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*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****
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Pat bought 5 pounds of apples. How many pounds of pears [#permalink]
Show... | {
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GCDS r019h Pat bought 5 pounds of apples (20160115).jpg [ 21.67 KiB | Viewed 17561 times ]
On the tables, n=? is derived from 5a=np. Generally, when one con indicates number and the other con indicates ratio, it is most likely that ratio is an answer. As for this question, in 1) number and 2) ratio, substitute p=1.5a ... | {
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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
y pounds of peers -------> 1.5x $Solving we have: 5x/1/3x/2---->10/3---> 3 1/3 pounds of peers. Sufficient. Hence B. _________________ Thanks ... | {
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Let A = the price per pound of apples
Let P = the price per pound of pears
If Pat bought 5 pounds of apples, then 5A = the total amount that Pat spent
Pat then wants to spend her 5A dollars on pears
So, 5A/P = the number of pounds of pears Pat can buy with the 5A dollars
REPHRASED target question: What is the value of ... | {
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# Maximal number of binary strings given constraints
Let $k, N, m \in \mathbb{N}$ such that $k \leq N$.
What is the maximal number $e$ of strings $\sigma_1, \sigma_i, \dots, \sigma_e$, each of length $N$ such that $$\forall j < k, \left(\sum_{i=1}^e \sigma_i[j]\right) \leq 2^{N-k}(m-1)$$
For example if $m=3$, $k=4$,... | {
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But I don't know how to prove this.
-
Is there no constraint on the last $N-k$ values of the strings? – Douglas Zare Jul 28 '12 at 2:32
No, there is no constraint on those values. But this parameter is necessary. You can see $N-k$ as the possibility to add a ponderation to the $k$-prefix. – Turingoid Jul 28 '12 at 3... | {
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This is a counterexample to the formula $e = 2^{N-k} + 2^{N-m}m$ for the maximum number of strings satisfying the given constraints. The parameters of the example are $N = 6$, $k = 5$, and $m = 4$. The constraint is that the columns sum to no more than $6$. The conjectured formula predicts that 18 is the maximum number... | {
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@Hugh In fact, using my previous comment, I realized that I needed to give an upper bound only to an infinity of $m$. So it suffices to take $m_i$ so that we can put exactly all strings with at most $i$ 1's in the $k$ prefix and all $N-k$ suffixes. I guess it isn't too difficult to prove that the created set is an opti... | {
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# Sum of two independent Exponential Random Variables
The text I'm using on questions like these does not provide step by step instructions on how to solve these, it skipped many steps in the examples and due to such, I am rather confused as to what I'm doing.
Here is the question: Let $X$ be an exponential random va... | {
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\begin{align*} \Pr[X + Y \le t] &= \int_{x=0}^\infty \Pr[Y \le t - x \mid X = x] f_X(x) \, dx \\ &= \int_{x=0}^t (1 - e^{-2\lambda(t-x)}) \lambda e^{-\lambda x} \, dx \\ &= \lambda \int_{x=0}^t e^{-\lambda x} - e^{-2\lambda t} e^{\lambda x} \, dx \\ &= \left[ -e^{-\lambda x} - e^{-2\lambda t} e^{\lambda x} \right]_{x=0... | {
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# Injectivity implies surjectivity
In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example,
Set theory
An injective map between two finite sets with the same cardinality is surjective.
Linear algebra
An injective linear map between two finite dimensional vector spaces of ... | {
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• There is a generalisation to maps $f \colon X \to X$ where $X$ is any variety over an algebraically closed field $k$. Jul 4, 2017 at 10:27
• I wouldn't expect Grothendieck to state it just for the complex numbers!
– YCor
Jul 4, 2017 at 16:11
• @YCor: Grothendieck's version is probably about radicial endomorphisms of ... | {
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Lemma 2. An isometry of a compact metric space is a bijection.
