how to calculate coefficient of coincidence and interference how to calculate coefficient of coincidence and interference

How to interpret this type of confidence interval. 0000000928 00000 n Analytical cookies are used to understand how visitors interact with the website. 4. The coefficient of coincidence is typically calculated from recombination rates between three genes. Share Your PDF File The following are to links of Friction formula and design data. how to calculate coefficient of coincidence and interference 2022-06-21T19:02:13+00:00 how to get a towing contract with geico university of west london ranking world We select a random sample of 30 residents and find the following information: Here is how to find a 95% confidence interval for the population correlation coefficient: Let zr = ln((1+r) / (1-r)) / 2 = ln((1+.56) / (1-.56)) / 2 = 0.6328, Let L =zr (z1-/2 /n-3) = .6328 (1.96 /30-3) =.2556, Let U =zr + (z1-/2 /n-3) = .6328 + (1.96 /30-3) = 1.01, Confidence interval = [(e2(.2556)-1)/(e2(.2556)+1), (e2(1.01)-1)/(e2(1.01)+1)] =[.2502, .7658]. What is the distance between the genes? Don't use plagiarized sources. If your correlation coefficient is based on sample data, you'll need an inferential statistic if you want to generalize your results to the population. Drosophila females of genotype a+a b+b c+c were crossed with males of genotype aa bb cc. And there are 81 + 23 + 27 + 89 = 220 progeny showing recombination between genes B and C. Thus the expected rate of double recombination is (350 / 1000) * (220 / 1000) = 0.077, or 77 per 1000. The # of recombinant offspring / total # of offspring x 100% = recombination frequency, Recombination frequency = map units = centiMorgan (cM). High negative interference has been reported in bacteriophage T4 (e.g. Muller in 1916 for estimating interference on the basis of dividing the number of double crossovers observed by the number of double crossovers expected by the probability of single crossover frequencies. Step 1. startxref This value is then divided by the product of standard deviations for these variables. In genetics, the coefficient of coincidence (c.o.c.) Step 1: the global luminance logarithmic transformation should be performed on the color information, and then, the HSV color information should be optimized based on a genetic algorithm to calculate the color information gradient. Expected double crossovers = Product of two single recombination values = 19.92 x21.75/100 = 4.33 % ADVERTISEMENTS: Coefficient of coincidence = 0.75/4.33 x 100 = 17.32% The c.o.c. Genetics. Sometimes, these two parts do not stay together. They can then join to another chromosomal half from either the egg or the sperm to become a whole pair. The locations of the genes on the chromosomes are loci. Welcome to BiologyDiscussion! = actual double recombinant frequency / expected double recombinant frequency Interference is then defined as follows: [1] interference = 1 c.o.c. If a diatomic ion dissociates very rapidly (before it has time to rotate) the fragments fly apart along the line of the atoms at the instant of ionization. would expect in 1% of cases that a double 1 of 5. a. The map distance (4 m.u.) 1958 Aug;6(1):115-28. This cookie is set by GDPR Cookie Consent plugin. Give an example. In contract, the double crossover offspring will be the least abundant, because the double crossover events between the genes of interest are more rare than single crossovers. The distance between D and E is 29.7 m.u. Calculate the map distance between loci given the phenotypes of offspring or predict phenotypes of offspring given the recombination frequency between loci. It is generally the case that, if there is a crossover at one spot on a chromosome, this decreases the likelihood of a crossover in a nearby spot. The greater the coincidence, lesser will be the interference and vice versa. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. If interference is zero, this means that the frequency of double crossovers is as expected and that a crossover in one region occurs independently of a crossover in an adjacent region. So, to capture this uncertainty we can create a confidence interval that contains a range of values that are likely to contain the true correlation coefficient between height and weight of residents in this county. Interference is then defined as follows: interference = 1 c.o.c. Use a testcross. It does not store any personal data. PMID 9223494, Anderson JA, Teufel RJ 2nd, Yin PD, Hu WS. Virology. The up-to-date simultaneous analyzers, organized by the type of simultaneously measured properties are collected in the following tables. . The coefficient of coincidence is calculated by dividing the actual frequency of double recombinants by this expected frequency:[1], Interference is then defined as follows:[1]. There are 23 + 152 + 148 + 27 = 350 progeny showing recombination between genes A and B. (With Methods)| Industrial Microbiology, How is Cheese Made Step by Step: Principles, Production and Process, Enzyme Production and Purification: Extraction & Separation Methods | Industrial Microbiology, Fermentation of Olives: Process, Control, Problems, Abnormalities and Developments. {n,1J9 JBr mj R0X_*5,XZV. This indicates that B is located between A and C as given below: Coefficient of coincidence = Observed double crossovers/Expected double crossovers x 100, 1. Observed double crossovers = 9/1200 x 100 = 0.75% 2. Friction Design, Formula and Calculations. 