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In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, need not have a unique rank if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as Lp spaces.)
Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.[1]
Modules over commutative rings can be generalized in a different direction: take a ringed space (X, OX) and consider the sheaves of OX-modules (see sheaf of modules). These form a category OX-Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring OX(X).
One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S, the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science.
Catalog Contents: Rings,ideals, isomorphism theorems, group rings, localization, factor rings. Modules,submodules, direct products, factor modules. Homomorphisms, classicalisomorphism theorems. The endomorphism ring of a module. Free modules, freegroups. Tensor product of modules. Finitely generated modules over a principalideal domain. 
Then by assumption $R/I$ is free. Assume for contradiction that $I$ is not equal to $0$ or $R$. First, since $R/I \neq 0$, $R/I$ is not spanned by the empty set. So it must have a basis $\\overliner_1, \ldots \overliner_k\$ which is nonempty. But since $I \neq 0$, we can choose some $i \in I\setminus \0\$, and then $i\cdot \overliner_1 = 0$. Thus, the "basis" is actually linearly dependent.
Modules over a ring RRR are called RRR-modules, for short. An RRR-module is called left if it arises from a left action, and right otherwise. As for vector spaces, we could unfold this definition into a list of axioms, but this would obfuscate the real purpose of modules: Many mathematical objects happen to be rings, and modules allow us to study rings by their action on a set (much like we can study groups via their representations).
@articleJukl1997,abstract = In this paper the problem of construction of the canonical matrix belonging to symplectic forms on a module over the so called plural algebra (introduced in [

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As a linear model, the QuantileRegressor gives linear predictions$\haty(w, X) = Xw$ for the $q$-th quantile, $q \in (0, 1)$.The weights or coefficients $w$ are then found by the followingminimization problem:
We performed stepwise logistic regression in the UK cohort, by randomly dividing it into training and test sets (ratio: 80:20) to identify independent symptoms most strongly correlated with COVID-19, adjusting for age, sex and BMI. A combination of loss of smell and taste, fatigue, persistent cough and loss of appetite resulted in the best model (with the lowest Akaike information criterion). We therefore generated a linear model for symptoms that included loss of smell and taste, fatigue, persistent cough and loss of appetite to obtain a symptoms prediction model for COVID-19:
Baseline characteristics are presented as the number (percentage) for categorical variables and the mean (standard deviation) for continuous variables. Multivariate logistic regression adjusting for age, sex and BMI was applied to investigate the correlation between loss of smell and taste and COVID-19 in 15,368 UK users of the symptom tracker app who were also tested in the laboratory for SARS-CoV-2 (6,452 UK individuals tested positive and 9,186 tested negative). The results were replicated in 726 US individuals who tested positive and 2,037 US individuals who tested negative. We then randomly split the UK sample into training and test sets with a ratio of 80:20. In the training set, we performed stepwise logistic regression combining forward and backward algorithms, to identify other symptoms associated with COVID-19 independent of loss of smell and taste. We included in the model ten other symptoms (fever, persistent cough, fatigue, shortness of breath, diarrhea, delirium, skipped meals, abdominal pain, chest pain and hoarse voice) as well as age, sex and BMI, and chose as the best model the one with the lowest Akaike information criterion. We then assessed the performance of the model both in the test set and via tenfold cross-validation in the entire UK sample of 15,638 individuals using the R package cvAUC13. We further validated the prediction model in the US cohort. 7ad9723583
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OK. Well, the idea of this first video is to tell you what's coming, to give a kind of outline of what is reasonable to learn about ordinary differential equations. And a big part of the series will be videos on first order equations and videos on second order equations. Those are the ones you see most in applications. And those are the ones you can understand and solve, when you're fortunate.
So first order equations means first derivatives come into the equation. So that's a nice equation that we will solve, we'll spend a lot of time on. The derivative is-- that's the rate of change of y-- the changes in the unknown y-- as time goes forward are partly from depending on the solution itself. That's the idea of a differential equation, that it connects the changes with the function y as it is.
And here is a nonlinear equation. The derivative of y. The slope depends on y. So it's a differential equation. But f of y could be y squared over y cubed or the sine of y or the exponential of y. So it could be not linear. Linear means that we see y by itself. Here we won't. Well, we'll come pretty close to getting a solution, because it's a first order equation. And the most general first order equation, the function would depend on t and y. The input would change with time. Here, the input depends only on the current value of y.
