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https://www.presentica.com/search/proof
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math
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Search Results for 'Proof'
The Basics of Proof Writing: Defining, Identifying, and Solving with Theorems" In this lesson, you will learn what a proof is and how to construct a proof for a given hypothesis and conclusion. You will also become familiar with
Fermat's Last Theorem: Infinite Descent
Paradox: The Illusion of Falsidical Proof
The Importance of Establishing a Prima Facie Case under USC in USPTO Proceedings
Potential Applications of Nanogenerator Technology
Garfield's Proof of the Pie Thagorean Theorem
Chapter 2: Self Stabilization Techniques and Paradigms
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https://www.bsbgofast.com/product-page/bsb-heavy-bearing-birdcage
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math
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BSB #8023 Heavy Bearing Birdcage
Left rear heavy birdcage adds drive to car in slick.
This is a good setup and will make car tighter in both conditions.
Weight is 28lbs total and comes complete.
Comes complete and bolted together.
#8021-10 4 link plate left
#8023-2 Shock plate
#8023-11 1/4" top plate left
#8023-12 1/4" bottom plate left
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https://www.reference.com/world-view/equation-x-squared-y-squared-r-squared-7dcbd4b129c66f43
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What Is the Equation “x Squared + Y Squared = R Squared?”
“X squared + y squared = r squared” is the formula also known as the definition of a circle, where r represents the radius. If the formula was “x squared + y squared = 4,” then the circle would have a radius of 2 because 2 squared equals 4.
A circle is a set of all points on a plane that are all a fixed distance from the center. If the circle was plotted on graph paper, the center would be located at (a,b) and the location of one point of the circumference is (x,y). It is possible to make a right-angled triangle with these two co-ordinates and a third unspecified one. Using Pythagoras’ theorem, “a squared + b squared = c squared,” the standard form for the equation is “(x-a) squared + (y-b) squared = r squared.”
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https://cs.nyu.edu/pipermail/fom/2018-September/021219.html
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math
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[FOM] "Mere" correctness of a proof
Timothy Y. Chow
tchow at math.princeton.edu
Thu Sep 6 14:19:23 EDT 2018
Harvey Friedman wrote:
> A) the first digit of pi is 3.
> B) the sixth digit of pi is 9.
> (recall pi ~ 3.14159)
> This is an interesting arena to test the idea of "explanation of proof".
Another possible arena is computational complexity. As longtime FOM
participants know, I am fond of citing the theorem "Checkers is a draw."
It is hard to imagine an "explanatory proof" of this theorem that does not
involve brute-force calculation. Assuming that there is some upper bound
on how long a proof can be before it ceases to be "explanatory," any
EXPTIME-complete problem furnishes a family of short theorems which cannot
all have short proofs.
"Checkers is a draw" could of course be criticized as an unnatural or
uninteresting mathematical theorem. But I see no reason why there could
not exist a short and beautiful theorem for which there does not exist an
explanatory proof. (Of course it might be practically impossible to show
that no explanatory proof exists, so we might be tempted to keep searching
for one.) In that event, would Weil have said that God exists because the
theorem is true, but the Devil exists because we cannot explain why it is
More information about the FOM
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https://itprospt.com/num/21072029/in-the-markovnikov-addition-of-hbr-to-trans-2-butene-the
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math
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Hello, everyone. My name is Mohammed Ali. Now we will answer the question. Considers a reaction. It's uto glass pr to for me to Italy are the graph shows a concentration off br toe as function off time. Use a graft to calculate each quantities averaging. So from the reading off the first bar, this question is related to the rape, so must knows his relation before answers. This question. The 1st 1 is the real, uh, he's a reed is equal minus sheen. Their concentration off B R. Two with Stein we cause a B R to his reactor. Also negative chain in EJ toe with Stein and both Steve off boasted because we will speak now about H B R. And it should be our is broader and for two more. So now this is a relation. You must know it's mine. When we want to know the rate off reaction, we can write it by other message that the reed is equal. The read off, decreasing pro mean and read off, decreasing on regime and real or formation h b r. But we must dry half, so you must know these relations before we going toe. Answer the question from this questioning Want to you to use the graph so we must know the scale of grab zero and the's to Auntie 30 40 And here 50 saw Each line is represented. Why in the year 0.2 you're going for and from in your twosies is represent or 0.0 point 1.2 points three upon for upon five upon sex 4.7 point eight No, we would answer the first question. The first question he wanted. You can clean the average rate of reaction between zero and 25. So here is given to me the concentration off for me. So we will use these relations. Sad saree is equal negative there the concentration off B R two at 25 2nd negative concentration off B R to N zero divided by change off time So 25 negative zero This question It will be negative. Concentration off poor mean 25 stand This is 20. So here is 25 Go to the girl. It will be here at the middle, So it will be all 0.75 Answer concentration off bro mean add zero It will be go up, It will be here. So it will be mom divided by 25 negative zero. So by calculation and the result will be, Oh, Boeing or Mom, This is the answer off. The first bar question on now we are going toe. Answer the second question off the bar A which you want to know the instant you a straight off reaction at 25 seconds to know the Amis tenuous. It's continuous ring. We must reaches a 25 by the curb which cut it as this point. And from this point we must drew attention. So here I reaches a 0.25 to the care and in hydro danger. Then, after droning their danger, we choose any two point off concentration and the Bosnian bhai's attention and the show. What is the related time? So here he want to know the rate of reaction and we said before the read off reaction is represent the change in concentration of premier divided by time. So from the line, we will change it. Choose this point which he is all point A that got ends a time 20 and we will choose this point which is here and it represent opening 55 and connected it here. It will got a 50. So I choose now that toe point that constant attention which is and 50 and 20 type at 50. It card at concentration. 0.55 and And Wendy, it got at oh, boy in E. So by calculating a we found the reed is equal. Oh, point or or a three. So this is answer off. Read every action at 25 seconds. Let's go to the survey question off the bark a. The certain question you want to know there any stenting you strain but now for the formation. But now is four formation off h b R. So it should be are we must related it by premier because this question is answer for permit. So if we go to for the first part, the rate off women is haves a rate off h e r. So we can right now. If we know the re off b r two, we will times it by toe to know the raid off h p. R. So we will solve it it as we measures Ari off br toe at 52nd Then we can know the read off, Actually, Are there it off? B R two is equal Negative change off the concentration on B R two divided by time and and we want at 50 seconds for gay to solve is is you will reach point So the curve a 50 then you will grow danger There we will it choose any two points from concentration and connected them to attention and see the corresponding time which we want to know the slow as we calculate the slope off the stage. So here which shoes there. Ah, time 30 In time 40 We found that at times 30 it cut at boy seven and at a time 40 It got at time at concentration opening 65 So now which was a poin which are 40 negative 30 at fort is a concentration is open 65 from grab and a 30. It's opening seven from grab win Cut the danger So now we can calculate the rail which is all point or fine. But don't forget that this is a read off for me. So to know they read off which we are We must time it by two So raid off H B R. Well, the O'Brien or Mom, This is the answer off the search bar Off the first question. Let's go to the second bar for the second. The bar. He wanted you to draw a roof sketch once a meaning off roofs sketch roof. Sketch me. We wantto only you draw a quick without calculation. Exact off curve represents the concentration off HB oris function off time. So now he wants you to represent how the concentration off the product H e B R will be increased. In fact is entropy are start from zero and it will be increased quickly. Zen the decreasing in B R. Two Because two moons are for me and we found it. It's halftime, so it will increase quickly than the decreasing in B R. To which I mean that they're one amount is decrees in 100 seconds, but here it will increase for the job or the Simone at 50 seconds. So this is a sketch for HB on. And this is the answer off this question. Thank you.
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http://openstudy.com/updates/5568d6e2e4b0c4e453f91007
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math
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i need to know this but i don't
1 solution: Different slopes No solution: Same slope, different y-intercept Infinite solution: Same slope and same y-intercept I'm writing this on the assumption that you are talking about lines, and not parabolas and such.
If you can solve for x, there is one solution. If both sides of the equation are equal, for example 9x+3=9x+3, there are infinitely many solutions. If the solution is untrue, for example 7=8, 9=3, etc, there are no solutions.
Sorry, it doesn't necessarily have to be x. It can be any variable.
how about one solution
If you can work out the equation and find the value of the variable normally, it has one solution.
if like 6=6 does that have 1 solution
That would have infinitely many solutions, because both sides of the equation are the same.
how about 2=3 or 1=7
2 does not equal 3, nor does 1 equal 7. Both of the equations are false, so they have no solution.
how about this prob.5(6x+2)=3(10x−2)−2x
First you would use the distributive property to solve the parenthesis on each side. Then you would continue working out the equation until you come to a solution, whether it be finding the value of x or finding that there are an infinite amount of solutions or no solutions.
how many solutions is there/
\[5(6x+2)=3(10x-2)-2x\]Distributive property \[5(6x)+5(2)=3(10x)+3(-2)-2x\]Multiply \[30x+10=30x-6-2x\]Combine like terms on right side to get the two equations\[30x+10=28x-6\]Get the x's on the left side\[2x+10=-6\] Subtract the 10 over\[2x=-16\] Divide to get\[x=-8\] If you want to find the point where they meet, plug the x into one of the original equations.\[y=30(-8)+10\]\[y=-240+10\]\[y=-230\] So the point they meet would be \[(-8,-230)\]
LegendarySadist was faster. But anyway, obviously it has one solution.
He left before I got mine in =(
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https://encyclopedia2.thefreedictionary.com/Liouville%27s+Theorem
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math
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Also found in: Wikipedia.
Liouville's theorem[′lyü‚vēlz ‚thir·əm]
(1) In mechanics, a theorem asserting that the volume in phase space of a system obeying the equations of mechanics in Hamiltonian form remains constant as the system moves. Liouville’s theorem was established in 1838 by the French scientist J. Liouville.
The state of a mechanical system defined by the generalized coordinates q1, q2, . . ., qN and the canonically conjugate generalized momenta p1, p2, . . ., pN (where N is the number of degrees of freedom of the system) can be considered as a point with rectangular Cartesian coordinates q1, q2, ..., qN; p1, p2, . . ., pN is a 2N-dimensional space called the phase space. The evolution of the system in time is represented as the motion of this phase point in the 27V-dimensional space. If phase points entirely fill some region of the phase space at the initial moment of time and pass over into another region of the space in the course of time, then the corresponding phase volume, according to Liouville’s theorem, will be the same. Thus the motion of the points that represent the state of the system in the phase space resembles that of an incompressible fluid.
Liouville’s theorem permits introduction of a distribution function of the particles of the system in phase space and is the basis of statistical physics.
REFERENCESSynge, J. L. Klassicheskaia dinamika. Moscow, 1963. (Translated from English.)
Gibbs, J. Osnovnye printsipy statisticheskoi mekhaniki. Moscow, 1946. (Translated from English.)
Leontovich, M. A. Statisticheskaia fizika. Moscow-Leningrad, 1944.
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https://www.physicsforums.com/threads/imaginary-part-of-dielectric-constant.410659/
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math
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in ac fields permittivity becomes complex quantity and has real and imaginary parts. in metals (may be few exceptions but i dont know) imaginary part is always positive and represents loss factor or energy absorbed. why the plot of imaginary part of dielectric constant as function of energy is exponentially decaying curve (it decreases with the increase in energy)?secondly img. part of dielectric constant is also related to conductivity so can we infer that energy lost to the metals appears as conductivity? but then it is directly proportional to conductivity but conductivity plots (as function of energy again) show structures. finally permittivity becomes complex because ac fields are complex or their is any other reason? i am studying electrodynamics of continuous media of landau and lifgarbagez.
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https://avesis.yildiz.edu.tr/yayin/2ab8f44b-4e2a-4101-bb72-7ecd5edc7863/rings-and-modules-which-are-stable-under-automorphisms-of-their-injective-hulls
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math
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It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) R is right self-injective if R-R is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphism-invariant are investigated. (C) 2013 Elsevier Inc. All rights reserved.
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https://www.devoir-de-philosophie.com/echange/statistics-i-introduction-statistics-branch-of-mathematics-that-deals-with-the-collection-organization-and-analysis-of-numerical-data-and-with-su-1
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Statistics I INTRODUCTION Statistics, branch of mathematics that deals with the collection, organization, and analysis of numerical data and with such problems as experiment design and decision making. II HISTORY Domesday Book Compiled in 1086 under the direction of William the Conquerer, the Domesday Book was a meticulous survey of feudal estates in England. Public Record Office, Surrey, England Simple forms of statistics have been used since the beginning of civilization, when pictorial representations or other symbols were used to record numbers of people, animals, and inanimate objects on skins, slabs, or sticks of wood and the walls of caves. Before 3000 BC the Babylonians used small clay tablets to record tabulations of agricultural yields and of commodities bartered or sold. The Egyptians analyzed the population and material wealth of their country before beginning to build the pyramids in the 31st century BC. The biblical books of Numbers and 1 Chronicles are primarily statistical works, the former containing two separate censuses of the Israelites and the latter describing the material wealth of various Jewish tribes. Similar numerical records existed in China before 2000 censuses to be used as bases for taxation as early as 594 BC. BC. The ancient Greeks held See Census. The Roman Empire was the first government to gather extensive data about the population, area, and wealth of the territories that it controlled. During the Middle Ages in Europe few comprehensive censuses were made. The Carolingian kings Pepin the Short and Charlemagne ordered surveys of ecclesiastical holdings: Pepin in 758 and Charlemagne in 762. Following the Norman Conquest of England in 1066, William I, king of England, ordered a census to be taken; the information gathered in this census, conducted in 1086, was recorded in the Domesday Book. Registration of deaths and births was begun in England in the early 16th century, and in 1662 the first noteworthy statistical study of population, Observations on the London Bills of Mortality, was written. A similar study of mortality made in Breslau, Germany, in 1691 was used by the English astronomer Edmond Halley as a basis for the earliest mortality table. In the 19th century, with the application of the scientific method to all phenomena in the natural and social sciences, investigators recognized the need to reduce information to numerical values to avoid the ambiguity of verbal description. At present, statistics is a reliable means of describing accurately the values of economic, political, social, psychological, biological, and physical data and serves as a tool to correlate and analyze such data. The work of the statistician is no longer confined to gathering and tabulating data, but is chiefly a process of interpreting the information. The development of the theory of probability increased the scope of statistical applications. Much data can be approximated accurately by certain probability distributions, and the results of probability distributions can be used in analyzing statistical data. Probability can be used to test the reliability of statistical inferences and to indicate the kind and amount of data required for a particular problem. III STATISTICAL METHODS How Polls Predict Professional pollsters typically conduct their surveys among sample populations of 1,000 people. Statistical measurements show that reductions in the margin of error flatten out considerably after the sample size reaches 1,000. © Microsoft Corporation. All Rights Reserved. The raw materials of statistics are sets of numbers obtained from enumerations or measurements. In collecting statistical data, adequate precautions must be taken to secure complete and accurate information. The first problem of the statistician is to determine what and how much data to collect. Actually, the problem of the census taker in obtaining an accurate and complete count of the population, like the problem of the physicist who wishes to count the number of molecule collisions per second in a given volume of gas under given conditions, is to decide the precise nature of the items to be counted. The statistician faces a complex problem when, for example, he or she wishes to take a sample poll or straw vote. It is no simple matter to gauge the size and constitution of the sample that will yield reasonably accurate predictions concerning the action of the total population. In protracted studies to establish a physical, biological, or social law, the statistician may start with one set of data and gradually modify it in light of experience. For example, in early studies of the growth of populations, future change in size of population was predicted by calculating the excess of births over deaths in any given period. Population statisticians soon recognized that rate of increase ultimately depends on the number of births, regardless of the number of deaths, so they began to calculate future population growth on the basis of the number of births each year per 1000 population. When predictions based on this method yielded inaccurate results, statisticians realized that other limiting factors exist in population growth. Because the number of births possible depends on the number of women rather than the total population, and because women bear children during only part of their total lifetime, the basic datum used to calculate future population size is now the number of live births per 1000 females of childbearing age. The predictive value of this basic datum can be further refined by combining it with other data on the percentage of women who remain childless because of choice or circumstance, sterility, contraception, death before the end of the childbearing period, and other limiting factors. The excess of births over deaths, therefore, is meaningful only as an indication of gross population growth over a definite period in the past; the number of births per 1000 population is meaningful only as an expression of the proportion of increase during a similar period; and the number of live births per 1000 women of childbearing age is meaningful for predicting future size of populations. IV TABULATION AND PRESENTATION OF DATA Frequency-Distribution Table A frequency-distribution table summarizes data. For example, there were 1200 grades received on 4 examinations by 10 sections of 30 students each. The first column lists the ten intervals into which the grades were grouped. The second column lists the midpoints of these intervals. The third column lists the number of grades in each interval, that is, their frequency. (There were 20 grades between 0 and 10.) The fourth column lists the proportion of grades in each interval, that is, their relative frequency. (.017 of the 1200 grades were between 0 and 10.) The fifth column lists the number of grades in an interval and all intervals below it, that is, their cumulative frequency. (35 grades were in or below the interval between 10 and 20.) The sixth column lists the proportion of grades in or below an interval, that is, their relative cumulative frequency. (0.029 of the 1200 grades were in or below the interval 10 to 20.) © Microsoft Corporation. All Rights Reserved. The collected data must be arranged, tabulated, and presented to permit ready and meaningful analysis and interpretation. To study and interpret the examinationgrade distribution in a class of 30 pupils, for instance, the grades are arranged in ascending order: 30, 35, 43, 52, 61, 65, 65, 65, 68, 70, 72, 72, 73, 75, 75, 76, 77, 78, 78, 80, 83, 85, 88, 88, 90, 91, 96, 97, 100, 100. This progression shows at a glance that the maximum is 100, the minimum 30, and the range, or difference, between the maximum and minimum is 70. In a cumulative-frequency graph, such as Fig. 1, the grades are marked on the horizontal axis and double marked on the vertical axis with the cumulative number of the grades on the left and the corresponding percentage of the total number on the right. Each dot represents the accumulated number of students who have attained a particular grade or less. For example, the dot A corresponds to the second 72; reading on the vertical axis, it is evident that there are 12, or 40 percent, of the grades equal to or less than 72. In analyzing the grades received by 10 sections of 30 pupils each on four examinations, a total of 1200 grades, the amount of data is too large to be exhibited conveniently as in Fig. 1. The statistician separates the data into suitably chosen groups, or intervals. For example, ten intervals might be used to tabulate the 1200 grades, as in column (a) of the accompanying frequency-distribution table; the actual number in an interval, called the frequency of the interval, is entered in column (c). The numbers that define the interval range are called the interval boundaries. It is convenient to choose the interval boundaries so that the interval ranges are equal to each other; the interval midpoints, half the sum of the interval boundaries, are simple numbers, because they are used in many calculations. A grade such as 87 will be tallied in the 80-90 interval; a boundary grade such as 90 may be tallied uniformly throughout the groups in either the lower or upper intervals. The relative frequency, column (d), is the ratio of the frequency of an interval to the total count; the relative frequency is multiplied by 100 to obtain the percent relative frequency. The cumulative frequency, column (e), represents the number of students receiving grades equal to or less than the range in each succeeding interval; thus, the number of students with grades of 30 or less is obtained by adding the frequencies in column (c) for the first three intervals, which total 53. The cumulative relative frequency, column (f), is the ratio of the cumulative frequency to the total number of grades. The data of a frequency-distribution table can be presented graphically in a frequency histogram, as in Fig. 2, or a cumulative-frequency polygon, as in Fig. 3. The histogram is a series of rectangles with bases equal to the interval ranges and areas proportional to the frequencies. The polygon in Fig. 3 is drawn by connecting with straight lines the interval midpoints of a cumulative frequency histogram. Newspapers and other printed media frequently present statistical data pictorially by using different lengths or sizes of various symbols to indicate different values. V MEASURES OF CENTRAL TENDENCY After data have been collected and tabulated, analysis begins with the calculation of a single number, which will summarize or represent all the data. Because data often exhibit a cluster or central point, this number is called a measure of central tendency. Let x1, x2, ..., xn be the n tabulated (but ungrouped) numbers of some statistic; the most frequently used measure is the simple arithmetic average, or mean, written ?, which is the sum of the numbers divided by n: If the x's are grouped into k intervals, with midpoints m1, m2, ..., mk and frequencies f1, f2, ..., fk, respectively, the simple arithmetic average is given by with i = 1, 2, ..., k. The median and the mode are two other measures of central tendency. Let the x's be arranged in numerical order; if n is odd, the median is the middle x; if n is even, the median is the average of the two middle x's. The mode is the x that occurs most frequently. If two or more distinct x's occur with equal frequencies, but none with greater frequency, the set of x's may be said not to have a mode or to be bimodal, with modes at the two most frequent x's, or trimodal, with modes at the three most frequent x's. VI MEASURES OF VARIABILITY The investigator frequently is concerned with the variability of the distribution, that is, whether the measurements are clustered tightly around the mean or spread over the range. One measure of this variability is the difference between two percentiles, usually the 25th and the 75th percentiles. The p th percentile is a number such that p percent of the measurements are less than or equal to it; in particular, the 25th and the 75th percentiles are called the lower and upper quartiles, respectively. The p th percentile is readily found from the cumulative-frequency graph, (Fig. 1) by running a horizontal line through the p percent mark on the vertical axis on the graph, then a vertical line from this point on the graph to the horizontal axis; the abscissa of the intersection is the value of the p th percentile. The standard deviation is a measure of variability that is more convenient than percentile differences for further investigation and analysis of statistical data. The standard deviation of a set of measurements x1, x2, ..., xn, with the mean ? is defined as the square root of the mean of the squares of the deviations; it is usually designated by the Greek letter sigma (?). In symbols The square, ?2, of the standard deviation is called the variance. If the standard deviation is small, the measurements are tightly clustered around the mean; if it is large, they are widely scattered. VII CORRELATION When two social, physical, or biological phenomena increase or decrease proportionately and simultaneously because of identical external factors, the phenomena are correlated positively; under the same conditions, if one increases in the same proportion that the other decreases, the two phenomena are negatively correlated. Investigators calculate the degree of correlation by applying a coefficient of correlation to data concerning the two phenomena. The most common correlation coefficient is expressed as in which x is the deviation of one variable from its mean, y is the deviation of the other variable from its mean, and N is the total number of cases in the series. A perfect positive correlation between the two variables results in a coefficient of +1, a perfect negative correlation in a coefficient of -1, and a total absence of correlation in a coefficient of 0. Intermediate values between +1 and 0 or -1 are interpreted by degree of correlation. Thus, .89 indicates high positive correlation, -.76 high negative correlation, and .13 low positive correlation. VIII MATHEMATICAL MODELS Distribution of IQ Scores The distribution of scores (commonly called IQ scores) on the Wechsler Adult Intelligence Scale follows an approximately normal curve, an average distribution of values. The test is regularly adjusted so that the median score is 100--that is, so that half of the scores fall above 100, and half fall below. © Microsoft Corporation. All Rights Reserved. A mathematical model is a mathematical idealization in the form of a system, proposition, formula, or equation of a physical, biological, or social phenomenon. Thus, a theoretical, perfectly balanced die that can be tossed in a purely random fashion is a mathematical model for an actual physical die. The probability that in n throws of a mathematical die a throw of 6 will occur k times is in which (À is the symbol for the binomial coefficient ) The statistician confronted with a real physical die will devise an experiment, such as tossing the die n times repeatedly, for a total of Nn tosses, and then determine from the observed throws the likelihood that the die is balanced and that it was thrown in a random way. In a related but more involved example of a mathematical model, many sets of measurements have been found to have the same type of frequency distribution. For example, let x1, x2, ..., xN be the number of 6's cast in the N respective runs of n tosses of a die and assume N to be moderately large. Let y1, y2, ..., yN be the weights, correct to the nearest 1/100 g, of N lima beans chosen haphazardly from a 100-kg bag of lima beans. Let z1, z2, ..., zN be the barometric pressures recorded to the nearest 1/1000 cm by N students in succession, reading the same barometer. It will be observed that the x's, y's, and z's have amazingly similar frequency patterns. The statistician adopts a model that is a mathematical prototype or idealization of all these patterns or distributions. One form of the mathematical model is an equation for the frequency distribution, in which N is assumed to be infinite: in which e (approximately 2.7) is the base for natural logarithms (see Logarithm). The graph of this equation (Fig. 4) is the bell-shaped curve called the normal, or Gaussian, probability curve. If a variate x is normally distributed, the probability that its value lies between a and b is given by The mean of the x's is 0, and the standard deviation is 1. In practice, if N is large, the error is exceedingly small. IX TESTS OF RELIABILITY The statistician is often called upon to decide whether an assumed hypothesis for some phenomenon is valid or not. The assumed hypothesis leads to a mathematical model; the model, in turn, yields certain predicted or expected values, for example, 10, 15, 25. The corresponding actually observed values are 12, 16, 21. To determine whether the hypothesis is to be kept or rejected, these deviations must be judged as normal fluctuations caused by sampling techniques or as significant discrepancies. Statisticians have devised several tests for the significance or reliability of data. One is the chi-square (c 2) test. The deviations (observed values minus expected values) are squared, divided by the expected values, and summed: The value of c2 is then compared with values in a statistical table to determine the significance of the deviations. X HIGHER STATISTICS The statistical methods described above are the simpler, more commonly used methods in the physical, biological, and social sciences. More advanced methods, often involving advanced mathematics, are used in further statistical studies, such as sampling theory, inference and estimation theory, and design of experiments. Contributed By: James Singer Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.
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http://idcards-uk.info/online-casino-best/statistics-expected-value-formula.php
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math
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Simple explanations for the most common types of expected value formula. Includes video. Hundreds of statistics articles and vidoes. Free help. In this video, I show the formula of expected value, and compute the expected value of a game. The final. The formula for the expected value is relatively easy to compute and involves several multiplications and additions. If the expected value exists, this procedure estimates the true expected value in an unbiased manner and has the property of minimizing the sum of the squares of the residuals the sum of the squared differences between the observations and the estimate. Der bedingte Erwartungswert spielt eine wichtige Rolle in der Theorie der stochastischen Prozesse. Ist die Summe nicht endlich, dann muss die Reihe absolut konvergieren , damit der Erwartungswert existiert. Click an empty cell. Given this information, the calculation is straightforward: The expected value of a random variable is just the mean of the random variable. How many tosses can we expect until the first heads not including the heads itself? Expected value formula for continuous random variables. Follow Us Facebook Twitter Pinterest. Since it is measuring the mean, it should come as no surprise that this formula is derived from that of the mean. More specifically, X will be the number of pips showing on the top face of the die after the toss. Herzblatt spiel section explains how to figure out the expected value for a single item like purchasing a single raffle ticket and what to do if you have multiple items. Die kumulantenerzeugende Funktion slot spiele gratis fur nokia lumia Zufallsvariable cube crash definiert als. Http://hospitalnews.com/novel-program-for-problem-gamblers-matches-the-intervention-to-the-gamblers-reasons-for-gambling/ the expected value of free slots lucky 7 random spiderman top trumps including the expected value for multiple events using this online expected value play easter eggs. The last equality used the formula for a geometric progression. Interaction Help About Shogun symbol Community portal Recent changes Contact page. For galgen spiel three coin toss, you could get anywhere from 0 to 3 heads. In der Physik findet die Bad wolf video Verwendung.
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https://www.ipl.org/essay/Sir-Isaac-Newton-And-Newtons-Laws-Of-F3L66U7ESCP6
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math
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Newton developed this law of motion has significant mathematical and physical elucidation that are needed to understand the motion of objects in our universe. Newton introduced the three laws in his book Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), which is generally referred to as the Principia. He also introduced his theory of universal gravitation, thus laying down the entire foundation of classical mechanics in one volume in 1687. These laws define the motion changes, specifically the way in which those changes in motion are related to force and mass. There are three laws of motion which were introduced by Sir Isaac Newton which are Newton’s First Law , Newton’s Second Law and Newton’s
This supports my hypothesis, where I predicted this proportionality. From Newton’s Second Law F=ma, I derived an equation isolating a: a= (1/m)F - (1/m)fric. I solved the equation for the line of best fit, and used the y-intercept to figure out the friction, which was the only remaining unknown in the equation. Once the friction was known, I was then able to plug in measured values and verify my hypothesis. I proved with this equation, using examples from my data, that the force is in fact directly proportional to
Electrohydrodynamic Electrohydrodynamic Phenomena The EHD phenomena involve the interaction of electric fields and flow fields in a dielectric fluid medium. This interaction can result in electrically induced fluid motion and interfacial instabilities which are caused by an electric body force. The electric body force density acting on the molecules of a dielectric fluid in the presence of an electric field consists of three terms (1): f_e=ρ_e E ̅-1/2 E^2 ∇ε+1/2 ∇[ρE^2 (δε/δρ)_T ] (1) The three terms in Eq (1) stand for two primary force densities acting on the fluid. The first term represents the force acting on the free charges in the presence of an electric field and is known as the Coulomb force. The second and third terms represent
Introduction In 1687, Newton put forward the Newton's Second Law of Motion-Force and Acceleration in the book “Philosophiae Naturalis Principia Mathematica”. According to Newton’s second law, , this is integrated over position from an initial position (i) to a final position (f). (Wang,1) . Therefore, we can get the work-energy theorem, . W is the work done by the net force on the object, which equals to the change in kinetic energy according to the equation.
MOJICA, Aselle Joyce G. Group no. 3 PHY13l/A3 Seat no. 3-3 ANALYSIS In PART 1 of the experiment which the is Magnetic Field of Permanent Magnets, we used two different magnets; two bar magnets and two U- magnets in order to see clearly what would be the result when magnets are placed in different orientations. For PART 1A, the bar magnets were oriented with like poles (N-N) facing each other. As a result after putting and scattering iron filings, each field line of the magnetic field from the north pole of the two magnets went away from each other which simply prove that like poles repel.
In Newtonian gravity (which was the classical theory of gravity), the source of gravity is the mass. In general theory of relativity, the mass turns out to be part of a more general quantity called the energy-momentum tensor (Tμυ), which includes both the energy and momentum densities. The field equation for gravity includes this tensor. The energy-momentum tensor is divergence free where its covariant derivative in the curved space-time is zero (∇^μ Tμυ= 0). By finding a tensor on other side which is divergence free, this yields the simplest set of equations which are called Einstein's (field) equations.
which is represented as, p/ρg+v^2/2g+z=constant, here z is height. Therefore, Bernoulli’s eq. of motion is defined as, In an ideal, steady flow of a fluid, the total energy at any random point of the fluid is constant. The total energy consists of kinetic energy, potential energy, and pressure
Aerodynamics is a branch of dynamics to the study of air movement together. It is a subfield of fluid dynamics and gas, and the term "drag" is often used to refer to the gas dynamics. The earliest records of the basic concepts of aerodynamics on the work of Aristotle and Archimedes in the third and second centuries BC, but the efforts to find a quantitative theory of airflow develop until the 18th century, beginning in 1726 was Isaac Newton as one of the first in modern aerodynamics mind when he developed a theory of air resistance, which was later verified for low flow rates. Air resistance experiments were carried out by researchers in the 18th and 19th centuries, with the aid of the construction of the first wind tunnel in 1871 In 1738
He is called the father of the clockwork universe because of the theories he invented, universal gravity, three laws of motion ("Physics is the study of your world and the world and universe around you." (Holzner 2006, p. 7-15) 3.1 USE OF DIFFERENTIAL EQUATIONS IN PHYSICS Differential equation is one of the examples of mathematical equations that is associated with functions and its derivatives. The functions determine the physical quantities (position, velocity, acceleration and forces acting on the object), the
For example, in classical mechanics, F=ma is a formula that tells us that the net force acting on a body is given by the mass of the body times its acceleration. That constitutes a very precise statement, and when we plug the numbers into the formula, we get a precise result, that is expressed in specific units, in our case
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https://hwhshop.com/qa/what-if-the-sample-size-is-less-than-30.html
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math
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- When should you increase sample size?
- What happens when a sample size is not big enough?
- What is considered a low sample size?
- What happens when the sample size decreases?
- Is 30 of the population a good sample size?
- What is the minimum sample size for t test?
- Does sample size affect validity?
- Does sample size affect bias?
- Does small sample size increase Type 2 error?
- When the sample size increases the population mean decreases?
- What is the minimum sample size for Anova?
- Why must sample size be greater than 30?
- Does increasing sample size increase confidence level?
- How does increasing sample size increase power?
- How small is too small for a sample size?
- Why is the minimum sample size 30?
- Which is a test of significance for sample size less than or equal to 30?
- Why is a small sample bad?
When should you increase sample size?
Higher sample size allows the researcher to increase the significance level of the findings, since the confidence of the result are likely to increase with a higher sample size.
This is to be expected because larger the sample size, the more accurately it is expected to mirror the behavior of the whole group..
What happens when a sample size is not big enough?
Sampling. The most obvious strategy is simply to sample more of your population. Keep your survey open, contact more potential participants, or consider widening the population.
What is considered a low sample size?
Generally, for any inferential statistic, a sample size of less than 500 may not be adequate.
What happens when the sample size decreases?
The population mean of the distribution of sample means is the same as the population mean of the distribution being sampled from. … Thus as the sample size increases, the standard deviation of the means decreases; and as the sample size decreases, the standard deviation of the sample means increases.
Is 30 of the population a good sample size?
Sampling ratio (sample size to population size): Generally speaking, the smaller the population, the larger the sampling ratio needed. For populations under 1,000, a minimum ratio of 30 percent (300 individuals) is advisable to ensure representativeness of the sample.
What is the minimum sample size for t test?
10 Answers. There is no minimum sample size for the t test to be valid other than it be large enough to calculate the test statistic.
Does sample size affect validity?
The use of sample size calculation directly influences research findings. Very small samples undermine the internal and external validity of a study. Very large samples tend to transform small differences into statistically significant differences – even when they are clinically insignificant.
Does sample size affect bias?
Increasing the sample size tends to reduce the sampling error; that is, it makes the sample statistic less variable. However, increasing sample size does not affect survey bias. A large sample size cannot correct for the methodological problems (undercoverage, nonresponse bias, etc.) that produce survey bias.
Does small sample size increase Type 2 error?
Type II errors are more likely to occur when sample sizes are too small, the true difference or effect is small and variability is large. The probability of a type II error occurring can be calculated or pre-defined and is denoted as β.
When the sample size increases the population mean decreases?
With “infinite” numbers of successive random samples, the mean of the sampling distribution is equal to the population mean (µ). As the sample sizes increase, the variability of each sampling distribution decreases so that they become increasingly more leptokurtic.
What is the minimum sample size for Anova?
3Is there a minimum sample size to run an ANOVA? In theory, it is 3. You need two populations, so that’s 2, but you need two samples to get a variance estimate. If you assume equal variances, you only need the estimate from one population so that’s 3 total.
Why must sample size be greater than 30?
As a general rule, sample sizes equal to or greater than 30 are deemed sufficient for the CLT to hold, meaning that the distribution of the sample means is fairly normally distributed. Therefore, the more samples one takes, the more the graphed results take the shape of a normal distribution.
Does increasing sample size increase confidence level?
A higher confidence level requires a larger sample size. Power – This is the probability that we find statistically significant evidence of a difference between the groups, given that there is a difference in the population. A greater power requires a larger sample size.
How does increasing sample size increase power?
The price of this increased power is that as α goes up, so does the probability of a Type I error should the null hypothesis in fact be true. The sample size n. As n increases, so does the power of the significance test. This is because a larger sample size narrows the distribution of the test statistic.
How small is too small for a sample size?
The numbers behind this phenomenon are kind of complicated, but often a small sample size in a study can cause results that are almost as bad, if not worse, than not running a study at all. Despite these statistical assertions, many studies think that 100 or even 30 people is an acceptable number.
Why is the minimum sample size 30?
One may ask why sample size is so important. The answer to this is that an appropriate sample size is required for validity. If the sample size it too small, it will not yield valid results. … If we are using three independent variables, then a clear rule would be to have a minimum sample size of 30.
Which is a test of significance for sample size less than or equal to 30?
If the sample sizes in at least one of the two samples is small (usually less than 30), then a t test is appropriate. Note that a t test can also be used with large samples as well, in some cases, statistical packages will only compute a t test and not a z test.
Why is a small sample bad?
Small samples are bad. Why? If we pick a small sample, we run a greater risk of the small sample being unusual just by chance. Choosing 5 people to represent the entire U.S., even if they are chosen completely at random, will often result if a sample that is very unrepresentative of the population.
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https://www.semanticscholar.org/author/B-F-Caviness/2021770
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math
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- Full text PDF available (7)
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Journals and Conferences
In this paper we give an extension of the Liouville theorem [RISC69, p. 169] and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone. Our main result generalizes Liouville's theorem by allowing, in addition to the elementary functions,… (More)
This paper deals wi th the simplification problem of symbolic mathemat ics . The notion of canonical form is defined and presented as a well-defined al ternat ive to the concept of simplified form. Following Richardson it is shown tha t canonical forms do not exist for sufficiently rich classes of mathemat ica l expressions. However, wi th the aid of a… (More)
In this paper new algorithms are given for Gaussian integer division and the calculation of the greatest common divisor of two Gaussian integers. Empirical tests show that the new gcd algorithm is up to 5.39 times as fast as a Euclidean algorithm using the new division algorithm.
It is a well-known empirical result that differentiation, especially higher order differentiation, of simple expressions can lead to long and complex expressions. In this paper we give some theoretical results that help to explain this phenomenon. In particular we show that in certain representations there exist expressions whose representations require… (More)
In this paper we discuss the problem of simplifying unnested radical expressions. We describe an algorithm implemented in MACSYMA that simplifies radical expressions and then follow this description with a formal treatment of the problem. Theoretical computing times for some of the algorithms are briefly discussed as is related work of other authors.
This paper gives a corollary to Schanuel's conjecture that indicates when an exponential or logarithmic constant is transcendental over a given field of constants. The given field is presumed to have been built up by starting with the rationals Q with π adjoined and taking algebraic closure, adjoining values of the exponential function or of some fixed… (More)
We show that in the ring generated by the integers and the functions x, sinxn and sin(x · sinxn) (n = 1, 2, . . .) defined on R it is undecidable whether or not a function has a positive value or has a root. We also prove that the existential theory of the exponential field C is undecidable. 1. Let S denote the class of expressions generated by the rational… (More)
In this paper the Brown-Collins modular greatest common divisor algorithm for polynomials in Z[x<subscrpt>1</subscrpt>,...,x<subscrpt>v</subscrpt>], where Z denotes the ring of rational integers, is generalized to apply to polynomials in G[x<subscrpt>1</subscrpt>,...,x<subscrpt>v</subscrpt>], where G denotes the ring of Gaussian integers, i.e., complex… (More)
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http://earsiv.arel.edu.tr/xmlui/handle/20.500.12294/193/discover?filtertype=rights&filter_relational_operator=equals&filter=info%3Aeu-repo%2Fsemantics%2FopenAccess
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math
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Now showing items 1-10 of 12
High-order harmonic generation from confined Rydberg atoms
(IOP Publishing, 2015)
We report results from our simulations of High Harmonic Generation (HHG) from a confined atom in a Rydberg state. We find that for the n = 2 excited state of H the cut-off of the harmonic spectrum is substantially extended ...
Inclusions and the approximate identities of the generalized grand Lebesgue spaces
Let (Omega, Sigma, mu) and (Omega, Sigma, upsilon) be two finite measure spaces and let L-p(),theta )(mu) and L-q),L-theta (upsilon) be two generalized grand Lebesgue spaces [9,10] , where 1 < p, q < infinity and theta >= ...
Raman frequencies calculated at various pressures in phase I of benzene
(KCS Publications, 2013)
We calculate in this study the pressure dependence of the frequencies for the Raman modes of A (Ag), B (Ag, B2g) and C (B1g, B3g) at constant temperatures of 274 and 294K (room temperature) for the solid phase I of benzene. ...
Recombination rate coefficients of boron-like Ne
(The American Astronomical Society, 2013)
Recombination of Ne5+ was measured in a merged-beam type experiment at the heavy-ion storage ring CRYRING. In the collision energy range 0–110 eV resonances due to 2s22p › 2s2p2 (?n = 0) and 2s22p › 2s23l (?n = 1), core ...
Calculation of the soft-mode frequency for the alpha – beta transition in quartz
The ? – ß structural transition occurs in quartz at TC = 846 K. The frequency of the soft mode associated with the volume increase, decreases with increasing temperature as the transition temperature is approached. In this ...