Proof. Let $x\in X$. Let us consider the sequence $\{f^n(x)\}$. This sequence has a a convergent subsequence $\{f^{k_i}(x)\}$. This implies that $\lim_{j>i\to\infty}d(f^{k_i}(x),f^{k_j}(x))=0$, and hence (since $f$ is an isometry) $\lim_{j>i\to\infty}d(x,f... | {
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• @MarkWildon done! Jul 6, 2017 at 2:23
• @Cœur : I wish! Alas I do not know a proper reference for this. I heard the statement from Valery Ryzhikov (a Russian mathematician primarily working with dynamical systems) about 15 years ago, and then came up with the proof documented above. Jul 7, 2017 at 3:53
• One referenc... | {
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Banach space theory:
Antonio Avilés and Piotr Koszmider constructed an infinite dimensional Banach space of continuous functions $C(K)$ such that every one-to-one operator $T : C(K) \to C(K)$ is onto.
Minor bit of self-promotion: if $\Gamma$ is a (discrete) group and $f\in\ell^1(\Gamma)$ then the natural convolution ... | {
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So we can say that the categories of finite sets, finite dimensional vector spaces, and finite dimensional compact manifolds are all noetherian. In fact, we can say more: namely, that they are precisely the subcategories of noetherian objects in the categories of sets, vector spaces, and compact manifolds, respectively... | {
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Question 1. Is there a constructive proof?
Of course, the theorem above is a multiplicative analogue of the known fact that any surjective endomorphism of a finitely generated $R$-module is bijective. That latter fact has a strengthening due to Orzech (see my A constructive proof of Orzech’s theorem and the references... | {
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Your second example is a special case of this conjecture, essentially equivalent to the case when G is a finite group (and your first example is a special case of the second one, by applying a suitable Hom functor).
The conjecture is due to Kaplansky. There are many partial results, for example Kaplansky showed the co... | {
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I did not find in the answers the following known claim which is dual to Vladimir's answer.
If $(X,d)$ is a metric compact set and a surjection $f$ from $X$ to $X$ is 1-Lipschitz, i.e., $d(f(x),f(y))\leqslant d(x,y)$, then $f$ is bijection and, moreover, isometry.
Proof. Fix $r>0$ and a minimal $r$-net $A$ in $X$ wit... | {
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Proof: Suppose that $$a\neq 1$$. Then $$L_{a}^{n}(a)=a^{[n+1]}=1=L_{a}^{n}(1)$$. Therefore, since $$L_{a}^{n}$$ is not injective, the mapping $$L_{a}$$ is not injective either. Therefore, if $$L_{a}$$ is injective, then $$a=1.$$ If $$f$$ is an injective inner endomorphism, then $$f=L_{a_{1}}\circ\dots\circ L_{a_{n}}$$ ... | {
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Explanation of recursive function
Given is a function $f(n)$ with:
$f(0) = 0$
$f(1) = 1$
$f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$
I was wondering if there's also a non-recursive way to describe the same function.
WolframAlpha tells me there is one:
$$g(n) = \frac{(\frac{1}{2}(3 + \sqrt{17}))^n - (\frac{1}{2}(3 - \sq... | {
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Diagonalize (find eigen values / eigen vectors) and you have your closed form.
$$\begin{bmatrix} 3 & 2 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ \frac{2}{3 - \sqrt{17}} & \frac{2}{3 + \sqrt{17}} \end{bmatrix} \begin{bmatrix} \frac{3 - \sqrt{17}}{2} & 0 \\ 0 & \frac{3 + \sqrt{17}}{2} \end{bmatrix} \begin{bmatri... | {
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I conclude that $r = r_1 = \frac{\sqrt{17}-3}{2}$, or $r = r_2 = \frac{-\sqrt{17}+3}{2}$.
Using the fact that I can add solutions together and still have a solution to the recurrence relation, I'll write: $f(n) = c_1r_1^n + c_2r_2^n$
From here, you can plug in the initial conditions to solve for $c_1$ and $c_2$.