3. 2023 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. 1958 Aug;6(1):115-28. The three pulses entering the DE2 can align in many ways; thus, we incorporate this by using a coefficient of 32. If so, calculate the coefficient of coincidence and the interference value.. . The two genetic copies that recombine are called chromatids. In the presence of coexisting interference and noninterference phases of crossing over, however, lack of change in the coefficient of coincidence may show only that spo16 reduces crossing over in the two phases by a similar factor. If interference is zero, this means that the double crossovers are occurring as predicted and that a crossover in one region occurs independently of a crossover in an adjacent region. With this, our condition for constructive interference can be written: R 1 R 2 = 0 + n l . Identify the parental offspring (the most abundant). However, there are actually only 23 + 27 = 50 double recombinants. The coefficient of coincidence is calculated by dividing the actual frequency of double recombinants by this expected frequency: [1] c.o.c. To use this coefficient of variation calculator, follow the below steps: Enter the comma separated values (,) in the input box Select the option of population dataset or Sample dataset according to your problem. The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". The coefficient of coincidence is calculated by dividing the actual frequency of double recombinants by this expected frequency: c.o.c. In this case, the maximum crossing over % is between gene A and C (40.3%). So, as surface unit-cell vectors one can also choose A 51 Which part of the male reproductive system store the sperm? BrainMass Inc. brainmass.com March 3, 2023, 5:11 pm ad1c9bdddf, Estimation of phenotypes by interference and gene map data, Problem solution for genetic mapping of linked genes, Chromosome mapping and coefficient of coincidence. D Question 6 2 pts You are given a PM as shown below: START READ READ2 ACCEPT ADD a READ, ADDD Using the approach discussed in the lecture video or follow the example in the textbook from page 459 to page 461, trace the paths of the following input strings on this PM: "aabbbb". The answer depends on how far apart they are! between the regions AB and BC can be calculated from the rate of double recombination. xy = Cov(x,y) xy x y = Cov ( x, y) x y. where, An interference fit (press fit & shrink fit) is a frictional shaft-hub connection. This page was last edited on 24 October 2022, at 22:04. Recombination 4. CV = (0.05) / (0.13) x 100 = 0.38 x 100 = 38%. R|naGy=$3#@(Y[Vi$`.T^AJY{6Mq^ejRIUfa,xn>vl0-Ni8wAC. These might make the number of observed recombinants different from expected, but we will not consider these factors at this time. The formula to create this type of confidence interval. Coefficient of Correlation ranges between -1 and 1. r = -1 : Perfect Negative Correlation. We use the following steps to calculate a confidence interval for a population correlation coefficient, based on sample sizenand sample correlation coefficientr. The final confidence interval can be found using the following formula: Confidence interval = [(e2L-1)/(e2L+1), (e2U-1)/(e2U+1)]. is a measure of interference in the formation of chromosomal crossovers during meiosis. Using the formula, she evaluates: CV = standard deviation / sample mean x 100 =. What do you mean by permeability of membrane? An organism with chromosomes Be / bE could produce only gametes Be and bE (50% each). Aconfidence interval for a correlation coefficientis a range of values that is likely to contain a population correlation coefficient with a certain level of confidence. 61 0 obj <>stream recombinants that one would expect. How do geneticists know if recombination has occurred? Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors. 0000001009 00000 n A correlation coefficient is a bivariate statistic when it summarizes the relationship between two variables, and it's a multivariate statistic when you have more than two variables. Interference. This cookie is set by GDPR Cookie Consent plugin. Determine the recombination frequency between the other locus and the middle locus. It is compared to the Hadley Centre sea surface temperatures (HadSST4, in gray) and the land and ocean surface temperature dataset (HadCRUT5, in orange).Happer, clearly states that the satellites are measuring a different temperature and have consistent, nearly global coverage. Introduction to Statistics is our premier online video course that teaches you all of the topics covered in introductory statistics. . Why clients love us. Coefficient of coincidence = Observed double crossovers/Expected double crossovers x 100 1. Coefficient of Correlation is denoted by a Greek symbol rho, it looks like letter r. To calculate Coefficient of Correlation, divide Covariance by Standard Deviation of two variables (Sx, Sy). If interference is 1, this means that interference is complete and that no double crossovers are observed because a crossover in one region eliminates the likelihood of a crossover in an adjacent region. phpWebsite video link: video goes through a few examples of gene mapping problems and genetic recombination frequency calculations. Single crossover between A and B will alter the position of two genes, viz., B and C (Fig. Statology Study is the ultimate online statistics study guide that helps you study and practice all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. { "4.5.01:_Linkage_and_Mapping" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.5.02:_GWAS" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "4.01:_Meiosis" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.02:__Mendelian_Genetics" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.03:_Pedigrees" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.04:_Exceptions_to_autosomal_inheritance" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.05:__Linkage" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.06:__Exceptions_to_simple_dominance" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.07:_Gene_Interactions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Chapter_4_Review_Questions_(draft)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "showtoc:yes", "authorname:swleacock" ], https://bio.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fbio.libretexts.org%2FCourses%2FUniversity_of_Arkansas_Little_Rock%2FGenetics_BIOL3300_(Fall_2022)%2FGenetics_Textbook%2F04%253A_Inheritance%2F4.05%253A__Linkage%2F4.5.01%253A_Linkage_and_Mapping, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), Loci are locations of genes on chromosomes, Effect of recombination on gamete possibilities, status page at https://status.libretexts.org. In Figure \(\PageIndex{1}\), why is a chromosome with Bb or Ee not shown as a possible arrangement? Coefficient of coincidence and interference can be calculated with the help of the number of progeny obtained from a back cross or test cross..To continue with the answer click on mycoursewriter.com/ Nice to meet you! In this article we will discuss about:- 1. The concept is that given specific recombination rates in two adjacent chromosomal intervals, the rate of double-crossovers in this region should be equal to the product Coefficient of coincidence and interference can be calculated with the help of the number of progeny obtained from a back cross or test cross. )), The Law on Obligations and Contracts (Hector S. De Leon; Hector M. Jr De Leon), Principles of Managerial Finance (Lawrence J. Gitman; Chad J. Zutter), Calculus (Gilbert Strang; Edwin Prine Herman), Conceptual Framework and Accounting Standards (Conrado T. Valix, Jose F. Peralta, and Christian Aris M. Valix), Theories of Personality (Gregory J. Feist), Auditing and Assurance Services: an Applied Approach (Iris Stuart), Intermediate Accounting (Conrado Valix, Jose Peralta, Christian Aris Valix), The Tragedy of American Diplomacy (William Appleman Williams), Science Explorer Physical Science (Michael J. Padilla; Ioannis Miaculis; Martha Cyr). 4.2 Description of Two-Dimensional (2D) Lattices and Structures taken from panel (c), the coincidence is also in every 5th substrate unit cell in a 1 1 = 5 a1 = A 151 and direction. If a crossover in one region does affect a crossover in another region, that interaction is called interference. age, due to the interference of the surrounding environment (site construction, vehicle driving, etc. However, there are actually only 23 + 27 = 50 double recombinants. 0000002245 00000 n This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here! Expected double crossovers = Product of two single recombination values, Coefficient of coincidence = 0.75/4.33 x 100 = 17.32%, Coefficient of interference = 1 0.1732 = 0.8268 or 82.68%, Biology, Genetics, Crossing-Over, Three Point Test Cross. This is a question and answer forum for students, teachers and general visitors for exchanging articles, answers and notes. Instead, we might select a simple random sample of residents and simply gather information about them. xY]o}( The female will produce eggs with an AC chromosome. Whether you need help solving quadratic equations, inspiration for the upcoming science fair or the latest update on a major storm, Sciencing is here to help. PMID 9445017, https://handwiki.org/wiki/index.php?title=Biology:Coefficient_of_coincidence&oldid=2249720. The coefficient of coincidence is therefore 50 / 77 = 0.65. The extent of interference is measured by