Inclusions and Noninclusions of Spaces of Multipliers of Some Wiener Amalgam Spaces
(Inst Math & Mechanics, 2019)
The main purpose of this paper is to study inclusions and noninclusions among the spaces of multipliers of the Wiener amalgam spaces. M. G. Cowling and J. J. F. Fournier in , L. Hormander in and G. I. Gaudry in ...
A note on the order bidual of f-algebras
The paper deals with the Arens Multiplication which we accomplished in four steps in the order bidual ~ ~ X . It is shown that if f is an element of order dual ~ X of X with ? ( f )?0 and , + x ? X then f .x = 0 implies f (x)= .
On the weighted variable exponent amalgam space W(L-P(X) , L-M(Q))
In , a new family W(L-p(x), L-m(q))of Wiener amalgam spaces was defined and investigated some properties of these spaces, where local component is a variable exponent Lebesgue space L-p(x) (R) and the global component ...
Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces
Let ?1, ?2 be slowly increasing weight functions, and let ?3 be any weight function on Rn. Assume that m(? ,?) is a bounded, measurable function on Rn × Rn. We define Bm(f, g)(x) = Rn Rnˆ f(? )gˆ(?)m(? ,?)e2?i?+?,x d? d? ...
High-order harmonic generation from Rydberg states at fixed Keldysh parameter
(Institute for Scientific Information, 2013)
Because the commonly adopted viewpoint that the Keldysh parameter ? determines the dynamical regime in strong field physics has long been demonstrated to be misleading, one can ask what happens as relevant physical parameters, ...
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https://community.victronenergy.com/questions/193626/what-is-the-difference-between-120v230v-multiplusq.html
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math
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I would like to install two multiplus or quattro units in parallel to get split phase 240V (120V line to neutral, 240V line to line. USA install). However, some units are labeled 120V, while others are labeled 230V. Does this impact which units can be used to produce the desired split phase 240V output?
Also, is it true that the 8kVA and up multis cannot be connected in parallel, while this limitation does not apply to any of the Quattros?
What are the largest two units that I could connect in parallel to achieve a 120V/240V?
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https://www.enotes.com/homework-help/what-important-about-y-ax2-bx-c-467637
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math
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What is so important about `y=ax^2+bx+c?`
The equation `y=ax^2+bx+c` is a means of describing the quadratic function.
If a quadratic function is equal to zero, the result will be a quadratic equation with roots, `x` . The x-values are the roots (or zeros) of `f(x) = 0.`
The graph of a quadratic function is called a parabola. In the equation, a change in `a` will change the shape of the parabola.
A change in `b` will change the placement of the vertex or turning point and
a change in `c` will move the parabola up or down.
Distance, speed and time, etc can be measured using quadratic equations. A practical real life example is throwing a ball. A parabola will show you how high the ball goes and when it will hit the ground. Quadratic functions can also be used to determine profit and the best price for maximum profit. a b and c therefore measure changes.
The black graph represents `y=x^2` where `b=0` and `c=0`
The red graph represents `y=x^2 +2` where `b=0`
the green graph represents `y=x^2 +2x +2`
Ans: The equation `y=ax^2+bx+c` is important as it is necessary for practical applications in many instances of mathematics, science and technology.
It enables people to use this formula in order to find missing variables
` ` `y=ax^2 + bx +c`is the original function for a parabola. You can change the shape and location of this by increasing the a, b, and c values.
This equation can also be factored to the form:`y = (x+-n)(x+-m)` .
The family function of a parabola is `y=x^2` where the vertex is on the origin.
You can also use the quadratic equation from this equation:
`x = (-b +- sqrt(b^2 -4ac))/(2a)`
You use this equation when you can't factor easily.
the equation describes a quadratic function and from it the quadratic formula can be derived.
`y = ax^2 + bx +c` is important because it is known as the quadratic formula. The quadratic formula is used when trying to find the variable x. The a,b, and c variable can be used to find x through the quadratic formula `x = (-b+-(b^2 -4ac))/(2a)` . Where you would just plug in a,b, and c in order to find x
It is known as the quadratic formula, this formula is used to find missing variable and can help with a lot of struggling when you cant find what a variable might be.
The equation you just described is called the quadratic equation. Its so very important because it enables people to solve functions to the second degree without factoring (many equations can't be factored so this method is especially useful for them) and without tediously completing the square (which has its own benefits especially in finding integrals in calculus). Now, if the question is: why is it important we're able to solve such equations; just look at at any STEM career/application. (Science, Technology, Engineering, Mathematics).
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http://www.biomedcentral.com/1472-6785/8/6/figure/F3
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math
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Correlation between mean density estimate against known density for all data sets. Line shows complete agreement between known and estimated density. Spearman's correlation coefficient shown in parentheses. Symbols denote spatial pattern of data set: Uniform – filled circle, Poisson – filled triangle, Clumped – open circle.
White et al. BMC Ecology 2008 8:6 doi:10.1186/1472-6785-8-6
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https://demonstrations.wolfram.com/topic.html?topic=Graph+Theory&limit=20
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math
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Demonstrations 1 - 20 of 346
Multiple Nets for a Cube
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Multiple Nets of an Octahedron
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Gauss-Bonnet and Poincaré-Hopf for Graphs
Deforming Nets of Polyhedra
Constructing a Steiner Tree for Five Points
Automata Generative Networks 2
Mazes in a Rectangular Solid
Graph of a Six-Cube and Skeleton of a Rhombic Triacontahedron
Stationary States of Maximal Entropy Random Walk and Generic Random Walk on Cayley Trees
Nonlinear Wave Resonances
A Rhombic Dodecahedron in the Skeleton of a Tesseract
Kowalewski's Settlement Problem for Triacontahedron
U.S. Presidential Interconnections and Exclusions
U.S. Presidential Interconnections
Enumerating Cycles of a Directed Graph
Tours through a Graph
Perfect 1-Factorizations of Graphs
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https://cafedoing.com/motion-graphs-worksheet/
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math
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Motion Graphs Worksheet
Motion graphs m. httpscienceclass.net the graphs below represent the motion of a car. match the descriptions with the graphs. explain your answers. descriptions. the car is stopped. the car is traveling at a constant speed. the speed of the car is decreasing.
the car is coming back.I use this worksheet to help teach motion graphs. this graph worksheet includes six graphs to show constant speed, acceleration, deceleration, no motion, and combinations of these. combine this worksheet with my mo worksheet on graphs to compare graphs Motion graphs kinematics worksheet.
List of Motion Graphs Worksheet
The graph below describes the motion of a fly that starts out going right. time s. a.Motion graphs displaying top worksheets found for motion graphs. some of the worksheets for this concept are motion graphs, work motion graphs name, motion graphs work with answers, physics b review, date pd constant velocity particle model work, skill and practice work, physics notes, name kinematics motion graphs.
Gradient of a graph the gradient or slope of a graph gives the the gradient m, of a linear straight line graph is the rise divided by the run. that is, m rise run. of the object. consider the graph above. from a to b the gradient of the graph is where rise run change in velocity change in time Questions refer to graph of a carts motion.
1. 7 Distance Time Graphs Ideas Graphing Worksheets
2. Quiz Created Grade Teach Classroom Support Facilitation Part Force Motion Graphs Geometry Worksheets
3. Motion Graphs Worksheet Answer Key Graph Analysis Arch Persuasive Writing Prompts Complex Sentences Worksheets
4. Motion Graphs Worksheet Answers Fresh Graphing Linear Grade Text Structure Worksheets
Student instruction the graph for motion of two bodies a and b is as shown. read the graph carefully and answer the following questions The graph below describes a journey that has several parts to it each represented by a different straight line. interpret distance time graphs as if they are pictures of.
5. Motion Graphs Worksheet Answers Fresh Interpreting Grade Line Graph Worksheets
Question a ball is dropped vertically from a height h above the ground.it hits the ground and bounces up vertically to a height h.neglecting subsequent motion and air resistance,its. graphs graphs vel.c, cc.b page base your answers to questions and on the graph below, which represents the motion of a car during a second time interval.
6. Motion Graphs Worksheet Answers Inspirational Velocity Acceleration Worksheets Graphing
The object moves toward the origin at a steady speed for, then stands still for. there are possibilities a in red object moves toward origin date in seconds y t a car travels between two sets of traffic lights. the diagram represents the graph of the car.
7. Motion Graphs Worksheet Answers Review Distance Time Worksheets
What is the acceleration of the car at t. seconds. ms. ms. ms. ms.Related posts of motion graphs worksheet answer key equations of lines worksheet answer key prior to preaching about equations of lines worksheet answer key, please are aware that knowledge is usually each of our factor to a more rewarding next week, as well as finding out wont only stop once the education bell rings.
8. Motion Review Worksheet Distance Time Graphs Worksheets Physical Science Lessons
9. Page 2 Distance Time Graph Motion Graphs Graphing Worksheets
10. Physics Constructing Velocity Time Graphs Position 1 High School Science Teaching
Displacement addresses the issue of is the overall change in position. for example gabby starts at point a and walks to the left and arrives at point b.Jun, free body diagram worksheet answers from distance and displacement worksheet answer key, sourcelivinghealthybulletin.
11. Physics Graphing Motion Ideas Middle School Science Teaching
12. Physics Graphs Unit 1 Ideas Graphing Physical Science
13. Position Time Graph Graphing Physics Mathematics
Since these rely on our choices for the final velocity, multiple valid answers are possible. some of the worksheets displayed are slope from a graphing quadratic name answer key baseball bar graph name reading and interpreting graphs work motion graphs bar graph work line graphs.
14. Printable Worksheet Kids
15. Real Life Graphs Maths Worksheets Distance Time Algebra Resources
It consists of a circle and two line segments. use the graph and your knowledge of motion to answer the following questions. a at what time, t hours, is the speed Distance vs time graph worksheet. part ii answers to the first problems a helicopter left the landing pad at the top of a skyscraper and then quickly flew downwards towards the ground and maintained a foot distance above the ground for a while before it had to fly up worksheets.
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17. Real Life Graphs Worksheet Practice Questions Distance Time Graphing Math
Oct, constant velocity particle model worksheet position vs. time and velocity vs. time graphs. robin, rollerskating down a marked sidewalk, was observed at the following positions at the times listed below a. plot the position vs. time graph for the skater.
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B. what do you think is happening during the time interval t to t how do.Position time graph for constant velocity in physics, the displacement or change in position of an object with regards to time w.r.t. is often shown in a graphical form. these position time graphs give us a visual representation of an objects motion which makes it much easier to understand.
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20. Real Life Graphs Worksheets Maths Graphing Distance Time
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23. Speed Time Graph Scenarios Card Match Activity Lab Teachers Pay Graphing Physical Science Middle School Learn Physics
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25. Speed Velocity Acceleration Revision Physics Forces Motion Force Mathematics
26. Story Distance Time Graphs Worksheets Biology Lessons
27. Story Motion Graphs Distance Time Graph Writing Activity Graphing Linear Equations
28. Velocity Time Graphs Concept Builder Interactive Exercise Challenges Learner Identify Motion Graphing
29. Worksheet Graphing Distance Displacement Running Wolf Time Graphs Interactive Science Notebook
30. Motion Graphs Ksheet Answer Key Inspirational Distance Time Ksheets
Most users should use child support guidelines worksheet if you need to save a partially form, you may choose to use the alternative form. alternative child support guidelines worksheet. this alternative version of the form allows you to May, child support worksheet b instructions.
31. Motion Graph Analysis Worksheet Inspirational Quiz Qualitatively Describing Graphs Persuasive Writing Prompts
You can use a calculator but you must show all of the steps in the spaces provided. a roller coaster car rapidly picks up speed as it rolls down a slope. as it.A kg car traveling south at. speed, velocity, acceleration worksheets and online activities.
32. Analyzing Motion Graphs Calculating Speed 1 Graphing
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Whole numbers, percents, and fractions are used to represent the data.Title word analyzing graphs of quadratic function ws.doc author created date referring to analyzing graphs worksheet, remember to know that instruction is definitely our step to a much better another day, and understanding wont just end when the institution bell rings.
of which being mentioned, all of us offer you a assortment of easy nonetheless informative content plus design templates manufactured ideal for every academic purpose.Analyzing quadratic graphs displaying top worksheets found for this concept. some of the worksheets for this concept are analyzing graphs of polynomial functions, lesson of for problem solving and data analysis, table of contents analyzing structure pacing days, calculus work p, understanding quadratic functions and solving quadratic, exponential functions and their.
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Motion worksheet interpreting graphs linear motion. , learning target i can identify the parts of the scientific method. review physics pretest review scientific method. movements i make. i can predict a graph based on the movements i make. moving man online activity graphing motion review.
Gradient of a graph the gradient or slope of a graph gives the the gradient m, of a linear straight line graph is the rise divided by the run. that is, m rise run. of the object. consider the graph above. from a to b the gradient of the graph is where rise run change in velocity change in time Displaying top worksheets found for physics motion graphs.
36. Calculating Graphing Speed Tortoise Hare Motion Graphs Lesson Plans
As the slope goes, so goes the velocity. review. categorize the following motions as being either examples of or acceleration.Positiontime graph practice you must answer the questions to out of of the graphs this is activity and needs to be completed in your notebook plotting graph activity.
37. Calculating Speed Time Distance Graphing Motion Graphs
The distance time graphs below represent the Id language school subject science grade,, and age main content speed graphs distance vs time other contents add to my workbooks download file embed in my website or blog add to google answer when there is a positive slope on the graph, the acceleration of.
the ball increases. when there is a negative slope on the graph, the acceleration of the ball. decreases. when there is a slope of zero on the graph, the ball is moving at a constant speed and it is. not speeding up or slowing down.youwillbegivenagraphofspeedvs.
38. Distance Time Graph Worksheet 8 Speed Graphs Sped Math Graphing
39. Distance Time Graph Worksheet Differentiated Graphs Worksheets Graphing
Interpreting graphics answer key chapter., , is media groups answer to, and fox. liquid water. chemistry chapter interpreting graphics answer key free download.Prentice hall chemistry chapter interpreting graphics answer key.rar download, interpreting graphics chemistry answers mar.
40. Distance Time Graphs Scenarios Card Sort Graphing Interpreting Motion
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44. Domain Range Graph Worksheet Answers Fresh Algebra State Graphing Worksheets Interpreting Motion Graphs
Explain your answers. descriptions. the car is stopped. the car is traveling at a constant speed. the car is accelerating. the car is slowing down.Motion graphs. the graph below describes the motion of a fly that starts out going right. time s.Velocity vs time graphs and displacement worksheet and graphs worksheet by unit ii additional practice graphs and motion maps for velocity time graph worksheet and answers by worksheet.
position vs time graph worksheet. grass, science accelerated physics semester The worksheet contains examples of motion graphs. students describe the motion in each of the graphs. the graph is a triple line graph of the motion of cars. the students are required to calculate the speed of each car based on the information given in the graph.
45. Education Motion Graphs Graphing Constant Speed
Plot the corresponding graph of acceleration as a function of time. t s d m . kinematics motion graphs cc page base your answers to questions and on the graph below which represents the motion of a car during a second time interval.Motion graphs worksheet along with valuable contents.
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46. Force Motion Interactive Notebook Pages Notebooks Middle School Science
Worksheet motion graphs answers physics fundamentals. in which sections is the cart accelerating.Worksheets. graphing motion worksheet. atidentity.com free. worksheets graphing motion worksheet speed acceleration. chapter motion in a line. quiz worksheet free fall practice problems study.
com print free fall physics practice problems worksheet. speed vs. velocity both describe motion download graphs.Describing motion with graphs position vs. time graphs graphs are commonly used in physics. they give us much information about the concepts and we can infer many things.
47. Motion Graph Analysis Worksheet Elegant Time Distance Velocity Acceleration Graphs Physics Teaching Ideas
On a velocity time graph it is not possible to determine how far from the detector the object is located.Consider the position vs. time graph below for cyclists a and b. there are no calculations b. looking at graph, do the cyclists start at the same position how do you know if not, which one is further to the right.
48. Force Motion Quiz Elementary School Science Activities
49. Force Motion Worksheet Graphs Elementary School Science
This worksheet is great to use in your classroom to help students understand an extremely complicated concept force and motion. through this activity students are asked to put themselves into imagined but relatable experiences in order to learn how inertia, gravity, friction, and force affect speed and velocity.
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50. Formative Assessment Lessons Beta School Worksheets Distance Time Graphs Graphing
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52. Graphing Data Tables Plots
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Day instructions use the molar mass of gallium to answer questions about calculating. chemistry day worksheet interpreting and drawing graphs.Lessoninterpreting graphs worksheet answer key. the graph below shows the relationship between students quiz averages over a semester and their final exam grades.
54. Graphing Motion Graphs Distance Time
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Velocity time graphs worksheet. split the graph up into distinct sections these can be seen in the image as a b c and d. videos worksheets a day and much more. velocity time graph answer key displaying top worksheets found for this concept.And so what i have constructed here is known as a position time graph, and from this, without an animation, you can immediately get an understanding of how the things position has changed over time.
56. Graphing Speed Worksheet Answers Distance Time Graph Uniform Motion Kids Worksheets Grade Graphs
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Love the moment at the end. a sense of surprise in math class is a rare and pretty wonderful thing.As the name suggests, graph it is a math worksheet that requires kids to draw a simple chart. giving kids just the basics to start off with, this graphs worksheet requires them to count the number of various objects and mark it correctly on the chart.
58. Interpreting Motion Graphs Worksheet Answers Text Structure Worksheets
Use the graph to answer the questions. a when was the object stopped b when was the object moving with a positive velocity c when was the object moving with a negative velocity d when did the object have a positive displacement e when did the object have a negative graphs.
the graph below describes the motion of a fly that starts out going right. time s.Physics. worksheet four time graphs velocity ms name review of graphs of motion the graph for a journey is shown. a b calculate the acceleration for each section. calculate the distance travelled in the first seconds.
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Dec, motion graphs worksheet along with best force and motion images on. the sixth is a good thing to use if you want to show important keywords of your graphic. you can place them in order of importance to the graphic that you are using. people can easily see the important key words in the background of the graphic.
Match the description provided about the behavior of a cart along a linear track to its best graphical representation. remember that velocities are positive when the graph is in i quadrant i velocities are negative when the graph is in quadrant iv graphs sloping towards the represent losing speed graphs sloping away from the represent gaining graphs and derivatives worksheet.
61. Worksheet Great Addition Force Motion Unit Beginners Students Analyze Basic Graph Graphs Graphing Calculating Speed
In which sections is the cart accelerating. in which sections is the cart not moving. in which sections is the cart moving backwards. in which sections is the carts instantaneous velocity at any time equal to its average velocity. When velocity is zero, the graph should be horizontal.
since the acceleration is constant, the graph will always be a parabola, the graph will always be straight, and the graph will always be horizontal. when acceleration is positive, the graph should have a positive slope and the graph should bend upward.
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CC-MAIN-2021-25
| 17,765 | 103 |
https://www.enotes.com/homework-help/binomial-probability-distribution-question-4-312783
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math
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binomial probability distribution question #4
A student is taking a multiple choice exam in which each question has 4 choices. Assuming that she has no knowledge of the correct answers to any of the questions, she has decided on a strategy in which she will place 4 balls (marked A, B, C and D) into a box. She randomly selects one ball for each question and replaces the ball in the box. The marking on the ball will determine her answer to the question. There are 5 multiple choice questions on the exam. What is the probability that she will get 5 questions correct?
You need to use binomial probability formula to determine what is the probability that the student to get 5 questions out of 5 correct, hence:
`P(r) = C_n^r*p^r*q^(n-r)`
Notice that the 5 questions are 4 choice questions, hence the probability to get one correct choice out of 4 is: `p = 1/4 = 0.25` .
You need to remember that `p+q=1 =gt q = 1 - 1/4 = 3/4 = 0.75`
Hence, the probability not to get a correct answer is of `3/4=0.75` .
Evaluating the probability that the student to get 5 correct answers out of 5 such that:
`P(5) = C_5^5*(0.25)^5*(0.75)^(5-5)`
`P(5) = 1*0.0009*1 =gt P(5) = 0.0009`
Hence, the probability that the student to get 5 answers correct is of 0.0009.
The student determines the correct answer by choosing a ball from a box that has 4 balls marked with the options A, B, C and D. The probability that a randomly picked ball has the right answer is 0.25.
In a test that has 5 questions the probability that using the system gives all answers right is (0.25)^5 = 9.765*10^-4
The probability that she gets all five answers right using her system is 9.765*10^-4
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| 1,653 | 14 |
http://songho.ca/dsp/system/systems.html
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math
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Variety of Systems
Systems that satisfy both homogeneity and additivity are considered to be linear system.
Homogeneity (scalar rule) means that as the strength of input signal is increased (scaled), then the strength of output signal will be also increased (scaled) with same amount.
Additivity denotes that the output of system can be computed as sum of the responses resulting from each input signal acting alone.
Taking these two rules together is called the principle of superposition. Therefore, the linear system can be presented as;
If input signal can be decomposed as a weighted sum of basis signals, the output of linear system is;
Determine if it is linear.
∴ linear system
∴ non-linear system
A system is time-invariant if a time shift in input signal causes an identical time shift in output signal.
A system is causal if the output at any time depends on values of the input at only the present and past times. In other words, the causal system does not anticipate future values of input.
For example, the output y(t0) depends on input x(t) for t ≤ t0.
This is an example of non-causal system, because the output responds ahead at t = t0 before input is defined.
A causal system is characterized by an impulse response h(t) that is zeros for t < 0.
Causal, because t ≥ t-2
Non-Causal, because t t+1
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| 1,321 | 16 |
https://www.originlab.com/doc/Quick-Help/Change-Plot-Type
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math
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Last Update: 7/30/2015
The quickest way to change a plot type is to right click on the plot and select Change Plot to and further select the plot type you want to switch to. However, the number of available plot types is limited.
In the Plot Setup dialog, you can change any valid plot type freely. To change the plot type:
Keywords:Change, Plot Type, Plot Setup
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| 362 | 4 |
https://www.mathematik.uni-wuerzburg.de/fluidmechanics/aktuelles/single/news/seminarreihe-structure-preserving-numerical-methods-for-hyperbolic-equations-im-oberseminar-mathem-23/
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math
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Seminarreihe "structure preserving numerical methods for hyperbolic equations" im Oberseminar Mathematische Strömungsmechanik: Min Tang, Asymptotic preserving scheme for the nonlinear radiation transport MHD equation
|17.12.2020, 09:30 - 10:15 Uhr
This talk is part of the seminar series "structure preserving numerical methods for hyperbolic equations", click here for more details
Asymptotic preserving scheme is poropsed for the nonlinear radiation transport MHD equation. Both the non equilibrium and equilibrium radiation diffusion MHD limit can be captured by the scheme. The advantages of our scheme are that
1) The space and time steps do not depend on the speed of the light;
2) Only macroscopic quantities, i.e. the radiation temperature, the fluid temperature have to be solved nonlinearly, while the radiation density flux can then be updated by solving a small linear system on each space grid.
3) The scheme has hyperbolic time step constraint whose CFL number does not depend on the speed of light.
via Zoom video conference (request the Zoom link from [email protected])
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s3://commoncrawl/crawl-data/CC-MAIN-2024-18/segments/1712296816864.66/warc/CC-MAIN-20240414033458-20240414063458-00705.warc.gz
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CC-MAIN-2024-18
| 1,100 | 8 |
http://lib.mexmat.ru/books/120344
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math
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Нашли опечатку? Выделите ее мышкой и нажмите Ctrl+Enter
Название: Diffusion on Random Lattices
Авторы: Wang F., Cohen E.G.D.
Journal of Statistical Physi~w, 1/ol. 84. Nos. 1/2, 1996, p. 233-261.
We study the motion of a point particle along the bonds of a two-dimensional random lattice, whose sites are randomly occupied with right and left rotators, which scatter the particle according to deterministic scattering rules. We consider both a Poisson (PRL) and a vectorized random lattice (VRL) and fixed as well as flipping scatterers. On both lattices, Ibr fixed scatterers and equal concentrations of right and left rotators the same anomalous diffusion of the particle is obtained as before for the triangular lattice, where the mean square displacement is ~t, the diffusion process non-Gaussian, and the particle trajectories exhibit scaling behavior as at a percolation threshold. For unequal concentrations the particle is trapped exponentially rapidly. This system can be considered as an extreme case of the Lorentz lattice gases on regular lattices discussed before or as an example of the motion of a particle along cracks or (grain or cellular) boundaries on a two-dimensional surface.
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CC-MAIN-2019-09
| 1,248 | 5 |
http://zumbahungary.eu/book-of-ra/expected-value-computation.php
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math
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Anticipated value for a given investment. In statistics and probability analysis, expected value is calculated by multiplying each of the possible outcomes by the. The formula for the expected value is relatively easy to compute and involves several multiplications and additions. In probability theory, the expected value of a random variable, intuitively, is the long-run .. This is because an expected value calculation must not depend on the order in which the possible outcomes are presented, whereas in a conditionally.
Determine the probability of each possible outcome. I agree with the other post that it was hard to figure out at first, but after practicing over and over it finally came to me. In the above proof, the treatment of summation depends on absolute convergence , which assumes existence of E X. This result can be a useful computational shortcut. Also recall that the standard deviation is equal to the square root of the variance.
Expected value computation - Einsteiger machen
ACM Transactions on Information and System Security. X n having a joint density f: Embed code Affiliate embed. The expected value of this scenario is: Cookies make wikiHow better. Assign a value to each outcome. If a random variable X is always less than or equal to another random variable Y , the expectation of X is less than or equal to that of Y:. The expected value of this scenario is: We then add these products to reach our expected value. A6 is the actual location of your x variables and f x is the actual location of your f x variables. If you were to roll a six-sided die an infinite amount of times, you see the average value equals 3. I am going to look at a different example. According to this formula, we take each observed X value and multiply it by its respective probability. You toss a fair coin three times. If is a random variable and is another random variable such that where and are two constants, then the following holds: If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. The expected value formula for a discrete random variable is: Multiply the gains X in the top row by the Probabilities P in the bottom row. Hence, if is integrable, we write. Hypothesis Real play poker Lesson 9: Text is available under the Http://www.med1.de/Forum/Psychologie/557833/ Commons Attribution-ShareAlike Club one casino fresno ; live wetten angebot terms may apply. Expected value for a discrete random variable. Perform the steps exactly as. More specifically, X rossmann adventskalender be the number of pips showing on the top face of the http://www.bizdb.co.uk/company/blue-skies-addiction-centre-ltd-08575087/ after the toss. Tempel spiele continuous variable situations, integrals apphack be used. The expected value formula changes a little if you have a series of trials dresden vs leipzig example, a series http://www.reviersport.de/138225---bochum-hans-walitza-interview-zum-65.html coin tosses.
Expected value computation Video
Expected Value and Variance of Discrete Random Variables
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CC-MAIN-2018-34
| 3,099 | 6 |
https://www.asknumbers.com/square-kilometer-to-square-feet.aspx
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math
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How many square feet in a square kilometer?
There are 10763910.417 square feet in a square kilometer. To convert square kilometers to square feet, multiply the square kilometer value by 10763910.417.
For example, to find out how many square feet in a square kilometers and a half, multiply 1.5 by 10763910.417, that makes 16145865.625 square feet in a square kilometer and a half.
1 Square Kilometer = 10763910.417 Square Feet
How many square kilometers in a square foot?
1 Square foot is equal to 0.00000009290304 square kilometer. To convert square feet to square kilometers, multiply the square foot value by 0.00000009290304 or divide by 10763910.417.
1 Square Foot = 0.00000009290304 Square Kilometer
What is a Square Kilometer?
Square kilometer (kilometre) is a metric system area unit and defined as a square with all sides are one kilometer in length. 1 km2 = 10763910.417 ft2. The symbol is "km²".
What is a Square Foot?
Square Foot is an imperial and United States Customary systems area unit and defined as a square with all sides are one foot in length. 1 ft2 = 0.00000009290304 km2. The symbol is "ft²".
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| 1,118 | 11 |
https://www.twirpx.com/file/2261810/
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math
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2nd Edition. — Cambridge University Press, 2013. — 516 p. — (Cambridge Studies in Advanced Mathematics 136) — ISBN 978-0-521-51363-0.One can take the view that local cohomology is an algebraic child of geometric parents. J.-P. Serre’s fundamental paper ‘Faisceaux alg´ebriques coh´erents’ represents a cornerstone of the development of cohomology as a tool in algebraic geometry: it foreshadowed many crucial ideas of modern sheaf cohomology. Serre’s paper, published in 1955, also has many hints of themes which are central in local cohomology theory, and yet it was not until 1967 that the publication of R. Hartshorne’s ‘Local cohomology’ Lecture Notes (on A. Grothendieck’s 1961 Harvard University seminar) confirmed the effectiveness of local cohomology as a tool in local algebra. In the fifteen years since we completed the First Edition of this book, we have had opportunity to reflect on how we could change it in order to enhance its usefulness to the graduate students at whom it is aimed. As a result, this Second Edition shows substantial differences from the First. The main ones are described as follows. Contents The local cohomology functors Torsion modules and ideal transforms The Mayer–Vietoris sequence Change of rings Other approaches Fundamental vanishing theorems Artinian local cohomology modules The Lichtenbaum–Hartshorne Theorem The Annihilator and Finiteness Theorems Matlis duality Local duality Canonical modules Foundations in the graded case Graded versions of basic theorems Links with projective varieties Castelnuovo regularity Hilbert polynomials Applications to reductions of ideals Connectivity in algebraic varieties Links with sheaf cohomology
Чтобы скачать этот файл зарегистрируйтесь и/или войдите на сайт используя форму сверху.
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CC-MAIN-2019-43
| 1,878 | 2 |
https://stacks.math.columbia.edu/tag/04QQ
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math
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Definition 35.20.1. Germs of schemes.
A pair $(X, x)$ consisting of a scheme $X$ and a point $x \in X$ is called the germ of $X$ at $x$.
A morphism of germs $f : (X, x) \to (S, s)$ is an equivalence class of morphisms of schemes $f : U \to S$ with $f(x) = s$ where $U \subset X$ is an open neighbourhood of $x$. Two such $f$, $f'$ are said to be equivalent if and only if $f$ and $f'$ agree in some open neighbourhood of $x$.
We define the composition of morphisms of germs by composing representatives (this is well defined).
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Hope & pie has one of the best pizzas I've ever enjoyed. Beer braised brisket, carmalized onions and house made mozzarella. It make your mouth water as it approaches your table. The service was amazing! Our server Frankie was polite and attentive. The ambience of the restaurant is very relaxed and artistic. They have several abstract paintings and dim lighting. Try their ceaser salad with homemade dressing and anchovies. Don't forget to try one of their 50 beers on tap or in a can. #EscalatedEats#DenverPizza#Denver#foodporn#foodie#pizza#beer#hopsandpie#TennysonFood#ambiance#DenverFood#denverfoodies
A little taste of the Willett tonight.
Willett 3 Year Rye, 110 proof. So very tasty. I think this is my favorite rye that I can afford ($43'ish as I drunkenly recall). Is there a better value Rye out there? I think not, but admittedly Rye is not my bailiwick. Feel free to discuss/recommend below!
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https://truefire.com/jazz-guitar-lessons/shades-of-jazz/half-step-down-lydian-sound-demonstration-4/v2416
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Watch the Half Step Down: Lydian Sound online guitar lesson by Kenny Wessel from Shades of Jazz
Another way to deal with playing over these Lydian chords is to play the blues. Since we're using minor pentatonic (blues) scales, why not use our blues licks, bends, slides, inflections, etc. to get into the guitar's wheelhouse a bit? Again, we're playing over the same 16-bar progression: G maj7#11, Bb maj7#11, C maj7#11, to Eb maj7#11. Use your bag of tricks, when you play!
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http://mathhelpforum.com/calculus/3033-integration-problem-my-textbook-wrong.html
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I think I might have found a typo in my calculus text, "Forgotten Calculus". It would not be the first. Anyway, if it is I and not the author who is wrong, then I'm missing something fundamental, and would appreciate guidance.
The problem asks to find the integral of
(10x - 3) [(5x^2 - 3x + 17)^(1/7)] dx
My solution is
(7/8) [(5x^2 - 3x + 17)^(8/7)] + C.
The author's is
(7/8) [(5x^2 - 3x - 17)^(8/7)] + C.
Note that we differ only over whether one should add or subtract 17.
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https://marcoslab.com/2020/11/new-publication-in-chemical-engineering-science-congratulations-dr-yao/
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Link to article: https://doi.org/10.1016/j.ces.2020.116254
We present a systematic procedure to analyze the interaction between two spheres under a uniform electric field in a porous medium. We solve Laplace’s equation in bispherical coordinates to obtain electric potential distribution. Such electric potential is utilized to obtain the slip velocities on the two spheres and the flow field around them is solved based on stream function. The solution is solved semi-analytically by considering geometric and electric parameters. The detailed flow field in the vicinity of the two spheres is investigated and our results reveal that the sphere interaction is dependent on the separation distance, sphere radii, zeta potential and normalized radius with respect to permeability. The increase of normalized radius with respect to permeability reduces the interaction between the two spheres. This is particularly prominent when normalized radius with respect to permeability is relatively small (αa <50).
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http://forum.bebac.at/mix_entry.php?id=18414
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Posting: # 18414
Because I was A) not able to find any post that even remotely dealed with this issue and B) had some discussion lately that might also betide anybody else and C) have some spare time and D) was bewildered that this issue caused so much discussion, I would like to show a simple example why in BE/BA the fancy stuff is not necessarily the correct approach.
May be boring for the experienced biometrician/statistician, but was enlightening for a lot of my colleagues.
Remember, you can stop reading at any time, just saying .
We got involved in discussing the evaluation of an endogenous substance (including a pre-dose profile for baseline correction), where we criticized that no baseline correction was implemented at all and, therefore, their conclusion on the compared products was not valid .
But people said, an ANCOVA was used, as recommended by the "Guideline on adjustment for baseline covariates in clinical trials", so this approach should suffice as a baseline correction.
From our point of view, this is not correct; as as a matter of fact, the use of a covariate should be considered if there actually is some impact of the starting value on the outcome. Likely fine for clinical endpoints and some PD parameters, but what should be the mechanistical concept in case of an AUC?
So, we did not agree and were able to enforce a "proper" baseline correction by subtraction. This was finally implemented and ... resulted in the exact same results . By closer examination it was revealed that the same model was applied, i.e. the ANCOVA was conducted considering the values after baseline-correction. Nice try...
As a little illustration to be used when such a discussion comes up consider these values:
Easy to see, we have a pure difference of 50 for T-R and 40 if baseline is considered. Hint: these are not real data.
Now, whatever software you use, the evaluation should resemble something like this:
where "Baseline" is used in case of inclusion of the covariate.
So what results do we get in which evalution (point estimates and 95%CI):
As you can see, use of the ANCOVA approach gives us results differing from what we get from the "expected" calculation. And as is to be expected due to the concept of an ANOVA it does not matter, whether you use the change from baseline or the end value. So, in particular in those cases, where officially the baseline-correction in accordance with the guidelines was implemented, but an ANCOVA was conducted...).
And good luck finding a medical writer who will recognize this in the SAS code or Phoenix output or...
Why is this important? Well, in the case that started our discussion, the improper ANOVA shifted the point estimate and allowed to conclude on a statistically significant difference. That is, it allowed to avoid crossing the 100% threshold. Could have been 125% as well.
In the presented case above on the other hand, the improper ANCOVA markedly increased the variance (the baseline values are admittedly a little bit one-sided), so hiding a difference might be possible.
As always, please do not hesitate to correct, add and challenge, if there is something wrong.
Edit: Tabulators changed to spaces and BBcoded; see also this post #6. [Helmut]
Posting: # 18424
Dealing with endogenous compounds is tricky and here are some more thoughts you may find helpful
This could be considered as change from baseline problem and you may have a look at Stephen Senns work on this topic relating to ANCOVA (e.g. Statist. Med. 2006; 25:4334–4344. https://doi.org/10.1002/sim.2682 )
You may find also this article of interest addressing adjustment of endogenous levels in PK modeling: Bauer, A. & Wolfsegger, M.J. Eur J Clin Pharmacol (2014) 70: 1465. https://doi.org/10.1007/s00228-014-1759-x
Best regards & hope this helps
Posting: # 18425
I read your post so many times now and I am somewhat confused.
What were you actually trying to prove or disprove?
Inclusion of a covariate one way or another makes an implicit assumption of a relationship that can be said to be linear between the covariate and the response (in the presence of the factors).
If the variance goes full Tasmanian devil on you when you include the covariate then perhaps this assumption is...well... of a nature that has the potential to cause some degree of debate. And then that is where the problem truly is.
In contrast to classical anovas where an additional factor will always decrease the unexplianed variance (or leave it unchanged, academically), the inclusion of a covariate is not necessarily having this effect.
Help me, please, I really wish to understand what this is all about.
"(...) targeted cancer therapies will benefit fewer than 2 percent of the cancer patients they’re aimed at. That reality is often lost on consumers, who are being fed a steady diet of winning anecdotes about miracle cures." New York Times (ed.), June 9, 2018.
Posting: # 18426
» I read your post so many times now and I am somewhat confused.
And I read it over so many times exactly to avoid being too confusing. Sorry for failing.
» What were you actually trying to prove or disprove?
Uh, nothing, really, I only wanted to share my experience with this discussion in a BA-setting as I found it difficult to find anything that was just a simple statement or experience shared. And in favour of our position (Baselines are not(!) a good covariate in PK and will potentially result in a misleading result).
» Inclusion of a covariate one way or another makes an implicit assumption of a relationship that can be said to be linear between the covariate and the response (in the presence of the factors).
» If the variance goes full Tasmanian devil on you when you include the covariate then perhaps this assumption is...well... of a nature that has the potential to cause some degree of debate. And then that is where the problem truly is.
Nothing to add here. Back at university, I essentially learned that covariates
Remembering the qualities of the two teachers we enjoyed I will just say that statistics is not the most important issue for some university degrees.
» In contrast to classical anovas where an additional factor will always decrease the unexplianed variance (or leave it unchanged, academically), the inclusion of a covariate is not necessarily having this effect.
» Help me, please, I really wish to understand what this is all about.
Again, I am sorry. I thought it might be helpful for others who happen to come across the discussion whether or not to implement a baseline as a covariate in a PK evaluation to show in a simple made-up example how this has an impact and that it is not an appropriate idea.
Plzeň, Czech Republic,
Posting: # 18459
» But people said, an ANCOVA was used, as recommended by the "Guideline on adjustment for baseline covariates in clinical trials", so this approach should suffice as a baseline correction.
I also think they took wrong cookbook.
You know EMA 1401 (page 9):
For endogenous substances, the sampling schedule should allow characterisation of the endogenous baseline profile for each subject in each period. Often, a baseline is determined from 2-3 samples taken before the drug products are administered. In other cases, sampling at regular intervals throughout 1-2 day(s) prior to administration may be necessary in order to account for fluctuations in the endogenous baseline due to circadian rhythms (see section 4.1.5).They done it Ok. but according to EMA 1401 section 4.1.5: Endogenous substances
If the substance being studied is endogenous, the calculation of pharmacokinetic parameters should be performed using baseline correction so that the calculated pharmacokinetic parameters refer to the additional concentrations provided by the treatment. ...the baseline correction must be done before PK analysis (subtraction of AUC, i.e. whole profile, each sample time has own baseline from predose conc. e.g. day before, or subtraction of mean of several predose concentrations).
Remember, you can restart reading at any time, just saying .
In the study you described there wasn't planned to do baseline correction before PK analysis. But then, theoretically in situation with the circadian rhythms, the maximum concentration from raw uncorrected data and the maximum concentration from baseline corrected data (i.e. after subtraction of predose profile) can be in different time - so different concentration would enter into the calculation.
(Not happened in the study ... results were completly the same when calculation without/with baseline correction.)
» Hint: these are not real data.
I have such data too. x)
For simplicity my data are little bit parallel (and as in parallel design but it could be made more complicated ...).
Artificial data example:
This example is perfectly linear and slopes of both treatment in Regression Analysis are the same (although in real study I would not expect the linearity much).
For R users the data are:
There is always one direction (slope) of "mean correction" in ANCOVA (something between slopes of linear regression of T and R - in this example slopes are the same). So means are "corrected" in the direction to the mean baseline (dashed line) as ilustrated in figures if it is keep simple (not complicated e.g. with missings - not balanced sequences). Left side raw data, right side ln-data (of course the same because both axes were ln transformed - that's why I didn't used your data with baseline=0 for R).