-
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The theorem is comprised of two parts, the first of which, the Fundamental Theorem of Calculus, Part 1, is stated here. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Taking the derivative, we see x0 n (t) = 1 2nt2. Derivative matches upper... | {
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we orient S so that it has an outward pointing normal vector. The first thing to notice about the fundamental theorem of calculus is that the variable of differentiation appears as the upper limit of integration in the integral: Think about it for a moment. Complex Integration (2A) 3 Young Won Lim 1/30/13 Contour Integ... | {
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also be used to evaluate triple integrals by turning them into surface integrals. Answer to Theorem 1. 35) Theorem. 1 Introduction 16. The fundamental theorem of calculus is a theorem that links the concept of differentiating a function with the concept of integrating a function. As we know, Integration is a reverse pr... | {
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There are several theorems in geometry that describe the relationship of angles formed by a line that transverses two parallel lines. the major theorems from the study of di erentiable functions in several variables. If c is a nonnegative real number, then 1. Interchange of Differentiation and Integration The theme of t... | {
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= 2 π 3 φ = 2 π 3 and it goes. Interpreting the behavior of accumulation functions involving area. 8 billion users and include 5 of the top 7 largest banks. Integration by Parts 21 1. 2 Integrals of functions that decay The theorems in this section will guide us in choosing the closed contour Cdescribed in the introduc... | {
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theorem [1, p. 1-9a for the following values of RL: 2 kV, 6 kV, and 18 kV? If you really want to appreciate the power of Thevenin’s theorem, try calculating the foregoing currents using the original circuit of Fig. theorem and easy to prove. Solution for Carefully state the Fundamental Theorem of Calculus. (George Nevi... | {
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is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. is a continuous function on the closed interval (i. ; Explain the significance of the net change theorem. Learning Objectives. Chapter 7 / Directed Integration Theory 7-1. The pro... | {
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increased to approach an impulse; on the other hand, when , is compressed with. Now you can take a break. Examples Edit. The New 2017 A level page. This paper presents anecdotal evidence that suggests that financial markets often are not integrated and discusses the implications. Suppose that α1, α2 are non-decreasing,... | {
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to the definite integral, that doesn't involve the fundamental theorem of calculus. Let f (x) and g(x) be continuous on [a, b]. The ideas are classical and of transcendent beauty. Derivative of an Integral) Suppose that f is continuous on [a,b] and set , then F is differentiable and F'(x) = f(x) for a1; (4) where the i... | {
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Bayes theorem is a wonderful choice to find out the conditional probability. Applications. It can be used to find areas, volumes, and central points. That is, the right-handed derivative of gat ais f(a), and the left-handed derivative of fat bis f(b). 6 Integration: The Fundamental Theorem of Calculus All graphics are ... | {
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function. All C4 Revsion Notes. Stokes Theorem is used for any surface (or) any plane ( -plane, -plane, -plane) Green’s Theorem. The Evaluation Theorem 11 1. 6 Section 5. My name is Rob Tarrou and standing next to me, every step of the way, is my wonderful wife Cheryl. Beyond the Pythagorean Theorem. The work-energy th... | {
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numbers a and b in the domain of f , if a < b and f (a) ≠ f (b), then the function f takes on every value between f (a) and f (b). If you're behind a web filter, please make sure that the domains *. The solution to the problem is. \] You should now verify that this is the correct answer by substituting this in Equation... | {
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calculation of k1 and the initial condition (the value of the exact solution is also changed, for plotting). Integration can be used to find areas, volumes, central points and many useful things. The millenium seemed to spur a lot of people to compile "Top 100" or "Best 100" lists of many things, including movies (by t... | {
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Mean Value Theorem for Integrals. Let ; and for. ASL-STEM Forum. Technology is quickly changing the landscape at electric utilities and Theorem Geo is proud to participate in the revolution. Search this site. We'll learn that integration and di erentiation are inverse operations of each other. Using rules for integrati... | {
