail when the key isn’t found. And @yha first approach I’d say is conceptually simpler. Your solution is closer, although it would fail when passed `:a, :b, :c` if `:b`doesn’t exist already.\n\nCan be fixed like this:\n\n``````dig(x) = x\ndig(x::Nothing, keys...) = nothing\ndig(d::AbstractDict, key, keys...) = dig(get(d, key, nothing), keys...)\n``````\n\nBut maybe I shouldn’t be trying to navigate through empty keys on the first place…\n\n1 Like\n\nThis is just what I was looking for. One nice thing about Ruby’s `dig` is it can be used to access nested arrays as well as dicts. This is handy for working with responses from Elasticsearch. So I would expand @Amval’s answer to:\n\n``````dig(x) = x\ndig(x::Nothing, keys...) = nothing\ndig(d::AbstractDict, key, keys...) = dig(get(d, key, nothing), keys...)\ndig(a::AbstractArray, i::Integer, keys...) = checkbounds(Bool, a, i) ? dig(a[i], keys...) : nothing\n\n``````\n\nThen it can do:\n\n``````julia> nested = Dict(\n:a => [\nDict(:b => \"b\"),\nDict(:c => \"c\"),\nDict(:d => \"d\")\n]\n)\n\njulia> dig(nested, :a, 2, :c)\n\"c\"\n``````\n1 Like\n\nYou may be interested in Setfield.jl which implements a “lens” API that looks similar to what you’re doing.\n\n2 Likes" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9086488,"math_prob":0.9304401,"size":1050,"snap":"2022-40-2023-06","text_gpt3_token_len":272,"char_repetition_ratio":0.08221798,"word_repetition_ratio":0.0,"special_character_ratio":0.25428572,"punctuation_ratio":0.17073171,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97657484,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-03T15:16:11Z\",\"WARC-Record-ID\":\"<urn:uuid:f4359231-ce59-4220-94e9-f648fce5c9f7>\",\"Content-Length\":\"36491\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3058d95d-2c1f-4d8a-9bf6-dad5c71376ec>\",\"WARC-Concurrent-To\":\"<urn:uuid:ca6f0752-2fed-47a8-a777-4c346a79f18f>\",\"WARC-IP-Address\":\"64.71.144.205\",\"WARC-Target-URI\":\"https://discourse.julialang.org/t/replicating-rubys-dig-in-julia/20551\",\"WARC-Payload-Digest\":\"sha1:WPSOPNCA3WA5NIO53XNGRRBTRQNJFELG\",\"WARC-Block-Digest\":\"sha1:ITMHZK4EJ2KQ5IZYASQ2KMVNXHXK57HA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337421.33_warc_CC-MAIN-20221003133425-20221003163425-00797.warc.gz\"}"}
https://uk.mathworks.com/help/stats/nonparametric-methods.html	[ "## Nonparametric Methods\n\n### Introduction to Nonparametric Methods\n\nStatistics and Machine Learning Toolbox™ functions include nonparametric versions of one-way and two-way analysis of variance. Unlike classical tests, nonparametric tests make only mild assumptions about the data, and are appropriate when the distribution of the data is non-normal. On the other hand, they are less powerful than classical methods for normally distributed data.\n\nBoth of the nonparametric functions described here will return a `stats` structure that can be used as an input to the `multcompare` function for multiple comparisons.\n\n### Kruskal-Wallis Test\n\nThe example Perform One-Way ANOVA uses one-way analysis of variance to determine if the bacteria counts of milk varied from shipment to shipment. The one-way analysis rests on the assumption that the measurements are independent, and that each has a normal distribution with a common variance and with a mean that was constant in each column. You can conclude that the column means were not all the same. The following example repeats that analysis using a nonparametric procedure.\n\nThe Kruskal-Wallis test is a nonparametric version of one-way analysis of variance. The assumption behind this test is that the measurements come from a continuous distribution, but not necessarily a normal distribution. The test is based on an analysis of variance using the ranks of the data values, not the data values themselves. Output includes a table similar to an ANOVA table, and a box plot.\n\nYou can run this test as follows:\n\n```load hogg p = kruskalwallis(hogg) p = 0.0020 ```\n\nThe low p value means the Kruskal-Wallis test results agree with the one-way analysis of variance results.\n\n### Friedman's Test\n\nPerform Two-Way ANOVA uses two-way analysis of variance to study the effect of car model and factory on car mileage. The example tests whether either of these factors has a significant effect on mileage, and whether there is an interaction between these factors. The conclusion of the example is there is no interaction, but that each individual factor has a significant effect. The next example examines whether a nonparametric analysis leads to the same conclusion.\n\nFriedman's test is a nonparametric test for data having a two-way layout (data grouped by two categorical factors). Unlike two-way analysis of variance, Friedman's test does not treat the two factors symmetrically and it does not test for an interaction between them. Instead, it is a test for whether the columns are different after adjusting for possible row differences. The test is based on an analysis of variance using the ranks of the data across categories of the row factor. Output includes a table similar to an ANOVA table.\n\nYou can run Friedman's test as follows.\n\n```load mileage p = friedman(mileage,3) p = 7.4659e-004```\n\nRecall the classical analysis of variance gave a p value to test column effects, row effects, and interaction effects. This p value is for column effects. Using either this p value or the p value from ANOVA (p < 0.0001), you conclude that there are significant column effects.\n\nIn order to test for row effects, you need to rearrange the data to swap the roles of the rows in columns. For a data matrix `x` with no replications, you could simply transpose the data and type\n\n`p = friedman(x')`\n\nWith replicated data it is slightly more complicated. A simple way is to transform the matrix into a three-dimensional array with the first dimension representing the replicates, swapping the other two dimensions, and restoring the two-dimensional shape.\n\n```x = reshape(mileage, [3 2 3]); x = permute(x,[1 3 2]); x = reshape(x,[9 2]) x = 33.3000 32.6000 33.4000 32.5000 32.9000 33.0000 34.5000 33.4000 34.8000 33.7000 33.8000 33.9000 37.4000 36.6000 36.8000 37.0000 37.6000 36.7000 friedman(x,3) ans = 0.0082```\n\nAgain, the conclusion is similar to that of the classical analysis of variance. Both this p value and the one from ANOVA (p = 0.0039) lead you to conclude that there are significant row effects.\n\nYou cannot use Friedman's test to test for interactions between the row and column factors." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8971346,"math_prob":0.96060777,"size":3988,"snap":"2020-45-2020-50","text_gpt3_token_len":919,"char_repetition_ratio":0.13202812,"word_repetition_ratio":0.049535602,"special_character_ratio":0.23520562,"punctuation_ratio":0.106355384,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9926399,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-28T03:43:03Z\",\"WARC-Record-ID\":\"<urn:uuid:d3149df1-0ad5-49b4-ba7e-03a3bf55aa2d>\",\"Content-Length\":\"67838\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a6136d6f-737f-46b9-9d6e-0f52cdb336d0>\",\"WARC-Concurrent-To\":\"<urn:uuid:8b382492-3f58-441d-96fb-b35bd2cdceac>\",\"WARC-IP-Address\":\"23.212.144.59\",\"WARC-Target-URI\":\"https://uk.mathworks.com/help/stats/nonparametric-methods.html\",\"WARC-Payload-Digest\":\"sha1:DAPCK3SB42JPHX5TCLFVFC3OZ3PMC7CY\",\"WARC-Block-Digest\":\"sha1:64Z5JRLTDED5ZQ7PEBNJYZYGWYAHUSAZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107896048.53_warc_CC-MAIN-20201028014458-20201028044458-00555.warc.gz\"}"}