Of course, it is not expected to have different baseline for T and R in randomized BE study, so... it's only artificial example (as well as your data).
Anyway the differences of means of T and R:
From the graphical interpretation of simple example of ANCOVA, if mean(Baseline_conc)ofT = mean(Baseline_conc)ofR = mean(Baseline_conc) then no correction is applied (means are the same) and PE from ANCOVA = PE from ANOVA.
But with more and more difference which is depending on the luck/misfortune of the randomization of subjects we can get more "corrected" means. It seems that we could conclude then something as "Treatments are equivalent ...; evident differences observed by simple comparison of mean of T versus mean of R are caused by different baseline values"?
So ANCOVA does not look as the correct baseline correction in this artificial example.
Moreover some burning points which are not to be answered:
How the sample size was calculated for the BE with ANCOVA evaluation.
Ignoring additional assumptions for ANCOVA.
PE is called in the guideline as GMR (for ANOVA it can be "tolerated" but for ANCOVA, GMR could be far away)
Acceptance BE limits still 0.8-1.25 (90% CI for PE from ANCOVA is different than 90% CI for PE from ANOVA with these limits set in guidelines).
(I would bet that 90% CI from ANCOVA would be always(?) wider ... so then this method would not be the best choise for sponsors.)
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https://www.physicsforums.com/threads/uniform-magnetic-field-in-a-circular-coil.356373/
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A 138 turn circular coil has a diameter of 2.00 cm and resistance of 53.0 Ω. The plane of the coil is perpendicular to a uniform magnetic field of magnitude 1.00 T. The direction of the field is suddenly reversed. a.) Find the total charge that passes through the coil. b.) If the reversal takes 0.100 s, find the average current in the coil. c.) Find the average emf in the coil. I am using stuff for a solenoid is that the same as a circular coil? l=lenght n= # of turns per unit length N=n*l total # of turns A=π*r^2 Magnetic Flux=B*A (threw one turn) B=mu_0*n*I (magnetic field inside solenoid) Inductance=((N*(B*A))/I)=mu_0*n^2*A*l E=IR I don't know what to do. Any suggestions ?
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http://www.tradingcardcentral.com/forums/index.php?s=d493b356253ccd5395ad2bf33e81894c&showtopic=147151
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QUOTE(nflsam79 @ Jun 1 2008, 04:29 PM)
i can buy for 8 bucks delivered which is about 2 bucks more then there selling elsewhere lol i also have these lj gu
2003 ud finite rookie gu
2006 lcm fotg serial# to 75 i also have 2-3 other 06-07 lj gu but there not serial# or rookie gu lmk thanx
Do you have a scan of the LJ's?
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https://www.physicsforums.com/threads/proof-about-relatively-prime-integers.749634/
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1. The problem statement, all variables and given/known data Prove that if you have n+1 integers less than or equal to 2n then at least 2 are relatively prime. 3. The attempt at a solution the book say integers but im pretty sure this will only work in the natural numbers. there are n even numbers between 0 and 2n okay and none of those are relatively prime but when we pick another number it will be odd and next to and even number. We know that consecutive integers are relatively prime because if they shared common factors it should divide their difference but the difference is (n+1)-n=1. so 1 is their only common factor. and picking n even integers is the most you pick that share common factors because multiples of 2 occur more frequently than any other multiple of a prime, because 2 is the smallest prime. Im just wondering how would i connect this to Ramsey theory.
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https://discourse.mcneel.com/t/curved-beam-ifc-export-issue/57118
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I have a question regarding exported curved elements, specifically beams, into revit. As shown in the revit 3d view, the beam generates a broken geometry face. The geometry is generated through grasshopper with an interpolated curve which produces a clean, single beam. Otherwise, the a polyline curve that determines the path of the beam creates individual segmented beams, not accounting for the difference between the angles for the segments. See images for reference.
Any ideas for solving this issue?
Any help is very much appreciated.
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https://www.teacherspayteachers.com/Product/Graphing-Systems-of-Linear-Inequalities-1441883
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These guided notes include step-by-step instructions for graphing linear inequalities. Students should know how to graph lines (including horizontal and vertical) and go from standard to slope-intercept form prior to completing these notes. Including in the document are teacher examples as well as student practice problems. These notes are most effective (and more fun!) when completed using multiple colors, demonstrating the solution set as the area where the colors overlap, as shown in the solutions. Happy graphing! :) Created by Math with Mrs. Holst!
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https://www.hindawi.com/journals/isrn/2012/471653/
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- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
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- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Volume 2012 (2012), Article ID 471653, 23 pages
Dynamics of Single-City Influenza with Seasonal Forcing: From Regularity to Chaos
Centre for Nutrition Modelling, Department of Animal and Poultry Science, University of Guelph, Guelph, ON, Canada N1G 2W1
Received 7 September 2011; Accepted 17 October 2011
Academic Editors: C. B-Rao and I. Rogozin
Copyright © 2012 John H. M. Thornley and James France. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Seasonal and epidemic influenza continue to cause concern, reinforced by connections between human and avian influenza, and H1N1 swine influenza. Models summarize ideas about disease mechanisms, help understand contributions of different processes, and explore interventions. A compartment model of single-city influenza is developed. It is mechanism-based on lower-level studies, rather than focussing on predictions. It is deterministic, without non-disease-status stratification. Categories represented are susceptible, infected, sick, hospitalized, asymptomatic, dead from flu, recovered, and one in which recovered individuals lose immunity. Most categories are represented with sequential pools with first-order kinetics, giving gamma-function progressions with realistic dynamics. A virus compartment allows representation of environmental effects on virus lifetime, thence affecting reproductive ratio. The model's behaviour is explored. It is validated without significant tuning against data on a school outbreak. Seasonal forcing causes a variety of regular and chaotic behaviours, some being typical of seasonal and epidemic flu. It is suggested that models use sequential stages for appropriate disease categories because this is biologically realistic, and authentic dynamics is required if predictions are to be credible. Seasonality is important indicating that control measures might usefully take account of expected weather.
1.1. Background and Objectives
Both seasonal influenza and influenza epidemics continue to cause, despite prophylactic and therapeutic efforts, considerable morbidity, mortality, and financial loss across the world (e.g., [1, 2]). Further concern is generated by connections between human influenza with avian influenza (; see below), which, in its own right, gives rise to economic damage and distress . Models provide a method of summarizing current ideas about mechanisms of disease development and propagation, understanding the contributions from the different processes, and exploring possible consequences and interventions [5–7]. Because of the world-wide relevance of influenza, human and avian, there continue to be many publications on the topic. The recent outbreak of H1N1 swine influenza, and especially the varied and conflicting prognoses from experts, reveals how thinly based much of our knowledge still is.
Our paper is partly a review, partly sets out in detail a particular approach to influenza modelling, and partly has some original content on seasonality, forcing, and the resulting predictions. The particular objectives are to construct a model with multiple sequential pools in various disease categories (Figure 1) so that the underlying biological dynamics is realistic, to validate the model by applying it to data describing an influenza epidemic (Figure 5), to suggest a possible mechanism for the direct effects of environment (season) on influenza dynamics (Figure 6), and last to apply various levels of environmental forcing to demonstrate how the model, without reparameterization, can give rise to a wide range of dynamics, ranging from regular (e.g., twice yearly, annual, biennial, etc.) epidemics to chaos, where sometimes predictions resemble some of the great influenza pandemics (Figures 7 and 8).
Before proceeding with this agenda, a brief review of some of the issues and modelling approaches relevant to influenza is given, and our contribution is discussed in this wider context.
1.2. Brief Review of Influenza Issues and Modelling Approaches
1.2.1. Avian Influenza
One of the reasons for the ongoing concern with influenza is the continuing threat from avian flu. Avian influenza is endemic in many parts of the world and, from time to time, there are outbreaks of highly pathogenic strains, such as the recent outbreak of the H5N1 strain of the virus . The virus can be transferred to other animals, including horses, pigs, and humans, frequently with fatal consequences . Because of antigenic drift (a high rate of immunologically significant mutations) and shift (reassortment between different strains of influenza within a single host), coupled to other factors such as loss of immunity, there is a continuing threat to the human population [10, 11]. This threat depends on possible changes in the to-date limited capacity of the highly pathogenic avian strains for human-to-human transmission [8, 9]. This risk makes it important that key factors determining virus evolution, epidemic occurrence, and cross-species transmission are well understood, so that effective strategies for containment and control might be designed [12–14].
There are also models dealing with the avian-human influenza nexus. For example, presents an ordinary differential equation model which combines an SI (susceptible, infected) avian model with an SIR (susceptible, infected, resistant) human model. They suggest that measures such as both extermination (of avians) and quarantine (for humans) could be needed to avoid a pandemic of influenza.
1.2.2. Two-Strain Influenza and Influenza in a Single Person
An interesting paper extends what might be called the complex models of simple influenza by presenting a simple deterministic model of a more complex situation: they treat the dynamics of two-strain influenza, focussing on competition and cross-immunity. Isolation period and crossimmunity are critical parameters. Some of their results are similar to those reported here (with variable interepidemic periods from 2 to 10–13 years), although the model and mechanisms are quite different.
At a more detailed level, the cell level in a single person, a model of the immune response to the influenza virus which treats innate and adaptive immunity has been proposed . The model has 10 ordinary differential equations, representing interferon, T-cells, killer cells, antibodies, and other states. They explore the impact of initial viral load on disease progression. When this is small, the disease progresses asymptomatically. The model builds on .
1.2.3. Network Models and Stochastic Simulations
Some recent influenza models are based on networks and stochastic simulations [11, 18–22]. The model of is of considerable interest: it includes an individual level (age, treatment, vaccination status) as well as a community level (household, workplace, supermarkets, schools, etc.). Within a city, most contacts occur in a few locations. Interventions at these locations can be expected to be more effective than less-targeted interventions. Isolation and quarantine (e.g., ) are possible treatments, as is antiviral use for both prophylaxis and therapy. A stochastic calculation with a half-day time step is applied. Their model gives valuable quantitative indications of how epidemics/pandemics may be prevented or controlled. These highly articulated models may be well suited to predictions of outcomes from interventions for particular situations. However, stochastic models are highly demanding, of computing resources, high-resolution data on populations, contacts, transmission, age-specific characteristics, and so forth. Moreover, their predictions are often highly sensitive to initial values and settings.
1.2.4. Deterministic Models
Deterministic models of influenza are also numerous, and the model presented here is of this type. While biological variability is a reality, there is a feeling amongst some modellers, especially those with a background in engineering or physics or applied mathematics, that a deterministic approach gets closer to the real science, to understand what is going on, than a stochastic approach. Reflecting this are many quotations, such as “God does not play dice,” and “if you need statistics, then do a better experiment.” Deterministic models are easier to build, easier to understand and use, easier to falsify, and easier for tracing cause and effect. With a deterministic model, a clearly labelled box diagram tells the reader most of what he needs to know. In the next two paragraphs, we refer to two examples of the genre. Although many models have the potential to represent directly seasonal effects of environment, we have not found one that actually does this. Some models represent places where people assemble and interact such as schools and clinics which affect contact rates: indirectly, these can represent a seasonality. But here, effects of weather variables on model parameters are represented directly.
The model of is a typical example of a simple deterministic influenza model. The model has seven significant state variables and ordinary differential equations. From our point of view, their use of a single stage to represent each category (e.g., latent, infectious, asymptomatic, hospitalized, etc.) gives a biologically unreasonable representation of the transit time distribution through that category. Since here we represent day-to-day effects of weather on influenza dynamics, it is important that the model should represent the dynamics of influenza realistically. This means using several serial stages for progress through each disease category.
A typical large deterministic influenza model is in . This comprises over 1000 differential equations and allows for many demographic and clinical parameters (such as risk, age, four levels of sickness, treated or untreated at home, and treated or untreated in hospital) so that it is useful in planning. Their model does employ multiple stages for the different disease categories, and therefore the model is able to be much more dynamically realistic than that of . The model has been used to explore the consequences of pharmaceutical and nonpharmaceutical interventions by . The interested reader might find it useful to start with [27, Figure 7] and proceed to [26, Tables 2, 3, and 4]. The model has not been applied to direct seasonal effects, although effects such as school closures giving a decrease in contact rates are included. There is no mention of chaos resulting from such forcing.
addresses generally for SIR and SEIR models some of the issues covered here: the inclusion of more realistic distributions; the destabilizing consequences of this so that lower levels of forcing are required to give chaos; the conclusion that the assumptions made in formulating the model have a major impact on its dynamical properties.
1.2.5. Simplicity versus Complexity in Influenza Modelling
There are numerous other examples of models of the deterministic ordinary differential equation type. Many parameters of these models are uncertain; predictions can be highly sensitive to initial conditions. The model may be applied as a large regression equation, irrespective of the known biological lacunae and the well-documented dangers of such procedures . Pertinent to this dilemma, it has been remarked that “phenomenological approaches are deficient in their lack of attention to underlying processes; individual-based models, on the other hand, may obscure the essential interactions in a sea of detail . The challenge then is to find ways to bridge these levels of description, ….” Others (e.g., ) have argued the importance of modelling the dynamics of influenza surveillance data, in order to provide early predictions of epidemic events; to this end, they apply purely statistical methods. Cogently, it has been suggested that “as a general policy in preparing for an outbreak of a disease whose parameters are not yet known, it would be better to use a general compartment model using relatively few parameters and not depending critically on the particular as yet unknown setting” . We concur with this view. Such models are easier to construct and explore. They are better suited for elucidating general properties of these systems, as is done here.
Seasonality has long been implicated in influenza incidence and severity, although the basis for this is not understood . Also, it has arguably been given little detailed attention by epidemiologists. Seasonality is a significant factor in mortality from several causes including influenza in temperate countries, with more people dying in the winter (approximately, November to March in the northern hemisphere) than in the summer [34, 35]. The contribution of influenza to these excess deaths is disputed [35, 36], as vaccination against influenza protects against deaths from other conditions . However, summarizes with “our findings are compatible with the hypothesis that the cause of winter-season excess mortality is singular and is most likely to be influenza.” This conclusion agrees with an earlier European study . Any model which covers many community levels (e.g., household, workplace, supermarkets, schools, as in ), offers many possibilities for applying seasonality by altering mixing patterns. Often seasonal forcing is represented empirically, by adding a sinusoid to the infection rate parameter [39, 40]. This approach to forcing leads to annual, biennial, and multiple cycles including chaos. It has been suggested that large seasonal oscillations in incidence can result from an amplification of very small seasonal changes in influenza transmission . Large amplification occurs when the driving frequency is close to the natural frequency of the unforced system. A two-state variable SIRS model (S + I + R = constant) is applied. A two compartment model with linear transfers (giving rise to negative exponentials) can only give a limited representation of the biological dynamics. They do not explore their model other than to support their suggestion of dynamical resonance and make no mention of chaos.
1.3. Current Model and Its Contributions
Here a simple model for epidemic influenza in a single city with seasonal forcing is constructed and evaluated. We are not aware of existing work which treats directly seasonal effects (but see , where the contact rate is reduced by a factor of ten for the 6-month nonepidemic season). The focus is on the essential biology of the problem using traditional scientific reductionism. The model is of the compartmental deterministic type with homogeneous mixing (but see ), and is without age or any other non-disease-status stratification. Various categories are represented, but, recognizing the importance of mechanistically realistic dynamics, and at variance with usual practice in SIR models, each category is represented by three or more stages. The reason for using several sequential stages (spelt out in Appendix A) to represent a given category is that this allows a more credible gamma-function progression through the category . With a single-stage category, the most probable time a person spends in that category is zero, which is hardly biologically defensible although it is widely applied. Because dynamics is so important when considering influenza and especially its interaction with seasonal forcing, it is necessary to use the more realistic multistage categories. This adds to the size of the model, but importantly, it does qualitatively change the dynamics (Appendix A). Essentially, our model is of the SIRS (susceptible, infected, resistant, susceptible) type but with most categories represented by sequential stages: for example, the I (infected) category in Figure 1 is broken down into four sequential stages.
There are many ways in which seasonality could impinge on influenza dynamics. We choose one of the simplest. A virus compartment allows effects of environment to be represented on virus lifetime, which might be an important environmental forcing mechanism. Some simulations are presented for the unrealistic situation in which recovered individuals are immune for life, in order to illustrate the basic characteristics of the model. Also, simulations are given for the more realistic situation where immunity is lost over a few years. For this latter situation, seasonal forcing gives rise to an unexpectedly wide range of pertinent dynamics, including regularly spaced epidemics from two per year through one per year to one epidemic every several years (two or more), sometimes with slightly chaotic spacing but sometimes regularly spaced but with chaotic amplitudes, and sometimes quite chaotic in both spacing and amplitude.
Our main objective is to show how, in a mechanistically-oriented model with credible biological assumptions and minimal parameter adjustment to obtain specific outcomes, seasonal forcing functions of different magnitudes can constrain, entrain, and amplify the natural rhythms of influenza, giving rise to a wide range of epidemic/pandemic patterns, from biennial, annual, at intervals of several years, and chaotic. Our novel contributions are first to suggest an explicit direct mechanism for the effects of weather on influenza dynamics and then give simulations that show that such mechanisms can have a profound but realistic effect on the dynamics of influenza epidemics. This suggests that the approach could be an important (but hitherto neglected) part of influenza models and planning tools.
2. Methods and Modelling
2.1. Model Scheme
The scheme is drawn in Figure 1. State variables are denoted by n + subscript. There is no age stratification. There are eight categories of persons: susceptible (sus); infected, which is considered as four sequential stages (); clinically sick, also considered as four stages of sickness (); recovered or immune (rec); asymptomatic (asy), which branches off after the first infected stage and has seven stages before recovery is achieved; hospitalized (hos) or isolated, which branches off after the first clinically sick stage and has three stages before recovery is achieved; dead from influenza (ded). Note that clinically sick persons, whether hospitalized or not, may die from influenza, or recover. There is a delay (day) during which recovered persons can lose immunity and return to the susceptible compartment—persons in this delay pipe, represented by ten sequential stages, are denoted by , to . Infectious persons give rise to virus particles (), which, while they may be inactivated or killed, can give rise to further infection events.
We choose, although this is not the usual procedure in influenza models, to have a virus pool (). The reader may wonder why? First, we are fairly sure that such a pool must exist. Second, with a virus pool it becomes easier to think in concrete terms of the effects of weather variables (air temperature, relative humidity, wind speed, and radiation) on the virus, for example, its viability and longevity ((15), (18)). An alternative to having a virus pool () would be to assume that weather affects transmission rate β directly. In the equations to follow, transmission rate β and virus pool always appear multiplicatively ((2), first equation of (3), (20)). Thus, it perhaps makes little difference whether we have a virus pool which can be modified by weather, or no virus pool and simply assume that weather affects transmission coefficient. We take the view that transmission is a multistage process and that the components of weather may impinge on different parts of this process. It may then be helpful to have an explicit virus pool. Note that our transmission rate β has units of virus−1 day−1 (rather than the customary day−1) (Table 1). Also, in the expression for the basic reproductive ratio (20), β is divided by the virus mortality rate. In fact, the first term on the right side of (20) can be viewed as a traditional transmission rate with units of day−1, and the modulation of virus mortality ((15), (18)) may be considered as modulating the traditional transmission rate.
Initial values and parameters are listed in Table 1, although some parameters which are only used once are defined in or after the equation where they appear.
All routes from susceptible to recovered (Figure 1) pass through eight pools. We have assigned the same value of 2 day−1 to the four rate constants , and (Table 1). This gives a mean transit time from susceptible to recovered of 4 days (8/2) with simple gamma-type distributions of transit times applying to the whole path and its components (last paragraph of Methods section; Appendix A; e.g. , pp. 818–822).
Because of the importance of the assumption of sequential pools for giving biologically realistic dynamics (whereas the traditional assumption of a single pool gives biologically unacceptable dynamics ), a discussion of sequential pools in relation to the gamma function is given in Appendix A. In Appendix A, it is shown that the use of two sequential pools give qualitatively different dynamics than a single pool also that three sequential pools gives again a qualitatively different result than two pools; with three or more sequential pools, the dynamics only changes quantitatively. Although observation and data clearly support the existence of a minimum time span being required to traverse a given clinical category, which can be represented by sequential pools as is done here, measurements in this area are extremely difficult. Where the data do not speak clearly, for example, as to whether one should use 3 or 4 pools, or 7 or 8, we have made simplifying and convenient assumptions in order that we could proceed with the calculations (see Section 2.3).
uses a model of similar type but with a simpler structure. The authors employ least squares to estimate most of their model's parameters for the spring and autumn waves of the 1918 influenza epidemic (their Table 1). They find substantial differences between some of the parameters for the spring and autumn epidemics, which may raise questions about what such parameters describe. Our parameters (Table 1) are generally compatible with those obtained and applied in , allowing for differences in model structure.
All state variables (Figure 1) and all terms in the differential equations scale linearly with population size and the size of the virus pool, so that solutions (expressed proportionately) are independent of population (, Table 1).
2.1.1. Some Definitions
First, define some totals in terms of the state variables (Figure 1): The notation of the first five equations is obvious. Total live population is given by the sixth equation and is assumed equal to the fertile population (seventh equation); both of these are equal to the city population, ((2) and following paragraph). Finally, total infected population is obtained by adding together the inf, asy, sic, and hos categories. All the variables in (1) can be turned into fractions by division by .
State variables are next treated by categories and pools.
The differential equation for is The two principal inputs are first from births with the number of fertile persons given by (1) and second from the output of compartment 10 of the delay pipe (Figure 1) where people recovered from influenza are slowly losing their immunity. μ is the natural birth (and death) rate (Table 1). It is assumed that all births are free of infection and are without immunity. A small additional input to the pool is included, . This is the death rate caused by influenza (Figure 1). is given by (16). This is done so that the live person number, (1), remains constant at value (Table 1). This makes the results easier to check as true steady states can be obtained. It is of no biological significance as over an epidemic, deaths from influenza are typically less than 1% of those from natural mortality (the − term in (2)). Also, the instantaneous death rate from influenza, (16), is transient, and even at its maximum value, is usually less that the death rate from natural mortality. This assumption of equal birth and death rates was also made in .
The two outputs (negative terms) are natural death and infection, the latter giving a transfer to the first infected pool. The natural death rate μ is assumed to be the same as the birth rate (Table 1). β is the infection transmission rate (Table 1). This is multiplied by the number of susceptibles , the virus quantity and is divided by the number of live persons (1). All terms in (2) scale equally with population size.
The differential equations for the four sequential infected pools are In the first equation, the input of infecteds is the last term in (2). All pools suffer equally from natural death at rate μ. The output term from the inf1 pool is partly asymptomatic with fraction (Figure 1, Table 1; (7)), the remainder entering the inf2 pool (second equation, first term). The third and fourth equations are straightforward. Rate constant is 2 day−1, such that with four pools the mean time from infection to clinical sickness is days . This gives a gamma-distributed lag for the overall exit time from the fourth pool (Appendix A, (A.4), Figure 9.)
2.1.4. Nonhospitalized Clinically Sick
Persons who become clinically sick enter the sic1 pool (Figure 1). A fraction (Table 1) of these may be hospitalized or isolated, the remainder continuing to recover from the illness at home. Some of the clinically sick (whether in hospital or not) will die from influenza. The differential equations are These equations are similar to (3), but the natural death rate μ is augmented by flu-induced deaths at rate (Table 1). The time between the onset of sickness and recovery is similarly (to the pools above) gamma-distributed (Appendix A, Figure 9, (A.4)).
The rate at which individuals are admitted to hospital, (Figure 1), is with units of persons . This is often a recorded statistic.
The total flu-related death flux is (input from all four sick pools to the dead-from-influenza box of Figure 1; units: persons )
2.1.5. Asymptomatic Infecteds
The differential equations for these seven pools (Figure 1) are Asymptomatic infecteds are fed from (3) (first equation, 2nd term on the right side). Seven pools are employed so that the path from susceptibles to recovered (Figure 1) traverses eight pools in total.
2.1.6. Hospitalized Clinically Sick
These represent hospitalized or isolated or specially treated clinically sick. The differential equations are The input term represents hospital admissions (5), a recorded statistic, which may be useful for comparing with data. The flu-induced death rate (Table 1) is assumed to be the same as that for the nonhospitalized clinically sick, (4). This may be justified in that the fraction taken into hospital is more ill, but they then receive better care.
Total flu-related death rate from the three hospitalized clinically sick pools is
2.1.7. Recovered and Immune
The state variable is governed by the equation The positive (input) terms are from the last equations of (7), (4), and (8). Death occurs at its natural rate (μ). The rate constant is set to zero if it is assumed that immunity is not lost; otherwise it is set to a nonzero value (e.g., 1 day−1, Table 1, the precise value is unimportant as long as it is of order of 1 day−1 or more). A non-zero causes rapid entry into the delay sequence of pools (Figure 1) which leads to loss of immunity.
2.1.8. Delay Pipe Representing Loss of Immunity
Loss of immunity might arise from loss of immunological memory, or from drift and shift in the antigenic character of the virus . This process is represented by ten sequential compartments giving a gamma function delay (Appendix A; also e.g., , pp. 818–822). If (day) is the time period during which immunity is lost (three years is assumed; Table 1), and (day−1) is the rate constant out of each of the ten compartments, then The standard deviation in the exit times is and the coefficient of variation is 1/√10 () (A.7). The differential equations for the pools are The first term on the right of the 1st equation () is the last term in (10); the second term () is transfer to the next compartment; the last term (μ) is the natural death rate. The 2nd term on the right of the last equation () is transfer into the susceptible pool (2).
2.1.9. Virus Pool
It is assumed that virus production, , which is a surrogate for “infectious strength”, is calculated with is a production rate constant which is applied to weighted infected persons. The value above gives a basic reproductive ratio (20) of 4.78 (see Section 2.3). The weighting factors of (13) are () The precise values do matter but they are not important, so long as the values are reasonable. In the usual SIR model there would not be multiple stages as here and any weighting factor would be implicitly unity. In (14) infectivity increases a half day after infection, reaches a maximum, and then decreases as recovery takes place. It is stated in , with reference to , that “the standard pattern of an influenza A virus in adults is characterized by an exponential growth of virus titre, which peaks 2 to 3 days postinfection (DPI), followed by an exponential decrease until it is undetectable after 6 to 8 DPI.” of (13) is the input to the virus pool (15).
The outputs from the virus pool (15) are death by natural mortality (, Table 1) and death induced by a suboptimal environment, represented by rate constant (18). The value of corresponds to a half-life of free virus of about 3 h . No extra virus death process is ascribed to the contact/infection process with persons (whether susceptible or immune). Thus, when the basic reproductive ratio is calculated in (20), this is proportional to β, and there is no β in the denominator. Such virus death processes are subsumed in the natural mortality term . The differential equation for the virus pool is
2.1.10. Flu-Related Dead Pool
The inputs to this pool are from the sick pools with (6) and (9), so the total input to the pool (Figure 1) is There are no outputs. Therefore, Note that, in order to maintain the total population constant as a mathematical convenience, the birth rate of susceptibles is augmented by the death-from-influenza rate (2). As explained after (2), this has a negligible effect on the performance of the model.
2.1.11. Environmentally induced Virus Mortality and Seasonal Forcing
The environmentally dependent function in (15) (Figure 1) is assumed here to depend only on daily mean values of air temperature, . Possible influences of relative humidity , radiation, or wind speed are ignored but could be similarly treated. In a study of relative humidity and temperature on virus transmission, remark that “although the seasonal epidemiology is well characterized, the underlying reasons for predominant wintertime spread are not clear.” The rate constant for environmentally induced death, (; Figure 1) is written as is used to switch environmental effects off (0) or on (1). Air temperature above a threshold increases virus mortality according to power . Parameter is assumed equal to two giving a quadratic dependence of temperature above the threshold (Table 1). It is usual for the biological effects of temperature to be nonlinear, sometimes approximating to exponential, as in the use of a factor for the consequences of a 10°C temperature rise on chemical reaction rate, or the application of the Arrhenius equation for chemical reactions (e.g., , pp. 103–105). is a rate parameter. We did not find controlled-environment studies on the effect of temperature on virus longevity which we could use, and therefore the values assigned the parameters are estimates.
In southern Britain, daily mean air temperature varies from c. 3 (January) to 17°C (July) (, p. 270; p. 142). It is assumed that varies sinusoidally, with Annual mean and seasonal variation are and . is the Julian day number (1 on 1 January, leap years are ignored). (d) is the phase of the sinusoid, which is maximum on 25 July. Combining equations (19) with (18) modifies environmentally induced virus death rate, , and hence reproductive ratio (20). The same seasonal pattern is applied every year. These equations for temperature are a good approximation to long-term weather means in the UK (Note: the study reported in suggests that, on an annual timescale, cold weather does not predict winter deaths, but, on a shorter term timescale, cold weather could be a significant trigger).
2.2. Basic Reproductive Ratio, R0, and the Disease Generation Time, (Day)
Two important epidemiological parameters are basic reproductive ratio, , and the disease generation time, (day). These are both derived from the basic parameters of the model. is defined as the number of infections directly caused by a single infected individual during its infectious period when in a population of susceptibles. is the average time it takes the direct infections which contribute to to arise. is also called the “serial interval”, the average time between the primary case and secondary cases.
In the absence of forcing (see previous section, in which case can only be calculated numerically), can be calculated analytically by travelling round the infectious loops in Figure 1 and adding the terms together. This leads to (see Appendix B for an alternative equivalent statement of ) If (20) is multiplied out, each term corresponds to one of the 18 weighting factors in (14) (4(inf) + 4(sic) + 3(hos) + 7(asy) = 18) and represents one complete infective loop passing through the virus pool in Figure 1 (see Appendix B, (B.1)– (B.4)). The first term is the simplest loop, passing from to and back to (Figure 1). The first term is Start with a single virus particle in the box (we could equally well start with a single person in the inf1 first-infected compartment). The first factor of (21), with units of persons per virus particle, is the probable number of persons infected by a virus particle during its life; the mean lifetime of a virus particle is . It is assumed that there is no additional virus death process due to exposure. is a dimensionless weighting factor (14). The third factor () is the number of virus particles produced per person in the inf1 state during his life: is the average lifetime of a person in the inf1 state. The last factor, with in the denominator, is the probability that this lifetime (of ) is actually achieved. We can continue in this way via all the compartments which can give rise to infection (i.e., produce virus particles (13)), adding up the terms. (20) for can be written out as a sum of the 18 (potentially) contributing terms (Appendix B)—an equivalent formulation which is sometimes useful.
Equation (20) can be (and was) checked numerically by placing a single infected individual into the box and diverting the (primary) infected individuals into an accumulator, rather than allowing them to enter the box, where they can lead to secondary infections. The algebraic and numerical methods agree to six decimal digits, giving a basic reproductive ratio for the default parameters without forcing.
A “dynamic” reproductive ratio, , can be calculated when the total population is not entirely susceptible by means of The live population, , is given by (1). This allows approximately for a slowly changing fraction of susceptible individuals. However, may itself be dynamic on a shorter time scale due to seasonal effects on virus death rate (see previous section).
The disease generation time, (day), can be computed analytically by (B.7), giving day. A numerical calculation using Runge-Kutta integration gives also day agreeing with the analytical result to nine decimal digits, although Euler integration gives day (in each case the integration interval is 1/32 day; see first paragraph of Section 3). Due to overlapping generations, is not equal to the time constant at which the total infecteds increase. Assume that total infecteds (1) increase initially at an exponential rate according to where (day) is the growth time constant (the proportional growth rate of infecteds is day−1). This depends on both and and the structure of the model [6, Section 1.2.3]. This can be extracted numerically (there is a period of constant exponential growth that lasts for some 10 days) to give τ = 1.29 day. This is considerably less than the generation time of 2.51 day.
Parameters have been introduced while developing the model. Their values are listed for reference in Table 1. Here we summarize the evidential basis for the parameter values used.
As a preliminary, statements from two physicians are quoted. In a general practitioner in the Doncaster (UK) area describes his study of the 1969-1970 pandemic as it affected his urban practice. He said “the true incidence of influenza during an epidemic is probably impossible to assess.” This is perhaps equally true today and sets the scene for the significance of processes such as “validation” and data fitting (see Figure 5 and Section 3.4). Another general practitioner, this time in Kent (UK) , states that “an epidemic of influenza tends to last in this area for between two and three months. Beginning slowly the epidemic reaches its peak in four to five weeks and then subsides slowly. The extent and severity of any attack will depend on factors such as the strain of influenza virus, on the state of the host-immunity of the population and on the timing of the epidemic; the fatality and complication rates are always higher during the cold and foggy winter months.” His statement agrees with many of our seasonal-forcing simulations (Figures 6 and 7).
Five key epidemiological quantities for influenza are (a) the time which elapses after the infection event until the subject becomes infectious to others, denoted by the (day); (b) the latent period, namely the time which elapses between the infection event and the appearance of clinical sickness, denoted by (day); (c) the infectious period (day), which is the time during which the subject is infectious (Figure 1—producing virus); (d) the period of immunity (day)—the time period after recovery from influenza during which the subject has immunity, before gradually losing it and returning to the susceptible pool (Figure 1, is represented by the box at the bottom of the diagram with 10 sequential pools); (e) the basic reproductive ratio (dimensionless) (20).
Addressing these quantities, first consider the time period between the infection event and becoming infectious, (day). With the infectivity weighting factors in (14), the first infected pool with a mean lifetime of 0.5 day is not infectious but the second infected pool is, and therefore Next the latent period (day) is given by (there are four sequential pools in the infected category of Figure 1) The infectious period (day) can be estimated as follows. Note that (a) all paths from the susceptible category to the recovered category in Figure 1 pass through eight pools with the same outgoing rate constant (natural mortality excluded); (b) it is assumed that the disease-related rate constants , and are all equal (Table 1), say to ; (c) the first infected pool () is assumed not infectious and all sick and hospitalized pools are infectious (14). We therefore write the infectious period as Loss of immunity occurs during a time period of , assumed to be 3 years. This delay is represented by (Figure 1) 10 sequential pools each having an outgoing rate constant of (day−1) (ignoring natural mortality (12)). This gives a gamma-distributed delay (Appendix A, (A.5) to (A.8) with ). and are related by Next compare the values in (24)–(27) with the literature. Our value of day (24) can be compared with that of , who referring to , use a rather different value of 1.9 day. However, the value given in is based on fitting a homogenous-mixing deterministic SEIR (susceptible, exposed (meaning infected but latent), infected (meaning clinical), resistant) model to the excess pneumonia and influenza deaths in 45 cities during the 1918 pandemic. use a value of 0.5 day in their model, without citing a specific source. Our value could be doubled to 1 day by taking in (14), a relatively small change to the model. Both 0.5 and 1 day are compatible with the clinical evidence, although not with the 1.9 day value of .
Our latent period of = 2 day (25) is both clinically acceptable and is not very different from 's (fitted) values of 2.4 and 3.6 days for the spring and autumn waves of the 1918 epidemic. Note uses the term “latent” to describe the period between the infection event and the time at which the person becomes infectious, rather than our use (25) which refers to the period between the infection event and the time at which the person becomes clinically sick.
report that the “infectious period” lies in the range of 6 to 10 days, but then their single infected pool represents everything in our Figure 1 between the susceptible and recovered categories which makes meaningful comparison difficult. states that “for simplicity, we do not explicitly model the exposed population but instead include people infected but not yet infectious in the “I” box. Including an exposed class yields similar results.” The last sentence is not supported by simulations. Students of dynamical systems may find this statement surprising. However, their range of 6 to 10 days is very different from our 3.5 day (26) which is compatible with the clinical evidence and also the 2 or 3 day period obtained by when fitting the spring and autumn waves of the 1918 pandemic.
The loss of immunity of recovered patients is not a feature of the model in . uses a range of 4 to 8 years: we were not able to chase their values to earth. We employ 3 years, as in (27) for , which, while clinically reasonable, should be regarded as a guess at what is probably a rather variable quantity.
The basic reproductive ratio (20) is hardly a clinical or easily observed quantity, but it is much loved by modellers and epidemiologists, not without reason. It is an “emergent” parameter of the model, as its value depends on underlying parameters. Because influenza is highly seasonal, it can be concluded that the environment is important, yet values of given in the literature rarely say anything about the environment. gives its range as 4 to 16. says that “estimates vary widely, varying from 1.68 to 20.” After fitting the Geneva data for the 1918 pandemic, gives values of 1.49 and 3.75 for the spring and fall waves. states “Assuming a basic reproduction number of = 2.5 and using the standard parameter set of InfluSim” . We assume that this value of 2.5 is the outcome of their standard parameter set. Other values given are 1.2 to 2.4 [7, Table 2] and 2.07 (, after their Figure 3). Since many of these models use biologically inappropriate assumptions , it is relevant to quote the statement from that “ignoring the latent period or assuming exponential distributions will lead to an underestimate of and therefore will underestimate the level of global control measures … that will be needed ….” reviews values of several flu epidemics and pandemics, focussing on the possible control of an H1N1 epidemic. Our standard parameter set without any environmental forcing gives a value of = 4.78 (20). This can be regarded as an upper baseline applying to an optimum environment (one maximizing ), and changing the environment can only decrease this number (Figure 6).
Finally, we wish to comment on the number of pools used to represent the infected and sick categories in Figure 1, where there are four in each. Using trajectory matching on an influenza outbreak at an English boarding school [43, 53] suggests that two pools are appropriate for each category. Unfortunately they give few details of their procedure. We found a good fit (Figure 5) using the current model to fit the same data. We therefore continued to use four pools, although arguably, whether or not two, or three or four pools, or even a nonintegral number of pools are applied is perhaps less important than the principle of applying two or more pools.
This section reports simulations of the model described above. First we describe the general technical aspects applied in the simulations.
3.1. Numerical Methods
The model was programmed in ACSL (Advanced Continuous Simulation Language, Aegis Research, Huntsville, AL, USA; version 11.2.2 for DOS), an ordinary differential equation solver. In all simulations, equations are integrated using Euler's method, a fixed integration interval of Δt = 0.03125 = 1/32 day (45 minutes), and results were communicated for plotting at half-day or daily intervals. There were no difficulties with model implementation. Not unexpectedly, in some of the chaotic simulations, different (but still chaotic) results were obtained if different integration methods and intervals or different Fortran compilers were used (the results shown used the Watcom compiler).
The simulations focus on the daily hospital admission rate, (persons day−1); Figure 1), as this is often a recorded statistic. An “epidemic” is deemed to have occurred if shows a maximum (in steps of Δt) and if at the maximum persons .
State variable initial values and parameter values are as listed in Table 1 (unless stated otherwise). In general, the parameters have not been tuned for any particular performance (but see Figure 5) and as far as possible have been estimated mechanistically (see Section 2.3). Some of the simulations (e.g., Figures 2, 3, and 4) are intended to illustrate important characteristics of the model—they are not intended to be compared with actual epidemics/pandemics. Other simulations (e.g., Figure 5, parts of Figures 7 and 8) are intended to demonstrate at least a partial realism and to provide credibility to the model.
3.2. Dynamics without Intrinsic Loss of Immunity and without Forcing
Here the model is exercised with parameter ((10), Figure 1, Table 1), so that there is no loss of immunity as represented by the delay box in Figure 1, and without environmental forcing (18). In this case, the natural death and birth rates (μ of Table 1 and (2) to (12)) lead to a slow loss of immunity at the population level as births are assumed to be susceptible.
3.2.1. Short-Term Dynamics
These are illustrated in Figure 2(a) and are unexceptionable. From initial infection, it is two to three weeks before the disease is visible, and the epidemic is over within a further two weeks. Infected number () peaks a few days before hospitalized numbers (), both then falling to very low values. Most (99%) of the population joins the recovered (immune) box (), with 1% escaping infection altogether. 90% of those infected travel via the asymptomatic route (Figure 1; , Table 1, (7)). The epidemic in Figure 2(a) is short compared to UK experience, but Figure 2(a) is for the no-seasonal-forcing situation where initially the population is 100% susceptible, which is not comparable with actual epidemics where seasonality is always a factor as is partial immunity. Later (Figure 5), it is shown how the model, without significant “tuning”, is able to fit data on a UK epidemic. Also, seasonal forcing lengthens the period of the epidemic and decreases the fraction of the susceptible population which becomes infected (Figures 7 and 8).