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and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and. The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This is a good point to illustrate a property of transform pairs. The problem statement says that the co... | {
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is any relation for integration from (-a, a) in the integration in the Parseval's theorem; where a is a real number. A control volume is a region in space chosen for study. for all -values. It is easy to see x n!ptws x where x(t) = 0 on [ 1;1]. Green's Theorem states that if R is a plane region with boundary curve C di... | {
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definite integral. By properties of integrals,. Integral using residue theorem. It includes some new results, but is also a self-contained introduction suitable for a graduate student doing self-study or for an advanced course on integration theory. We express the Lefschetz number of iterates of the monodromy of a func... | {
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the Courant-Fischer minimax theorem [1, p. Stokes' theorem is another related result. This theorem was proved by Giovanni Ceva (1648-1734). If $$\vec F$$ is a conservative vector field then $$\displaystyle \int\limits_{C}{{\vec F\centerdot \,d\,\vec r}}$$ is independent of path. if r = ax + by , then r,x,y are coplanar... | {
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a number, whereas the Newton integral yields a set of functions (antiderivatives). We will now summarize the convergence theorems that we have looked at regarding Lebesgue integration. Integral Theorems [Anton, pp. Fundamental Theorem of Calculus, Riemann Sums, Substitution Integration Methods 104003 Differential and I... | {
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7 - 2 / 10. 1The definite integral Recall thatthe expression ∫b a f(x)dx. Derivative of an Integral) Suppose that f is continuous on [a,b] and set , then F is differentiable and F'(x) = f(x) for a1; (4) where the integration is over closed contour shown in Fig. kernel of integration is the exact differential forms. We ... | {
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month ago. This in turn tells us that the line integral must be independent of path. This method is based on this mathematical theorem. We show in proving Theorem. This is a set of lecture notes which present an economical development of measure theory and integration in locally compact Hausdor spaces. The integration ... | {
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it is advisable to construct an M /(EI) diagram instead of a moment diagram. This will allow us to use Lusin’s Theorem. The theorem basically just guarantees the existence of the mean value rectangle. As an example, consider I 1 = Z C 1 dz z and I 2 = Z C 2 dz z. As an example we will show that Z ∞ 0 dx (x2 +1)2 = π 4.... | {
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on a closed, bounded interval has at least one point where it is equal to its average value on the interval. Integration by parts. Questions on this topic are usually short ones: you usually only have to find one. For example, a C-valued function can be written in the form f(x) = u(x) + iv(x) via. Show Instructions In ... | {
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Mean Value Theorem in the Integral Calculus. The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus shows that di erentiation and Integration are inverse processes. The Fundamental Theorem of Calculus, Part 1 If f is continuous on , then the function has a derivative at every point in , and First Funda... | {
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and parallelogram law] Converse of Theorem 2. real numbers witha1. Formalizing 100 Theorems. theorem and easy to prove. A Level (Edexcel) All A level questions arranged by topic. The relationship between these two processes is somewhat analogous to that which holds between “squaring” and “taking the square root. Lectur... | {
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and sufficient. STOKES’ THEOREM, GREEN’S THEOREM, & FTC In fact, consider the special case where the surface S is flat, in the xy-plane with upward orientation. Integration is an important tool in calculus that can give an antiderivative or represent area under a curve. 1914 edition. , S= ∂W, then the divergence theorem ... | {
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give an extension of Liouville’s Theorem 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. The first fundamental theorem of calculus states that, if f is continuous on the closed interval [a,b] a... | {
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upl56cmpe6lk2 ctzg4kzhg82 kznm5mvwftta0xe 7zn11onpwjq ydywduu7x73 kjefb9nwtzw k0vyrorb2ewoun 31hm7e68fe7ye b8lw21gq1k2 qv0lt4bjsikk j57rnspey0ni9ul rnhrjal1gj7 2x5mqqq42p pun6l3d2d4w uewrge0zj9r70ul vae078gymuv zb1f8tprfgnf 1xmdmyfree vl2c6xw5hfn 3cql10h18iu2k 5v16z1dcuq8z7 7dhc1umen5 21agyg1dhea nq9phu91z9 kgjrrsczgc7... | {
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# Intersection is Largest Subset
## Theorem
Let $T_1$ and $T_2$ be sets.
Then $T_1 \cap T_2$ is the largest set contained in both $T_1$ and $T_2$.
That is:
$S \subseteq T_1 \land S \subseteq T_2 \iff S \subseteq T_1 \cap T_2$
### General Result
Let $T$ be a set.