So that's a big equation. And let me just say, at this point, we let things be nonlinear. And we had a pretty good chance. If we get these to be non-linear, the chance at second order has dropped. And the further we go, the more we need linearity and maybe even constant coefficients. m, b, and k. So that's really the problem that we can solve as we get good at it is a linear equation-- second order, let's say-- with constant coefficients. But that's pretty much pushing what we can hope to do explicitly and really understand the solution, because so linear with constant coefficients. Say it again. That's the good equations.
And I think of solutions in two ways. If I have a really nice function like a exponential. Exponentials are the great functions of differential equations, the great functions in this series. You'll see them over and over. Exponentials. Say f of t equals-- e to the t. Or e to the omega t. Or e to the i omega t. That i is the square root of minus 1.
So again, linear. Constant coefficients. But several equations at once. And that will bring in the idea of eigenvalues and eigenvectors. Eigenvalues and eigenvectors is a key bit of linear algebra that makes these problems simple, because it turns this coupled problem into n uncoupled problems. n first order equations that we can solve separately. Or n second order equations that we can solve separately. That's the goal with matrices is to uncouple them.
One of the options may be the favorite. ODE for ordinary differential equations 4 5. And that is numbers 4, 5. Well, Cleve Moler, who wrote the package MATLAB, is going to create a series of parallel videos explaining the steps toward numerical solution.
So that's a parallel series where you'll see the codes. This will be a chalk and blackboard series, where I'll find solutions in exponential form. And if I can, I would like to conclude the series by reaching partial differential equations.
And I would like to also include a the Laplace equation. Well, if we get there. So those are goals for the end of the series that go beyond some courses in ODEs. But the main goal here is to give you the standard clear picture of the basic differential equations that we can solve and understand. 7ad9723583
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This lab manual accompanies the text of the same name. It covers the theory and application of operational amplifiers and other linear integrated circuits. It is appropriate for Associate and Bachelors degrees programs in Electrical and Electronic Engineering Technology, Electrical Engineering and similar areas of study.
This text covers the theory and application of operational amplifiers and other linear integrated circuits. It is appropriate for Associate and Bachelors degrees programs in Electrical and Electronic Engineering Technology, Electrical Engineering and similar areas of study. Topics include negative feedback, comparators, voltage amplifiers, summing and differencing amplifiers, high speed and high power devices, non-linear circuit applications, regulators, oscillators, integrators and differentiators, active filters and AD/DA conversion. A companion laboratory manual is available. 7ad9723583

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This chapter is an overview of the sorts of problems that can be addressed using the methods from this book. It also discusses the major differences between mathematical modeling and mathematics, and reviews some basic terminology that is used throughout the book. The appendix provides a much more in-depth review of engineering mathematics. This book is meant to be self-contained in the sense that only prior knowledge of college-level calculus, linear algebra, and differential equations is assumed. Therefore, if it is read sequentially and something does not make sense, then the appendix most likely contains the missing piece of knowledge. Standard references on classical mathematics used in engineering and physics include [2, 5], which also can be consulted to fill in any missing background. 7ad9723583

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Chapter Outline 1.0 Introduction 1.1 A First Regression Analysis 1.2 Examining Data 1.3 Simple linear regression 1.4 Multiple regression 1.5 Transforming variables 1.6 Summary 1.7 For more information
This web book is composed offour chapters covering a variety of topics about using SAS for regression. We shouldemphasize that this book is about "data analysis" and that it demonstrates howSAS can be used for regression analysis, as opposed to a book that covers the statisticalbasis of multiple regression. We assume that you have had at least one statisticscourse covering regression analysis and that you have a regression book that you can useas a reference (see the Regression With SAS page and our Statistics Books for Loan page for recommended regressionanalysis books). This book is designed to apply your knowledge of regression, combine itwith instruction on SAS, to perform, understand and interpret regression analyses.
This first chapter will cover topics in simple and multiple regression, as well as thesupporting tasks that are important in preparing to analyze your data, e.g., datachecking, getting familiar with your data file, and examining the distribution of yourvariables. We will illustrate the basics of simple and multiple regression anddemonstrate the importance of inspecting, checking and verifying your data before acceptingthe results of your analysis. In general, we hope to show that the results of yourregression analysis can be misleading without further probing of your data, which couldreveal relationships that a casual analysis could overlook.