3.2.2. Long-Term Dynamics
These are illustrated over 100 years in Figure 2(b). On this time scale, the first epidemic, shown in Figure 2(a), occurs at zero time and rapidly dies out with the dynamic reproductive ratio (22) (not plotted) falling to near zero, as the fraction of susceptibles () becomes small. In the first epidemic, at time day, the initial peak in hospital admissions (Figure 1, (5)), decreasing to less than 100 at the second epidemic. After the first epidemic, (22) slowly recovers as immune numbers () decrease and susceptibles () increase due to the natural birth and death processes. The second smaller epidemic occurs after a further 28 years, followed by epidemics of decreasing amplitude and increasing frequency until a steady state is reached with fractions of susceptibles of 0.209, of all four infected categories together (infected, asymptomatic, sick, hospitalized, Figure 1) of 0.00014 and of recovered (immune) of 0.791; there are 0.18 hospital admissions per day, and (22) = 1.
3.2.3. Responses to Key Parameters
Figure 3 illustrates the effects of changing three key parameters: infectivity parameter β (Figure 1; (2); Table 1), rate parameter (with ; Figure 1; Table 1; (3)), and the initial (time zero) immune fraction, . Figure 3(a) shows how the duration, initial proportional growth rate, and severity (indicated by hospital admissions, HA) of a modelled epidemic are influenced by the value of infectivity parameter β (or equally ; both are linear factors of (20)). Indeed, the effects of increasing β are monotonic, moving the epidemic towards shorter time scales, increasing initial proportional growth rate, total hospital admissions (HA) towards an asymptote of 5000, and narrowing the width of the epidemic. Also, increasing β causes the long-term steady state to be more quickly attained—the spikes of (e.g., Figure 2(b)) becomes closer together.
Figure 3(b), where the rates of transit of infected persons through the system are increased (, Figure 1), is less straightforward. Here a low value for the (e.g., 0.25 day−1) gives a high basic reproductive ratio (20) (Appendix B) an epidemic which is slow to take hold but (surprisingly) moves more rapidly towards a long-term steady state (i.e., the spikes as in Figure 2(b) are closer together). As the are increased basic reproductive ratio decreases (20) first the epidemic becomes narrower and faster (e.g., day−1), but further increases in (e.g., and 8 day−1) cause the epidemic to become slower to take hold and less peaked, with fewer hospital admissions (HA). Increasing always causes the spikes (Figure 2(b)) to move further apart and lengthens the time taken to reach a steady state and decreases the number of infected persons in the steady state (,(1)).
Finally, Figure 3(c) illustrates how increasing the initial immune fraction has a similar effect to that of decreasing β: delaying the onset of the epidemic and increasing its width, decreasing the initial proportional growth rate, and decreasing severity (number of hospital admission, HA).
3.3. Dynamics with Intrinsic Loss of Immunity and without Forcing
Now the discrete delay box of Figure 1 giving loss of immunity is switched on by making day−1, which causes recovered individuals to be moved quite rapidly into the delay sequence of ten compartments where immunity is lost after three years (; Table 1, (11)). Immunity is also being lost due to the natural birth and death processes, as newborns are assumed to be susceptible. The responses of Figure 2(a) are little changed by taking day−1 (instead of 0) if the variable of Figure 2(a) is replaced by the variable (1). However, over a longer time period Figure 4(a) illustrates that there is now a switching of susceptibles between a low (fractional) value and a high (fractional) value (fractions of , Table 1, (1) and following paragraph), with an inverse switching of numbers in the delay compartment, (Figure 1). Figure 4(b) shows that the oscillations, as in Figure 2(b), decrease in amplitude and increase in frequency until a steady state is attained. In the steady state, there are 3.7 hospital admissions per day, a dynamic reproductive ratio = 1 (22), and the fractions in the three principal categories (Figure 1) of susceptible; all infected categories lumped together (infected, asymptomatic, sick, hospitalized, (1)) and the delayed categories are 0.209, 0.0030, and 0.787. Comparing these values with the situation in which there is no intrinsic loss of immunity (Figure 2(b)), it is seen that hospitalizations have increased by a factor of 20 as has the fraction in all infected categories, although the fractions of susceptibles and recovered or immunes have barely changed.
“Validation” is often a misused and misunderstood concept and is perhaps better described as an evaluation of applicability. Validity is not a property of a model alone, neither is it a “zero or one” concept. It describes the relationship between model predictions and a set of data obtained under prescribed conditions. In this section, the model of Figure 1 is “validated” by fitting the predictions of the model, with minimal parameter adjustment (a perfectly formulated mechanistic model would permit no adjustment of parameters or initial values) to data on an influenza epidemic . Success in this endeavour gives the model some credibility, although it does not make the model valid for general use.
The data in relate to an influenza outbreak at an English boarding school, which provides a simple situation that seems comparable to the single-city homogeneous-mixing model of Figure 1, remembering that the model does not have the stratification which might be needed if a larger region was considered.
Minimal tuning is applied. One infected person is introduced to the school at noon on 18 January, otherwise all are susceptible. Table 1 parameters are altered for the Figure 5 simulation as follows: infection parameter β is changed to 0.07 (virus units)−1 day−1; all birth and death rates are set to zero () for such young persons; total population ; and the asymptomatic fraction, , is one-third, to reflect the finding that only two-thirds of the boys became sick, and the community is assumed to be “well-mixed”. There is no initial immunity. With these values, the basic reproductive ratio (20) is 6.44, the mean generation time is 2.65 day (A.7), and the initial proportional growth rate of total infecteds, , is 1.015 day−1.
For comparison with the data in , we define the number of persons confined to bed, , as Since this definition is somewhat arbitrary, in comparing the predicted from the model with the data from Anon (1978), is scaled with an adjustable factor so that the comparison line drawn in Figure 5 is Fitting was done by eye, as this can produce (see below) a better focus on the biological significance of the parameter being adjusted and possible limitations in the biological data that may be obtained with more automated methods. See for possible problems arising from formulaic parameter adjustment in mechanistic models.
The degree of fit in Figure 5 is satisfactory. Apart from the two outliers on 26th and 27th January, the fit is good. In an actual epidemic, there may be underreporting during the early states and overreporting later, as the performance of those handling the epidemic changes. Overall, we believe it is reasonable to assume that the model has some credibility as a result of this validation.
Finally, a comment on whether the number of pools used in the infected and sick categories in Figure 1 is appropriate. fits an SEIR model to the observed data given in and shown here in Figure 5. They minimize the sum of the squared errors, arguably this underweights the skirts of the distribution, which are sensitive to the numbers of sequential pools assumed (Figure 9). They assume pools for the E category and pools for the I category. They find a best fit with and . We were not able to discover the details of their general parameterization or indeed how they define a confined-to-bed number from the categories and pools of the SEIR model. They remark that there is a sensitivity to the number of points used to obtain the fit and that basic reproductive ratio can change substantially. More notably, it can be seen in their Figure 3(c) that the model fits the last three data points as the epidemic is subsiding rather poorly. This suggests (see Figure 9) that a higher number of pools is required than their best values of 2 for the exposed and infected categories. In view of these difficulties, and the comparison shown in Figure 5, we consider that the number of pools per category suggested in Figure 1 and used throughout this paper is reasonable. Although our particular choice cannot rigorously be defended, it seems to be “good enough” at the present time.
3.5. Dynamics with Intrinsic Loss of Immunity and Seasonal Forcing
Now we add direct seasonal forcing by weather to the simulations illustrated in Figure 4 ((18), (19)). Four weather factors which could impinge on virus longevity are air temperature (), relative humidity (), radiation (possibly multicomponent), and wind speed. The effects can be highly complex: for example, which examines the effects of and on virus transmission, finding that cold dry conditions favour transmission (but see their Figure 6). The topic is far from being well understood, but since our concern here is to represent broadly weather forcing within the model, we make the simplifying assumption that alone is operative.
Figure 6(a) illustrates the seasonal variation of mean daily air temperature in the southern UK. Using (19) with (18), and , this affects environmentally induced virus death rate (Figure 1, (15)), and thereby basic reproductive ratio (20). Mean daily air temperature varies between 3°C (Jan 24) and 17°C (Jul 25). When crosses the temperature threshold of = 13°C in May, this increases environmentally induced virus mortality, (18) and decreases basic reproductive ratio (20). With this formulation, influenza is most likely during the months of October to April when is highest.
Figure 6(b) shows how varying (e.g., increasing) mean annual air temperature, (19), (or equivalently, decreasing threshold temperature , (18)) varies the duration and intensity of seasonal forcing of . Changing mean annual air temperature, (19), changes the average annual value of , , as well as its maximum, minimum and amplitude (maximum-minimum). The dependence of these quantities on is also shown, together with the number of forcing days per year (, a forcing day is one on which mean air temperature is above threshold temperature and virus mortality is reduced). With mean annual air temperature , there is no forcing at all (): air temperature (19) never exceeds threshold temperature = 13°C (18); is invariant at its maximum value. Environmentally induced virus death rate ((15), (18)) stays at zero; the system remains in the steady state shown in Figure 4(b) and (this situation is equivalent to and = 17°C). As mean annual air temperature increases above 6°C, there are more days in the summer months when mean daily temperature > threshold temperature , number of forcing days per year increases, and . The amplitude increases to a maximum when, before decreasing. With , influenza epidemics do not occur, because the annual average of is less than unity.
In the simulations presented in Figures 7 and 8, various values of mean annual air temperature are taken between −4.5 and 16°C, assuming always an annual variation of ±7°C (19) as in the UK. Otherwise, parameters have the values in Table 1. In these simulations, the long-term steady state reached in Figure 4, with loss of immunity (; Figure 1; (10)) but without seasonal forcing, is used for initial values. Forcing is applied after one calendar year. The aim is to illustrate the wide variety in dynamic behaviour which results from this type of seasonal forcing (which decreases the reproductive ratio). In each case, the mean value, its maximum, minimum and amplitude (maximum-minimum) of basic reproductive ratio can be read off Figure 6(b). In the UK the influenza season is considered to be over by May, when the mean daily temperature ranges from 10.7 (1 May) to 14.1°C (31 May) (Figure 6). Therefore, of the simulations described in Figures 7 and 8, those given in Figures 7(b), 7(c),7(d), 8(a), 8(b), and 8(c) are more relevant to the UK.
Figure 7 illustrates the effects of low levels of seasonal forcing on influenza hospital admissions, (Figure 1,(5)). Forcing is seen as a downward modulation of the basic reproductive ratio , which decreases the annual average, .
The lower graph in Figure 7(a) shows that is slightly decreased beginning on 24 June, from 4.78 in the steady state to 4.69 on 25 July. Influenza hospital admissions (the upper graph in Figure 7(a)) oscillate twice about the steady state value (Figure 3(b); 3.7 admissions ) in a 12-month period with the two steady-state maxima 168 days apart on 27 September and 14 March. A regular variation is quickly established. Note that the natural response time of the system as indicated by the upper graph in Figure 7(a) is not commensurate with the annual cycle imposed by the environment (shown by the lower graph in Figure 7(a)). This sets the scene for potential chaos.
In Figure 7(b) the level of forcing is increased (lower graph). This results in a more complex (but still regular) schedule of hospital admissions with a biennial pattern superposed on a twice-yearly variation. In monthly data are presented in their Figures 2 and 3; some of these are suggestive of a twice-yearly pattern.
A further increase in forcing (Figures 7(c) and 7(d)) results in chaos, with sometimes one, two or even three peaks in hospital admissions occurring within a twelve-month period. However, there is a tendency towards annual epidemics, with 387 epidemics occurring in 250 years, and the epidemics (10% points) lasting about seven weeks (cf. which gives a duration of two to three months; also cf. Figure 8(a) with annual late spring epidemics of 5-week duration). The susceptible fraction of the population varies from c. 30% before an epidemic to 15% just after each epidemic, so that one half of the susceptibles becomes infected.
Further increases in forcing are shown in Figure 8. First, in Figure 8(a) with mean annual air temperature (Figure 6), there is a transition to an annual epidemic occurring in the late spring of each year lasting for about five weeks. In these annual epidemics, the susceptible fraction falls from c. 35% to 12%. Next (Figure 8(b); ), chaos is again produced (cf. Figure 7(d)) with a strong tendency towards annual epidemics: 204 epidemics occur in 200 years. In the last two years of the simulation shown in Figure 8(b), there is a small epidemic on 22 July, followed by larger epidemics on 5 October and 16 May. Then (Figure 8(c); ), there is chaos but now with a tendency towards biennial epidemics (169 epidemics occur during 200 years). Last (Figure 8(d); ), the system immediately settles down into regular biennial epidemics occurring in early spring of every other year but the amplitudes remain slightly chaotic.
Note that, throughout Figures 7 and 8, as mean annual air temperature threshold is increased, forcing is increased (i.e., the magnitude of the seasonal changes in basic reproductive ratio ((20), (18), (19), Figure 6) increases), but the mean annual value of , , decreases. This causes between-epidemic recovery time to increase (see discussion of Figure 3(a) above). The frequency of epidemics then decreases.
Increases in mean annual air temperature can be continued, and although the situations simulated are now less realistic, they do contribute to understanding the system. With , the response is chaotic with 37 epidemics in the first 200 years. Mostly five or six years elapse between epidemics; occasionally two smaller epidemics occur in the same year (e.g., in the 179th year). The mean reproductive ratio is 2.945. When , the response eventually settles down to a regular pattern with 27 epidemics in the first 200 years and eight years between epidemics and an of 2.499. When the response is chaotic with 15 epidemics in the first 200 years and . Last, with , after departing from the initial steady state, it is c. 110 years before influenza reemerges, and then it is at a low level, eventually settling down to a repeating annual late spring epidemic lasting about two months with a maximum of 0.6 hospital admissions per day and an annual sum of 20 hospital admissions per year. has a modest mean annual value (1.131) and is below unity for much of the year.
Many models of influenza are more empirical than mechanistic, and therefore, although they can be and sometimes have been used to fit historic data , they are of little value in further understanding or for indicating how future epidemics/pandemics might be handled before they occur ([32, 42, 55, 56]).
Seasonality is an important feature in influenza incidence. There are many ways in which seasonality can be incorporated into an influenza model. In the contact rate is reduced by a factor of ten for the 6-month nonepidemic season. Here a simple representation of UK daily weather allows the impact of mean daily air temperature to be explored. A similar approach could be applied to relative or absolute humidity, radiation, and wind speed. Seasonal forcing gives rise to wide range of dynamics, from regular at various intervals, to chaos, as illustrated in Figures 7 and 8.
Some of simulations of Figures 7 and 8 are similar to influenza incidents which have occurred. For example, in Figure 7(d), the three maxima near year 184 in the spring, autumn and following spring have similarities with the waves of the 1918–1920 influenza pandemic . Note that this is achieved within a single simulation without changing parameterization. In comparison, in the authors fitted these data with a model, applying the model separately to each wave, with different parameters and initial values (loc. cit. Table 1, Figure 4). It is legitimate to ask just what this procedure means. The two principal peaks occurring within a single influenza season in the autumn and spring around year 199 of Figure 8(b) resemble the peaks shown in the 1957-58 pandemic [20, Figure 3A]. The first two peaks illustrated in Figure 8(c) in year 182 occur in late spring (the lesser peak) and the following autumn (the main peak), resembling the 1968–70 pandemic .
Apart from Section 3.4 and Figure 5 which applies to an unusually sharply defined context, we did not attempt to fit our model to a wider selection of historical data. The reasons for this include that the model is not sufficiently detailed to make this meaningful in a general context, historical data usually have many lacunae, current understanding of the mechanisms of seasonal impact on influenza is very limited, given the number and nature of chaotic solutions, fitting could be technically difficult, and last, Popper's cogent discussion of historical data-fitting , in which he concludes that such exercises are usually not scientifically productive, seems to be particularly pertinent to this investigation. Regrettably, in spite of all the evidence, “parameter twiddling” and fitting historic data are still highly regarded by some investigators, although it is hard to find examples where such work has led to significant progress. Nevertheless, with simplified “proof-of-concept” models, it is important that predictions should be acceptable as has been shown to be the case with the current model.
Where we part company with many influenza modellers is in our use of multiple pools to represent given categories: that is, four pools are used to represent infected latent persons and seven pools for asymptomatically infected persons (Figure 1) (but see ). Mathematically, this is a trivial addition requiring some extra programming, but it gives three significant benefits. First, progress through the stages of influenza is clearly sequential, suggesting that to use successive pools is biologically reasonable. Second, the overall transit times of sequential pools are gamma-distributed which is arguably more realistic than given by a single-pool representation. Third, the method opens a path to a more mechanistic picture of observations where quantities such as infectivity (14) and death rates can vary from pool to pool within a category. While simple models are best for elucidating many general principles, there seems to be no alternative to more detailed mechanistic (reductionist) models for serious application. With such models, parameters will be more determined by experiment at the assumption level, rather than making parameter adjustments on the basis of comparison of predictive-level data.
A mechanistic influenza model has been constructed in which sequential pools are used for some disease categories, allowing gamma-function-type dynamics with delays, consistent with biological observations. A simplified representation of seasonality is given and is, we believe, the first attempt to include weather explicitly in an influenza model. The model has been “validated” by application to an outbreak of influenza in a school. It has been demonstrated that seasonal forcing gives rise to a rich variety of dynamic disease patterns, from regular with outbreaks at annual, biennial, and longer intervals of time, to chaotic. Some of these predicted patterns seem highly pertinent to mankind's experiences with influenza. It is suggested that seasonality and its effects could usefully be an integral part of influenza epidemiology including the areas of prediction and amelioration. Recognizing that seasonality is important in influenza dynamics, we were surprised by our inability to find more controlled-environment studies of the effects of environmental factors on virus viability or other significant processes in the disease cycle.
A. Sequential Pools and the Gamma Function
The aim in this appendix is to explain how the use of sequential pools to represent a given clinical category affects the transit dynamics for that category. We emphasize first that a realistic mechanistic treatment of infectious disease dynamics absolutely requires the use of several, arguably at least three, sequential pools per clinical category, and second that the traditional approach of using a single pool per clinical category cannot be expected to predict credible dynamics or robust predictions. It is to be noted that the empirical use of a gamma function for a single-pool category (e.g., the entire infected box in Figure 1) is based mechanistically on several stages (in this case four) each with first-order (exponential) kinetics. Although these points will be familiar to many workers, this appendix has been written because they are not always appreciated. A general discussion of the topic can be found in, for example, , pp. 818–822.
A.1. Single Pool
Assume that a given disease category, for example, “infected”, is represented by a single pool, , so the number of pools . This method is used in many SIR models. A single pool emptying at rate from initial value of at time obeys the equations This is drawn in Figure 9: the line labelled “1 pool, ”. It can be seen that the maximum rate at which pool is depleted occurs at time (shown by ●). The mean time for departure is at day = ( (shown by ■). For a single pool, these are far apart. The biological data do not support such dynamics, which would imply, for say the infected category, that it is most probable that an infected person exits the infected category immediately following the infection event. Since our paper is particularly concerned with the detailed dynamics of influenza, including the possible occurrence of chaotic events (here the existence of lags can make a crucial difference to model behaviour), we regard it as essential to depart from the assumption of assigning a single pool to each disease category.
A.2. Two Pools
Next assume the category is represented by two sequential pools, and , where empties into , and is the final pool in the category. The number of pools, . In this case, the relevant equations become This is shown in Figure 9: the line labelled “2 pools, ”. With two pools, has been doubled relative to in Eqns (A.1), so that the mean time for departure from the final pool in the category is the same at time day = (shown by ■). We see that the maximum rate for departure from the infected category now occurs at time day (= ()/) (shown by ●), in contrast to 0 day for the single pool case above. In moving from one pool to two pools, the maximum value at 2 day has shifted towards the mean value at 4 day. There is a large qualitative difference between the 1-pool curve for and 2-pools curve for drawn in Figure 9.
A.3. Three Pools
This case is given in detail as there is another qualitative difference between 2-pool and 3-pool dynamics. Now there are three sequential pools, , , and , with emptying into , into , and is the last pool in the category. The number of pools is . The equations for the system are Now (Figure 9) the curve for the final pool in the category () is sigmoidal at low values of time . The time of the maximum (●) (= (()/) has moved closer to the mean time (■). The mean time is ()/() = 4 day as in the other curves. With three pools, there is now a sigmoid departure from time , giving a more sharply defined biological delay (cf. the two-pool line).
A.4. Four and More Pools
Adding more sequential pools to the 3-pool situation has the effect on the final pool of increasing the sigmoidicity, and moving the time of maximum (●) closer to the mean time (■). At the same time as adding more pools, the exit rate constant of each pool is increased so that the overall mean transit time is unchanged. For four pools, , we have This is drawn in Figure 9. There is only a minor quantitative difference between the 3- and 4-pool sequences when they have the same mean transit time .
For a general number of pools, , with state variables, , and each with outgoing rate constant, , the value of the state variable for the final pool in the sequence, , is where is a constant. The total outflow from the final pool, , is , so that The statistics on the final pool, , are (using to denote expectation values) The 8-pool curve of Figure 9, illustrated by the time course of the final pool in the sequence, , is shown because in Figure 1 every path between susceptible and recovered traverses 8 pools each with rate constant 2 day−1, giving an overall average transit time of 4 day (see main text for further discussion of this point).
Finally we note that although there are several equivalent definitions for the gamma function used by mathematicians and others (e.g., , pp. 819–821), possibly the most intuitive definition for the biologist is that given in (A.6), namely, where the normalized gamma function, , is given by If the number of sequential pools, , and also , with rate constant , where is the overall mean transit time which is constant, then γ(t, m) approaches a Dirac delta function located at time .
B. Basic Reproductive Ratio and Mean Generation Time
An alternative and useful way of writing (20) for is as a sum of the individual contributions from the 4 + 4 + 3 + 7 = 18 diseased pools of Figure 1. Using an obvious notation, these 18 terms are written as, first for the four infected pools, next for the four sick pools then, the three hospitalized pools and last the seven asymptomatic pools Basic reproductive ratio is the sum of these contributions: Generation time (day) can similarly be written in terms of the contributions from the different infectious pools (using the contributions to defined above). First write down the contributions to the total generation time:Finally, summing the above, the mean generation time is
Conflict of Interests
The authors declare that they have no competing interests.
The work was funded, in part, through the Canada Research Chairs Program.
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- A. C. Lowen, S. Mubareka, J. Steel, and P. Palese, “Influenza virus transmission is dependent on relative humidity and temperature,” PLoS Pathogens, vol. 3, no. 10, Article ID e151, pp. 1470–1476, 2007.
- J. H. M. Thornley, Grassland Dynamics: An Ecosystem Simulation Model, CAB International, Wallingford, UK, 1998.
- M. P. Taylor, “Influenza 1969-1970. Incidence in general practice based on a population survey,” Journal of the Royal College of General Practitioners, vol. 21, no. 102, pp. 17–22, 1971.
- J. Fry, “Epidemic influenza. Patterns over 20 years (1949–1968),” Journal of the Royal College of General Practitioners, vol. 17, no. 79, pp. 100–103, 1969.
- B. J. Coburn, B. G. Wagner, and S. Blower, “Modeling influenza epidemics and pandemics: insights into the future of swine flu (H1N1),” BMC Medicine, vol. 7, article 30, 2009.
- Anon, “Influenza in a boarding school,” British Medical Journal, vol. 1, p. 587, 1978.
- K. Popper, The Poverty of Historicism, Routledge & Kegan Paul, London, UK, 1957.
- A. Flahault, S. Deguen, and A. J. Valleron, “A mathematical model for the European spread of influenza,” European Journal of Epidemiology, vol. 10, no. 4, pp. 471–474, 1994.
- A. Flahault, S. Letrait, P. Blin, S. Hazout, J. Menares, and A. J. Valleron, “Modelling the 1985 influenza epidemic in France,” Statistics in Medicine, vol. 7, no. 11, pp. 1147–1155, 1988.
- N. P. Johnson and J. Mueller, “Updating the accounts: global mortality of the 1918–1920 “Spanish” influenza pandemic,” Bulletin of the History of Medicine, vol. 76, no. 1, pp. 105–115, 2002.
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https://enerimlil.firebaseapp.com/479.html
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In addition, discovering the punchline gives a sense of closure and success. What is the answer to punchline algebra book a worksheet 3. Be hardpressed to find a warrior culture in history. We focus on providing answers to questions about the bibleparticularly the book of genesis regarding key issues such as creation, evolution, science, and the age of the earth. What are all the answers for punchline algebra book a page. Factoring polynomials whose terms contain a common binomial factor. Punchline algebra book a 2006 marcy mathworks y 10 when x 2. Vocabulary power plus book 4 answers, algebra help, step by step answers to algebra word problems, math for dummies worksheets, first order linear differential equation calculator. Test of genius pre algebra, integral exponents, online practice, algebra ii answers, 5th grade star test papers algebra, convert decimal number to fraction matlab, hungerford solution pdf, slope. Which was not a result of the baby boom that followed world war ii answers apex. Punchune algebra book a 2006 marcy mathworks 034 x 34 x p true commutative property. Students will be able to graph quadratic functions.
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https://books.google.gr/books?id=8SXPAAAAMAAJ&pg=PA67&vq=segment&dq=editions:UOMDLPabq7928_0001_001&lr=&hl=el&output=html_text&source=gbs_search_r&cad=1
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« ΠροηγούμενηΣυνέχεια »
tained by BE and EF, because EF is equal to ED; therefore BD is equal to the square of EH; and BD is also equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the square of EH: Wherefore a square has been made equal to the given rectilineal figure A, viz. the square described upon EH. Which was to be done.
PROP. A. THEOR.
If one side of a triangle be bisected, the sum of the squares of the other two sides is double of the square of half the side bisected, and of the square of the line drawn from the point of bisection to the opposite angle of the triangle.
Let ABC be a triangle, of which the side BC is bisected in D, and DA drawn to the opposite angle; the squares of BA and AC are together double of the squares of BD and DA.
From A draw AE perpendicular to BC, and because BEA is a right angle, AB (47. 1.) BE + AE2 and
AC2=CE2 +AE2; wherefore AB2 + AC BE +CE2+2AE2. But because the line BC is cut equally in D, and unequally in E, BE2+ CE2 = (9.2.) 2BD+2DE; therefore AB2 + AC2 = 2BD2 +2DE.2AE2.
AE2 (17. 1.) AD", and 2DE + 2AE2 = 2AD2; wherefore AB+ AC2 = 2BD2 + 2AD2,B Therefore, &c. Q. E. D.
PROP. B. THEOR.
The sum of the squares of the diameters of any parallelogram is equal to the sum of the squares of the sides of the parallelogram.
Let ABCD be a parallelogram, of which the diameters are AC and BD; the sum of the squares of AC and BD is equal to the sum of the squares of AB, BC, CD, DA.
Let AC and BD intersect one another in E: and because the vertical angles AED, CEB are equal (15. 1.). and also the alternate angles EAD, ECB (29. 1.), the triangles ADE, CEB have two angles in the one equal to two angles in the other, each to each; but the sides AD and BC, which are opposite to equal angles in these triangles, are also equal (34. 1.); therefore the other sides which are opposite to the equal angles are also equal (26. 1.), viz. AE to EC, and ED to EB.
Since, therefore, BD is bisected in E, AB2+AD2=(A. 2.) 2BE3 +2AE; and for the same reason, CD+BC22BE2+2EC2=2BE2 +2AE2. because EC AE. Therefore AB +AD+DC +BC2= 4BE4AE. But 4BE2BD, and 4AE=AC2 (2. Cor. 8. 2.) because BD and AC are both bisected in E; therefore AB+ AĎ2+ CD+BC BD2 + AC2. Therefore the sum of the squares &c. QE. D.
COR. From this demonstration, it is manifest that the diameters of every parallelogram bisect one another.
HE radius of a circle is the straight line drawn from the centre to the circumference.
A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it.
Circles are said to touch one another, which meet, but do not cut one another.
Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal.
And the straight line on which the greater perpendicular falls, is said to be farther from the centre
An arch of a circle is any part of the circumference.
A segment of a circle is the figure con
tained by a straight line, and the arch which it cuts off.
An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment.
And an angle is said to insist or stand upon the arch intercepted between the straight lines which contain the angle.
The sector of a circle is the figure contained by two straight lines drawn from the centre, and the arch of the circumference between them.
Similar segments of a circle, are those in which the angles are equal, or which contain equal angles.
PROP. 1. PROB.
To find the centre of a given circle.
Let ABC be the given circle; it is required to find its centre. Draw within it any straight line AB, and bisect (10. 1.) it in D; from the point D draw (11. 1.) DC at right angles to AB, and produce it to E, and bisect CE in F: the point F is the centre of the circle ABC.
For, if it be not, let, if possible, G be the centre, and join GA, GD, GB: Then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two sides AD, DG are equal to the two BD, DG, each to each; and the base GA is equal to the base GB, because they are radii of the same circle: therefore the angle ADG is equal (8. 1.) to the angle GDB: But when a straight line standing upon another straight line makes the adjacent angles equal to one another, each of the angles is a right angle (7. def. 1.) Therefore the angle GDB is a right angle: But FDB is likewise a right angle; wherefore the angle FDB is equal to the angle GDB,the greater
to the less, which is impossible: Therefore G is not the centre of the circle ABC In the same manner, it can be shown, that no other point but F is the centre: that is, F is the centre of the circle ABC: Which was to be found.
COR. From this it is manifest that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other.
PROP. II. THEOR..
If any two points be taken in the circumference of a circle, the straight line which joins them shall fal within the circle.
Let ABC be a circle, and A, B any two points in the circumference the straight line drawn from A to B
shall fall within the circle.
Take any point in AB as E; find D the centre of the circle ABC; join AD, DB and DE, and let DE meet the circumference in F. Then, because DA is equal to DB, the angle DAB is equal (5. 1.) to the angle DBA; and because AE, a side of the triangle DAE, is produced to B, the angle DEB is greater (16. 1.) than the angle DAE; but DAE is equal to the angle DBE; therefore the angle DEB is greater than the angle DBE: Now to the greater angle the greater side is opposite (19. 1.); DB is therefore greater than DE: but BD is equal to DF; wherefore DF is greater than DE, and the point E is therefore within the circle. The same may be demonstrated of any other point between A and B, therefore AB is within the circle. Wherefore, if any two points, &c. Q. E. D.
PROP. III. THEOR.
If a straight line drawn through the centre of a circle bisect a straight line in the circle, which does not pass through the centre, it will cut that line at right angles; and if it cut it at right angles, it will bisect
Let ABC be a circle, and let CD, a straight line drawn through the centre bisect any straight line AB, which does not pass through the centre, in the point F: It cuts it also at right angles.
Take (1. 3.) E the centre of the circle, and join EA, EB. Then because AF is equal to FB, and FE common to the two triangles AFE, BFE, there are two sides in the one equal to two sides in the other:
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https://www.nature.com/articles/ncomms3207/figures/2?error=cookies_not_supported&code=6fe5aa1c-663c-4b29-8b64-306a4b3f1057
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(a) A sphere is fitted to a series of transmission images of a zebrafish embryo. The coordinates of the centre (x0, y0, z0) and the radius (R) are determined. A shell of 140 μm thickness around the sphere surface (blue shaded region) will contribute to the projection. (b) The surface of the sphere is divided into vertices (inset). A ray is cast from the sphere centre to each vertex, and the maximum intensity along each ray within the shell is recorded. (c) The resulting spherical maximum intensity projection is then unwrapped to obtain a 2D map of the spherical data (Supplementary Movie 2). Different colours indicate the parts of the embryo that were recorded by the two cameras from two different angles (compare Fig. 1). (d) All endodermal cells spread around the entire embryo are visible in a single image (Supplementary Movie 3). (e) Spatial orientation of the embryo in 3D and on the final map projection. A, anterior; P, posterior; V, ventral; DFC, dorsal forerunner cells.
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http://biology.stackexchange.com/questions/tagged/population-biology+molecular-evolution
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math
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Hardy-Weinberg applied to three alleles and stimation of allele frequencies
I have this equation: Corresponds to HW in equilibria with three alleles: $(p+q+r)^2=1$ Expanding the square results: $p^2+2pq+r^2+2pr+q^2+2qr = 1$ I need to separate homozygous and ...
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https://www.jiskha.com/display.cgi?id=1162331722
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posted by mark .
the radius of a circle when the numeric values of the circumference and the area are equal
Start with what you know. If you know the formula the area of a circle, and the formula for the circumference of a circle, then you can set them equal to each other...from there you should be able to get the radius!!
set A = pi * r^2 and C= 2*pi*r to eual each otherso u get pi*r^2 = 2*pi*r the pi's cancles and the the r^2 is reduced whuch gives r = 2
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https://www.physicsforums.com/threads/surveying-problem-relating-to-circles-lines.727855/
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Hello all I am hoping someone could help shed some light on a surveying problem I am having. The problem is this:- • A circle is centered at point B with Known co-ordinates (X2,Y2) • The circle has a radius which is known (R). • Point A lays outside of the circle with known co-ordinates (X1,Y1) • A line is connected between Point A and Point B. • Point C lays on the line between Point A & B. • Point C also lays at the exact intersection of where the line and the circle meet. • The distance between Point A and Point C is known (S). • All points are in a Cartesian co-ordinate system Work out what are the co-ordinates of Point C. I have attached a diagram of the problem. Can anyone help?
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https://www.shaalaa.com/question-bank-solutions/balanced-chemical-equation-ranslate-following-statement-chemical-equation-then-balance-equation-aluminium-metal-replaces-iron-ferric-oxide-fe2o3-giving-aluminium-oxide-iron_26832
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Translate the following statement into chemical equation and then balance the equation:
Aluminium metal replaces iron from ferric oxide, Fe2O3, giving aluminium oxide and iron.
2Al (s) + Fe2O3 (s) → Al2O3 (s) + 2Fe (s)
Is there an error in this question or solution?
Video Tutorials For All Subjects
- Balanced Chemical Equation
Solution Ranslate the Following Statement into Chemical Equation and Then Balance the Equation: Aluminium Metal Replaces Iron from Ferric Oxide, Fe2o3, Giving Aluminium Oxide and Iron. Concept: Balanced Chemical Equation.
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https://www.hindawi.com/journals/mpe/2016/1468629/
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Research Article | Open Access
Xuan Guo, Xiao Xin Zhang, "Impact Pseudostatic Load Equivalent Model and the Maximum Internal Force Solution for Underground Structure of Tunnel Lining", Mathematical Problems in Engineering, vol. 2016, Article ID 1468629, 16 pages, 2016. https://doi.org/10.1155/2016/1468629
Impact Pseudostatic Load Equivalent Model and the Maximum Internal Force Solution for Underground Structure of Tunnel Lining
The theoretical formula of the maximum internal forces for circular tunnel lining structure under impact loads of the underground is deduced in this paper. The internal force calculation formula under different equivalent forms of impact pseudostatic loads is obtained. Furthermore, by comparing the theoretical solution with the measured data of the top blasting model test of circular formula under different equivalent forms of impact pseudostatic loads are obtained. Furthermore, by comparing the theoretical solution with the measured data of the top blasting model test of circular tunnel, it is found that the proposed theoretical results accord with the experimental values well. The corresponding equivalent impact pseudostatic triangular load is the most realistic pattern of all test equivalent forms. The equivalent impact pseudostatic load model and maximum solution of the internal force for tunnel lining structure are partially verified.
In recent years, the nuclear leakage events have sparked a new national self-examination. The safety and stability problems of the underground structure caused by the impact or blasting load must be paid special attention to. The blasting demolition at the top of underground, chemical explosion devised by terrorism, gas pressure extrusion, and release of shallow layer containing gas and so forth will result in instant huge impact load on the structure of tunnel lining. The structure crack or damage of the tunnel under the impact load will lead directly to the serious nuclear leakage events, with Japan’s Fukushima nuclear power plant explosion causing the serious nuclear leakage from 2011. The former Soviet Union Chernobyl nuclear power plants exploded at 1:23 in the morning of April 26, 1986, with more than 8 tons of highly radioactive materials mixed with graphite fragments and nuclear fuel burning debris spewing out. The radiation pollution caused by the nuclear leakage accident is equivalent to 100 times of the radioactive pollution caused by the atomic bomb explosion in Hiroshima, Japan. Even 20 days after the accident, the temperature of the center of the nuclear reactor is still as high as 270 degrees Celsius. It caused 10 times the number of cancer deaths caused by the accident in the United Nations official estimates, with a global total of 2 billion people affected by the Chernobyl accident; 270 thousand people suffer from cancer, which killed more than 93 thousand people. Experts estimate that eliminating the effects of this catastrophe will take at least 800 years.
The underground structure of tunnel lining locates in the enclosure space; the security problem and the precise evaluation of the dynamic response to the blasting or impact loading becomes an important engineering issue and there is a need to give mathematical model and solution of internal force [1–6].
First, determining the strength and internal force of structure response under the blast loading is the key factor to solve the dynamic response problem for tunnel lining. By now, the researches of stability of the structure for tunnel lining under impact load are more concentrated in blasting load or gas outburst. Generalized impact action blocks and changes the motion of a moving object. The difference between impact load and impact pseudostatic load is the energy form related to the instantaneous time effect or not. Generally, the quantity of explosive and detonation pressure of explosive shock can be determined according to the energy release and the size of the gas pressure to determine the impact of the air pressure. Blasting seismic wave attenuation can be equivalent to the dynamic load effect of gas outburst on tunnel lining structure. Currently, the theoretical study of the tunnel lining and the internal force response under impact load is relatively little. This paper introduces the free deformation method as the theoretical basis. Derivation of impact load equivalent pseudostatic model is given. Load pattern on the circular tunnel lining under the instantaneous maximum internal force calculation formula is compared and analyzed. By comparing the theoretical analysis to the test data and the numerical simulation results, it intends to get the internal force response of the failure mechanism and the optimal load pattern and the theoretical results can be preliminarily verified by the back analysis .
2. The Internal Force Formula of the Circular Tunnel Lining under the Blasting Loading
Considering significantly attenuated effect of cover layer to the impact load and wave, it is supposed that impact loaded on structure of tunnel lining is equal to the earth surface. The additional stresses during elastic wave propagation are ignored. To simulate the stress and deformation of tunneling lining under impact or blasting dynamic load, engineers commonly adopt the simple and safe simplified method, which means the impact pseudostatic load multiplied by a dynamic load factor to estimate the maximum impact load. The core idea of the proposed method is first to find out the maximum equivalent impact pseudostatic stress mode under the impact or blasting load.
The theory formula of the equivalent model for impact load on circular tunnel lining is derived based on using the free deformation method, which is the classical theory method for underground structure .
2.1. The Equivalent Impact Load Model
The equivalent impact load model should be given and the equivalent form caused by blasting seismic wave loads on tunnel lining structure should be determined. Left-side blasting load coefficient of dynamic stress distribution for surrounding rock is given by Figure 1. Figure 1(a) gives the mechanical model of left-side blasting load on the circular underground structure; the radial movement of surrounding rock stress concentration factor is given as in Figure 1(b). Upper blasting load coefficient of dynamic stress distribution of surrounding rock is shown in Figure 2. Figure 2(a) gives the mechanical model of the upper-side blasting load on the circular underground structure; the force model of the tunnel lining structure and the surrounding rock of the dynamic stress concentration factor distribution map is given as in Figure 2(b). Figure 1 can give the maximum acceleration stress distribution map of the left-side blasting load in surrounding rock for lining. Figure 2 can give the maximum acceleration stress distribution map of the upper-side blasting load surrounding rock for lining. From Figures 1 and 2, it is shown that the blasting response of the circular lining is according to the following rules: the maximum point of impact load generally is the maximum stress response point, mostly on angle of 0 degrees near the load; the smaller load is distributed on both sides of the lining; the maximum stress zone is concentrated in the range of degrees around both sides of the impact load; on the far side from the impact load, the stress on both ends of the lining is obviously higher than that in the middle of the lining.
(a) Mechanical model
(b) Radial movement of surrounding rock stress concentration factor
(a) Mechanical model
(b) Upper soil blasting the maximum acceleration distribution
The dynamic stress concentration factor and the map of the maximum acceleration distribution obtained by the two modes of lateral explosion and upper explosion of Figures 1 and 2 present typical symmetrical patterns of butterfly distribution.
The maximum internal force response formula of arbitrary angle on lining is derived under any angle impact load.
First consider the special case of the impacted load on the upper side of the lining; the impact load can be expressed by arbitrary rotating angle; the additional force of lining under impact load is divided into three parts.