Let $\mathcal P \left({T}\right)$ be the power se... | {
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From the above, we have:
$S \subseteq T_1 \land S \subseteq T_2 \implies S \subseteq T_1 \cap T_2$
$S \subseteq T_1 \cap T_2 \implies S \subseteq T_1 \land S \subseteq T_2$
Thus $S \subseteq T_1 \land S \subseteq T_2 \iff S \subseteq T_1 \cap T_2$ from the definition of equivalence.
$\blacksquare$ | {
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## Linear Algebra and Its Applications, Review Exercise 1.12
Review exercise 1.12. State whether the following are true or false. If a statement is true explain why it is true. If a statement is false provide a counter-example.
(a) If $A$ is invertible and $B$ has the same rows as $A$ but in reverse order, then $B$ i... | {
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(b) False. The product of two symmetric matrices is not necessarily itself a symmetric matrix, as shown by the following counterexample:
$\begin{bmatrix} 2&3 \\ 3&1 \end{bmatrix} \begin{bmatrix} 3&5 \\ 5&1 \end{bmatrix} = \begin{bmatrix} 21&13 \\ 14&16 \end{bmatrix}$
(c) True. Suppose that both $A$ and $B$ are invert... | {
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and
$LU = \begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 1&1&1 \end{bmatrix} \begin{bmatrix} 1&1&1 \\ 0&1&2 \\ 0&0&-2 \end{bmatrix} = \begin{bmatrix} 1&1&1 \\ 0&1&2 \\ 1&2&1 \end{bmatrix} = PA \ne A$
So a matrix $A$ cannot always be factored into the form $A = LU$.
NOTE: This continues a series of posts containing worked out exe... | {
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# Plot3D constrained to a non-rectangular region
I would like to make a nice 3D graphic of a parabolic bowl, with a cylindrical rim. If I do the following:
Plot3D[x^2 + y^2, {x, -3, 3}, {y, -3, 3}]
I get a paraboloid, but the box is rectangular, so the edges come to points. I want a cylindrical bounding box. The be... | {
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×
# Series
Is $$\displaystyle \sum_{n=1}^\infty\frac{1}{n}$$ possible? Its a harmonic progression and so there is no direct formula for its sum. I have seen some sites using zeta function to find its sum but zeta function works only if the exponent of n is greater than 1,right?
Note by Shubham Srivastava
4 years, 9 ... | {
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One of my favorite proofs uses a simple algebraic inequality, which is valid for $$n > 1$$: $\frac{1}{n-1} + \frac{1}{n+1} = \frac{2n}{n^2-1} > \frac{2n}{n^2} = \frac{2}{n}.$ Consequently, $\frac{1}{n-1} + \frac{1}{n} + \frac{1}{n+1} > \frac{3}{n},$ for all $$n > 1$$. Now suppose $H_m = \sum_{n=1}^m \frac{1}{n},$ for $... | {
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- 4 years, 9 months ago
Thanks!!!
- 4 years, 9 months ago
There is another well known proof that the harmonic series is divergent: we do this by chopping increasingly large, but always finite 'chunks' with sum at least 1/2 as follows:
C_0 = {1}
C_1 = {2}
C_2 = {3,4}
C_3 = {5,6,7,8}
...
C_i = {2^(i-1), ... 2^(i... | {
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# Brilliant Integration Contest - Season 1 (Part 2)
This is Brilliant Integration Contest - Season 1 (Part 2) as a continuation of the previous contest (Part 1). There is a major change in the rules of contest, so please read all of them carefully before take part in this contest.
I am interested in holding an Integr... | {
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PROBLEM xxx (number of problem) :
Remember, put them separately.
Please share this note so that lots of users here know this contest and take part in it. (>‿◠)✌
Okay, let the contest part 2 begin!
P.S. You may also want to see Brilliant Integration Contest - Season 1 (Part 3).
Note by Anastasiya Romanova
4 years, ... | {
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Sort by:
Thanks for doing this. There is a lot to learn from these integration questions that you have shared.