We will not go into all of the details of this output. Note that there are 400observations and 21 variables. We have variables about academic performance in 2000and 1999 and the change in performance, api00, api99 and growthrespectively. We also have various characteristics of the schools, e.g., class size,parents education, percent of teachers with full and emergency credentials, and number ofstudents. Note that when we did our original regression analysis it said that therewere 313 observations, but the proc contents output indicates that we have 400observations in the data file.
We see that among the first 10 observations, we have four missing values for meals. It is likely that the missing data for meals had something to do with thefact that the number of observations in our first regression analysis was 313 and not 400.
Another kind of graph that you might want to make is a residual versus fittedplot. As shown below, we can use the plot statement to make thisgraph. The keywords residual. and predicted. in this contextrefer to the residual value and predicted value from the regression analysisand can be abbreviated as r. and p. .
Finally, as part of doing a multiple regression analysis you might be interested inseeing the correlations among the variables in the regression model. You can do thiswith proc corr as shown below.
Earlier we focused on screening your data for potential errors. In the nextchapter, we will focus on regression diagnostics to verify whether your data meet theassumptions of linear regression. Here, we will focus on the issueof normality. Some researchers believe that linear regression requires that the outcome (dependent)and predictor variables be normally distributed. We need to clarify this issue. Inactuality, it is the residuals that need to be normally distributed. In fact,the residuals need to be normal only for the t-tests to be valid. The estimation of theregression coefficients do not require normally distributed residuals. As we areinterested in having valid t-tests, we will investigate issues concerning normality.
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I'm fond of baby Rudin: elegant presentation, "clever" proofs,certain terseness, difficult exercises, etc. I'm working up my enthusiasm to seek similar textbooks in other areas of math, e.g. linear algebra, abstract algebra, ODE, number theory. Below are some books I want to exclude:
Linear Algebra -- Lang's Linear Algebra is often overshadowed by his much more popular book "Algebra", but it is an amazingly written text on linear algebra of arbitrary vector spaces. I learned linear algebra from the 2nd edition, and it was unbelievably well written and rigorous. It is not as difficult as his book on abstract algebra, but it is by no means easy. (Note: The later editions changed the writing style up a bit so I would recommend looking at one of the first few editions).Also, Greub's Linear Algebra (the predecessor of his Multilinear Algebra book discussed below) is an extremely comprehensive, and tersely written, treatment, which is accompanied by difficult exercises
Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.
This inexpensive text covers the basics of linear algebra and serves as an excellent introduction to proof-based mathematics. This is an essential read for scientists of all disciplines; the applications are immediate and uniform across the sciences.
Differential geometry is a general and powerful psychological aid to not only physics in curved space, but calculus in flat space with arbitrary curving coordinates. Its utility is apparent during Landau 2, when justifying the formulae for changes of coordinates of tensor quantities. The first chapter serves as an advanced mini course on multilinear algebra, extending the material from Shilov. The remainder of the book constructs geometric quantities from space rather than numerically, and produces a theory of integration, differentiation, optimization which re-instantiate into familiar calculus when represented locally with coordinates. The material then shifts to consider lie groups and their basic uses for spatial structure and calculus, to be regarded as the polytopes of the continuous differentiable world.
There are essentially three books here. Although there never seems to be a good time to start algebra, it quickly becomes needed either for simplification or explanation of naturally arising structures everywhere. When your need for algebra is strong, consider picking the appropriate and helpfully self-contained sections. The basic objects of algebra, algebraic equations, representation theory of finite groups and related linear algebra. Applications are numerous and deserve a separate blog post.
Algebraic Geometry is foremost about the geometries of solutions to algebraic equations and other locally ringed topological spaces whose functions also behave like such solutions. The subject is most easily studied in analogy to differential geometry, and insights cross feed since both kinds of spaces (manifolds and schemes) can be constructed as dual to certain kinds of rings. One fundamental analogy worth noting is that both classes of geometries have a notion of fundamental groups and covering spaces, where in the algebraic case, fundamental groups are a general case of Galois groups in algebra. At the center of the abstract geometric formalism are still concrete cases of solutions to equations over algebraically closed fields like the complex num

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