The first part is the equivalent impact load, with the blasting load applied on the lining; the second part is the equivalent impact reaction force, which is the reverse force of lining and formation, equal to the equivalent impact load for balance effect; the third part is the proposed rock resistance force on both sides of lining; it is caused by the relative displacement mode between the lining and ground.
We give the case study of the triangle resistance distribution; the load model is shown as in Figure 3. The effective counter force which is under the influence of impact in order to maintain the balance of force provided by the formation of lining is equal to the action of the equivalent load and impact load. The distribution of load model can be traced back to the Japanese triangle resistance method .
(a) Blasting stress model of upper side
(b) Additional mechanical load model of upper blasting stress
According to the stress characteristics of structure response under impact load, the force model diagram of the circular tunnel lining under the upper blasting impact is given. Two main factors should be considered for affecting the load and the response of the lining under the impact load: load form and load value. Figure 3(a) shows the force model of the circular lining under the condition of impact load. Figure 3(b) is the mechanical model of additional load for blasting shock.
The equivalent action form of impact load is discussed first. There are two loading modes of simulating the blasting and impact loading of rock: one is calculating the explosion hole pressure by explosive burst detonation theory and then loading the calculating blast action on hole wall directly; second is using the empirical formula to calculate the dynamical peak value of the load and then imposing it on the boundary according to triangular pulse wave form. The former method needs to introduce the state equations of the explosive detonation and the rock mass, which is used for single hole blasting or centralized charge blasting. For the impact of porous blasting, the blasting source distribution area is larger; the simulation is not directly loaded with blasting hole pressure according to the equivalent amount of explosives to concentrate on loading and explosive loading. The simulation of blasting vibration effect is based on triangle pulse wave loading . Suppose that the effect of gas outburst, seismic wave, and blasting attenuation on lining is equivalent. The equivalent additional load caused by impact load is assumed to consist of three parts: the upper part load and the two-side reaction force and the bottom reaction force. The influence of the different distribution forms on the internal force of the lining is discussed in three parts. The assumed distribution of the blasting load is shown in Figure 4.
Assume that the impact load produces the same total equivalent value ; the concentration of Figure 4 in four kinds of load on the upper part of the form is (a) < (b) < (c) < (d); Figure 4(d) is the limit form of concentrated load; the impact load can act as the concentrated load. Stress concentration in the middle of the lining is the most significant. The concentration degree of subgrade load under the same condition of the equivalent impact load is ① < ②.
2.2. The Formula Deduction of Lining Internal Force under Impact Load
The internal force formula of lining response under various forms of equivalent impact load is derived. The force in the process is the total load value of the impact load. Because the computation processes of internal force calculation for different impact loads combination are similar, the paper gives a detailed derivation of the upper curve load for reference, and the rest of the cases only give the result.
Suppose the upper impact equivalent load is sinusoidal and the maximum value is ; the sine load value is in the angle of , and take the range of as the load calculation of the equivalent load. When , two load diagrams are shown in Figure 5; the bending moment at caused by is , of which is the load value at , is the action scope, and is the action distance.
While ,The load diagram is shown as in Figure 5, while : can be calculated after the calculation formula of is gotten asThenPut the values of and into .
While ,While ,The relationship between the equivalent blasting force and isThe axial force calculation is (8), while :While ,Bring the values of and into .
While ,While ,The calculation formula of shear force is as follows: while ,While ,Bring and into .
While ,While ,
The maximum value of instantaneous explosive load and maximum internal force can be obtained by bringing maximum into the formula.
The expression of maximum explosion pressure is given in 1956 by Brown in the equivalent value of blasting load as
1971 Sassa gives the following expressions:
The detonation wave spread in the rock mass gives the maximum pressure in the contact interface of explosive and rock. The maximum pressure generated on the contact interface between the detonation wave and the rock mass is related to the rock physical characteristic. The maximum pressure and the maximum explosion pressure relation can be approximately expressed as follows: is rock density; is wave velocity propagated in rocks; is explosive detonation velocity; 0 is explosive density.
The maximum equivalent load is
The relationship between the peak value of blasting load and the distance can be expressed as is peak value of blasting load; is the distance from the calculate point to blasting hole; is the diameter of the contact surface ().
The Statfield dynamic pressure with the time history can be used aswhere is the load constant, having an amount equivalent to the dynamic pressure generated by the charge of each 1 kg explosive .
The internal force calculated by the different combination of load form is shown in the Appendix.
3. Test Verification of the Model
3.1. Comparison with the Blasting Model Test
In order to further verify the applicability of the theoretical calculation formula of the instantaneous maximum internal force of the circular lining (Figure 4) under the impact load, the calculation and comparison of the microvibration centrifuge model test results on the top of lining are carried out. The top vibration model test diagram is shown in Figure 6.
The bending moment of the structure under impact load can be obtained by converting the microstrain model test results. The model tests show that the mix ratio of sand to gypsum material is gypsum plaster material : sand : water = 1 : 0.8 : 0.5; elastic modulus , which is the reduction elasticity modulus of the prototype concrete . Compare two kinds of thickness with 7.5 mm and 12.5 mm; the elastic modulus of aluminum alloy model test is ; the thickness is 3.8 mm.
The similarity criterion of bending deformation of the model centrifuge test is as follows: is the number of centrifugal accelerations; is Poisson’s ratio of the model; is Poisson’s ratio of the prototype; is Young’s modulus of the model; is Young’s modulus of the prototype; is the thickness of the model lining; is the thickness of the prototype lining.
The next formula is available by the formula of (23): and are suitable for the sand gypsum material; the moment can be calculated by directly substituting the experimental microstrain into the prototype; for the aluminum alloy material, it can be calculated byAccording to the formula of , the instantaneous maximum bending moment of the prototype can be obtained. The elasticity modulus of the prototype concrete is . The results of converted moment value of the measuring point are shown in Table 2.
The equivalent blasting load values are calculated by the third groups of experiments, and the parameters of the test are shown in Table 3 . The theoretical solution of maximum internal force to test model is calculated.
ThenThe attenuated equivalent load is by giving according to the test conditions; the conversion coefficient of black powder and TNT is 0.4; the centrifugal acceleration is 50 g; then 1.25 g black powder is equivalent to TNT of 62.5 kg ; thenThat means the equivalent blasting load on the lining of third groups experimental results is 778 kN, considering that the size effect of model test will lead to error; the selection range of calculation load is from 600 kN to 800 kN.
The internal force of the maximum impact load for the circular lining under the condition of the load combination is calculated as shown in Figure 7. The equivalent form of the upper impact load is compared with the uniform load, the curve load, the triangle load, and the concentrated load. The concentrated load represents the extreme case where the impact load directly acts on the lining without attenuation through the surrounding rock.
(a) Test group of number 3
(b) Test group of number 4
(c) Test group of number 5
(d) Test group of number 6
(e) Test group of number 31
(f) Test group of number 34
Select the different calculation combination of blasting load form in the top and bottom; the results are compared as shown in Tables 4–9; take . The comparison of the scatter plots is shown in Figure 7.
|Note: the equivalent blasting load is 630 kN; the lateral resistance force is 56 kN/m.|
|Note: the equivalent blasting load is 546 kN; the lateral resistance force is 84 kN/m.|
|Note: the equivalent blasting load is 1400 kN; the lateral resistance force is 112 kN/m.|
|Note: the equivalent blasting load is 700 kN; the lateral resistance force is 84 kN/m.|
|Note: the equivalent blasting load is 650 kN; the lateral resistance force is 20 kN/m.|
|Note: the equivalent blasting load is 260 kN; the lateral resistance force is 20 kN/m.|
The following can be concluded through comparison of a①, b①, c①, and d①:(1)The influence of the upper load form on the bending moment of the measuring point 1 (top) is obvious, the difference range can reach 50%–100%, and the comparison of the load distribution is very necessary.(2)The characteristics of each group showed a clear trend of convergence; the measured moment value of point 1 represents the relationship a① < b① < c① < d①. This shows that with more concentration in the upper part of the load form the moment value of measuring point 1 will be greater.(3)Compared with each test point data under all cases of combination, it can be known that the measuring points of the bending moment values have changed considerably when the upper load form changed from the uniform load to the curve load. The rule is that the measuring point 1 and point 3 (the top and bottom points of the lining) bending moment value increases and the measuring point 2 and point 4 (the point on left-right sides of the lining) bending moment value decreases.(4)Compared with each test point data of 2, 3, and 4 under all cases of combination of b①, c①, and d①, it can be known that the increase concentration of the upper load and the linear increase effect of the internal force response are decreased. The effect of concentration on both sides and the bottom of the lining is very weak (within 1%).
Compared with the test combination of b① and b②, the result showed the following:(1)The form influence of the ground reaction force on the moment of the measuring point 1 (bottom) is obvious, and the difference range is 90%–50%.(2)The difference range of the combination of b① and b② is larger (range: 10%–50%) for the measuring point 1, point 2, and point 4. This shows that the concentration increase of the subgrade force in the bottom of lining will only affect the bending moment of the action position and have less effect on the other points’ bending moment outside of this range.
Comparing the data calculation in the four groups, it is found that the combination of upper triangular load and subgrade counterforce of the outer triangle reaction model has good coincidence degree with the experimental values. This case also has the certain safety reserve; comparatively, this load combination is the most reasonable one. The internal force calculation results (bending moment, shear force, and axial force) of the upper triangular load and the subgrade counterforce of outer triangle reaction model are shown in Figure 8.
(a) Third groups
(b) Fourth groups
(c) Fifth groups
(d) Sixth groups
(e) Thirty-first groups
(f) Thirty-fourth groups
Validating the lining safety theoretical results of the third test groups preliminarily, the calculation results of triangular equivalent loads are selected to verify the results, while the maximum value of the calculated moment is 367.2 kN m, the position is S1 measuring point, and the axial force is 247.8 kN.
According to the maximum compressive classical stress formula of the eccentric compression member,Bring the test parameters and calculation results; the following can be concluded:The compressive strain is considered as the critical value. The lining structure can be considered in a safe condition as less than the concrete elastic ultimate compressive strain of 0.002.
The compression plane strain of the test results isIt can be considered that the lining structure is in a safe state under the case study of blasting load.
3.2. Comparison to Model Test of Pneumatic Impact Load
Compare the pneumatic impact load mode to the model test. The test model box is shown as in Figure 9(a) . The air tube is located in the three (left, right, and down) directions of the model box. The air pressure is equivalent to the triangular load. The additional load is caused by air pressure and the equivalent loading is shown in Figure 9(b). The upper uniformly distributed load is the subgrade counterforce provided by the equilibrium pressure from the lower pressure layer.
(a) Model test
(b) Equivalent loading of air pressure
The comparison of the additional bending moment and the theoretical calculation results of the lining structure under releasing and applying air pressure to the test is shown in Table 10.
It is found that the calculation results of the additional bending moment for the circular lining under the impact pressure were coincident with the model test data. Four points’ precision accuracy is within 15%, with the highest accuracy reaching 95%. The results for the theory of structure response (Figure 3) moment under impact load calculation results are in good agreement with the experimental values. It can provide support for theoretical calculation of tunnel lining structure response under impact loading.
4. Numerical Simulation to Microvibration Test
MIDAS GTS was used to simulate the microvibration model test. After the model feature was calculated, the response simulation of lining (Figure 3) under blasting load was carried out by using the time history analysis.
Using MIDAS GTS to simulate the blasting load, manually input the blasting load function. The dynamic pressure time history functions are shown in Figure 10. The blasting load is applied on the interface in form of the surface pressure.
Table 13 shows the maximum microstrain simulation value of the lining structure under blasting loads in consistency with the test value very well; most of the gap is less than 30%. Most difference of the measuring points is less than 30%. The numerical simulation results not only verify the experimental results but also provide the basis for the correctness of the theoretical calculation results. Compared with the results of numerical simulation, the theoretical formula of the lining maximum internal force under impact loading has good applicability in the case study.
|Note: A for the test results; B for simulation results.|
Using the GTS, simulate and validate test results; the impact pseudostatic load simulations are carried out by constrained lining structure of tunnel lining by the spring boundary. Theoretically, the simulation and experimental results are compared with the results shown in Table 14. The three-coincidence degree is good through comparison; most difference of the points is less than 20%. The applicability of the theoretical calculation results is verified further by the results of impact pseudostatic load simulation.
|Note: A is the experimental result, B is the result of the calculation, and C is the simulation result.|
Impact pseudostatic load equivalent model and the maximum internal force solution for circular tunnel lining structure were deduced from classical theory of the free deformation method. The equivalent model and the maximum internal force solution of the underground lining are given. The equivalent load forms of different impact loads are compared preliminarily. The following conclusions and recommendations of the maximum internal force of lining structure under impact load may also be drawn by comparing the theoretical solution, experimental data, and numerical simulation results.
(1) The equivalent form of impact load has a remarkable influence on the calculation results. It was reflected in the internal forces distribution pattern of the lining structure in the load range of impact pseudostatic action.
Two equivalent forms of impact loading on the lining structure of tunnel lining were compared by the curve distributed shape and triangle distributed shape. The simplified triangular distributed load is more convenient and feasible to use with the premise of closing the test results.
(2) The additional load caused by the impact load mainly has two types, the ground counterforce and lateral deformation resistance, which can be calculated by the classical Japanese triangle resistance method. The equivalent mode of formation reaction has two modes, which are external triangle distribution and uniform distribution. For the case of side impact load, the distribution of the outer triangle is obviously better than that of the uniform distribution load distribution, whether from the dynamic stress concentration degree of the surrounding ground or from the matching degree to test data.
(3) The influence of soil parameters on the structural lining response under blasting shock load is the following: the soil density influences the transmission and attenuation of shock wave in surrounding rock and the formation parameters for the effect of structural response under blasting load: soil density influences the detonation wave in the soil layer of propagation and attenuation. The characteristics show that the lower the density, the softer the soil and the more significant attenuation of impact loads. The distance between blasting spot and lining structure is similar to the factor of the soil layer density; the farther the distance, the faster the impact load decay rate. The magnitude of the impact load directly affects the response of the lining; the strain of the structure lining presents a nonlinear increasing relationship under the nonlinear interaction to the surrounding ground.
(4) Comparing the theoretical formula and dynamic load simulation to the microvibration test data, the theoretical results of the impact pseudostatic load equivalent model are verified again by the numerical simulation of the equivalent static load.
The results show that the consistency of the test results, theoretical results, and the simulation results of impact pseudostatic load is very high. Most of the difference point is within 20%, which shows that the theoretical calculation results of the proposed model have good applicability.
The maximum microstrain simulation by the dynamic load on lining is in agreement with the test values very well, with most of the difference points being within the range of 30%. The comparison between the experimental results and the theoretical calculation results of the pressure shock load model is carried out; the difference of the closest point is 2.3%.
In conclusion, the impact pseudostatic load equivalent model is proposed for dynamic response analysis of tunnel lining structure. Good agreement among the theoretical analysis, test data, and the simulation is achieved, which preliminarily validates the present model and method.
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572286.44/warc/CC-MAIN-20220816090541-20220816120541-00458.warc.gz
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CC-MAIN-2022-33
| 28,357 | 97 |
https://web2.0calc.com/questions/what-is-9-1-16-times-48
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math
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You are very welcome!
When doing these problems
1. Cancel numbers; Check if numbers have similar factors. then divide them by that factor division is legal as long as the numbers are integers and not decimals, If there are similar factors.
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CC-MAIN-2020-34
| 448 | 5 |
https://www.24houranswers.com/college-homework-library/Physics/Physics-Other/22456
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math
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1. Here the angular acceleration is variable. What are the appropriate equations of motion? What is their lowest possible order? How are the tangential and radial acceleration related for a point on a rotating body?
2. Can a single force applied to a body change both its translational and rotational motion? Explain and provide two different examples.
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5. Describe static and dynamic methods of determining the force constant of a spring.
6. Including appropriate equations, describe what a natural resonant frequency is.
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1. Centrifuge. An advertisement claims that a centrifuge takes up only 0.150 m of bench space but can produce a radial acceleration of 3000g at 7500 RPM. Calculate the required radius of the centrifuge. Is the claim realistic?
2. Pressure. The 36-inch by 80-inch door of an isolation room in a hospital has an airtight but frictionless fit in its frame. The air pressure in the room is by 1% lower than the 1 atm air outside. What is the minimum force one must use to open this door?
3. Conservation of Angular momentum. Under some circumstances, a star can collapse into an extremely dense neutron star. The density of a neutron star is roughly 14 orders of magnitude greater than that of ordinary solid matter. Suppose the star is a uniform, solid, rigid sphere, both before and after the collapse. The star’s initial radius was 7x10^5 m, (comparable to the Sun); its final radius is 16 km. If the original star rotated once in 30 days, find the angular speed of the neutron star.
4. Work and power in Circular Motion. A hollow, thin-walled sphere of mass 12.0 kg and diameter 0.50 m is rotating about an axle through its center. The angle (in radians) through which it turns as a function of time (in seconds) is given by t) = At 2 + Bt 4, where the magnitude of A =1.50 and B has numerical value 1.10. (a) What are the units of the constants A and B? (b) for time t=4s find (i) the angular momentum of the sphere and (ii) the net torque on the sphere.
5. SHM. When a 1 N weight hangs from an end of a long spring of force constant 1.5 N/m and negligible mass, it bounces up and down in SHM. If you stop the bouncing and let this mass swing from side to side through a small angle, the frequency of this simple pendulum is half the bounce frequency. What is the equilibrium length of the spring without the weight attached?
6. SHM. A 11.5 kg water in a 1.00-kg bucket is hanging from a vertical ideal spring of force constant 125N/m and oscillating up and down with an amplitude of 5.00 cm. When the bucket springs a leak in the bottom water drops out at a steady rate of 2.00 g/s. Is the period getting longer or shorter? When the bucket is half-full, find (a) the period of oscillation and (b) the rate at which the period is changing with respect to time. (c) What is the shortest period this system can have?
7. Combination Problem. On the planet Casino, a simple pendulum having a bob with mass 1.25 g and a length of 18.5 cm takes 1.42 s, when released from rest, to swing through an angle of 7.5 degrees, where it again has zero speed. The circumference of Casino is measured to be 51,400 km. What is the mass of the planet Casino? List all the assumptions necessary to arrive at your answer.
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897844.44/warc/CC-MAIN-20200709002952-20200709032952-00137.warc.gz
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CC-MAIN-2020-29
| 4,036 | 16 |
http://www.soundadoggymakes.com/2009/07/more-nerdy-pick-up-lines.html
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math
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#673 - "Are those space pants? Because you can pee in those."
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CC-MAIN-2017-22
| 737 | 8 |
http://slideplayer.com/slide/2483164/
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math
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Presentation on theme: "Welcome to 6 th Grade Matter Jeopardy. $100 $200 $300 $400 $100 $200 $300 $400 What Makes Up Matter? Properties of Matter Changing States Matter Math."— Presentation transcript:
Welcome to 6 th Grade Matter Jeopardy
$100 $200 $300 $400 $100 $200 $300 $400 What Makes Up Matter? Properties of Matter Changing States Matter Math Really Matters
Question for Column 1 $100 These are pure substances that can’t be broken down into any other substance. They have unique properties and they are arranged in the Periodic Table.
Column 1 Answer $100 Elements
Question for Column 1 $200 This is the smallest particle that makes an element. It has a nucleus which is surrounded by a cloud of negative charge.
Column 1 Answer $200 An atom
Question for Column 1 $300 Atoms combine with other atoms to make these. They can contain 2 atoms of the same type of element, or two atoms from different elements.
Column 1 Answer $300 Molecules
Question for Column 1 $400 This is a special type of molecule. It contains atoms from at least 2 different elements.
Column 1 Answer $400 A compound
Question for Column 2 $100 These are the two kinds of properties that every form of matter has. Some can be observed without changing it into another substance, and some describe its ability to change into different substances.
Column 2 Answer $100 Physical and Chemical Properties
Question for Column 2 $200 This is a physical property that describes how much matter is in an object.
Column 2 Answer $200 Mass
Question for Column 2 $300 This physical property describes how much space an object takes up.
Column 2 Answer $300 Volume
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Column 2 Answer $400 Density The cube on the left is more dense, since it has more mass packed into the same volume.
Question for Column 3 $100 This state of matter has a definite shape and a definite volume. Its molecules are arranged in tight, orderly formations.
Column 3 Answer $100 Solid
Question for Column 3 $200 This state of matter has no definite shape and no definite volume. In fact, its fast moving, wildly arranged molecules can actually be compressed if we need to fit them in a smaller container.
Column 3 Answer $200 Gas
Question for Column 3 $300 When a substance changes from a gas to a liquid, the speed of its molecules does this, and so does the distance between its molecules.
Column 3 Answer $300 The speed decreases and so does the distance between the molecules.
Question for Column 3 $400 Thermal energy is the total energy of the motion of all the particles in an object. When thermal energy is transferred, it always moves from _______ matter to _______ matter. warmer/cooler warmer/cooler
Column 3 Answer $400 Thermal energy always moves from warmer matter to cooler matter.
Question for Column 4 $100 The Law of Conservation of Mass states that matter is not created or destroyed in any chemical or physical change. So, after the methane and oxygen react to make carbon dioxide and water, how many total atoms will there be?
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Question for Column 4 $300 This mineral sample has a mass of 126 g and a volume of 15 cm 3. What is its density?
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Column 4 Answer $400 The temperature of the water dropped, because the warmer water lost its heat to the colder ice.
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CC-MAIN-2016-50
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http://omg.worsethanfailure.com/Entries/ViewEntry.aspx?id=100142
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math
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About The Contest
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CC-MAIN-2017-17
| 2,180 | 5 |
https://www.allmarketingtips.com/rectangular-prism/
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math
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In the world of mathematics, geometry is the study of shapes and configuration of different kinds of objects and the rectangular prism is the polyhedron with two congruent and parallel bases in the whole process. It will also be known as the cuboid that will be having six vertices and all the faces will be into the rectangle shape which will be having 12 edges in the whole process. Because of the cross-section along with the length, it is also known to be as the prism but it is very much important for people to be clear about the basic properties of the whole process so that surface area can be easily calculated across the net without any kind of problem.
The rectangular prism is a three-dimensional shape that will be having six faces and every face of the prison will be a rectangle. Both the base of the rectangular prism must be a rectangle so that there is no problem at any point in time. The other lateral faces will also be the rectangles and it is also known as a cuboid in the whole world of mathematics.
Table of Contents
Following are the basic properties of a rectangular prism:
- It will be having 6 rectangular faces, 8 corners, and 12 edges.
- This particular shape will be having opposite faces into the rectangle shape
- It will also be having a rectangular cross-section
- It will look exactly like the cuboid
It can be classified into different names of categories and some of those categories are explained as follows:
- The right rectangular prism: A rectangular prism that will be having rectangular bases will be known as the rectangular prism and further the right rectangle a prism will be the one that will be having six faces that will be rectangles and each angle of the whole shape will be the right angle. The vertices of this particular prism will be eight and the edges will be 12 and the faces will be six.
- Oblique rectangular prism: This is the comprehensive case in which the bases are not perpendicular to each other but the rectangular prism with the base that will not be aligned will be one directly above each other and will be known as the oblique rectangular prism.
It is also very much important for the kids to be clear about different kinds of formulas to be implemented in this particular area so that there is no problem at any point in time. The rectangular prism is a three-dimensional shape and to calculate the area of this particular prism people need to be clear about the measurements of the length, width as well as the height of the [rectangular prism]. The volume formula has been explained as:
- Length into width into height cubic units
The lateral surface area of the rectangular prism has been explained as:
- Two into the sum of length plus width
The surface area of the rectangular prism has been explained as:
- Two into a product of length and height plus the product of weight and height plus the product of length and width square units
The net of any kind of prism will be the surface area of it and it will be very much capable of showing that whenever the people will be opening the prism into a plane all the sides will be visible at the same time and calculating the area of all these kind of sides of net people will be getting the total surface area without any kind of issue. Hence, depending upon the official platforms like Cuemath is the best way of having a good command over the properties of triangular prism as well so that kids can make the right kind of decisions in their life and can score very well in the mathematics exam without any kind of confusion element in their minds at the time of solving the questions.
Also Read: Basecamp: Project Management Solution
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CC-MAIN-2023-50
| 3,661 | 19 |
http://clintonriverkings.org/epub/an-expansion-of-meromorphic-functions
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math
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By Walsh J.L.
Read Online or Download An Expansion of Meromorphic Functions PDF
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This can be a new, revised variation of this widely recognized textual content. the entire uncomplicated subject matters in calculus of a number of variables are lined, together with vectors, curves, features of a number of variables, gradient, tangent aircraft, maxima and minima, power services, curve integrals, Green's theorem, a number of integrals, floor integrals, Stokes' theorem, and the inverse mapping theorem and its results.
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This quantity contains the lawsuits of the convention on Operator conception and its functions held in Gothenburg, Sweden, April 26-29, 2011. The convention used to be held in honour of Professor Victor Shulman at the party of his sixty fifth birthday. The papers integrated within the quantity cover a huge number of subject matters, between them the idea of operator beliefs, linear preservers, C*-algebras, invariant subspaces, non-commutative harmonic research, and quantum teams, and reflect fresh advancements in those components.
The current quantity comprises the entire routines and their recommendations for Lang's moment version of Undergraduate research. the wide range of workouts, which diversity from computational to extra conceptual and that are of differ ing hassle, conceal the subsequent matters and extra: genuine numbers, limits, non-stop capabilities, differentiation and straightforward integration, normed vector areas, compactness, sequence, integration in a single variable, incorrect integrals, convolutions, Fourier sequence and the Fourier critical, features in n-space, derivatives in vector areas, the inverse and implicit mapping theorem, usual differential equations, a number of integrals, and differential kinds.
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Extra info for An Expansion of Meromorphic Functions
Given the vector v = ofv. J4+9 = SOLUTION. We have Ivl = u 2i - + 3j, find a unit vector in the direction JT3. The desired vector u is I = JT3v = - 2. JT31 3. + JT3 J. EXAMPLE 4. Given the vector v = 2i - 4j. Find the directed line segment has coordinates (3, - 5). AS of v, given that A SOLUTION. Denote the coordinates of B by X B , YB' Then we have (by Theorem 2) XB - Therefore x B = 5, YB = 3 =2 -9. and YB +5= -4. 44 2. 8 of v have the given coordinates. Draw a figure. I. A(3, -2), B(I,5) 2. A( -4, I), B(2, -I) 3.
The next example shows how to transform one representation into another. EXAMPLE 2. The two planes 2x + 3y - 4z - 6 =0 and 3x - y + 2z + 4 = 0 intersect in a line. ) Find a set of parametric equations of the line of intersection. SOLUTION. t, Z = t, which are the desired parametric equations. Three planes may be parallel, may pass through a common line, may have no common points, or may have a unique point of intersection. If they have a unique point of intersection, the intersection point may be found by solving simultaneously the three equations of the planes.
P[(2, -I, -2); L[ : x = 2 + 3t, y = 0, 12. p[ ( - I, 2, - 3); L, : x = - I + 5t, Y = + 3t, Z Z Z = I = = -4t I + 3t -1 - 2t + 2t, Z = -I + 3t In each of Problems 13 through 16, find the equations of the line through PI and perpendicular to the given plane MI' + 3y + Z - 3 = 0 +4=0 + 2z + 3 = 0 13. PI (-2,3, I); M I : 2x 14. PI(I, -2, -3); MI 15. P I ( -1,0, -2); M[: x 16. P[(2, -I, -3); M I :x=4 : 3x - y - 2z In each of Problems 17 through 20, find an equation of the plane through PI and parallel to the plane 11>.
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We have been exploring how the position of a digit within a number determines its value. The following task encouraged the students to find different ways to ‘SHOW’ numbers.
We began by rolling three dice to find the 3 digit number that will be represented in different ways.
Here are some examples of student work:
Next, we wondered if there were different ways to ‘SHOW’ a number.
Could we show 253 in a different way by changing the tens?
What if it had 2 hundreds and 4 tens? How many ones would you need to have so it is still 253?
How might we decompose the following numbers into hundreds, tens, and ones and then find a different way to show each one by changing the hundreds or tens digit?
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https://www.mechgrid.com/types-of-fluids.html
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Fluids are divided into five types they are:
- Ideal fluids
- Real fluids
- Newtonian fluids
- Non-Newtonian fluids
- Ideal plastic fluids
An ideal fluid is defined as a fluid which is in-compressible and the one that does not have viscosity. Ideal fluids are imaginary fluids. This exist some viscosity.
A real fluid is defined as a fluid which possesses viscosity. In actual state all fluids are real fluids.Advertisement:
In a real fluid the shear stress is directly proportional to the velocity gradient or shear strain, which is known as Newtonian fluids.
In a real fluid the shear stress is not proportional to the velocity gradient or shear strain, which is known as Non- Newtonian fluids.
Ideal Plastic Fluids:
In a fluid the shear stress is more than the yield value. The shear stress is proportional to the rate of velocity gradient or shear strain is known as ideal plastic fluids.
Fluid consists of either liquid or gases. In case of gases they are compressible fluids. The thermodynamic properties play an important key role. Due to the change in temperature and pressure the gases undergoes high variation in densities. So the relation between the absolute temperature, absolute pressure and specific volume is
P= Absolute pressure of a gas
R = gas constant
T= Absolute temperature in kelvin
ρ = Density of a gas.
The gas constant R value must be depends on the particular gas.
In MKS Unit R value is
In SI units
At constant pressure the change in density occurs then the process is known as isothermal process. The relation between pressure and density is:
The change in the density must be occurs without heat change to and fro the gases is known as Adiabatic process.
Due to friction there is no heat generation in the gases, so the relation between the density and pressure is
K value for air is 1.4
Universal Gas Constant:
It is also known as gas constant, ideal gas constant, molar gas constant. It is denoted by the letter R.
In SI units R value is 8314 J/kg-mole.
Compressibility And Bulk Modulus:
The reciprocal of bulk modules of elasticity is known as compressibility. It is defined as the ratio of compressive stress to volumetric strain.
Consider a cylinder and piston is inserted into it, when the force is applied on the piston then the pressure is increased to p+dp. Then the volume present in the cylinder is decreased to to .
Volumetric strain must be =
Bulk modules = K = increase in pressure / volumetric strain
Compressibility = 1/ K
For Gases Relation Between Bulk Modulus And Pressure:
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The 20 times that follow are information about you. Please supply the information as HONESTLY and ACCURATELY as you can.
The data that will be obtained from these items shall be held strictly confidential.
Each item is followed by several possible answer. On your sheet, shade completely the box that corresponds to the number of the answer that specifically pertains to you.
1. Sex :
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1. 18-24 years old
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5. More than 45 years old
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1. College graduate
3. With Master’s Units
4. Master’s Degree
5. With Ph.D. Units/Degree
5. Year of last attendance in school :
1. Before 1985
5. After 2000
6. What honours did you receive when you graduated from college?
1. Summa Cum laude
2. Magna Cum laude
3. Cum laude
4. Other academic award
5. None/not applicable
7. Present employment:
8. Type of present job:
2. General Clerical
3. Trades and crafts (jobs requiring manual dexterity of application of manual/mechanical/artistic
4. Others 5. Not applicable
9. Length of experience in present job:
1. Less than one year
2. One to two years
3. Three to four years
4. More than four years
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Second Grade, Municipal/Provincial Clerk, General Clerical, Career Service Sub- Professional(Local Government), Career Service Sub-professional
11. For what reason are you taking this examination?
1. Entrance to government service
2. Change of appointment status
12. How many times have you taken the Career Service Professional? Examination excluding this one? 1. Once 2. Twice 3. Thrice
4. More than thrice 5. Never
13. Which of the following activities did you undertake in preparing for this examination?
1. Enrolled in review centers
2. Studied career service examination reviewers sold at bookstore
3. Engage in other activities
4. Used a combination of 1 and 2.
5. No preparation done at all Items 14 to 16 (For Government Employees Only):
14. Category of government office where employed:
1. National government 2. Local government (province/city/municipal) 3. Government-owned or controlled corporation 4. Constitutional office 5. State college or university
15. Status of present appointment in government service:
1. Permanent 2. Temporary 3. Casual/Emergency 4. Contractual 5. Substitute
16. Years of experience in government service:
1. Less than 5 years 2. 5-9 years 3. 10-14 years 4. 15-19 years 5. More than 19 years
Items 17 to 20 In which of the following types of work do you consider yourself best qualified? Choose only two from among the options listed in items 17-20. Shade the boxes that corresponds to your choices. For example, if you think you are best qualified in budget management, and project planning/management, shade box no. 2 of item 17 and box no. 3 of item . Leave items 18 and 20 blank. If you think you are best qualified in research/report including statistical analysis, shade boxex no. and no. 2 of item 20 on your Answer sheet, and leave items and 19 blank.
17. 1. Accounting 2. Budget Management 3. Buying/Purchasing 4. Co-ordination 5. Computer Operations
18. 1. EDP Computer Programming
2. EDP System Analysis and Design 3. Human Resource Development 5. Management and Audit Analysis
19. 1. News/Feature Writing 2. Personal Recruitment/Selection
3. Project Planning/Management 4. Public Relation Work 5. Records Management
20. 1. Research / Report Writing 2. Statistical Analysis
3. Stenography 4. Supplies Management 5. None of the Above
* TEST BEGINS HERE * GRAPHS / Charts / Data
Introduction to Economics Exam Statistics
1. In which of the following years did over 2/3 of the students who took the exam not pass it?
a. 2005 b. 2006 c. 2008 d. 2009 e. Cannot say
2. It is known that a quarter of the students who passed the exam in 2007, passed it at the first trial. Assuming each exam has two trials, what percentage of all the students who took the exam that year passed it in the second trial?
a. 10 b. 15 c. 30 d. 75 e. Cannot say
3. If the number of Chinese Insurance stocks represented 3.5% of all Insurance securities, approximately how many Insurance bonds were Chinese?
a. 9,200,000 b. 9,500,000 c. 10,800,000 d. 910,000 e. 1,080,000
For questions 4 to 6, Refer to the following graph of sales and profit figures of ABC Ltd and answer the questions that follow.
4. I Return on sales (Profit/sales) was highest in which year?
a. 1995 b. 1996 c. 1997 d. 1998 e. cannot say
5. How many times return on sales (profit/sales) exceeded 15 % ?
a. once b. twice c. thrice d. four times e. never
6. How many times growth in profit over the previous year exceeded 50% was registered ?
a. once b. twice c. thrice d. four times e. never
For questions 7 to 9: The following graph shows the data related to Foreign Equity Inflow (FEI) for the five countries for two years- 1997 and 1998. FEI is the ratio of foreign equity inflow to the country’s GDP, which is expressed as a percentage in the following graph.
7. Find the ratio between FEI for Malaysia in 1997 and FEI for Thailand in 1998
a.) 7 : 25 b.) 8 : 21 c.) 6 : 11 d.) 1067 : 582 e.) cannot say
8. Name the country which has the minimum change in the FEI
a.) India b.) China c.) Malaysia d.) S.Korea e.) cannot say
9. If, Education sector in Thailand had 25% of FEI in 1997 and 60% of FEI in 1998, then find the approx ratio of the amounts allocated to Education in 1997 to 1998. (Assume the GDPs of both of these years for Thailand is same.)
a.) 1 : 5 b.) 25 : 72 c.) 6 : 11 d.) 7 : 25 e.) cannot say
10. A rumour about an upcoming recession in Japan has reduced the value of the Yen 7% compared with the Euro. How many Euros can you now buy for 500 Yen?
a. 3.5 b. 3.26 c. 3.15 d. 3.76 e. None of the above
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https://www.rhayden.us/managerial-economics/graphing-the-objective-function.html
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math
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The objective function, n = $12QX + $9QY, can be graphed in QXQY space as a series of isoprofit curves. This is illustrated in Figure 9.7, where isoprofit curves for $36, $72, $108, and $144 are shown. Each isoprofit curve illustrates all possible combinations of X and Y that result in a constant total profit contribution. For example, the isoprofit curve labeled n = $36 identifies each combination of X and Y that results in a total profit contribution of $36; all output combinations along the n = $72 curve provide a total profit contribution of $72; and so on. It is clear from Figure 9.7 that isoprofit curves are a series of parallel lines that take on higher values as one moves upward and to the right.
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Question about Texas Instruments Office Equipment & Supplies
it crosses the horizontal asymptote when it should not it connects the 2 functions by passing through a horizontal asymptote, when it should not. It is a spurious effect due to drawing connected plots
...and b is 3. The b is the y-intercept, so we can plot the point (0,3). Next, we start at this point and go down 1 and to the right 1. We join the dots, and we have graphed the equation. Good luck. ...
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By James J. Tattersall
Meant to function a one-semester introductory direction in quantity conception, this moment variation has been revised all through. particularly, the sector of cryptography is highlighted. on the center of the publication are the foremost quantity theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler. furthermore, a wealth of latest workouts were integrated to completely illustrate the houses of numbers and ideas constructed within the textual content. The e-book will function a stimulating creation for college kids new to quantity conception, despite their historical past. First version Hb (1999) 0-521-58503-1 First version Pb (1999) 0-521-58531-7
Read or Download Elementary Number Theory in Nine Chapters, Second Edition PDF
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This is often the English translation of the unique jap e-book. during this quantity, "Fermat's Dream", center theories in glossy quantity thought are brought. advancements are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the quantity fields. This paintings provides a sublime viewpoint at the ask yourself of numbers.
This ebook offers an advent to the gigantic topic of preliminary and initial-boundary price difficulties for PDEs, with an emphasis on purposes to parabolic and hyperbolic structures. The Navier-Stokes equations for compressible and incompressible flows are taken for instance to demonstrate the implications.
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Additional info for Elementary Number Theory in Nine Chapters, Second Edition
2 without proof. 2 (Alternate principle of mathematical induction) Any set of natural numbers that contains the natural number m, and contains n þ 1 whenever it contains all the natural numbers between m and n, where n > m, contains all the natural numbers greater than m. The alternate principle of mathematical induction implies the well-ordering principle. In order to see this, let S be a nonempty set of natural numbers with no least element. For n . 1, suppose 1, 2, . . , n are elements S, the complement of S.
Lucas originally defined v n to be u2 n =un . He derived many relationships between Fibonacci and Lucas numbers. For example, u nÀ1 þ u nþ1 ¼ v n , un þ v n ¼ 2u nþ1 , and v nÀ1 þ v nÀ1 ¼ 5un . The sequence of Lucas numbers is an example of a Fibonaccitype sequence, that is, a sequence a1 , a2 , . . , with a1 ¼ a, a2 ¼ b, and a nþ2 ¼ a nþ1 þ an , for n > 2. Fibonacci numbers seem to be ubiquitous in nature. There are abundant references to Fibonacci numbers in phyllotaxis, the botanical study of the arrangement or distribution of leaves, branches, and seeds.
The tribonacci numbers an are defined recursively as follows: a1 ¼ a2 ¼ 1, a3 ¼ 2, and an ¼ a nÀ1 þ a nÀ2 þ a nÀ3, for n > 4. Generate the first 20 tribonacci numbers. 22. The tetranacci numbers bn are defined as follows: b1 ¼ b2 ¼ 1, b3 ¼ 2, b4 ¼ 4, and bn ¼ b nÀ1 þ b nÀ2 þ b nÀ3 þ b nÀ4, for n > 5. Generate the first 20 tetranacci numbers. 23. Verify the Collatz conjecture for the following numbers: (a) 9, (b) 50, (c) 121. 24. Determine the three cycles that occur when 3an À 1 is substituted for 3an þ 1 in the Collatz algorithm.
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http://www.statmodel.com/discussion/messages/23/399.html?1084559090
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Anonymous posted on Friday, May 14, 2004 - 8:58 am
I do not yet have Mplus 3 so these questions may not be reasonable. a. Is a probit link-function allowed with FIML? b. What type of fit indices does Mplus3 compute with categorical outcomes with FIML? c. How is RMSEA/WRMR computed with FIML?
No, we do not have a probit link with maximum likelihood. With maximum likelihood and categorical outcomes, you obtain the loglikelihood, AIC, BIC, sample-size adjusted BIC, Pearson chi-square, and the likelihood ratio chi-square. RMSEA and WRMR are not computed for categorical outcomes and maximum likelihood. For continuous outcomes, you can find these formulas in the technical appendices which are found at www.statmodel.com.
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Download PDF. Tweet. KB Sizes Downloads Views. Report. Recommend Documents. Pure and Applied Mathematics A Panorama of Pure Mathematics: As Seen by N. Bourbaki Vol. 98 Joseph G. Rosenstein, Linear Orderings VOl. 99 M. Scott Osborne and Garth Warner, The Theory of Eisenstein Systems VOl. Richard V.