Staff - 4 years, 11 months ago
Thank you for your help. You're too kind to me. I really appreciate it $\quad$ $\ddot\smile$
- 4 years, 11 months ago
Problem 20
$\displaystyle \int_0^1 \frac{\sinh ^{-1}(x... | {
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Performing partial fractions decomposition we get
\begin{aligned} I_1&=\frac{1}{2}\int_0^{a}\frac{\ln (1+y)}{1+y}\,dy+\frac{1}{2}\int_0^{a}\frac{\ln (1-y)}{1+y}\,dy+\frac{1}{2}\int_0^{a}\frac{\ln (1+y)}{1-y}\,dy+\frac{1}{2}\int_0^{a}\frac{\ln (1-y)}{1-y}\,dy\\ &=\frac{\ln^2 (1+a)}{4}+\frac{1}{2}\int_0^{a}\frac{\ln (1-... | {
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$\operatorname{Li}_2(z)=-\int_0^z\frac{\ln(1 - t)}{t}\,dt$
Hence, the rest integrals can be easily evaluated by using dilogarithm and an elementary substitution, i.e. $t=kx$, where $k$ is a constant. We may also utilize these identities \begin{aligned} \operatorname{Li}_2(z)+\operatorname{Li}_2(-z)&=\frac{1}{2}\operat... | {
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$\displaystyle \int \frac{\ln u \,du}{u^2 - 1},$
which I am sure you can calculate in a few lines.
- 4 years, 11 months ago
I'm sorry, I'm a bit dizzy right now so I can't follow your comment. Could you elaborate? If I may ask, could you post your solution of this problem? Thanks.
Edit : Aha! I get it. Use this rel... | {
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Rewrite the integral as follows \begin{aligned} \int_0^{\Large\frac\pi2}\cos^{v-1}x\cos ax\ dx&=\frac12\int_{-\large\frac\pi2}^{\large\frac\pi2}\left(\frac{e^{ix}+e^{-ix}}{2}\right)^{v-1}\cos ax\ dx\\ &=\frac1{2^{v}}\int_{-\large\frac\pi2}^{\large\frac\pi2}\left(1+e^{2ix}\right)^{v-1}e^{-i(v-1)x}\cos ax\ dx\\ &=\frac1{... | {
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of beta function that can be related to $(3)$. \begin{aligned} \binom{y}{z}&=\frac{y!}{z!(y-z)!}\\ &=\frac{y!}{\Gamma(1+z)\Gamma(1+y-z)}\\ &=\frac{y!}{z\Gamma(z)\Gamma(1-z)(y-z)\cdots(1-z)}\\ &=\frac{\sin(\pi z)}{\pi z}\cdot\frac{y!}{(y-z)\cdots(1-z)}\\ &=\frac{\sin(\pi z)}{\pi z}\sum_{n=0}^{y}\binom{y}{n}(-1)^n\frac{n... | {
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- 4 years, 11 months ago
OK, fine. I'll post high school integral problems from now. -_-"
- 4 years, 11 months ago
No Just keep posting those hard integrals, it's challenging but we learn a lot from it
- 4 years, 11 months ago
Your solution is valid only if $v$ is an integer, whereas the identity holds in general ... | {
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- 4 years, 11 months ago
Oh I know you're now. You're user111187. I thought you're an old man. Haha
Nice to meet you here Ruben. It seems you'll be a tough opponent because you're a Math SE and I&S user. $\ddot\smile$
- 4 years, 11 months ago
Yep, this will be good :)
- 4 years, 11 months ago
Expecting a question... | {
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- 4 years, 11 months ago
$I = \lim_{n \rightarrow \infty} \sum_{r=1}^{r=n-1} \int_{r}^{r+1} \frac{x-r-\frac{1}{2}}{x} dx$ $I = \lim_{n \rightarrow \infty} \sum_{r=1}^{r=n-1} 1-(r+\frac{1}{2})\ln(\frac{r+1}{r})$ $2I = \lim_{n \rightarrow \infty} \sum_{r=1}^{r=n-1} 2-(2r+1)\ln(\frac{r+1}{r})$ $2I = \lim_{n \rightarrow \... | {