the coefficient of coincidence ( C ) . Here, the variable n is used to specify an integer and can take on any value, as long as it is an integer. apart. The coefficient of coincidence is therefore 50 / 77 = 0.65. Coefficient of coincidence is the ratio of observed double cross over frequency and expected double cross over frequency .The level of Incidence I = 1 - C ___________________________________________________________________________In this video tutorial you will learn how to calculate the value of I and C very easily with the help of given problem ___________________________________________________________________________for any query / doubt or notes msg me : 8112215832___________________________________________________________________________link for three point test cross problem https://www.youtube.com/watch?v=wZlgYh9n5AA___________________________________________________________________________link for concept of linkage https://youtu.be/QaW4tavoo3M___________________________________________________________________________Link For Bateson Punnet Experiment:https://youtu.be/5_OZQQYTuYw___________________________________________________________________________Link for recombination and crossing over \u0026 Two point test cross gene mapping https://youtu.be/H9IcwamqXcs___________________________________________________________________________#interference_coincidence #coefficient_of_coincidence_ingenetics #Geneticsproblem #calulateinterferenceingeneticsproblem #three_point_testcross #Gene_mapping #Three_pointcross_over #three_point_cross_geneorder_and_mapping #Three_point_cross_problem #stephen_curry_crossover_three_point #genetics #gene_mapping_problem#cross_test #gene_mapping_class_12#genemappingcsirnet KQpl>< 94-7!e RB^a*|RZ5xoT+F{|1(QMT:Z%c&(332DQMykLv_m\uoZcJ3X6')};XiX#4jh$^jbK"G"i8]i {`+%eu/%r{Qq 4"{;|e(V%"+w%FTM"+!++ J0rs${~XE"D2Fm-UR Name the types of nitrogenous bases present in the RNA. Geneticists calculate the corresponding percentage to assign the degree of genetic linkage, which has units of centimorgans, or cM. The correlation coefficient, denoted by r, tells us how closely data in a scatterplot fall along a straight line. Coefficient of Coincidence (c) = observed double crossovers/expected double crossovers. 0000002547 00000 n The rate of double recombinants Determine whether the data below is consistent with your map. trailer 9.3). When wild-type Drosophila Females were mated to Males homozygous for two autosomal mutationsvestigial (vg) short wings, and black (b) body coloration. 0000000596 00000 n How can malaria be prevented and controlled? Date Package Title ; 2023-02-24 : actuaRE: Handling Hierarchically Structured Risk Factors using Random Effects Models : 2023-02-24 : appRiori: Code and Obtain Customized Planned These cookies will be stored in your browser only with your consent. if Z = X - Y, the order of genes is A-C-B; if Z = Y - X, the order of genes is B-A-C. What is radiative recombination coefficient? Disclaimer Copyright, Share Your Knowledge The coincidence maps clearly demonstrate correlations in positions and at the same time anti-correlations in momenta, as expected from the EPR state. This website uses cookies to improve your experience while you navigate through the website. The term interference refers to the fact that For example, a dihybrid BbEe can have one chromosome with both dominant alleles (BE) or one chromosome with a dominant allele for one gene and recessive allele for the other (Be for example). This page titled 4.5.1: Linkage and Mapping is shared under a not declared license and was authored, remixed, and/or curated by Stefanie West Leacock. Percent recombination frequency calculator can be a helpful tool for these students. The first gas-phase measurement of angular distributions from orientated ions was made by detecting photoelectrons in coincidence with ionic dissociation products . xb```f``Z- @1A9e]*L-500OtR]NWKI+\w,Mq*&&hP;65iK Especially for large chromosomes, multiple crossover events can occur on the same chromosome. In the given question we have to calculate the expected number of double recombinants. Recombination frequency is the percent of meioses in which homologous recombination exchanges two loci. = actual double recombinant frequency / expected double recombinant frequency Interference is then defined as follows: [1] interference = 1 c.o.c. PMID 9223494, Anderson JA, Teufel RJ 2nd, Yin PD, Hu WS. A double crossover occurs when fragments of the chromosome are exchanged in two places. Chase M, Doermann AH. CV = volatility / projected return x 100 =. J@((b|M ~@m@%NQf`g(ebc^0\cbJPp!Ss}_8#/#ZkA1-10) @ tXF The result suggest one cross over inhibited the occurrence of another nearby , a phenomenon called interference. Correlated template-switching events during minus-strand DNA synthesis: a mechanism for high negative interference during retroviral recombination. At the second point of contact, the chromatids can again separate and again exchange genetic