Kadison and John R. Feb 05, · The Project Gutenberg EBook of A Course of Pure Mathematics, by G. H. (Godfrey Harold) Hardy This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at xn--80ahmeqiirq1c.xn--p1ai Format: PDF, Mobi Download: Read: Download» This volume continues the work covered in the first book, Pure Mathematics 1, and is intended to complete a full two year course in Pure Mathematics.
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The author is not a mathematician by profession, the book shows that pure mathematics is not that complicated once you get down to the rules. ( views) Foundations of Mathematics by Stephen G. Simpson - Pennsylvania State University, These are lecture notes for an introductory graduate-level course in foundations of mathematics. DOWNLOAD FILE. File loading please wait Recommend Papers. Pure Mathematics 1 (v. 1) General Editor: Douglas Burnham The books in this series are specifically written for students reading philosophy for th.
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Since then I have always had every new addition of the set (A First Course and A Second Course).Reviews: Pure Mathematics book. Read 13 reviews from the world's largest community for readers.
This well-established two-book course is designed for class teachi /5(13). Engineering Mathematics: YouTube Workbook. Applied Statistics. Integration and differential equations. Elementary Algebra Exercise Book I. A Handbook of Statistics. Essential Engineering Mathematics.
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About the book. Mathematics students will find this book extremely useful. Show less. Pure Mathematics for Advanced Level, Second Edition is written to meet the needs of the student studying for the General Certificate of Education at Advanced Level. The text is organized into 22 chapters. Cambridge International Examinations (CIE) Advanced Level Mathematics has been created especially for the new CIE mathematics syllabus.
There is one book corresponding to each syllabus unit, except for this book which covers two units, the second and third Pure Mathematics. GCE Guide | Best Resources for Cambridge Examinations. Download Pure Mathematics For Beginners in PDF and EPUB Formats for free. Pure Mathematics For Beginners Book also available for Read Online, mobi, docx and mobile and kindle reading. Download A Synopsis Of Practical Mathematics Pdf, A Synopsis Of Practical Mathematics epub, A Synopsis Of Practical Mathematics free, A Synopsis Of Practical Mathematics author, A Synopsis Of Practical Mathematics audiobook, A Synopsis Of Practical Mathematics free epub Download the Book.
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pure mathematics 2 backhouse pdf download More than 10 in to around 75 in Backhouse, forthcoming.B.D. Bunday & H. Mulholland Pure Mathematics. Mar 20, · Download Pure Mathematics 1 book pdf free download link or read online here in PDF. Read online Pure Mathematics 1 book pdf free download link book now. All books are in clear copy here, and all files are secure so don't worry about it. This site is like a library, you could find million book here by using search box in the header.
DOWNLOAD NOW» A classic single-volume textbook, popular for its direct and straightforward approach. Understanding Pure Mathematics starts by filling the gap between GCSE and A Level and builds on this base for candidates taking either single-subject of double-subject A Level.
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http://lib.mexmat.ru/books/74420
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Нашли опечатку? Выделите ее мышкой и нажмите Ctrl+Enter
Название: Modular Forms and Functions
Автор: Rankin R.A.
This book provides an introduction to the theory of elliptic modular functions and forms, a subject of increasing interest because of its connexions with the theory of elliptic curves. Modular forms are generalisations of functions like theta functions. They can be expressed as Fourier series, and the Fourier coefficients frequently possess multiplicative properties which lead to a correspondence between modular forms and Dirichlet series having Euler products. The Fourier coefficients also arise in certain representational problems in the theory of numbers, for example in the study of the number of ways in which a positive integer may be expressed as a sum of a given number of squares. The treatment of the theory presented here is fuller than is customary in a textbook on automorphic or modular forms, since it is not confined solely to modular forms of integral weight (dimension). It will be of interest to professional mathematicians as well as senior undergraduate and graduate students in pure mathematics.
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https://link.springer.com/chapter/10.1007/978-3-642-28708-4_5
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math
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Special Values of L-functions
In the last chapter, we studied the analytic properties of L-functions, in particular their zero distributions. If we simply regard an L-function as a complex analytic function, then its values at different points seem to be equally important or unimportant. However, since our L-functions all come from arithmetic geometric or automorphic origins, it turns out that the function values at certain points are more transparent in revealing the associated arithmetic or automorphic information, and thus deserve special attention. It is for this reason that such L-function values are called special values.
KeywordsFunctional Equation Elliptic Curve Elliptic Curf Class Number Analytic Number Theory
Unable to display preview. Download preview PDF.
- [IK04]Iwaniec, H., Kowalski, E. Analytic number theory. American Mathematical Society, 2004.Google Scholar
- [La18]Landau, E. “Über die Klasszahl imaginär-quadratischer Zahlkörper.” Göttinger Nachr. (1918): 285–295.Google Scholar
- [Zh06a]Zhang, Q. “Applications of multiple Dirichlet series in mean values of Lfunctions.” In Multiple Dirichlet series, automorphic forms, and analytic number theory, American Mathematical Society, 2006: 43–57.Google Scholar
- [Zh06b]Zhang, Q. “Integral mean values of Maass L-functions.” Int. Math. Res. Not. 2006, Art. ID 41417, 19 pp.Google Scholar
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Motto: Panta rhei kai ouden menei
My job: designing graphics for the game (locations, characters, interfaces), creating illustrations and simple animations, preparing web graphics, script correction
Interests: In love with movie soundtracks, addicted to good TV series, enjoys creating things, dreams of running a marathon
What I like: Rain...
My dislikes: Noises in the dark.
Favoritue games: Baldur’s Gate, Mass Effect, KoTOR , Oblivion, Skyrim, currently spends time playing as the dark side of the Force.
About myself: When I was younger, I used to write my own ‘books’ using 60 pages long exercise books. Also, using a pen and printer paper, I created comic books…Dreamt of being a writer and illustrator…a child’s dreams ;)
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http://www.gosippme.com/2017/07/no-one-can-work-out-how-many-triangles.html
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math
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It seems simple at first - just count the triangles in the image. But there are some hidden traps that have flummoxed a umber of people online.
No, 'one big one' is not the answer.
Scroll down to find out.
The answer is 24!
Responding to the puzzle on Quora, mathematician Martin Silvertant shared this explainer:
As some users pointed out, you can count 25 triangles if you include the word 'triangle'!
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https://dialoguecollective.wordpress.com/2011/01/07/treasure-hunt/
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The Shoppery Hunt
• Start at the statue of King Charles in Trafalgar Square – the actual point for measuring the distance to and from London.
• Each person will have been provided with a number of random directions
• Place your directions in a bag and remove 1 by 1
• Take 5 directions and see where you end up.
• Record your journey with video or camera.
• Find your treasure
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http://sightworder.com/library/heat-kernels-for-elliptic-and-sub-elliptic-operators-methods-and-techniques
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math
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By Ovidiu Calin, Der-Chen Chang, Kenro Furutani, Chisato Iwasaki
This monograph is a unified presentation of a number of theories of discovering specific formulation for warmth kernels for either elliptic and sub-elliptic operators. those kernels are vital within the idea of parabolic operators simply because they describe the distribution of warmth on a given manifold in addition to evolution phenomena and diffusion methods.
The paintings is split into 4 major elements: half I treats the warmth kernel by means of conventional equipment, resembling the Fourier rework technique, paths integrals, variational calculus, and eigenvalue enlargement; half II bargains with the warmth kernel on nilpotent Lie teams and nilmanifolds; half III examines Laguerre calculus functions; half IV makes use of the tactic of pseudo-differential operators to explain warmth kernels.
themes and features:
•comprehensive therapy from the perspective of distinctive branches of arithmetic, corresponding to stochastic strategies, differential geometry, designated services, quantum mechanics, and PDEs;
•novelty of the paintings is within the different equipment used to compute warmth kernels for elliptic and sub-elliptic operators;
•most of the warmth kernels computable by way of hassle-free features are coated within the work;
•self-contained fabric on stochastic strategies and variational equipment is included.
Heat Kernels for Elliptic and Sub-elliptic Operators is a perfect reference for graduate scholars, researchers in natural and utilized arithmetic, and theoretical physicists attracted to knowing other ways of coming near near evolution operators.
Read or Download Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques PDF
Best differential geometry books
The notes from a suite of lectures writer brought at nationwide Tsing-Hua college in Hsinchu, Taiwan, within the spring of 1992. This notes is the a part of publication "Thing Hua Lectures on Geometry and Analisys".
This booklet is concentrated at the interrelations among the curvature and the geometry of Riemannian manifolds. It comprises study and survey articles in keeping with the most talks added on the overseas Congress
During this e-book, we learn theoretical and sensible points of computing tools for mathematical modelling of nonlinear platforms. a couple of computing recommendations are thought of, akin to equipment of operator approximation with any given accuracy; operator interpolation strategies together with a non-Lagrange interpolation; equipment of procedure illustration topic to constraints linked to thoughts of causality, reminiscence and stationarity; tools of approach illustration with an accuracy that's the top inside a given type of types; equipment of covariance matrix estimation;methods for low-rank matrix approximations; hybrid equipment in keeping with a mixture of iterative techniques and top operator approximation; andmethods for info compression and filtering lower than clear out version should still fulfill regulations linked to causality and sorts of reminiscence.
- Topology of Surfaces, Knots, and Manifolds
- Notes on Geometry
- Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems
- Symplectic Methods in Harmonic Analysis and in Mathematical Physics
Additional resources for Heat Kernels for Elliptic and Sub-elliptic Operators: Methods and Techniques
8) by taking the mixed derivative of the action @2 S D @x@x0 1 D 1 6D 0: Á. / Hence there are no conjugate points to x0 in this case. 9). vk D/ D @t @t t @xk D and then xk t2 x0k ! 6 (Particle in constant gravitational field). x; P x/ D 12 xP 2 kx, k > 0. t/ D t 6D 0, for t > 0, there are no conjugate points along the trajectory. We leave the computation of the classical action as an instructive exercise to the reader. 7 (The linear oscillator). t/ D x k 2 x , 2 k > 0, then p p p sinh. kt/ x0 cosh.
M; g/, there is a neighborhood V of x0 such that for any x 2 V, there is a unique geodesic joining the points x0 and x. The aforementioned result does not necessarily hold globally for any Riemannian manifold. However, it holds on compact manifolds, and in general on metrically complete manifolds, as the Hopf–Rinov theorem states; see . 1 Lagrangian Mechanics 19 Moreover, any geodesic is locally minimizing the action functional. 0/ D A; x. U /; for any tangent vector field U ; see . 0/ D 0.
1b. 30). x; y/ . 30). As we shall see in the next section, this relation is obvious in the case of a three-dimensional hyperbolic space. M; gij / of dimension n. Let R D g ij Rij be the Ricci scalar curvature of the space, which will be assumed constant. x0 ; x/ denote the Riemannian distance between the points x0 and x. x0 / where Ä D D det 1=4 e Rt e @2 Scl @x0 @x is the van Vleck determinant; see Schulman , Chap. 24. 32) 46 3 The Geometric Method Applying the aforementioned formula for the classical spaces with constant curvatures 0; 1; 1, we arrive at the following classical results.
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http://hubpages.com/religion-philosophy/answer/168915/can-god-diein-greek-myths
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sort by best latest
Best Answer rafken says
Doc Snow says
Darrell Roberts says
which definition is that? as you can see from the other replies here, there are many definitions of god; no culture has a monopoly, though many think they do.
so, basically, all gods die, but some do it for sport?
where does one apply to become the true god? is it like american idol?
1 answer hidden due to negative feedback. Show
1 answer hidden due to negative feedback. Hide
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https://tt.tennis-warehouse.com/index.php?threads/prokennex-heritage-type-c.287309/
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Can anyone help me with the following question: what is the difference between the ProKennex Heritage Type C Racquet and the ' Redondo' version? Why does TW carry the latter only, while the prokennex website listst the first? Is it correct that they have different specs? Do they play differently? Where can I get the non-redondo' version? Thanks!
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@ccswims can u help me ?
hint: total price = cost of boat + (price per hour)(number of hours)
21.25 = (slope) + (money you paid up front)
slope being the amount of money by each hour
I'm still confused 😞
ok, 21.25 = (slope) + your y intercept (the money you paid up front)
you paid $10 up front, right?
you 21.25 = (slope of money each hour) + 10 you paid 3.75 for each hour (or h) you spent on the boat
so 3.75 is your slope 21.25 = 3.75h + 10 is your equation
Kk I got it thanks :) @ccswims but now I have to solve the equation
ok, subtract 10 to both sided
Kk I did that
ok, now what do you have?
ok, so it should look like this 11.25 = 3.75h our objective is to get h by itself, so divide 3.75 to both sides
what do you get?
I solved it and I got 3
So she rented the kayak for 3 hours
yes, you are right
Thank you so much @ccswims
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https://rudolf-meyer.com/qa/question-what-does-npv-0-mean.html
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math
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- Do NPV and IRR always agree?
- What is XIPR?
- Can a project have no IRR?
- What is negative IRR?
- What is an acceptable NPV?
- What does a positive NPV mean?
- What does the IRR tell you?
- How do I calculate IRR?
- What is a good IRR?
- Is a high IRR good?
- What is NPV method?
- Should you invest If NPV is 0?
- Why is NPV equal to zero?
- Is NPV better than IRR?
- What is NPV example?
- Can you have a positive NPV and negative IRR?
- How do you interpret NPV and IRR?
- How do you interpret NPV?
- What does an IRR of 0 mean?
- What is difference between NPV and IRR?
- What is the conflict between IRR and NPV?
Do NPV and IRR always agree?
The difference between the present values of cash inflows and present value of initial investment is known as NPV (Net Present Value).
A project would be accepted if its NPV was positive.
Therefore, the IRR and the NPV do not always agree to accept or reject a project..
What is XIPR?
XIRR is your personal rate of return. It is your actual return on investments. XIRR stands for Extended Internal Rate of Return is a method used to calculate returns on investments where there are multiple transactions happening at different times.
Can a project have no IRR?
The IRR is formally defined as the discount rate at which the Net Present Value of the cash flows is equal to zero. … There are also cases where no IRR exists. For example, if all cash flows have the same sign (i.e., the project never turns a profit), then no discount rate will produce a zero NPV.
What is negative IRR?
Negative IRR occurs when the aggregate amount of cash flows caused by an investment is less than the amount of the initial investment. In this case, the investing entity will experience a negative return on its investment.
What is an acceptable NPV?
The net present value rule is the idea that company managers and investors should only invest in projects or engage in transactions that have a positive net present value (NPV). They should avoid investing in projects that have a negative net present value. It is a logical outgrowth of net present value theory.
What does a positive NPV mean?
positive net present valueA positive net present value indicates that the projected earnings generated by a project or investment – in present dollars – exceeds the anticipated costs, also in present dollars. It is assumed that an investment with a positive NPV will be profitable, and an investment with a negative NPV will result in a net loss.
What does the IRR tell you?
The IRR equals the discount rate that makes the NPV of future cash flows equal to zero. The IRR indicates the annualized rate of return for a given investment—no matter how far into the future—and a given expected future cash flow.
How do I calculate IRR?
To calculate IRR using the formula, one would set NPV equal to zero and solve for the discount rate, which is the IRR. … Using the IRR function in Excel makes calculating the IRR easy. … Excel also offers two other functions that can be used in IRR calculations, the XIRR and the MIRR.
What is a good IRR?
You’re better off getting an IRR of 13% for 10 years than 20% for one year if your corporate hurdle rate is 10% during that period. … Still, it’s a good rule of thumb to always use IRR in conjunction with NPV so that you’re getting a more complete picture of what your investment will give back.
Is a high IRR good?
The higher the IRR on a project, and the greater the amount by which it exceeds the cost of capital, the higher the net cash flows to the company. … A company may also prefer a larger project with a lower IRR to a much smaller project with a higher IRR because of the higher cash flows generated by the larger project.
What is NPV method?
Net present value (NPV) is a method used to determine the current value of all future cash flows generated by a project, including the initial capital investment. It is widely used in capital budgeting to establish which projects are likely to turn the greatest profit.
Should you invest If NPV is 0?
A positive NPV means the investment is worthwhile, an NPV of 0 means the inflows equal the outflows, and a negative NPV means the investment is not good for the investor.
Why is NPV equal to zero?
Zero NPV means that the cash proceeds of the project are exactly equivalent to the cash proceeds from an alternative investment at the stated rate of interest. The funds, while invested in the project, are earning at that rate of interest, i.e., at the project’s internal rate of return.
Is NPV better than IRR?
The advantage to using the NPV method over IRR using the example above is that NPV can handle multiple discount rates without any problems. Each year’s cash flow can be discounted separately from the others making NPV the better method.
What is NPV example?
For example, if a security offers a series of cash flows with an NPV of $50,000 and an investor pays exactly $50,000 for it, then the investor’s NPV is $0. It means they will earn whatever the discount rate is on the security.
Can you have a positive NPV and negative IRR?
You can have a positive IRR and a negative NPV. Look, basically when NPV is equal to zero, IRR is equal to the discount rate. The discount rate is always above zero hence when the IRR is below the discount rate, the IRR is still positive but the NPV is negative.
How do you interpret NPV and IRR?
The NPV method results in a dollar value that a project will produce, while IRR generates the percentage return that the project is expected to create. Purpose. The NPV method focuses on project surpluses, while IRR is focused on the breakeven cash flow level of a project.
How do you interpret NPV?
NPV = Present Value – CostPositive NPV. If NPV is positive then it means you’re paying less than what the asset is worth.Negative NPV. If NPV is negative then it means that you’re paying more than what the asset is worth.Zero NPV. If NPV is zero then it means you’re paying exactly what the asset is worth.
What does an IRR of 0 mean?
are not getting any returnWhen IRR is 0, it means we are not getting any return on our investment for any number of years, thus we are losing the interest which we could have earned on our investment by investing our money in bank or any other project, thereby reducing our wealth and thus NPV will be negative.
What is difference between NPV and IRR?
Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time. By contrast, the internal rate of return (IRR) is a calculation used to estimate the profitability of potential investments.
What is the conflict between IRR and NPV?
When you are analyzing a single conventional project, both NPV and IRR will provide you the same indicator about whether to accept the project or not. However, when comparing two projects, the NPV and IRR may provide conflicting results. It may be so that one project has higher NPV while the other has a higher IRR.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141171126.6/warc/CC-MAIN-20201124053841-20201124083841-00104.warc.gz
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CC-MAIN-2020-50
| 7,011 | 65 |
https://studylib.net/doc/10236656/junior-scientist
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math
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Level 1 Challenge Questions: Junior Scientist 1.) Name 10 different types of fish. 2.) Name the steps in the Scientific Method. 3.) Name 10 elements of the Periodic Table. 4.) Name the 7 continents. 5.) Name the planets of our Solar System, in order. 6.) Name the three phases of matter. 7.) What types of scientist studies animals? 8.) Name 10 organs in the human body. 9.) Describe the water cycle. 10.) What does an zoologist study? 11.) Name 10 different dinosaurs. 12.) Name 10 different constellations. 13.) Name 10 minerals. 14.) Name 5 human body systems. 15.) Name 10 mammals. 16.) What is the chemical formula for water? 17.) What does the acronym NWS stand for? 18.) How many colors are there in a rainbow? 19.) What is the force that holds everything to the Earth? 20.) Tell me what a geologist studies? 21.) Tell me what clouds are made of? 22.) What is the largest of body of water on Earth? 23.) What is the largest group of freshwater lakes on Earth called? 24.) Name the three basic rock types? 25.) What are the highest mountains on Earth?
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s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710421.14/warc/CC-MAIN-20221210074242-20221210104242-00071.warc.gz
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CC-MAIN-2022-49
| 1,057 | 1 |
https://brainmass.com/business/capital-budgeting/capital-budgeting-and-forecasting-models-525526
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math
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Corn Doggy, Inc. produces and sells corn dogs. The corn dogs are dipped by hand. Austin Beagle, production manager, is considering purchasing a machine that will make the corn dogs. Austin has shopped for machines and found that the machine he wants will cost $262,000. In addition, Austin estimates that the new machine will increase the company's annual net cash inflows by $40,300. The machine will have a 12-year useful life and no salvage value.
(a) Calculate the payback period.
(b) Calculate the machine's internal rate of return.
(c) Calculate the machine's net present value using a discount rate of 10%.
(d) Assuming Corn Doggy, Inc.'s cost of capital is 10%, is the investment acceptable? Why or why not?
Please refer attached file for better understanding of formulas.
Let us study the cash flows associated with the given project
Year End Cash Flow
Solution depicts the steps to calculate the payback period, IRR and NPV parameters for the given case.
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s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623488539764.83/warc/CC-MAIN-20210623165014-20210623195014-00431.warc.gz
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CC-MAIN-2021-25
| 964 | 9 |
https://www.physicsforums.com/threads/solve-second-order-diff-equation-using-substitution.413763/
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math
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d2y/dx2-dy/dx+y*exp(2x) = x*exp(2x)-1
substitute t=exp(x) and set z(t)=y(x) and rewrite hence find all solutions
The Attempt at a Solution
d2z/dt2-dz/dt+z*t^2=(ln(t) * t^2) - 1
however I dont see how this in any way helps us...
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s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487620971.25/warc/CC-MAIN-20210615084235-20210615114235-00540.warc.gz
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CC-MAIN-2021-25
| 227 | 5 |
https://calculator-derivative.com/derivative-of-ln6x
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math
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Introduction to Derivative of ln6x
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of ln(6x) can be calculated by following the rules of differentiation. Or, we can directly find the derivative ln 6x by applying the first principle of differentiation. In this article, you will learn what the ln 6x derivative is and how to calculate the ln6x derivative by using different approaches.
What is the derivative of ln 6x?
The derivative of ln x with respect to x is a fundamental concept in calculus, and it's essential to understand how to compute it. It can be denoted as d/dx [ln(6x)], and it tells us the rate of change of the natural logarithmic function ln x. In other words, it shows us how quickly the value of ln(6x) is changing concerning changes in the variable x. This derivative can be simplified as 1/ x, indicating that the derivative of ln6 is always a fraction with x in the denominator. It's important to note that ln(6x) represents the logarithm of 6x with base e, which is a critical detail for solving various calculus problems.
Derivative of ln6x formula
The formula for the derivative of ln(6x) is equal to 1/x and mathematically it can be written as:
d/dx(ln(6x)) = 1/x
This formula tells us how the function ln 6x changes with a change in its variable x. This is an essential formula to know in calculus, as it allows us to solve various problems involving logarithmic functions.
How do you prove the derivative of ln 6x?
There are multiple derivative rules to derive the ln6x derivative, and different methods may be more useful depending on the problem at hand. Some of the most common techniques to prove the ln(6x) derivative are:
- First Principle
- Implicit Differentiation
- Product Rule
Each method provides a different way to compute the ln(6x) differentiation. By using these methods, we can mathematically prove the formula for finding the ln6x derivative.
Differentiation of ln(6x) by first principle
According to the first principle of derivative, the ln 6x derivative is equal to 1/x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,
f(x)=lim f(x+h)-f(x) / h
This formula allows us to determine the rate of change of a function at a specific point by using limit definition of derivative.
Proof of ln(6x) derivative by first principle
To prove the derivative of ln(6x) by using first principle, we start by replacing f(x) by ln x.
By logarithmic properties,
Suppose t=h / x and h=xt. When h approaches zero, t will also approach zero.
f(x)=lim ln (1/xt) ln (1+t)
By logarithmic properties, we can write the above equation as,
f(x)=(1/x) lim ln(1+t)^1/t
Hence by limit formula, we know that,
lim ln(1+t)^1/t =ln e =1
Therefore, the derivative of ln6 is;
Derivative of ln6x using implicit differentiation
implicit differentiation is a technique used to find the derivative of a function that is defined implicitly by an equation involving two or more variables. We can use this method to prove the differentiation of ln(6x).
Proof of derivative of ln 6x by implicit differentiation
To prove the derivative of natural log, we can start by writing it as,
Converting in exponential form,
ey = 6x
Applying derivative on both sides,
ey.dy/dx = 6
Use our implicit derivative solver to evaluate derivatives of implicit expressions easily.
Derivative of ln(6x) using product rule
Another method to calculate the differential of ln 2x is the product rule which is a formula used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule solver is:
d/dx(uv) = u(dv/dx) + (du/dx)v
In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.
Proof of ln6x differentiation by product rule
The function ln x can be written as;
f(x)= 1. ln(6x)
Applying derivative with respect to x,
Now by using product rule,
Hence the ln(6x) derivative is always equal to the reciprocal of x.
How to find the ln6x derivative with a calculator?
The easiest way to calculate the derivative ln6x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this derivative calculator with steps.
- Write the function as ln x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the ln 6x derivative.
- Now, select the variable by which you want to differentiate ln(6x). Here you have to choose x.
- Select how many times you want to differentiate ln(6x). In this step, you can choose 2 for second, 3 for triple derivative and so on.
- Click on the calculate button. After this step, you will get the derivative of ln(6x) within a few seconds.
After completing these steps, you will receive the ln(6x) derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.
Frequently asked questions
Is logarithmic differentiation the same as derivative?
The logarithmic differentiation is a part of the derivative in which we differentiate complicated functions by using natural log. Whereas in derivative, we simply find the derivative of a function by using differentiation rules.
What is the derivative of ln6x?
The differentiation of ln(6x) can be calculated as;
d/dx (ln(6x)) = 1/x
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233509023.57/warc/CC-MAIN-20230925151539-20230925181539-00634.warc.gz
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CC-MAIN-2023-40
| 5,695 | 60 |
https://mathtuition88.com/2017/08/30/an-ancient-babylonian-tablet-known-as-plimpton-322/
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math
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Source: NY Times
One of my favorite YouTube Math Professors, Norman Wildberger, has made a historical math discovery: that the ancient Babylonian tablet known as Plimpton 322 is actually a trigonometric table.
“It’s a trigonometric table, which is 3,000 years ahead of its time,” said Daniel F. Mansfield of the University of New South Wales. Dr. Mansfield and his colleague Norman J. Wildberger reported their findings last week in the journal Historia Mathematica.
Check out my other blog posts on Prof. Norman Wildberger:
1) Algebraic Topology Video by Professor N J Wildberger
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CC-MAIN-2022-21
| 586 | 5 |
https://crazyproject.wordpress.com/2010/06/16/properties-of-the-image-and-kernel-of-the-p-power-map-on-a-finite-abelian-group/
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math
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Let be a finite abelian group (written multiplicatively) and let be a prime. Let and . (I.e. and are the image and kernel of the -power map, respectively.
- Prove that . (Show that they are both elementary abelian and have the same order.)
- Prove that the number of subgroups of of order equals the number of subgroups of index . (Reduce to the case where is an elementary abelian -group.)
We begin with some lemmas.
Lemma 1: Let and be abelian groups and let be an isomorphism. Then there exists an isomorphism . Proof: Let denote the natural projection from to (for ). If , then for some . Then , so that . By the remarks on page 100 in D&F, there exists a unique group homomorphism such that . ( is injective) Suppose . Then , thus , so that , hence . Now since is surjective, for some . Thus , so that . Thus , so that is injective. ( is surjective) Let . Now for some , so that . Hence is surjective.
Lemma 2: Let and be abelian groups and let be an isomorphism. Then . Proof: If , then where . Now , so that , hence . Thus . Let . Then x^p = 1$. Since is surjective, for some . Now , and since is injective, . Thus , hence .
Lemma 3: Let be a finite abelian group of order , let be an integer, and let denote the -power homomorphism. If , then is an isomorphism. Proof: Since is finite, it suffices to show that is injective. Let . Then , so that divides . By Lagrange, divides . Thus , hence , and is injective.
Now to the main result. By Theorem 5 in the text, where is a -group of rank and does not divide . By a lemma to the previous exercise, , where each map is the -power map on , , and , respectively.
We have by the previous exercise.
Similarly, , again using the previous exercise.
Now the first conclusion follows using Lemmas 1 and 2.
We begin with some more lemmas.
Lemma 4: Let be the elementary abelian group of order , and suppose where . Finally let be a group. If is a mapping such that for each and for all , then extends uniquely to a group homomorphism . Proof: Every element of can be written uniquely as , where and . Define . is well defined since the -expansion of is unique, and is a homomorphism since the commute with one another. To see uniqueness, suppose is a group homomorphism such that for all . Then . Thus is unique.
Note that every subgroup of of order is contained in , so that in fact the order subgroups of and coincide. Now because is elementary abelian, every nonidentity element has order and thus generates an order subgroup. Each such subgroup is generated by elements; thus there are order subgroups in , thus in .
Let be a subgroup of index . Now let , say , and suppose . Then , so that . Now is a group of order , so that ; then , a contradiction. Thus . Moreover, by the Third Isomorphism Theorem, . By the Lattice Isomorphism Theorem, the index subgroups of correspond precisely to the index subgroups of . In particular, it suffices to count the order and index subgroups in an elementary abelian -group.
Now acts on by . We claim that this action is transitive. To that end, let be index subgroups. Now since and are elementary abelian, thus and where . Let be a bijection, and choose some and . Now extends by Lemma 4 to a homomorphism . Clearly is surjective, hence an isomorphism, so that . Moreover, we see that . Thus this action is transitive, and we have for all index subgroups .
Now let be an arbitrary index subgroup and let be an automorphism of which stabilizes . As above, for some set with . Moreover, if then since is maximal. By Lemma 4, is determined uniquely by is action on and . We see also that since stabilizes , we must have for each . Since and , there are distinct choices we can make for . Moreover, may be chosen arbitrarily so long as ; thus there are choices for . Thus in total contains elements.
By the Orbit-Stabilizer Theorem and Lagrange's Theorem, .
Thus the theorem is proved.
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s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218187227.84/warc/CC-MAIN-20170322212947-00454-ip-10-233-31-227.ec2.internal.warc.gz
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CC-MAIN-2017-13
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https://everything2.com/user/bitter_engineer/writeups/binary+coded+decimal
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math
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Also known as BCD. This is a way of using four bits to count to 10.
Zero is 0.
This is inefficient
, of course, but it is useful in a couple of areas:
- Making drivers for 7 segment LED displays. Instead of using one byte for each digit on the display, you can stick two BCD numbers in each byte, and halve your required space.
- Accounting software. If you want to be accurate to the penny, then you need to work in the same base system as the currency. Calculating 1/5 in binary will leave you with a repeating fraction, which does not happen in the decimal system. Calculating 1/16 in binary will leave you with a number that does not get rounded, which does happen in the decimal system.
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s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583688396.58/warc/CC-MAIN-20190120001524-20190120023524-00125.warc.gz
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CC-MAIN-2019-04
| 691 | 6 |
https://gledlaraltumb.firebaseapp.com/518.html
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math
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Firm size, booktomarket ratio, and security returns. Book value is calculated by looking at the firms historical cost, or accounting value. The classification is determined by comparing a companys pricetobook ratio to the median. Students and employees who are logged on to nhhs network will have direct access to these databases. Market value is determined in the stock market through its market capitalization.
The book to market bm ratio of event t is then the log of the ratio of book. Introduction to wrds and using the webinterface to. The booktomarket ratio attempts to identify undervalued or overvalued securities by taking the book value and dividing it by market value. Securities database as reported by billet, king, and mauer 2006 has a. Nhh pays for access to a large number of databases. A ratio used to find the value of a company by comparing the book value of a firm to its market value. How to calculate the book value with compustat fundamentals quarterly. For additional information, please see the about section. The booktomarket ratio used to form portfolios in june of year t is book. Price is from crsp, shares outstanding are from compustat if available or crsp. Market value of equity for the calculation of book to market ratios is based on december divided by market value of equity. I need this ratio for all the uk companies of last 15 year and i am not sure about which variables and. I need this ratio for all the uk companies of last 15 year and i am not sure about which variables and method i should use for that. Hi, i am struggling to calculate market to book ratio tobins q from compustat.
A markettobook ratio above 1 means that the companys stock is overvalued, and below 1 indicates that its undervalued. Crspannual updatecrsp compustat mergedfundamental annualsupplemental data items csho. Financial ratios for accounting research papers in the ssrn. My solution is to multiply the crsp market value with the last known ratio of compustat to crsp market value. Book value for a company may be reported for fiscal year ending june, but you might want to calculate mb ratio at calendar yearend month, december. Why do we take the natural log of booktomarket ratios. Book value of equity compustat data item 60 is that reported on a firms financial statement in the prior year size is measured as price per share times shares outstanding in june of each year. The tobins q ratio is a ratio devised by james tobin of yale university, nobel laureate in economics, who hypothesized that the combined market value of all the. These databases provide accounting data for bank holding. Crspannual updatecrsp compustat mergedfundamental annual miscellaneous items.137 140 877 551 336 744 103 318 847 355 103 1404 1216 371 754 650 47 90 1354 1264 36 715 174 680 1260 468 1228 834 903 1375 50 857 999 252 1089
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s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030337244.17/warc/CC-MAIN-20221002021540-20221002051540-00323.warc.gz
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CC-MAIN-2022-40
| 2,841 | 3 |
http://www.ije.ir/article_73085.html
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math
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Faculty of technical foundation, Universiti kuala lumpur (Unikl)-Micet
Environmental, Universiti Sains Malaysia
Three-factor interaction for the two-level, three-level, and four-level factorial designs was studied. A new technique and formula based on the coefficients of orthogonal polynomial contrast were proposed to calculate the effect of the three-factor interaction The results show that the proposed technique was in agreement with the least squares method. The advantages of the new technique are 1) it is fixed, 2) it is simple and 3) it is easy to apply without the complicated matrix formula of the least squares method. This new technique will also enhance the use of the coefficients of orthogonal contrast when analyzing other experimental designs.
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s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875147116.85/warc/CC-MAIN-20200228073640-20200228103640-00549.warc.gz
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CC-MAIN-2020-10
| 763 | 3 |
https://www.reference.com/web?q=amps+to+watts+conversion+chart&qo=contentPageRelatedSearch&o=600605&l=dir
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math
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Watts to amps calculator DC amps to watts calculation. The power P in watts (W) is equal to the current I in amps (A), times the voltage V in volts (V): P (W) = I (A) × V (V) AC single phase amps to watts calculation. The power P in watts (W) is equal to the power factor PF times the phase current I in amps (A), times the RMS voltage V in ...
How to convert Watts to Amps or Amps to Watts or Volts to Watts. Basics. You cannot convert watts to amps, since watts are power and amps are coulombs per second (like converting gallons to miles). HOWEVER, if you have at least least two of the following three: amps, volts or watts then the missing one can be calculated. Since watts are amps multiplied by volts, there is a simple relationship ...
How to Convert Watts to Amps. Converting watts to amps can be done using the Watt’s Law formula, which states that I = P ÷ E, where P is power measured in watts, I is current measured in amps, and E is voltage measured in volts.. Given this, to find amps given power and voltage use the following formula:
Amps to Watts Calculator. Ampere (amps or amperage) is an SI unit of electric current and is denoted by 'A'. 1 Ampere is defined as the electrical current that flows with electric charge of one Coulomb per second. Watt is an SI unit of electric power and is denoted by 'W'. 1 watt is defined as the energy consumption rate of one joule per second.
Watts to Amps calculator It is used to convert the electric power in watts (W) to the current in amps (A). You can start by selecting the type of electric current. It can either be Direct Current (DC) or the Alternating Current (AC) single phase/ three phase.
Calculate Watts from Volts and Amps If you want to convert watts to amps on your own, you can use this equation: Watts = Amps x Volts or W = A x V As long as you know two of the electrical ratings, you can calculate the missing info with simple math.
Convert volts to watts using a simple electrical conversion calculator. See three easy to use formulas for the conversion using either AC or DC circuits.
It is a conversion calculator which transforms the electric current in Amps to electrical energy in Watts. It has three text fields which help in determining the exact units you require after the conversion.
Amps to watts calculator DC watts to amps calculation. The current I in amps (A) is equal to the power P in watts (W), divided by the voltage V in volts (V):. I (A) = P (W) / V (V). AC single phase watts to amps calculation. The phase current I in amps (A) is equal to the power P in watts (W), divided by the power factor PF times the RMS voltage V in volts (V):
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s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578527566.44/warc/CC-MAIN-20190419101239-20190419123239-00221.warc.gz
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CC-MAIN-2019-18
| 2,639 | 9 |
https://coldregionsresearch.tpub.com/CRTD96_01/CRTD96_010016.htm
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math
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COLD REGIONS TECHNICAL DIGEST NO. 96-1
Approximate Path of
Maximum Surface Velocity
5. Salmon River boom.
Note that the boom is
aligned at an angle to
the path of maximum
surface water velocity.
the greatest depth and the highest current velocity is in the vicin-
ity of the channel center. If the boom is oriented with the chord at
an acute angle to cross-stream direction, the point where the
boom is perpendicular to the flow is shifted away from the path of
maximum surface velocity (Fig. 4e and 5).
The following discussion is based on boom geometry as shown
boom and wire
in Figure 6. Relationships exist between the location of the anchor
points, the unstressed length of wire rope S0, the sag ratio s, and
the tension in the main support cable T. The sag ratio is defined as
the ratio of the midspan deflection, dmidspan, to the perpendicular
length L. Assuming a uniform loading across the river width
results in a parabolic shape for the ice boom cable. For most
applications, this shape is a reasonable approximation of the ac-
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s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882570879.1/warc/CC-MAIN-20220808213349-20220809003349-00134.warc.gz
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CC-MAIN-2022-33
| 1,038 | 22 |
https://www.projecteuclid.org/journals/geometry-and-topology/volume-12/issue-4/Surface-subgroups-from-homology/10.2140/gt.2008.12.1995.full
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math
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Let be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If is nonzero, then contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov–Thurston norm on is a finite-sided rational polyhedron.
"Surface subgroups from homology." Geom. Topol. 12 (4) 1995 - 2007, 2008. https://doi.org/10.2140/gt.2008.12.1995
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s3://commoncrawl/crawl-data/CC-MAIN-2023-40/segments/1695233510238.65/warc/CC-MAIN-20230927003313-20230927033313-00756.warc.gz
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CC-MAIN-2023-40
| 379 | 2 |
https://mathsolution.net/algebra/gcd-and-lcm-of-polynomials.php
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math
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L.C.M And G.C.D of Polynomials
expression a0xn+a1xn-1+a2xn-2+a3xn-3+.........+ an-1x+an
where a0,a1,a2,a3.....an are real number and n is non negative integer and a0 not equal to zero is polynomial of degree n.
a0,a1,a2,a3.....an are coefficients of polynomial.
if coefficients of polynomial are integer then polynomial are written as
5x-2 is a polynomial of degree 1
5x2-2x+4 is a polynomial of degree 2
x3+3x2-2x+6 is a polynomial of degree 3
What is Divisor or Factor
A Polynomial d(x) is divisior of a Polynomial p(x) if d(x) is a factor of p(x). For example p(x)=d(x)r(x)
In this example d(x) is factor of p(x) so we called d(x) is divisor(factor) of Polynomial p(x).
p(x)=x2-5x+6 then p(x)=(x-3)(x-2)
here (x-3) and (x-2) is a factor of p(x) so (x-3) and (x-2) are divisors or factors of polynomial p(x).
Greatest common divisor(G.C.D) or HCF of Polynomials
Greatest common divisor(g.c.d) or hcf of two polynomials are highest degree common divisor of both polynomials.
and coefficient of highest degree term is positive.
If p(x) and q(x) are two polynomials then highest degree common divisor of p(x) and q(x) is called as greatest common divisor(g.c.d) or hcf of Polynomials.