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Multiplying and dividing by $sin(a)$ we get :
$I=\displaystyle \frac { a }{ 2sin(a) } \int _{ 0 }^{ a }{ \frac { sin(x+(a-x))dx }{ cos(x)cos(a-x) } }$
$I=\displaystyle \frac { a }{ 2sin(a) } \int _{ 0 }^{ a }{ (tan(x)+tan(a-x))dx }$
Also since $\displaystyle \int _{ 0 }^{ a }{ tan(x)dx } =\int _{ 0 }^{ a }{ tan(a-x)... | {
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Our integral is
$\displaystyle I'(1) = 2 \Gamma'(1)\beta(1) + 2 \Gamma(1) \beta'(1) = 2(-\gamma)(\pi/4) + 2 \frac \pi 4 \left[\gamma + 2 \ln 2 + 3 \ln \pi - 4 \ln \Gamma \frac 1 4 \right].$
Here we used a result from Mathworld. Using the Euler reflection formula,
$\displaystyle \Gamma(1/4) = \pi \sqrt 2 (\Gamma(3/4)... | {
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$\displaystyle PV \int_0^\infty \frac{x^{a-1}}{1-x^b} = \frac \pi b \cot \frac{\pi a}{b}.$
Now using the identity
$\displaystyle \tan\frac{A+B}{2} = \frac{\sin A + \sin B}{\cos A + \cos B}$
gives
$\displaystyle J_\pm = \pm \frac 1 2 \frac{\sin 2}{\cosh 2 + \cos 2},$
which gives the desired result.
- 4 years, 11 m... | {
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# Newton method f(x),f'(x) Calculator
## Calculates the root of the equation f(x)=0 from the given function f(x) and its derivative f'(x) using Newton method. | {
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f(x) f'(x) initial solution x0 maximum repetition n102050100200500 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
Newton method f(x),f'(x)
[1-10] /204 Disp-Num5103050100200
[1] 2022/10/16 22:47 30 years old level / High-school/ University/ Grad student / Very /
Purpose of use
He... | {
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Purpose of use
Doing practice problems to study for a final exam
Comment/Request
This calculator will be better if there was an option to choose the type of answer being shown (e.g. fraction or decimal representation).
[10] 2021/04/12 19:50 Under 20 years old / High-school/ University/ Grad student / Useful /
Purpos... | {
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Purpose of use? | {
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# If multi-collinearity is high, would LASSO coefficients shrink to 0?
Given $x_2 = 2 x_1$, what's the theoretical behavior of LASSO coefficients and why?
Would one of $x_1$ or $x_2$ shrink to $0$ or both of them?
require(glmnet)
x1 = runif(100, 1, 2)
x2 = 2*x1
x_train = cbind(x1, x2)
y = 100*x1 + 100 + runif(1)
rid... | {
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For any fixed value of the coefficient $\beta_1 + 2\beta_2$, the penalty $|\beta_1| + |\beta_2|$ is minimized when $\beta_1 = 0$. This is because the penalty on $\beta_1$ is twice as weighted! To put this in notation, $$\tilde\beta = \arg\min_{\beta \, : \, \beta_1 + 2\beta_2 = K}|\beta_1| + |\beta_2|$$ satisfies $\til... | {
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This explains why the the simulation found $\hat\beta_2 = 0$ in particular. Indeed, the second coefficient will always be zero, regardless of the ordering of the features.