information. C) There is a very high coefficient of coincidence. To calculate the coefficient of variation in the bond for comparison, Jamila divides a volatility of 3% by a projected return of 15%. Coefficient of Coincidence (COC) = Observed double crossover frequency / Expected double crossover frequency = 0.0072 / 0.0096 = 0.75 Interference = 1-COC = 1-0.75 = 0.25 Step-by-step explanation Image transcriptions If the individual recombination rates (between A and B; and between B and C) are known, then the c.o.c. If a crossover in one region does affect a crossover in another region, that interaction is called interference. We also use third-party cookies that help us analyze and understand how you use this website. On the origin of high negative interference over short segments of the genetic structure of bacteriophage T4. Typically, your data will show an interference of between 0 and 1. r = 0 : No Correlation. This figure tells us how strongly a crossover in one of the DNA regions (AB or BC) interferes with the formation of a crossover in the other region. Let zr = ln ( (1+r) / (1-r)) / 2 Step 2: Find log upper and lower bounds. Enter your answer to 3 decimal places. Required fields are marked *. The cookie is used to store the user consent for the cookies in the category "Performance". trivial level, imagine three loci, each 10% The loci are the locations of the genes on chromosomes. Gene Sequence of Three Point Test Cross: 5. Problems such as Downs syndrome or other genetic disorders can be caused when genes combine incorrectly. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 0000001844 00000 n 0000001139 00000 n What is a trophic hormone? The extent of interference is measured by the coefficient of coincidence ( C. DNA segment with three genes, showing a double recombination event. The cookies is used to store the user consent for the cookies in the category "Necessary". Coefficient of Coincidence in Three point Test Cross: This cookie is set by GDPR Cookie Consent plugin. Values higher than zero but below one indicate that interference is occurring. between the regions AB and BC can be calculated from the rate of double recombination. If loci B and E in the above example (Figure \(\PageIndex{1}\)) were on two different chromosomes, we would expect to obtain four gamete genotypes (25% each): BE, Be, bE, and be, as observed by independent assortment. Homologous recombination occurs in a distinct retroviral subpopulation and exhibits high negative interference. Consider three genes on the same chromosome, geneD, geneE and geneF. v9!B)"xtZI/)e5U z%zW0e%4gxhI. The gene sequence is determined with the help of crossing over percentage between two genes. Use Promo Code: 20OFF Order with discount Calculate the price This online calculator calculates index of coincidence (IC, IOC) for the given text Here is the calculator, which calculates the index of coincidence, or IOC (IC) for the given text. 1958 May;43(3):332-53. At a Thus eight types of gametes are produced by F1 and only one type of gamete is produced by homozygous recessive parent. For example, suppose we want to estimate the correlation coefficient between height and weight of residents in a certain county. Recombination (%) of Three point Test Cross: 4. The homozygous recessive parent can only transmit an f g chromosome, so determine the percentage of f g gametes from the dihybrid parent. And there are 81 + 23 + 27 + 89 = 220 progeny showing recombination between genes B and C. Thus the expected rate of double recombination is (350 / 1000) * (220 / 1000) = 0.077, or 77 per 1000. Order Essay. The Watch & Learn video article provides an overview of cloud computing from Electronics Desktop and details the product licenses and subscriptions to ANSYS Cloud Service that are. frquencies of the ingle crosses overs. The way we would interpret a confidence interval is as follows: There is a 95% chance that the confidence interval of [.2502, .7658] contains the true population correlation coefficient between height and weight of residents in this county. 0000002873 00000 n %PDF-1.3 In genetic mapping, this number expresses distance in map units (m.u.) number of double recombinants to the For a given distance between two loci, one Most crossovers occur normally. Gene interference is a measure of the independence of crossovers from each other. Solution. Be careful. PMID 9445017, https://en.wikipedia.org/w/index.php?title=Coefficient_of_coincidence&oldid=1136217742, This page was last edited on 29 January 2023, at 08:36. High Negative Interference over Short Segments of the Genetic Structure of Bacteriophage T4. Alleles: Meaning, Features and Test | Genetics, Experiments Performed by Mendel: Back Cross and Test Cross (With Diagram). [2][3] ) and in human immunodeficiency virus (HIV) infections.[4][5]. Your email address will not be published. = actual double recombinant frequency / expected double recombinant frequency. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org.