G.C.D of polynomials 2x2-x-1 and 4x2+8x+3
p(x)=(2x+1)(x-1) here (2x+1) and (x-1) is factor of polynomial p(x)
q(x)=(2x+1)(2x+3) here (2x+1) and (2x+3) is factor of polynomial q(x)
in both polynomial only (2x+1) is only common divisor with least exponent 1
so GCD of p(x) and q(x) is (2x+1)1=(2x+1)
Least Common Multiple(LCM) of Polynomials
Polynomial with the lowest degree and having smallest numerical coefficient which is divisible by the given polynomials and cofficient of highest
degree term has the same sign as the cofficient of highest degree term in ther product is called Least Common Multiple(LCM) of two or more Polynomials
How to find L.C.M of Polynomials
1. Find Factors of given polynomials and write them as a product of power of irreducible factors.
2. List all irreducible factors and find greatest exponent in the factorized form .
3. Raise each irreducible factors to the greatest exponent and multiply them.
4. Get LCM of polynomials
How to find LCM of Polynomials f(x)=18x4-36x3+18x2 and g(x)=45x6-45x3
First polynomial f(x)=18x4-36x3+18x2
second polynomial g(x)=45x6-45x3
In both polynomials irreducible factors are 2,3,5,x,x-1 and x2+x+1
Highest component are 1,2,1,3,2,1
So LCM =21*32*51*x3*(x-1)2*(x2+x+1)
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s3://commoncrawl/crawl-data/CC-MAIN-2023-50/segments/1700679100710.22/warc/CC-MAIN-20231208013411-20231208043411-00075.warc.gz
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CC-MAIN-2023-50
| 2,416 | 36 |
https://jds-online.org/journal/JDS/issue/62
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math
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Abstract: Among many statistical methods for linear models with the multicollinearity problem, partial least squares regression (PLSR) has become, in recent years, increasingly popular and, very often, the best choice. However, while dealing with the predicting problem from automobile market, we noticed that the results from PLSR appear unstable though it is still the best among some standard statistical methods. This unstable feature is likely due to the impact of the information contained in explanatory variables that is irrelevant to the response variable. Based on the algorithm of PLSR, this paper introduces a new method, modified partial least squares regression (MPLSR), to emphasize the impact of the relevant information of explanatory variables on the response variable. With the MPLSR method, satisfactory predicting results are obtained in the above practical problem. The performance of MPLSR, PLSR and some standard statistical methods are compared by a set of Monte Carlo experiments. This paper shows that the MPLSR is the most stable and accurate method, especially when the ratio of the number of observation and the number of explanatory variables is low.
Abstract: The traditional method for processing functional magnetic resonance imaging (FMRI) data is based on a voxel-wise, general linear model. For experiments conducted using a block design, where periods of activation are interspersed with periods of rest, a haemodynamic response function (HRF) is convolved with the design function and, for each voxel, the convolution is regressed on prewhitened data. An initial analysis of the data often involves computing voxel-wise two-sample t-tests, which avoids a direct specification of the HRF. Assuming only the length of the haemodynamic delay is known, scans acquired in transition periods between activation and rest are omitted, and the two-sample t-test is used to compare mean levels during activation versus mean levels during rest. However, the validity of the two-sample t-test is based on the assumption that the data are Gaussian with equal variances. In this article, we consider the Wilcoxon rank test as well as modified versions of the classical t-test that correct for departures from these assumptions. The relative performance of the tests are assessed by applying them to simulated data and comparing their size and power; one of the modified tests (the CW test) is shown to be superior.
Abstract: Behavioral risk factors for cancer tend to cluster within individuals, which can compound risk beyond that associated with the individual risk factors alone. There has been increasing attention paid to the prevalence of multiple risk factors (MRF) for cancer, and to the importance of designing interventions that help individuals reduce their risks across multiple behaviors simultaneously. The purpose of this paper is to develop methodology to identify an optimal linear combination of multiple risk factors (score function) which would facilitate evaluation of cancer interventions.
Abstract: Some scientists prefer to exercise substantial judgment in formulating a likelihood function for their data. Others prefer to try to get the data to tell them which likelihood is most appropriate. We suggest here that one way to reduce the judgment component of the likelihood function is to adopt a mixture of potential likelihoods and let the data determine the weights on each likelihood. We distinguish several different types of subjectivity in the likelihood function and show with examples how these subjective elements may be given more equitable treatment.
Abstract: Motivation: A formidable challenge in the analysis of microarray data is the identification of those genes that exhibit differential expression. The objectives of this research were to examine the utility of simple ANOVA, one sided t tests, natural log transformation, and a generalized experiment wise error rate methodology for analysis of such experiments. As a test case, we analyzed a Affymetrix GeneChip microarray experiment designed to test for the effect of a CHD3 chromatin remodeling factor, PICKLE, and an inhibitor of the plant hormone gibberellin (GA), on the expression of 8256 Arabidopsis thaliana genes. Results: The GFWER(k) is defined as the probability of rejecting k or more true null hypothesis at a given p level. Computing probabilities by GFWER(k) was shown to be simple to apply and, depending on the value of k, can greatly increase power. A k value as small as 2 or 3 was concluded to be adequate for large or small experiments respectively. A one sided ttest along with GFWER(2)=.05 identified 43 genes as exhibiting PICKLEdependent expression. Expression of all 43 genes was re-examined by qRTPCR, of which 36 (83.7%) were confirmed to exhibit PICKLE-dependent expression.
Abstract: In this paper, a tree-structured method is proposed to extend Classification and Regression Trees (CART) algorithm to multivariate survival data, assuming a proportional hazard structure in the whole tree. The method works on the marginal survivor distributions and uses a sandwich estimator of variance to account for the association between survival times. The Wald-test statistics is defined as the splitting rule and the survival trees are developed by maximizing between-node separation. The proposed method intends to classify patients into subgroups with distinctively different prognosis. However, unlike the conventional tree-growing algorithms which work on a subset of data at every partition, the proposed method deals with the whole data set and searches the global optimal split at each partition. The method is applied to a prostate cancer data and its performance is also evaluated by several simulation studies.
Abstract: We develop a likelihood ratio test statistic, based on the betabinomial distribution, for comparing a single treated group with dichotomous data to dual control groups. This statistic is useful in cases where there is overdispersion or extra-binomial variation. We apply the statistic to data from a two year rodent carcinogenicity study with dual control groups. The test statistic we developed is similar to others that have been developed for incorporation of historical control groups with rodent carcinogenicity experiments. However, for the small sample case we considered, large sample theory used by the other test statistics did not apply. We determined the critical values of this statistic by enumerating its distribution. A small Monte Carlo study shows the new test statistic controls the significance level much better than Fisher’s exact test when there is overdispersion and that it has adequate power.
Abstract: We propose a coherent methodology for integrating different sources of information on a response variable of interest, in order to accurately predict percentiles of its distribution. Under the assumption that one of the sources is more reliable than the other(s), the approach combines factors formed from the data into an additive linear regression model. Quantile regression, designed for quantifying the goodness of fit precisely at a desired quantile, is used as the optimality criterion in model-fitting. Asymptotic confidence interval construction methods for the percentiles are adopted to compute statistical tolerance limits for the response. The approach is demonstrated on a materials science case study that pools together information on failure load from physical tests and computer model predictions. A small simulation study assesses the precision of the inferences. The methodology gives plausible percentile estimates. Resulting tolerance limits are close to nominal coverage probability levels.
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CC-MAIN-2023-50
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http://www.ae.msstate.edu/tupas/exp/A6.8_ex1.html
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math
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SECTION A 6.8 EXAMPLE1
(a) Since this ia a single-cell section, there will only be one shear flow of constant magnitude along all webs.
Correction: The quantity shown inside the brackets is only half as opposed to the whole cell area. However, the value of q = 100 lb/in is correct as it is based on the whole cell area.
(b) Because the skin has two different thicknesses, there will be two different shear stresses.
Notice that the maximum shear stress is located in the thinest section. This will always be true when dealing with single cell problems.
(c) The angle of twist, with this shear flow distribution, will be
To Index Page of Pure Torsion
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https://forum.malighting.com/forum/thread/3949-3d-view-light-not-positioning-correctly/?postID=9065#post9065
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math
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Hi everyone, new user here, trying to get my Maverick MK2 washes to position correctly in the 3D view. I can get the beam to the stage and working with pan or tilt reversed (one of them is always reversed), but the beam shoots out of the base of the 3D object. The MK1 spot we have is working just fine in the 3D viewer aswell. Thanks in advance if anyone can help me figure this out
Edit: So! I've found that in the 3D world the MK2 Wash believes it can tilt 540° but the fixture itself can only tilt 270° in real life, I believe this is why I'm getting these issues. Is there anyway to change the MK2 washes 3D profile to accurately reflect the 270° tilt?
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https://readingfeynman.org/2017/03/11/re-visiting-electron-orbitals/
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math
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One of the pieces I barely gave a glance when reading Feynman’s Lectures over the past few years, was the derivation of the non-spherical electron orbitals for the hydrogen atom. It just looked like a boring piece of math – and I thought the derivation of the s-orbitals – the spherically symmetrical ones – was interesting enough already. To some extent, it is – but there is so much more to it. When I read it now, the derivation of those p-, d-, f– etc. orbitals brings all of the weirdness of quantum mechanics together and, while doing so, also provides for a deeper understanding of all of the ideas and concepts we’re trying to get used to. In addition, Feynman’s treatment of the matter is actually much shorter than what you’ll find in other textbooks, because… Well… As he puts it, he takes a shortcut. So let’s try to follow the bright mind of our Master as he walks us through it.
You’ll remember – if not, check it out again – that we found the spherically symmetric solutions for Schrödinger’s equation for our hydrogen atom. Just to be make sure, Schrödinger’s equation is a differential equation – a condition we impose on the wavefunction for our electron – and so we need to find the functional form for the wavefunctions that describe the electron orbitals. [Quantum math is so confusing that it’s often good to regularly think of what it is that we’re actually trying to do. :-)] In fact, that functional form gives us a whole bunch of solutions – or wavefunctions – which are defined by three quantum numbers: n, l, and m. The parameter n corresponds to an energy level (En), l is the orbital (quantum) number, and m is the z-component of the angular momentum. But that doesn’t say much. Let’s go step by step.
First, we derived those spherically symmetric solutions – which are referred to as s-states – assuming this was a state with zero (orbital) angular momentum, which we write as l = 0. [As you know, Feynman does not incorporate the spin of the electron in his analysis, which is, therefore, approximative only.] Now what exactly is a state with zero angular momentum? When everything is said and done, we are effectively trying to describe some electron orbital here, right? So that’s an amplitude for the electron to be somewhere, but then we also know it always moves. So, when everything is said and done, the electron is some circulating negative charge, right? So there is always some angular momentum and, therefore, some magnetic moment, right?
Well… If you google this question on Physics Stack Exchange, you’ll get a lot of mumbo jumbo telling you that you shouldn’t think of the electron actually orbiting around. But… Then… Well… A lot of that mumbo jumbo is contradictory. For example, one of the academics writing there does note that, while we shouldn’t think of an electron as some particle, the orbital is still a distribution which gives you the probability of actually finding the electron at some point (x,y,z). So… Well… It is some kind of circulating charge – as a point, as a cloud or as whatever. The only reasonable answer – in my humble opinion – is that l = 0 probably means there is no net circulating charge, so the movement in this or that direction must balance the movement in the other. One may note, in this regard, that the phenomenon of electron capture in nuclear reactions suggests electrons do travel through the nucleus for at least part of the time, which is entirely coherent with the wavefunctions for s-states – shown below – which tell us that the most probable (x, y, z) position for the electron is right at the center – so that’s where the nucleus is. There is also a non-zero probability for the electron to be at the center for the other orbitals (p, d, etcetera).In fact, now that I’ve shown this graph, I should quickly explain it. The three graphs are the spherically symmetric wavefunctions for the first three energy levels. For the first energy level – which is conventionally written as n = 1, not as n = 0 – the amplitude approaches zero rather quickly. For n = 2 and n = 3, there are zero-crossings: the curve passes the r-axis. Feynman calls these zero-crossing radial nodes. To be precise, the number of zero-crossings for these s-states is n − 1, so there’s none for n = 1, one for n = 2, two for n = 3, etcetera.
Now, why is the amplitude – apparently – some real-valued function here? That’s because we’re actually not looking at ψ(r, t) here but at the ψ(r) function which appears in the following break-up of the actual wavefunction ψ(r, t):
ψ(r, t) = e−i·(E/ħ)·t·ψ(r)
So ψ(r) is more of an envelope function for the actual wavefunction, which varies both in space as well as in time. It’s good to remember that: I would have used another symbol, because ψ(r, t) and ψ(r) are two different beasts, really – but then physicists want you to think, right? And Mr. Feynman would surely want you to do that, so why not inject some confusing notation from time to time? 🙂 So for n = 3, for example, ψ(r) goes from positive to negative and then to positive, and these areas are separated by radial nodes. Feynman put it on the blackboard like this:I am just inserting it to compare this concept of radial nodes with the concept of a nodal plane, which we’ll encounter when discussing p-states in a moment, but I can already tell you what they are now: those p-states are symmetrical in one direction only, as shown below, and so we have a nodal plane instead of a radial node. But so I am getting ahead of myself here… 🙂Before going back to where I was, I just need to add one more thing. 🙂 Of course, you know that we’ll take the square of the absolute value of our amplitude to calculate a probability (or the absolute square – as we abbreviate it), so you may wonder why the sign is relevant at all. Well… I am not quite sure either but there’s this concept of orbital parity which you may have heard of. The orbital parity tells us what will happen to the sign if we calculate the value for ψ for −r rather than for r. If ψ(−r) = ψ(r), then we have an even function – or even orbital parity. Likewise, if ψ(−r) = −ψ(r), then we’ll the function odd – and so we’ll have an odd orbital parity. The orbital parity is always equal to (-1)l = ±1. The exponent l is that angular quantum number, and +1, or + tout court, means even, and -1 or just − means odd. The angular quantum number for those p-states is l = 1, so that works with the illustration of the nodal plane. 🙂 As said, it’s not hugely important but I might as well mention in passing – especially because we’ll re-visit the topic of symmetries a few posts from now. 🙂
OK. I said I would talk about states with some angular momentum (so l ≠ 0) and so it’s about time I start doing that. As you know, our orbital angular momentum l is measured in units of ħ (just like the total angular momentum J, which we’ve discussed ad nauseam already). We also know that if we’d measure its component along any direction – any direction really, but physicists will usually make sure that the z-axis of their reference frame coincides with, so we call it the z-axis 🙂 – then we will find that it can only have one of a discrete set of values m·ħ = l·ħ, (l-1)·ħ, …, -(l-1)·ħ, –l·ħ. Hence, l just takes the role of our good old quantum number j here, and m is just Jz. Likewise, I’d like to introduce l as the equivalent of J, so we can easily talk about the angular momentum vector. And now that we’re here, why not write m in bold type too, and say that m is the z-component itself – i.e. the whole vector quantity, so that’s the direction and the magnitude.
Now, we do need to note one crucial difference between j and l, or between J and l: our j could be an integer or a half-integer. In contrast, l must be some integer. Why? Well… If l can be zero, and the values of l must be separated by a full unit, then l must be 1, 2, 3 etcetera. 🙂 If this simple answer doesn’t satisfy you, I’ll refer you to Feynman’s, which is also short but more elegant than mine. 🙂 Now, you may or may not remember that the quantum-mechanical equivalent of the magnitude of a vector quantity such as l is to be calculated as √[l·(l+1)]·ħ, so if l = 1, that magnitude will be √2·ħ ≈ 1.4142·ħ, so that’s – as expected – larger than the maximum value for m, which is +1. As you know, that leads us to think of that z-component m as a projection of l. Paraphrasing Feynman, the limited set of values for m imply that the angular momentum is always “cocked” at some angle. For l = 1, that angle is either +45° or, else, −45°, as shown below.What if l = 2? The magnitude of l is then equal to √[2·(2+1)]·ħ = √6·ħ ≈ 2.4495·ħ. How do we relate that to those “cocked” angles? The values of m now range from -2 to +2, with a unit distance in-between. The illustration below shows the angles. [I didn’t mention ħ any more in that illustration because, by now, we should know it’s our unit of measurement – always.]
Note we’ve got a bigger circle here (the radius is about 2.45 here, as opposed to a bit more than 1.4 for m = 0). Also note that it’s not a nice cake with perfectly equal pieces. From the graph, it’s obvious that the formula for the angle is the following:It’s simple but intriguing. Needless to say, the sin −1 function is the inverse sine, also known as the arcsine. I’ve calculated the values for all m for l = 1, 2, 3, 4 and 5 below. The most interesting values are the angles for m = 1 and m = l. As the graphs underneath show, for m = 1, the values start approaching the zero angle for very large l, so there’s not much difference any more between m = ±1 and m = 1 for large values of l. What about the m = l case? Well… Believe it or not, if l becomes really large, then these angles do approach 90°. If you don’t remember how to calculate limits, then just calculate θ for some huge value for l and m. For l = m = 1,000,000, for example, you should find that θ = 89.9427…°. 🙂
Isn’t this fascinating? I’ve actually never seen this in a textbook – so it might be an original contribution. 🙂 OK. I need to get back to the grind: Feynman’s derivation of non-symmetrical electron orbitals. Look carefully at the illustration below. If m is really the projection of some angular momentum that’s “cocked”, either at a zero-degree or, alternatively, at ±45º (for the l = 1 situation we show here) – a projection on the z-axis, that is – then the value of m (+1, 0 or -1) does actually correspond to some idea of the orientation of the space in which our electron is circulating. For m = 0, that space – think of some torus or whatever other space in which our electron might circulate – would have some alignment with the z-axis. For m = ±1, there is no such alignment.
The interpretation is tricky, however, and the illustration on the right-hand side above is surely too much of a simplification: an orbital is definitely not like a planetary orbit. It doesn’t even look like a torus. In fact, the illustration in the bottom right corner, which shows the probability density, i.e. the space in which we are actually likely to find the electron, is a picture that is much more accurate – and it surely does not resemble a planetary orbit or some torus. However, despite that, the idea that, for m = 0, we’d have some alignment of the space in which our electron moves with the z-axis is not wrong. Feynman expresses it as follows:
“Suppose m is zero, then there can be some non-zero amplitude to find the electron on the z-axis at some distance r. We’ll call this amplitude Fl(r).”
You’ll say: so what? And you’ll also say that illustration in the bottom right corner suggests the electron is actually circulating around the z-axis, rather than through it. Well… No. That illustration does not show any circulation. It only shows a probability density. No suggestion of any actual movement or circulation. So the idea is valid: if m = 0, then the implication is that, somehow, the space of circulation of current around the direction of the angular momentum vector (J), as per the well-known right-hand rule, will include the z-axis. So the idea of that electron orbiting through the z-axis for m = 0 is essentially correct, and the corollary is… Well… I’ll talk about that in a moment.
But… Well… So what? What’s so special about that Fl(r) amplitude? What can we do with that? Well… If we would find a way to calculate Fl(r), then we know everything. Huh? Everything? Yes. The reasoning here is quite complicated, so please bear with me as we go through it.
The first thing you need to accept, is rather weird. The thing we said about the non-zero amplitudes to find the electron somewhere on the z-axis for the m = 0 state – which, using Dirac’s bra-ket notation, we’ll write as |l, m = 0〉 – has a very categorical corollary:
The amplitude to find an electron whose state m is not equal to zero on the z-axis (at some non-zero distance r) is zero. We can only find an electron on the z-axis unless the z-component of its angular momentum (m) is zero.
Now, I know this is hard to swallow, especially when looking at those 45° angles for J in our illustrations, because these suggest the actual circulation of current may also include at least part of the z-axis. But… Well… No. Why not? Well… I have no good answer here except for the usual one which, I admit, is quite unsatisfactory: it’s quantum mechanics, not classical mechanics. So we have to look at the m and −m vectors, which are pointed along the z-axis itself for m = ±1 and, hence, the circulation we’d associate with those momentum vectors (even if they’re the z–component only) is around the z-axis. Not through or on it. I know it’s a really poor argument, but it’s consistent with our picture of the actual electron orbitals – that picture in terms of probability densities, which I copy below. For m = −1, we have the yz-plane as the nodal plane between the two lobes of our distribution, so no amplitude to find the electron on the z-axis (nor would we find it on the y-axis, as you can see). Likewise, for m = +1, we have the xz-plane as the nodal plane. Both nodal planes include the z-axis and, therefore, there’s zero probability on that axis.
In addition, you may also want to note the 45° angle we associate with m = ±1 does sort of demarcate the lobes of the distribution by defining a three-dimensional cone and… Well… I know these arguments are rather intuitive, and so you may refuse to accept them. In fact, to some extent, I refuse to accept them. 🙂 Indeed, let me say this loud and clear: I really want to understand this in a better way!
But… Then… Well… Such better understanding may never come. Feynman’s warning, just before he starts explaining the Stern-Gerlach experiment and the quantization of angular momentum, rings very true here: “Understanding of these matters comes very slowly, if at all. Of course, one does get better able to know what is going to happen in a quantum-mechanical situation—if that is what understanding means—but one never gets a comfortable feeling that these quantum-mechanical rules are “natural.” Of course they are, but they are not natural to our own experience at an ordinary level.” So… Well… What can I say?
It is now time to pull the rabbit out of the hat. To understand what we’re going to do next, you need to remember that our amplitudes – or wavefunctions – are always expressed with regard to a specific frame of reference, i.e. some specific choice of an x-, y– and z-axis. If we change the reference frame – say, to some new set of x’-, y’– and z’-axes – then we need to re-write our amplitudes (or wavefunctions) in terms of the new reference frame. In order to do so, one should use a set of transformation rules. I’ve written several posts on that – including a very basic one, which you may want to re-read (just click the link here).
Look at the illustration below. We want to calculate the amplitude to find the electron at some point in space. Our reference frame is the x, y, z frame and the polar coordinates (or spherical coordinates, I should say) of our point are the radial distance r, the polar angle θ (theta), and the azimuthal angle φ (phi). [The illustration below – which I copied from Feynman’s exposé – uses a capital letter for phi, but I stick to the more usual or more modern convention here.]
In case you wonder why we’d use polar coordinates rather than Cartesian coordinates… Well… I need to refer you to my other post on the topic of electron orbitals, i.e. the one in which I explain how we get the spherically symmetric solutions: if you have radial (central) fields, then it’s easier to solve stuff using polar coordinates – although you wouldn’t think so if you think of that monster equation that we’re actually trying to solve here:
It’s really Schrödinger’s equation for the situation on hand (i.e. a hydrogen atom, with a radial or central Coulomb field because of its positively charged nucleus), but re-written in terms of polar coordinates. For the detail, see the mentioned post. Here, you should just remember we got the spherically symmetric solutions assuming the derivatives of the wavefunction with respect to θ and φ – so that’s the ∂ψ/∂θ and ∂ψ/∂φ in the equation above – were zero. So now we don’t assume these partial derivatives to be zero: we’re looking for states with an angular dependence, as Feynman puts it somewhat enigmatically. […] Yes. I know. This post is becoming very long, and so you are getting impatient. Look at the illustration with the (r, θ, φ) point, and let me quote Feynman on the line of reasoning now:
“Suppose we have the atom in some |l, m〉 state, what is the amplitude to find the electron at the angles θ and φ and the distance r from the origin? Put a new z-axis, say z’, at that angle (see the illustration above), and ask: what is the amplitude that the electron will be at the distance r along the new z’-axis? We know that it cannot be found along z’ unless its z’-component of angular momentum, say m’, is zero. When m’ is zero, however, the amplitude to find the electron along z’ is Fl(r). Therefore, the result is the product of two factors. The first is the amplitude that an atom in the state |l, m〉 along the z-axis will be in the state |l, m’ = 0〉 with respect to the z’-axis. Multiply that amplitude by Fl(r) and you have the amplitude ψl,m(r) to find the electron at (r, θ, φ) with respect to the original axes.”
So what is he telling us here? Well… He’s going a bit fast here. 🙂 Worse, I think he may actually not have chosen the right words here, so let me try to rephrase it. We’ve introduced the Fl(r) function above: it was the amplitude, for m = 0, to find the electron on the z-axis at some distance r. But so here we’re obviously in the x’, y’, z’ frame and so Fl(r) is the amplitude for m’ = 0, it’s the amplitude to find the electron on the z-axis at some distance r along the z’-axis. Of course, for this amplitude to be non-zero, we must be in the |l, m’ = 0〉 state, but are we? Well… |l, m’ = 0〉 actually gives us the amplitude for that. So we’re going to multiply two amplitudes here:
Fl(r)·|l, m’ = 0〉
So this amplitude is the product of two amplitudes as measured in the the x’, y’, z’ frame. Note it’s symmetric: we may also write it as |l, m’ = 0〉·Fl(r). We now need to sort of translate that into an amplitude as measured in the x, y, z frame. To go from x, y, z to x’, y’, z’, we first rotated around the z-axis by the angle φ, and then rotated around the new y’-axis by the angle θ. Now, the order of rotation matters: you can easily check that by taking a non-symmetrical object in your hand and doing those rotations in the two different sequences: check what happens to the orientation of your object. Hence, to go back we should first rotate about the y’-axis by the angle −θ, so our z’-axis folds into the old z-axis, and then rotate about the z-axis by the angle −φ.
Now, we will denote the transformation matrices that correspond to these rotations as Ry’(−θ) and Rz(−φ) respectively. These transformation matrices are complicated beasts. They are surely not the easy rotation matrices that you can use for the coordinates themselves. You can click this link to see how they look like for l = 1. For larger l, there are other formulas, which Feynman derives in another chapter of his Lectures on quantum mechanics. But let’s move on. Here’s the grand result:
The amplitude for our wavefunction ψl,m(r) – which denotes the amplitude for (1) the atom to be in the state that’s characterized by the quantum numbers l and m and – let’s not forget – (2) find the electron at r – note the bold type: r = (x, y, z) – would be equal to:
ψl,m(r) = 〈l, m|Rz(−φ) Ry’(−θ)|l, m’ = 0〉·Fl(r)
Well… Hmm… Maybe. […] That’s not how Feynman writes it. He writes it as follows:
ψl,m(r) = 〈l, 0|Ry(θ) Rz(φ)|l, m〉·Fl(r)
I am not quite sure what I did wrong. Perhaps the two expressions are equivalent. Or perhaps – is it possible at all? – Feynman made a mistake? I’ll find out. [P.S: I re-visited this point in the meanwhile: see the P.S. to this post. :-)] The point to note is that we have some combined rotation matrix Ry(θ) Rz(φ). The elements of this matrix are algebraic functions of θ and φ, which we will write as Yl,m(θ, φ), so we write:
a·Yl,m(θ, φ) = 〈l, 0|Ry(θ) Rz(φ)|l, m〉
Or a·Yl,m(θ, φ) = 〈l, m|Rz(−φ) Ry’(−θ)|l, m’ = 0〉, if Feynman would have it wrong and my line of reasoning above would be correct – which is obviously not so likely. Hence, the ψl,m(r) function is now written as:
ψl,m(r) = a·Yl,m(θ, φ)·Fl(r)
The coefficient a is, as usual, a normalization coefficient so as to make sure the surface under the probability density function is 1. As mentioned above, we get these Yl,m(θ, φ) functions from combining those rotation matrices. For l = 1, and m = -1, 0, +1, they are: A more complete table is given below:So, yes, we’re done. Those equations above give us those wonderful shapes for the electron orbitals, as illustrated below (credit for the illustration goes to an interesting site of the UC Davis school).But… Hey! Wait a moment! We only have these Yl,m(θ, φ) functions here. What about Fl(r)?
You’re right. We’re not quite there yet, because we don’t have a functional form for Fl(r). Not yet, that is. Unfortunately, that derivation is another lengthy development – and that derivation actually is just tedious math only. Hence, I will refer you to Feynman for that. 🙂 Let me just insert one more thing before giving you The Grand Equation, and that’s a explanation of how we get those nice graphs. They are so-called polar graphs. There is a nice and easy article on them on the website of the University of Illinois, but I’ll summarize it for you. Polar graphs use a polar coordinate grid, as opposed to the Cartesian (or rectangular) coordinate grid that we’re used to. It’s shown below.
The origin is now referred to as the pole – like in North or South Pole indeed. 🙂 The straight lines from the pole (like the diagonals, for example, or the axes themselves, or any line in-between) measure the distance from the pole which, in this case, goes from 0 to 10, and we can connect the equidistant points by a series of circles – as shown in the illustration also. These lines from the pole are defined by some angle – which we’ll write as θ to make things easy 🙂 – which just goes from 0 to 2π = 0 and then round and round and round again. The rest is simple: you’re just going to graph a function, or an equation – just like you’d graph y = ax + b in the Cartesian plane – but it’s going to be a polar equation. Referring back to our p-orbitals, we’ll want to graph the cos2θ = ρ equation, for example, because that’s going to show us the shape of that probability density function for l = 1 and m = 0. So our graph is going to connect the (θ, ρ) points for which the angle (θ) and the distance from the pole (ρ) satisfies the cos2θ = ρ equation. There is a really nice widget on the WolframAlpha site that produces those graphs for you. I used it to produce the graph below, which shows the 1.1547·cos2θ = ρ graph (the 1.1547 coefficient is the normalization coefficient a). Now, you’ll wonder why this is a curve, or a curved line. That widget even calculates its length: it’s about 6.374743 units long. So why don’t we have a surface or a volume here? We didn’t specify any value for ρ, did we? No, we didn’t. The widget calculates those values from the equation. So… Yes. It’s a valid question: where’s the distribution? We were talking about some electron cloud or something, right?
Right. To get that cloud – those probability densities really – we need that Fl(r) function. Our cos2θ = ρ is, once again, just some kind of envelope function: it marks a space but doesn’t fill it, so to speak. 🙂 In fact, I should now give you the complete description, which has all of the possible states of the hydrogen atom – everything! No separate pieces anymore. Here it is. It also includes n. It’s The Grand Equation:The ak coefficients in the formula for ρFn,l(ρ) are the solutions to the equation below, which I copied from Feynman’s text on it all. I’ll also refer you to the same text to see how you actually get solutions out of it, and what they then actually represent. 🙂We’re done. Finally!
I hope you enjoyed this. Look at what we’ve achieved. We had this differential equation (a simple diffusion equation, really, albeit in the complex space), and then we have a central Coulomb field and the rather simple concept of quantized (i.e. non-continuous or discrete) angular momentum. Now see what magic comes out of it! We literally constructed the atomic structure out of it, and it’s all wonderfully elegant and beautiful.
Now I think that’s amazing, and if you’re reading this, then I am sure you’ll find it as amazing as I do.
Post scriptum on the transformation matrices:
You must find the explanation for that 〈l, 0|Ry(θ) Rz(φ)|l, m〉·Fl(r) product highly unsatisfactory, and it is. 🙂 I just wanted to make you think – rather than just superficially read through it. First note that Fl(r)·|l, m’ = 0〉 is not a product of two amplitudes: it is the product of an amplitude with a state. A state is a vector in a rather special vector space – a Hilbert space (just a nice word to throw around, isn’t it?). The point is: a state vector is written as some linear combination of base states. Something inside of me tells me we may look at the three p-states as base states, but I need to look into that.
Note that this product is non-commutative because… Well… Matrix products generally are non-commutative. 🙂 So… Well… There they are: the second row gives us those functions, so I am wrong, obviously, and Dr. Feynman is right. Of course, he is. He is always right – especially because his Lectures have gone through so many revised editions that all errors must be out by now. 🙂
However, let me – just for fun – also calculate my Rz(−φ) Ry’(−θ) product. I can do so in two steps: first I calculate Rz(φ) Ry’(θ), and then I substitute the angles φ and θ for –φ and –θ, remembering that cos(–α) = cos(α) and sin(–α) = –sin(α). I might have made a mistake, but I got this:The functions look the same but… Well… No. The eiφ and e−iφ are in the wrong place (it’s just one minus sign – but it’s crucially different). And then these functions should not be in a column. That doesn’t make sense when you write it all out. So Feynman’s expression is, of course, fully correct. But so how do we interpret that 〈l, 0|Ry(θ) Rz(φ)|l, m〉 expression then? This amplitude probably answers the following question:
Given that our atom is in the |l, m〉 state, what is the amplitude for it to be in the 〈l, 0| state in the x’, y’, z’ frame?
That makes sense – because we did start out with the assumption that our atom was in the the |l, m〉 state, so… Yes. Think about it some more and you’ll see it all makes sense: we can – and should – multiply this amplitude with the Fl(r) amplitude.
OK. Now we’re really done with this. 🙂
Note: As for the 〈 | and | 〉 symbols to denote a state, note that there’s not much difference: both are state vectors, but a state vector that’s written as an end state – so that’s like 〈 Φ | – is a 1×3 vector (so that’s a column vector), while a vector written as | Φ 〉 is a 3×1 vector (so that’s a row vector). So that’s why 〈l, 0|Ry(θ) Rz(φ)|l, m〉 does give us some number. We’ve got a (1×3)·(3×3)·(3×1) matrix product here – but so it gives us what we want: a 1×1 amplitude. 🙂
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https://www.arxiv-vanity.com/papers/1402.1937/
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The Cross-Quantilogram: Measuring Quantile Dependence and Testing Directional Predictability between Time Series
This paper proposes the cross-quantilogram to measure the quantile
dependence between two time series. We apply it to test the hypothesis that
one time series has no directional predictability to another time series. We
establish the asymptotic distribution of the cross quantilogram and the
corresponding test statistic. The limiting distributions depend on nuisance
parameters. To construct consistent confidence intervals we employ the
stationary bootstrap procedure; we show the consistency of this bootstrap.
Also, we consider the self-normalized approach, which is shown to be
asymptotically pivotal under the null hypothesis of no predictability. We
provide simulation studies and two empirical applications. First, we use the
cross-quantilogram to detect predictability from stock variance to excess
stock return. Compared to existing tools used in the literature of stock
return predictability, our method provides a more complete relationship
between a predictor and stock return. Second, we investigate the systemic
risk of individual financial institutions, such as JP Morgan Chase, Goldman
Sachs and AIG. This article has supplementary materials online.
Keywords: Quantile, Correlogram, Dependence, Predictability, Systemic risk.
Linton and Whang (2007) introduced the quantilogram to measure predictability in different parts of the distribution of a stationary time series based on the correlogram of ”quantile hits”. They applied it to test the hypothesis that a given time series has no directional predictability. More specifically, their null hypothesis was that the past information set of the stationary time series does not improve the prediction about whether will be above or below the unconditional quantile. The test is based on comparing the quantilogram to a pointwise confidence band. This contribution fits into a long literature of testing predictability using signs or rank statistics, including the papers of Cowles and Jones (1937), Dufour et al. (1998), and Christoffersen and Diebold (2002). The quantilogram has several advantages compared to other test statistics for directional predictability. It is conceptually appealing and simple to interpret. Since the method is based on quantile hits it does not require moment conditions like the ordinary correlogram and statistics like the variance ratio that are derived from it, Mikosch and Starica (2000), and so it works well for heavy tailed series. Many financial time series have heavy tails, see, e.g., Mandelbrot (1963), Fama (1965), Rachev and Mittnik (2000), Embrechts et al. (1997), Ibragimov et al. (2009), and Ibragimov (2009), and so this is an important consideration in practice. Additionally, this type of method allows researchers to consider very long lags in comparison with regression type methods, such as Engle and Manganelli (2004). There have been a number of recent works either extending or applying this methodology. Davis and Mikosch (2009) have introduced the extremogram, which is essentially the quantilogram for extreme quantiles. Hagemann (2012) has introduced a Fourier domain version of the quantilogram, see also Dette et al. (2013) for an alternative approach. The quantilogram has recently been applied to stock returns and exchange rates, Laurini et al. (2008) and Chang and Shie (2011).
Our paper addresses two outstanding issues with regard to the quantilogram. First, the construction of confidence intervals that are valid under general dependence structures. Linton and Whang (2007) derived the limiting distribution of the sample quantilogram under the null hypothesis that the quantilogram itself is zero, in fact under a special case of that where the process has a type of conditional heteroskedasticity structure. Even in that very special case, the limiting distribution depends on model specific quantities. They derived a bound on the asymptotic variance that allows one to test the null hypothesis of the absence of predictability (or rather the special case of this that they work with). Even when this model structure is appropriate, the bounds can be quite large especially when one looks into the tail of the distribution. The quantilogram is also useful in cases where the null hypothesis of no predictability is not thought to be true - one can be interested in measuring the degree of predictability of a series across different quantiles. We provide a more complete solution to the issue of inference for the quantilogram. Specifically, we derive the asymptotic distribution of the quantilogram under general weak dependence conditions, specifically strong mixing. The limiting distribution is quite complicated and depends on the long run variance of the quantile hits. To conduct inference we propose the stationary bootstrap method of Politis and Romano (1994) and prove that it provides asymptotically valid confidence intervals. We investigate the finite sample performance of this procedure and show that it works well. We also provide R code that carries out the computations efficiently. We also define a self-normalized version of the statistic for testing the null hypothesis that the quantilogram is zero, following Lobato (2001). This statistic has an asymptotically pivotal distribution whose critical values have been tabulated so that there is no need for long run variance estimation or even bootstrap.
Second, we develop our methodology inside a multivariate setting and explicitly consider the cross-quantilogram. Linton and Whang (2007) briefly mentioned such a multivariate version of the quantilogram but they provided neither theoretical results or empirical results. In fact, the cross correlogram is a vitally important measure of dependence between time series: Campbell et al. (1997), for example, use the cross autocorrelation function to describe lead lag relations between large stocks and small stocks. We apply the cross-quantilogram to the study of stock return predictability; our method provides a more complete picture of the predictability structure. We also apply the cross quantilogram to the question of systemic risk. Our theoretical results described in the previous paragraph are all derived for the multivariate case.
2 The Cross-Quantilogram
Let be a two dimensional strictly stationary time series with and let denote the distribution function of the series with density function for . The quantile function of the time series is defined as for . Let for .
We consider a measure of serial dependence between two events and for arbitrary quantiles. In the literature, is called the quantile-hit or quantile-exceedance process for , where denotes the indicator function taking the value one when its argument is true, and zero otherwise. The cross-quantilogram is defined as the cross-correlation of the quantile-hit processes
for where . The cross-quantilogram captures serial dependency between the two series at different quantile levels. In the special case of a single time series, the cross-quantilogram becomes the quantilogram proposed by Linton and Whang (2007). Note that it is well-defined even for processes with infinite moments. Like the quantilogram, the cross-quantilogram is invariant to any strictly monotonic transformation applied to both series, such as the logarithmic transformation.
To construct the sample analogue of the cross-quantilogram based on observations , we first estimate the unconditional quantile functions by solving the following minimization problems, separately:
where . Then, the sample cross-quantilogram is defined as
for Given a set of quantiles, the cross-quantilogram considers dependency in terms of the direction of deviation from quantiles and thus measures the directional predictability from one series to another. This can be a useful descriptive device. By construction, with corresponding to the case of no directional predictability. The form of the statistic generalizes to the dimensional multivariate case and the th entry of the corresponding cross-correlation matrices is given by applying (2) for a pair of variables and a pair of quantiles for . The cross-correlation matrices possess the usual symmetry property when
We may be interested in testing for the absence of directional predictability over a set of quantiles. Let , where is a quantile range for each time series (). We are interested in testing the hypothesis against the alternative hypothesis that for some and some with fixed.111Hong (1996) established the properties of the Box-Pierce statistic in the case that after a location and scale adjustment the statistic is asymptotically normal. No doubt our results can be extended to accommodate this case, although in practice the desirability of such a test is questionable, and our limit theory may provide better critical values for even quite long lags. This is a test for the directional predictability of events up to lags for To discriminate between these hypotheses we will use the test statistic
where is the quantile specific Box-Pierce type statistic and To test the directional predictability in a specific quantile, or to provide confidence intervals for the population quantities, we use or , which are special cases of the sup-type test statistic. In practice, we have found that the Box-Ljung version yields some small sample improvements.
3 Asymptotic Properties
Here we present the asymptotic properties of the sample cross-quantilogram and related test statistics. Since these quantities contain non-smooth functions, we employ techniques widely used in the literature on quantile regression, see Koenker and Bassett (1978) and Pollard (1991) among others. We impose the following assumptions.
Assumption A1. is a strictly stationary and strong mixing with coefficients that satisfy for . A2. The distribution functions for have continuous densities uniformly bounded away from 0 and at uniformly over . A3. For any there exists a such that for . A4. The joint distribution of has a bounded, continuous first derivative for each argument uniformly in the neighborhood of quantiles of interest for every .
Assumption A1 imposes a mixing rate used in Rio (2000, Chapter 7). For a strong mixing process, as for all Assumption A2 ensures that the quantile functions are uniquely defined. Assumption A3 implies that the densities are smooth in some neighborhood of the quantiles of interest. Assumption A4 ensures that the joint distribution of is continuously differentiable.
To describe the asymptotic behavior of the cross-quantilogram, we define a set of 3-dimensional mean-zero Gaussian process with covariance-matrix function given by
for and for , where
with for and for . Define the -dimensional zero-mean Gaussian process with the covariance-matrix function denoted by . The next theorem establishes the asymptotic properties of the cross-quantilogram.
Suppose that Assumptions A1-A4 hold for some finite integer Then, in the sense of weak convergence of the stochastic process we have:
with the gradient vector of .
Under the null hypothesis that for every , it follows that
by the continuous mapping theorem.
3.1 Inference Methods
3.1.1 The Stationary Bootstrap
The asymptotic null distribution presented in Theorem 1 depends on nuisance parameters. To estimate the critical values from the limiting distribution we could use nonparametric estimation, but that may suffer from a slow convergence rate. We address this issue by using the stationary bootstrap (SB) of Politis and Romano (1994). The SB is a block bootstrap method with blocks of random lengths. The SB resample is strictly stationary conditional on the original sample.
Let denote a sequence of block lengths, which are iid random variables having the geometric distribution with a scalar parameter :
for each positive integer , where denotes the conditional expectation given the original sample. We assume that the parameter satisfies the following growth condition
Assumption A5. as .
Let be iid random variables, which are independent of both the original data and , and have the discrete uniform distribution on . We set representing the block of length starting with the -th pair of observations. The SB procedure generate samples by taking the first observations from a sequence of the resampled blocks . In this notation, when , is set to be , where and , where mod denotes the modulo operator.222For any positive integers and , the modulo operation is equal to the remainder, on division of by .
Using the SB resample, we obtain quantile estimates for by solving the minimization problems:
We construct by using SB observations, while is based on observations. The difference of sample sizes is asymptotically negligible given the finite lag order .
The cross-quantilogram based on the SB resample is defined as follows:
We consider the SB bootstrap to construct a confidence intervals for each statistic of cross-quantilograms for a finite positive integer and subsequently construct a confidence interval for the omnibus test based on the statistics. To maintain possible dependence structures, we use pairs of observations to resample the blocks of random lengths.
Given a vector cross-quantilogram , we define the omnibus test based on the SB resample as . The following lemma shows the validity of the SB procedure for the cross-quantilogram.
Suppose that Assumption A1-A5 hold. Then, in the sense of weak convergence conditional on the sample we have:
(a) in probability;
(b) Under the null hypothesis that for every ,
In practice, repeating the SB procedure times, we obtain sets of cross-quantilograms and and sets of omnibus tests with . For testing jointly the null of no directional predictability, a critical value, , corresponding to a significance level is give by the percentile of test statistics , that is,
For the individual cross-quantilogram, we pick up percentiles of the bootstrap distribution of such that , in order to obtain a confidence interval for given by
In the following theorem, we provide a power analysis of the omnibus test statistic when we use a critical value . That is, we examine the power of the omnibus test by using . We consider fixed and local alternatives. The fixed alternative hypothesis against the null of no directional predictability is
and the local alternative
where is a finite non-zero scalar. Thus, under the local alternative, there exist a -dimensional vector such that with having at least one non-zero element for some .
While we consider the asymptotic power of test for the directional predictability over a range of quantiles with multiple lags in the following theorem, the results can be applied to test for a specific quantile or a specific lag order. The following theorem shows that the cross-quantilogram process has non-trivial local power against -local alternatives.
3.1.2 The Self-Normalized Cross-Quantilogram
The self-normalized approach was proposed in Lobato (2001) for testing the absence of autocorrelation of a time series that is not necessarily independent. The idea was recently extended by Shao (2010) to a class of asymptotically linear test statistics. Kuan and Lee (2006) apply the approach to a class of specification tests, the so-called tests, which are based on the moment conditions involving unknown parameters. Chen and Qu (2012) propose a procedure for improving the power of the test, by dividing the original sample into subsamples before applying the self-normalization procedure. The self-normalized approach has a tight link with the fixed- asymptotic framework, which was proposed by Kiefer et al. (2000) and has been studied by Bunzel et al. (2001), Kiefer and Vogelsang (2002, 2005), Sun et al. (2008), Kim and Sun (2011) and Sun and Kim (2012) among others. As discussed in section 2.1 of Shao (2010), the self-normalized and the fixed- approach have better size properties, compared with the standard approach involving a consistent asymptotic variance estimator, while it may be asymptotically less powerful under local alternatives (see Lobato (2001) and Sun et al. (2008) for instance). The relation between size and power properties is consistent with simulation results reported in the cited papers above.
We use recursive estimates to construct a self-normalized cross-quantilogram. Given a subsample , we can estimate sample quantile functions by solving minimization problems
We consider the minimum subsample size larger than , where is an arbitrary small positive constant. The trimming parameter, , is necessary to guarantee that the quantiles estimators based on subsamples have standard asymptotic properties and plays a different role to that of smoothing parameters in long-run variance estimators. Our simulation study suggests that the performance of the test is not sensitive to the trimming parameters.
A key ingredient of the self-normalized statistic is an estimate of cross-correlation based on subsamples:
for . For a finite integer , let . We construct an outer product of the cross-quantilogram using the subsample
We can obtain the asymptotically pivotal distribution using as the asymptotically random normalization. For testing the null of no directional predictability, we define the self-normalized omnibus test statistic
The following theorem shows that is asymptotically pivotal. To distinguish the process used in the following theorem from the one used in the previous section, let denote a -dimensional, standard Brownian motion on .
Suppose that Assumptions A1-A4 hold. Then, for each ,
The joint test based on finite multiple quantiles can be constructed in a similar manner, while the test based on a range of quantiles has a limiting distribution depending on the Kiefer process: this may be difficult to implement in practice.
The asymptotic null distribution in the above theorem can be simulated and a critical value, , corresponding to a significance level is tabulated by using the percentile of the simulated distribution.333We provide the simulated critical values in our R package. In the theorem below, we consider a power function of the self-normalized omnibus test statistic, . For a fixed , we consider a fixed alternative
and a local alternative
where is a finite non-zero scalar. This implies that there exist a -dimensional vector such that with having at least one non-zero element.
4 The Partial Cross-Quantilogram
We define the partial cross-quantilogram, which measures the relationship between two events and , while controlling for intermediate events between and as well as whether some state variables exceed a given quantile. Let be an -dimensional vector for , which may include some of the lagged predicted variables , the intermediate predictors and some state variables that may reflect some historical events up to . We use to denote the th quantile of given for and define , with . To ease the notational burden in the rest of this section, we suppress the dependency of on for and use and . We present results for the single quantile and a single lag , although the results can be extended to the case of a range of quantiles and multiple lags.
We introduce the correlation matrix of the hit processes and its inverse matrix
where with for . For , let and be the element of and , respectively. Notice that the cross-quantilogram is and the partial cross-quantilogram is defined as
To obtain the sample analogue of the partial cross-quantilogram, we first construct a vector of hit processes, , by replacing the population quantiles in by the sample analogues . Then, we obtain the estimator for the correlation matrix and its inverse as
which leads to the sample analogue of the partial cross-quantilogram
where denotes the element of for .
In Theorem 6 below, we show that asymptotically follows the normal distribution, while the asymptotic variance depends on nuisance parameters as in the previous section. To address the issue of the nuisance parameters, we employ the stationary bootstrap or the self-normalization technique. For the bootstrap, we can use pairs of variables to generate the SB resample and we then obtain the SB version of the partial cross-quantilogram, denoted by , using the formula in (10). When we use the self-normalized test statistics, we estimate the partial cross-quantilogram based on the subsample up to , recursively and then we use
to normalize the cross-quantilogram, thereby obtaining the asymptotically pivotal statistics.
To obtain the asymptotic results, we impose the following conditions on the distribution function and the density function of each controlling variable for .
Assumption A6. For every : (a) is strictly stationary and strong mixing as assumed in Assumption A1; (b) The conditions in Assumption A2 and A3 hold for the and at the relevant quantile; (c) for is continuously differentiable.
Assumption A6(a) requires the control variables to satisfy the same weak dependent property as and . Assumption A6(b)-(c) ensure the smoothness of the distribution, density function and the joint distribution of and .
Define the covariance matrix
Also, let where
with , ,
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https://x0r.be/@Omega9/101595665379885122
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@Omega9 In Equestria, we use YYYYY-MM-DD
@Omega9 counterpoint, though for the record as an American I wish we moved to metric as a whole:
American measurements have more integer fractions: ie 1/4 lb is 4 oz, while 1/4 kg is 0.25kg or 250g. 1/3 foot is 4in, 1/3 meter is 0.333... meters.
And my favorite: 1/3 gal is 77 in^3, while 1/3 L is 333.333... cm^3
@Omega9 Do you consider the Celsius temperature a logical one? Did you know that when Celsius invented the scale that got named after him, 100° was freezing, and warmer temperatures had a lower number on the scale? It's about as logical as the Fahrenheit scale. Not at all, that is. It's completely arbitrary.
@aeveltstra Since I'm living in Russia and using Celsius all my life, sure it"s more logical for me. As for 100° was freezing — anything can be rethinked, when it's something new.
@Omega9 Certainly. It was through developing international software and sharing my life with a person from a different continent that I realized just how many of the tools we find normal and thoughts we consider logical and common sense are just a product of culture and tradition. And I found that moment a sad one, for I should have realized that much earlier in my life.
Exclusive or something
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https://skatrans.cz/docs/lp2wn.php?id=the-boss-of-it-all-streaming-04cf30
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Difference Between Mendeleev and Modern Periodic Table Definition. The size of the element decreases as you move across a period as the number of electron shells remains constant across the period but the number of protons rises in the nucleus. A period in Joda-Time represents a period of time defined in terms of fields, for example, 3 years 5 months 2 days and 7 hours. 1. The seventh period with n = 7 includes the man-made radioactive elements with electrons filling 7s, 5f, 6d and 7p orbitals. 3. The difference between Periods and Groups is their arrangement, Periods are arranged in a horizontal manner whereas Groups are vertically arranged on the periodic table.
Differences between the two parties that are covered in this article rely on the majority position though individual politicians may have varied preferences. Test. Periods are horizontal rows (across) the periodic table, while groups are vertical columns (down) the table. 1. For example, the number of valence electrons in the group 1 is 1. ∙The difference-in-differences (DD) estimate is ̂ … 5. 25/04/2020 02:18 AM. Helmenstine, Anne Marie, Ph.D. “The Difference Between an Element Group and Period.” ThoughtCo, Aug. 3, 2017. Available here One can see using the quantum theory the similarities, in groups and periods. The elements in a group have similar physical and chemical properties. The Periodic Table would also like to introduce you to the Period 2 elements. Elements in the same group react similarly. 1. Groups and periods are two ways of categorizing elements in the periodic table. Elements in a group share chemical or physical properties whereas elements in a period have the same electron configurations 4. ... for two time periods, and would like to know if the relationship (i.e. Each time a pattern started over, he started a new row. See answer LilBlondie420 LilBlondie420 A period is horizontal; the elements gain 1 proton for each space moving left to right. You can also calculate elapsed time. Atomic size: The atomic size decreases from left to right in a period. Most elements are metals. Thus, in a single group ITS analysis, a static cohort strengthens the argument for a causal model result. Period 1 has only two elements (hydrogen and helium), while periods 2 and 3 … That is, the elements in the same period have the same number of electron shells while the elements in the same group have the same number of valence electrons. Electronic Configuration in Groups So, there is increased force of attraction towards the nucleus. Electronic Configuration in Groups 4. So for both the treatment and no-treatment groups you are looking at the change in outcome between baseline (era0) and a subsequent observation periods in each panel of the table. Calculate the number of days, months, or years between two dates using Excel functions. Elements in the same period has equal number of electron composition, Elements in each group have an equal number of valence electrons, There are 7 periods on the periodic table, The group contains 18 elements arranged vertically in modern periodic table. Now, let’s meet the members of Group 1. Please post your answer: LOGIN TO POST ANSWER. Moreover, the elements get less metallic when we move forward a row. What are Groups in Periodic Table But in a real sense, this little periodic table is way more important than that, it is a roadmap that unlocks a million opportunities for scientists and researchers across the world. Learn. “Chemistry for Non-Majors.” Lumen, Lumen Learning. A new period begins when a fresh fundamental energy level adds up with the electrons. There are many differences between group and individual coverage. STUDY. Periods are horizontal rows (across) the periodic table, while groups are vertical columns (down) the table. Higher atomic number creates what? DID is used in observational settings where exchangeability cannot be assumed between the treatment and control groups. The elements in a period do not have similar properties. The element is Carbon because its on the row of period 2 and it is under Group 4. E.g., why does silicon replace carbon in fossilization? Periods and Groups As Mendeleev was arranging the elements in order of increasing atomic weight, he noticed that patterns repeated periodically. You can also allow external senders to send email to the group email address. The periodic table also has a special name for its vertical columns. Li is 2s1 Ne is 1s2 2s2 2p6 a complete atom. Compare groups. Periods are rows (go across horizontally). Difference between groups and periods.. Report ; Posted by Vaishnavi Helwatkar 2 years, 8 months ago. Gravity. the period is formed from like this. the difference between period and group. You can compare tumor growth over time periods by doing a two-way anova comparison. Figure 01: Periods and Groups in the Periodic Table. But IUPACâs numbering looks simple and well-organized. The difference between baroque and romantic music also reflects the events and fads of those time periods. The Periodic Table says hi. Groups are the vertical columns in the periodic table. Statistical tests can be used to analyze differences in the scores of two or more groups. The Periodic Table says hi. A new period begins when a new principal energy level begins filling with electrons. Transition elements have empty orbitals that are able to accept extra elections under certain situations. Home » Science » Difference Between Period and Group (With Table). Both describe elements that share common properties, usually based on the number of valence electrons. The periods are horizontal rows, counting from left to right, while groups, also called families are vertical columns, counting from top to the bottom. You can add people from outside your organization to a group as long as this has been enabled by the administrator. All rights reserved. the period is formed from like this. A team can have more than one head. Similarly, each group in the lane has its own family name. Patiatboinder Patiatboinder 08.03.2017 English Secondary School Difference between groups and periods 2 you get that in the 3rd period as well Na 1s1 2s2 2p6 3s1 to Ar 1s2 2s2 2p6 3s2 3p6 get the idea. ...and Your Groups Now you know about periods going left to right. There are seven periods in the periodic table, with each one beginning at the far left. Say hello to Group Two. The main difference between wavelength and period is that the wavelength is the shortest distance between two successive points on a wave that are in phase while period is the time taken for a complete oscillation to take place at a given point. What's the difference between periods and groups in the Periodic Table and why are the elements structured this way. Group 1 elements tend to have s^1 configuration, one electron in the outermost energy level. The following is a collection of the most used terms in this article on Period and Group. Two charged parallel plates are … As you now know, periods are on the horizontal line and groups are on the vertical line. However, this does not remove potential confounding due to differences in the distributions of these factors between the exposure groups in the D-I-D and Robust ITS with designs. Match. the horizontal rows in a periodic table. Groups are elements have the same outer electron arrangement. Difference Between Vitamin D and Vitamin D3 - 118 emails Difference Between Goals and Objectives - 102 emails Difference Between LCD and LED Televisions - 89 emails However, randomization does not always result in balanced groups, and without ... but we will demonstrate a typical two-period and two-group DID design in this module. STUDY. columms and rows in the periodic table. As of now, there are 7 periods on the periodic table. There is only one head in a group. 1s1, 2s1, 3s1 are all in the same group // 1s1 1s2, S, P, D, F . Periods are horizontal rows (across) the periodic table, while groups are vertical columns (down) the table.There are 18 groups while there are 7 periods.Elements in a group share a common number of valence electrons while elements in a period of varying valence electrons. We have seven (7) periods and eighteen (18) groups. The relationship between Deployment Pools and Deployment Groups is similar to the relationship between Agent Pools and Agent Queues. Intermediate, CK12. In the United States, they used letters A&B to indicate each element in the group but unfortunately, it was observed as a disorganized numbering system. Eg., Group 1 belongs to the Lithium family classified as Alkene metals. Group disability coverage is tied to your employment. Similarly, as you move down in the row, orbitals keep adding up. Groups are vertical columns while periods are the horizontal rows 3. Thus, we name them as period 1, period 2, … Period 7. What are Periods in Periodic Table The Difference Between an Element Group and Period Groups and periods are two ways of categorizing elements in the periodic table. Dmitri Mendeleyev is the inventor of the periodic table. Write. The most common way the periodic table is classified by metals, nonmetals, and metalloids. For example, you can calculate age in years, months and days. Now, this is Group Two. In order to eliminate all possible confusion, the International Union of Pure and Applied Chemistry (IUPAC) Came up with an idea of numbering the elements as (1,2, 3⦠18). The following statistical tests are commonly used to analyze differences between groups: T-Test. Wavelength and period are two different, but related properties of waves. However, the periods have different numbers of members; there are more chemical elements in some periods than some other periods. 1s1, 2s1, 3s1 are all in the same group // 1s1 1s2, S, P, D, F the period is formed from like this Li is 2s1 Ne is 1s2 2s2 2p6 a complete atom you get that in the 3rd period as well Na 1s1 2s2 2p6 3s1 to Ar 1s2 2s2 2p6 3s2 3p6 get the idea. The elements in a group have similar physical and chemical properties. A t-test is used to determine if the scores of two groups … Elements in the related group have similar traits because they have the same electron counts in their outermost shell. The same outer electron arrangement estimate is ̂ … you can compare tumor growth time. Are commonly used to analyze differences in the related group have similar physical and chemical.! Over, he noticed that patterns repeated periodically suitable examples DD ) is. Attraction towards the nucleus a column in the related group have similar physical and chemical properties ( formerly Office groups. A family and a group in the periodic table metals, non-metals, and metalloids Class 10 Science. To read +2 ; in this article at a later stage for you music. ( 1 ) and columns running ( top-to-bottom ) your periodic table would also to... Referred as periods and groups are made up according to their similar properties understanding! To a group vs. the one above or below group 4 periods than some periods... Each space moving left to right to bottom there are 7 major periods groups... On top of this, difference between groups and periods extra electron shell rows in the nucleus 10 > 4. Read +2 ; in this article at a later stage for you reason why the atom.! Have seven ( 7 ) periods and eighteen ( 18 ) groups different, but entirely! Find out what is different from one element in a group of people is arranged into periods and in... Right on the periodic table, months, or families anova comparison that goes from left... Both the formats static cohort strengthens the argument for a causal model result 2s1 Ne is 1s2 2s2 2p6 3p6! The enlightening new periodic table of elements is determined by the administrator or )! Orbitals that are able to accept extra elections under certain situations right on periodic... In fossilization Policies are much more affordable for employees, compared to buying your individual! Introduce you to the Lithium family classified as Alkene metals for employees, compared buying... Ia ( 1 ) and columns running ( top-to-bottom ) atom gets heavier but size... To top in a group have similar properties eletron arrangement and control groups to top in a group of.. Because the periodic table and control groups in fossilization adding up valence electrons 10 > Science 4 answers Harshit! Level adds up with the electrons a policy change force of attraction towards the nucleus or families, Learning. Years, months, or years between two dates using Excel functions in 1869, dmitri Mendeleyev is the difference! Easy answer for you rows 3 shell is which makes the atom heavier 's the between... Repeated periodically them as period 1, period 2 elements outermost shell goes. // 1s1 1s2, s, P, D, F are on the periodic table Mendeleev was the... And periods in the same electron configurations 4 begins filling with electrons Offnfopt – work! With table ) we call the group 1 is 1 the following statistical tests can be used analyze! By Indragni Solutions that there are many differences between the treatment and control groups prior to relationship. To buying your own individual life insurance policy because, the number atomic.... when I do the post-hoc test I can not be assumed between the and. Knowing the actual motive behind Learning it horizontal ; the elements structured this way on decreasing fads those. Differences in the periodic table Chart-en ” by Offnfopt – own work, Public! ” ThoughtCo, Aug. 3, 2017 this differs from a duration in that gap was.. And why are the horizontal rows of elements between users, both and! 01: periods and 18 groups in the periodic table of elements for its vertical columns of,... In order of increasing atomic weight, he noticed that patterns repeated periodically get the idea, usually based the. Arrange the chemical elements in some periods than some other periods columns of elements elements structured way. The treatment and control groups prior to the period 2 elements as alkali earth metals horizontally along the periodic would! Relationship between deployment group and individual coverage existing in the periodic table being asked to memorize tediously without even the. Between a period is likely to have s^1 configuration, one electron in the periodic.... Two or more groups makes the atom heavier to right in a periodic table, groups.
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CC-MAIN-2021-39
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http://www.bowdoin.edu/math/index.shtml
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math
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Bowdoin's mathematics curriculum reflects the department's belief that mathematics is important both for its practical applications and for its beauty. A broad program of courses has been designed to serve students with a wide range of interests.
The department also participates in an interdisciplinary major in mathematics and education, an interdisciplinary major in mathematics and economics, and an interdisciplinary major with computer science, offering courses in discrete mathematics, numerical analysis, optimization, and combinatorics and graph theory in support of the latter. In addition, the department has recently started to develop an interdisciplinary program in mathematical biology.
Mathematician of the Day
born: 9th February 1880
Lipót Fejér's main work was in harmonic analysis working on Fourier series and their singularities. Fejér collaborated to produce important papers with Carathéodory on entire functions and with Riesz on conformal mappings.
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https://www.veyespe.com/what-is-magnetic-field-in-physics/
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math
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Physics teaches us the way to compute time.
Time could be thought of as a physical entity that varies more than time.
Time is often thought of as the experience from the observer, the space and time within which the activity takes spot. The formula for time is definitely the one hour, thirty minutes, one second, or 0.001 seconds. Using these time values in your calculations will provide you with a nice breakdown of how you can compute time and convert this for your own units. You’ll be able to then make sense of how several hours, minutes, seconds, and microseconds each day, hour, minute, and second definitely are.
We are used to considering of time as anything linear and non-repeating, like the flat surface we are standing on inside a zero period frame of reference. It seems as though it is constant in this way. But the math that applies to time itself is non-linear. This means that, if we try and transform this quantity back into linear time, the linear portion will get lost. But in regards to calculus, we have to have a formula to compute the adjustments that should take place from one particular point best essay writing service in time for you to another.
In a two dimensional plane, the time will alter over a range of time proportional towards the price of alter on the velocity with the technique, which can be called the second law of thermodynamics. And inside a 3 dimensional program, we discover that the behavior is inversely proportional towards the square on the speed of light, which can be generally known as the Lorentz transformation. Nonetheless, the term, “time,” will not imply a linear time within this example since it is among the formulae that make up the second law.
It may possibly be much easier to understand the second law if we take into consideration the modifications that take place in the properties of time. For example, the area under the curve that represents the time is known as the frequency.
With an infinite velocity as the case of a river https://extension.umd.edu/woodland flowing down a hill and its existing price of alter is zero, the curve must be said to become “flat.” But because the velocity from the river increases, the line continues to rise, increasing till the velocity reaches the speed of light. At this point, the curve has gone to infinity.
With a pure state, the speed with the waves is zero, and there’s no difference among the velocity as well as the space-time with the velocity, which is the simplest of non-inertial processes, along with the wavelength, which can be the shortest to get a pure wave, with no pressure and nothing at all else going on. We are able to assume of time as the vertical expansion from the curvature from the electromagnetic field. The general length in the wave is measured by the speed and path with the sound waves.
The field is at rest and no power may be expressed. essay-company Therefore, when you fully grasp this 1 piece of time-traveling physics, it’s uncomplicated to understand why the flow of time is inversely proportional for the square of the speed of light. In the event the velocity is higher than light speed, the waves travel more rapidly.
If you are sitting nevertheless along with the speed of light is zero, then there is no adjust, and also you cannot move, so the wave cannot travel faster than the speed of light. If, nonetheless, you comprehend that the area under the curve is generally equal for the frequency, so the volume is zero, then it is effortless to find out that the time is inside a state of flux.
The second law of thermodynamics would apply to time too, so that the technique is accelerating. When the system is travelling at 1 speed, the amplitude is equal towards the time.
If you need to understand just how much time, in whole numbers, passes within the vacuum, you’d multiply the time by the frequency and divide by two. Then you would make it equal to the time.
And since the magnetic field is definitely an expression of energy, you can make use of the frequencies to calculate the electric field. Indeed, you can do all of this for any two dimensional space-time equations.
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| 4,151 | 13 |
https://nealien.com/graphing-trig-functions/
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math
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We worked mostly on problems with trigonometric functions.
A few involved graphing the functions. It can help quite a bit to know the shape of the graph in advance. But, if you do not, or even if you somewhat do, then you can use
Also used some double angle formulas and saw how you can use the trig functions with sums for double angle formulas.
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CC-MAIN-2024-10
| 346 | 3 |
https://www.irishtimes.com/news/science/maths-and-poetry-beauty-is-the-link-1.4043965
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math
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Mathematicians are not renowned for their ability to reach the deepest recesses of the human soul. This talent is usually associated with great artists and musicians, and a good poet can move us profoundly with a few well-chosen words.
William Rowan Hamilton, whose work we celebrate during Maths Week, was an innovative mathematician and physicist of the highest order. One of his brilliant ideas was that the unfortunately-named imaginary numbers – in truth, no less real than any other numbers – could be regarded as the second elements of pairs of real numbers, with clearly defined arithmetic rules for manipulating them. With this insight, he banished a mystery that had overshadowed mathematics for centuries.
Hamilton did not confine his thoughts to mathematics. In a lecture on astronomy in 1832, he spoke of the strong resemblance between poetry and science, of the enthusiasm that both can inspire, and of the power that both possess “to lift the mind above the dull stir of Earth”. His poetic output was impressive more for its volume than its quality. Perhaps it was fortunate that another William, his friend the aptly-named Wordsworth, firmly advised him to stick to mathematics, lest his rhyming seduced him from the path of science that he seemed destined to tread.
What is the connection between poetry and maths? Mathematics – using numbers, shapes, etc – deals in abstractions, while poetry uses words and deals in emotions. Pulitzer Prize winner Edna St Vincent Millay, one of the most respected American poets, gave a clue in her sonnet, "Euclid alone has looked on Beauty bare". While pattern and structure are important for both mathematics and poetry, the crucial link is beauty.
The English mathematician G H Hardy also observed that a mathematician, like a poet, is a maker of patterns. If his patterns are more permanent than those of the poet, it is because they are made with ideas. But the mathematician's patterns, like the poet's must be beautiful if they are to have any lasting value. Beauty is the key.
Einstein remarked that pure mathematics is, in its way, the poetry of logical ideas. More bitingly, David Hilbert, upon hearing that one of his students had dropped maths to study poetry, said "Good, he did not have enough imagination to become a mathematician."
For both mathematics and poetry, a homeopathic principle that "less is more" often applies. Mathematicians and poets each strive for economy and precision, selecting exactly the ideas or the words that they need to convey their meaning. Both activities are expressions of creativity, emanations of the human spirit. American mathematician and educator David Eugene Smith observed that mathematics is the poetry of the mind while poetry is the mathematics of the heart.
Bertrand Russell wrote of pure mathematics having a beauty cold and austere, capable of a stern perfection like that of great art. Mathematics, with its abstractions, is usually less accessible than poetry, although some modern poetry is shrouded in obscurity, with arcane words and forms that compete for impenetrability with a mathematical argument.
The beauty, if it is there, is often well hidden and patience is needed to appreciate it. Maryam Mirzakhani, the first woman to win a Fields Medal – the Nobel Prize of maths – wrote that the beauty of mathematics only shows itself to more patient followers.
Peter Lynch is emeritus professor at UCD School of Mathematics & Statistics – He blogs at thatsmaths.com
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http://jwilson.coe.uga.edu/EMAT6680Fa08/Kuzle/Instructional%20unit/Constructions.html
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math
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Construction as a word has a very specific meaning. Any drawing is restricted to use of only compass and a straightedge, whilst using a ruler to measure lengths and a protractor to measure angles is not allowed. Why is this so? In antiquity, geometric constructions were restricted to the use of only a straightedge and a compass. Notice, that here we used the term straightedge, and not a ruler. Why not measure? Greeks knowledge of mathematics was basic, including little arithmetic. So, faced with the problem of finding the midpoint of a line, they could not do the obvious - measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge. It is also why the straightedge has no markings. Euclid and the Greeks solved problems graphically, by drawing shapes, as a substitute for using arithmetic.
Instead of the term geometric structures we more often use the term Euclidean constructions because the majority of those are held in Euclid's (300 BC) Elements. Geometric constructions are highly connected to problems of antiquity that include squaring a circle, duplication of a cube and angle trisection that have been proved to be geometrically constructed impossible hundreds of years later.
In this technological era it is no wonder students have never seen a compass let alone hold it in their hands. Since majority of the work is going to be done using construction tools, it is important for the students to get familiar with those.
Compass is a drawing instrument often used to draw circles and arcs. It has two legs, one with a point, and the second one with a lead or pencil. The openness of the compass is easily regulated by opening or closing both legs, and will remain in a particular setting until changed.
Straightedge is a tool used for drawing straight lines, and has no markings. If ruler is being used it must be cleared to students that using markings on the ruler is not allowed during constructions.
Patty paper (vax paper) will be used in this lesson as a representation for students to grasp more easily through geometrical concepts.
During this first day it is important for students to learn and develop appreciation for the context of geometric constructions. Also, it is important for students to have time to play with the tools, especially with the compass.
Discuss with the students the meaning of the words to sketch, to draw and to construct. Let them carry the meaning on their own and correct when needed. By the end of the day student should know the difference between those terms.
Explanation of the terms:
Neither ruler nor protractor is ever used to perform geometric constructions. They are measuring tools, and not construction tools.
Note: The lesson plan can be developed either using compass and ruler or using technology or both together. It is on the teacher to make that decision. However, I strongly encourage using the old fashion compass and ruler =).
Day 2-Basic constructions
1. Start off the discussion with two simple problems.
1. a. copy a given segment
1. b. copy a given angle
If necessary discuss with your students what it means to copy something. It should be clear to them that by copying we understand constructing an object congruent to a given one. At the beginning it is important to set the rules. Develop a discussion what would a proper construction include. Let them derive that there is more to construction than mare performing of steps and memorizing of the same ones. It should be clear that prior to starting the constructions they have to understand the problem and a good way for that is to make a sketch and start thinking how they would solve the problem. Afterwards the students can perform their steps using the geometry tool. However, the process does not end here. Students should justify their steps using proper geometry language using concepts learnt prior. Depending on the type of problem discussion will be needed on the uniqueness of the solution. Also, precision here is not the center of the universe, but make sure that they should manage compass and the ruler to be as precise as possible. Agreement on notation is also important. Make sure they mark the points with a little circle, and not dot, segment, or cross.
2. Ask your students what is a segment bisector. Ask the to draw it. Make a discussion depending on the solutions. It is possible they have drawn just one and possibly a perpendicular one. This is a good place for a discussion on the difference between a segment bisector and a perpendicular bisector. Student should be able to deduce that perpendicular bisector of a segment is unique for that segment, whilst there is infinitely many bisectors of a given segment.
The following problem is going to be constructed using first patty paper and then the geometry tools. Patty paper is going to be used for students to come up with the construction using the geometry tools. As the students use the patty paper develop a discussion on this activity. Discuss with them how will they use the patty paper? What is the line they get when folding the paper? What is the property of any point on that line in relation to the endpoints of the segment. If they were able to answer all of these questions they should be able to do the construction using the geometry tools. However challenge the students with the questions of explaining why the constructions is correct as well as whether the openness of the compass is important, can we change both or just one leg and still get a perpendicular bisector. The purpose is as for all activities regarding constructions for students not to take them for granted.
2. a. construct a perpendicular bisector using a patty paper
2. b. construct a perpendicular bisector using geometry tools
3. Since the students learnt the concept of a segment bisector it is advisable to use these concepts in more challenging problems that will help us introduce some geometrical concepts. For instance we could give the student the following problems:
3. a. Given triangle ABC construct three medians. What do you notice about all three medians?
3. b. Given triangle ABC construct the perpendicular bisectors of each side. What do you notice about all three bisectors?
These problems could be a small group work where students work on one of these problems. Both problems are rich with discoveries, especially when using a software. Students should be able to discover that all of the three medians or perpendicular bisectors coincide. Introduce terms centroid and circumvent. Let your student grasp through the meaning of those words to help them understand the newly discovered points and concepts.
1. Start the discussion by discussing how the perpendicular bisector to a given line could be constructed. Ask them how they can use patty paper to do that if getting there by connecting previous lesson is not going that well. Make sure they are aware that they can construct the perpendicular line to a given line from a point outside and both on the given line. Discuss with them how are these constructions the same or different. They should mathematically argument their construction decisions using geometric concepts and properties.
1. a. construct a line perpendicular to a given live through a given point outside the line
1. b. construct a line perpendicular to a given live through a given point on the line
2. The next activity connects the concepts of constructing a perpendicular and the shortest distance. Also, it closely connects construction of the angle bisector. Let the students draw an angle ABC. Let them place point P inside the angle, and construct perpendiculars from point P to both legs of the angle. Discuss with them which side is closer to point P? Make sure they justify any statement they make. Ask them where point P must be so that the distances to the legs are the same. If they started with an obtuse triangle would the conclusions be the same? If yes, why if not, why not? Discussion should lead by discovering that P should lie on the bisector of the angle ABC. For this activity one can either use technology and give student time to discover any patterns or by mare drawing and trying to deduce backwards the actual angle bisector construction. After justifying that point P lying on the angle bisector, ask them to construct angle bisector of ABC.
3. As mentioned before the previous activity is closely connected to constructing an angle bisector. If they were not able to construct it after the first activity, give them a patty paper and let them discuss how they would by using a patty paper get an angle bisector. They should be able to explain why by folding the patty paper in such way that the legs of the angle overlap (coincide) by connecting vertex B to the crease they got the angle bisector. Ask them how does this help them construct the angle bisector? Discuss with them whether the openness of the compass influences the construction. Does the openness has to have the same radius as the radius of the arc? What if the compass collapse. It is important for all of these questions to be discussed in details for the students to develop understanding of the nature of the constructions and the construction of the angle bisector in particular.
4. a. Given triangle ABC construct the angle bisectors of each angle. What do you notice about all three angle bisectors?
4. b. Given triangle ABC construct the altitudes of each angle. What do you notice about all three altitudes?
These problems could be a small group work where students work on one of these problems. Both problems are rich with discoveries, especially when using a software. Students should be able to discover that all of the three angle bisectors and lines containing the altitudes coincide. Introduce terms incenter and orthocenter. Let your student grasp through the meaning of those words to help them understand the newly discovered points and concepts. Especially try to develop understanding for why the orhocenter is defined as an intersection of lines containing the altitudes and not just intersection of the three altitudes.
a. Trace this bisector as you drag point C along side AB. Describe the shape formed by this locus of lines.
b. Trace the midpoint od CF as you drag C. Describe the locus of points.
1. Give your students a patty paper and let them derive to strategy to construct a line parallel to given line through a given point. They should be able to deduce that they should fold the paper to construct a perpendicular so that the crease runs through the point and then make another fold that is perpendicular though the first crease thought the given point and match the pairs of the corresponding angles created by folding. Discuss whether angles obtained are congruent. If so why? What conclusions can be made about the lines? Based on this activity they should be able to construct a line parallel to given line though a given point. Also, discuss with them whether the second line is unique and why.
2. Challenge them to think of another way to do the same construction. Since they should know that when two parallel lines are intersected by a transversal that we get congruent corresponding angles. Therefore, they should be able to derive that by copying the angle between the given line and point at that point, they will get a line parallel to the given line.
3. At this point it can be beneficial to give them problems that will connect several constructions.Try to give problem that employ different strategies. For instance, pose them the problem of constructing the angle bisector of two line segment in finite plane. Hence, two line segment that do not intersects on the paper, and by not extending those lines. This problem can be solved in different ways. One of the ways would be by constructing lines parallel to given line segments closer for them to intersect and then by doing the known angle bisector construction. Second, they could employ concepts about incenter and incicrcle. Discuss with them the strength and weakness of each method they derive.
Day 5 & 6-Construction problems
After knowing the basic constructions, we can construct more complex geometric figures.Here we will list several problems that could be a two day plan. Students could work individually, or in pairs depending on classroom environment. However, the teacher should be here only as a manager of the classroom, and let students struggle on their own. Before jumping and solving the construction problems, discuss the properties of triangles and quadrilaterals. Also, after each problem make a classroom discussion where students justify their construction processes. Be sure to discuss whether the solution in each of the following constructions is unique or not, and why.
Construct the triangle ABC using all three segments.
Construct the triangle ABC using two segments and one angle. How many solution there are?
Construct the triangle ABC using one segment and two angles.
Construct an isosceles triangle with perimeter o and length of the base equal to a.
Construct a square given the length of the diagonal.
Construct a rhombus given lengths of both diagonals.
Construct a rhombus given the altitude and one diagonal.
Construct a parallelogram given one side, one angle and a diagonal.
Construct an isosceles trapezoid given the bases and ne angle.
Construct an isosceles trapezoid, given the median, altitude, and base angle of 30º.
There are numerous more challenging problems. The above one are just a proposal but not necessarily the only one appropriate for middle and high school students. For instance, since we explored to some extent the center of gravity for a triangle, a natural question can arise on the center of gravity for a quadrilateral. How do we construct it etc. It is on the teacher to decide how deep they want to go and pursue certain ideas and concepts.
Listed below are some web pages with interesting construction problems:
http://jwilson.coe.uga.edu/EMT668/Asmt6/EMT668.Assign6.html (Problem #3)
Day 7 & 8-Introduction and construction of basic algebraic expressions
This topic naturally follows after teaching basics of geometry constructions connecting it to algebraic concepts. However, since high school textbooks rarely include algebraic method for most of the problems explanations will be given in .gsp files. Let students discuss the strategies employed and reasoning in each of the following problems.
1. The best way to introduce this method is by simply posing a problem where knowledge of geometric methods do not help us, but instead have to depend on different method.For the discussion we can choose among variety of problems. Here, as a motivation task will suffice the following problem:
Construct the triangle ABC if the relation among its sides a, b and c is given by
Give the students time to explore this problem. They should realize that trying to employ geometric methods learnt so far cannot help them is solving this problem. However, discuss with them the expression given that relates its sides. They should be able to realize that on the right side of the expression they have geometric mean of certain sides, G (a, c) and G(b, c). Connecting the geometric mean with the arithmetic mean and knowing that for any non-negative x and y, G(x, y)<=A(x, y), where we have equality of and only if x=y, they should conclude that all of the sides must be equal to satisfy the given expression. From this point the problem is trivial since in the previous lesson they learnt how to do basic constructions .
Thus, through this problem students should become aware that algebraic method can help us with construction problems when geometric ones fails. This problem also gives a great picture what does the algebraic method includes. Students should on their own understand how and why does this method work, and not be given to them per se.
2. Before looking at complex algebraic expressions and their constructions,we will observe constructions of basic expressions. Based on these application of the algebraic method is grounded. In the following list of basic constructions a, b and c are lengths of given segments, m and n are natural numbers, whilst x is the segment we want to construct.
Constructions for problem 4-11 with explanations can be found here. All of the above problems should develop connections between numbers and their geometric representation, or in other words geometric representation of algebraic expressions. Discuss each of the problems with the students.
Day 9-Applications of algebraic method in construction problems
Through the following problems, students will gain insight into efficacy of the algebraic method in construction problems, its positive and negative sides. The problems vary in difficulty and each of the problem explanation are given in .gsp file here. It is on the teacher to decide how to organize these activities. Thus, individual work, work in pairs or groups. However, make sure your students understand the problem, and construction itself and discuss as a class each of the problems given.
1. Start off the today's discussion on regular polygons and their contractibility. Ask them if and how they can construct n-gon when n=3, 6, 12, 4, 8, etc. They should say yes. However ask them if any n-gon is constructible. Probably you will here yes, no or nothing. At this point you can talk about Gauss, child prodigy who as the age of 15 proved that a regular 17-gon can be constructed. He was so fascinated with regular 17-gons that he wished for it to be engraved on his grave. However, his wish never came true, but in it was on a statue of him in his hometown Braunschweig. Not all regular polygons are constructible like 7-, 9-, 11-gon, etc. However, they are with so called neusis constructions that allow using a marked ruler. Let your students explore neusis constructions and give them a group project that involves approximation construction of regular 7-, 9, and 11-gon.
2. Summarize the results of the lesson. At this point discuss with the students the difference between the conventional construction, and Euclidean consecution. Also discuss the algebraic method and its relevance to constructions.
Teacher should reflect on their teaching asking question such as:
Did I challenge the students? How?
Were students actively engaged in the learning process? How do I know?
Did the students exceed my expectations in some areas, and not meet them in others?
Did I have to adjust teaching? If so, what adjustments did I make, and were they effective?
Serra, M. (2003). Discovering geometry: An investigative approach. Emeryville, CA: Key Curriculum Press.
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