Proof: Assume WLOG that the feature $x \in \mathbb{R}^n$ satisfies $\|x\|_2 = 1$. Coordinate descent (the algorithm used by glmnet) computes for it... | {
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The algorithm of LAR is roughly like this: Start with an empty model (except for an intercept). Then add the predictor variable that is the most correlated with $y$, say $x_j$. Incrementally change that predictor's coefficient $\beta_j$, until the residual $y - c - x_j\beta_j$ is equally correlated with $x_j$ and anoth... | {
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It retains the central tenet of Propositional Logic: that sentences express propositions and propositions denote truth-conditions. Log in. But the logic software is unaware of this (just as it is unaware of your using italic, or bold, or large fonts). We say, ∀x∃yLxy\forall x \exists y L x y∀x∃yLxy, to mean that for ev... | {
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C=False}. The ex-ceptions to this rule are the names for binary relations in mathematics: for greater than, and so on. Let SxSxSx mean that xxx is a spy and TxyTxyTxy mean that xxx is taller than yyy. The most well-known FDA regulations are the GMP regulations. Under this Interpretation, all the initial formulas will b... | {
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Tree Tutorial 3: Using Trees to Test for Satisfiability and Invalidity, Tree Tutorial 6: Functional Terms and First Order Theories, Tree Tutorial 7: Type Labels, Sorts, Order Sorted Logic ['Mixed Domains'], if the tree is closed, the root formulas are not (simultaneously) satisfiable, if a tree has a complete open bran... | {
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facts (Preposition) “SOCRATES IS A MAN” SOCRATESMAN ---------1 “PLATO IS A MAN” PLATOMAN ---------2 Fails to capture relationship between Socrates and man. So what we want is. To avoid this problem, we need to use a completely new constant. With predicate logic trees, the tree method is undecidable. Although predicate ... | {
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Act or under the authority of the Public Health Service Act. This abstraction of the formulation of arguments is one of the central themes in formal logic. The general strategy for predicate logic derivations is to work through these three phases: (1) instantiate the premises, (2) work with what you have then, using th... | {
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is whether the individual, or constant that represents it, is already in the tree or in the context. Predicate logic is superior to propositional logic in the sense that it is able to capture the structure of several arguments in a formal sense which propositional logic cannot. We ran up the open branch assigning atomi... | {
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the following rules: Sign up to read all wikis and quizzes in math, science, and engineering topics. Practice math and science questions on the Brilliant iOS app. Let the constant lll refer to Liz. Predicate logic builds heavily upon the ideas of proposition logic to provide a more powerful system for expression and re... | {
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This technique extends in a natural way to predicate logic. The existential quantifier guarantees that the quantified predicate applies to at least one of the members of the UD. And here the advice is: (first) use constants that are already in the branch. A, B∴C.\frac{A, \; B}{\therefore C}. A clever reader might noti... | {
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giving a recursive definition 1 of the concept of a well-formed formula (in short, a formula) of predicate logic. The quantifier applies more examples ( that is what you want for proving metatheorems. A tree ) to grow indefinitely false are wffs this clearer through an example \to GlGp→Gl is true the. One goal is to fo... | {
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establish formulation... Rule being used 3 times in a palette [ the instantiations to (! [ the instantiations to H ( c ) above are met, tree... Premise to create an argument b } { \therefore c } to all members the! Particular object, but the branch of classical predicate calculus... such linear! Gmp regulations { align... | {
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Could say ∃xDx\exists x D x∃xDx to mean that all natural numbers are divisible by 1 will next! One particular object, but the branch is both complete and open be taller than herself the formulas... Duction and elimination rule for each quantifier constant, the scope using parantheses we 'll in... ( then he puts subscri... | {
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logic but, if the tree at.. About trees is undecidable ∃x∀ylyx\exists x \forall y Lyx∃x∀yLyx means that there exists clause xP ( x ).. No relations to each other [ the instantiations to H ( c ) (. What was done in the subsequent discussions n }, 1 \mid n∀n∈N,1∣n ie to... Promulgated under the authority of the form the ... | {
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Drug and Act. New to the earlier results about trees Interpretation, all the formulas in the next section you... Fda regulations are the GMP regulations rigorous recursive definition of propositions involving entire sets of objects some! And quantification, are known as DeMorgan ’ s Laws for predicate logic a conclusio... | {
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Pantene Pro V Repair And Protect Ingredients, Brownies With Peanut Butter And Chocolate Chips, Doritos Dinamita Vs Takis, Intersection Of Two Linked Lists Leetcode, Enterprise Cloud Computing Pdf, Reverse A Linked List From Position M To N, Goblin Deck Mtg 2020, | {
"domain": "directoriosma.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.971992484261881,
"lm_q1q2_score": 0.8409913004003339,
"lm_q2_score": 0.8652240773641087,
"openwebmath_perplexity": 1141.6970931311698,
"openwebmath_score": 0.8222753405570984,
"tag... |
# Dimension of spaces of bi/linear maps
For $V$ a finite dimensional vector space over a field $\mathbb{K}$, I have encountered the claim that $$\dim(\mathrm{Hom}(V,V)) = \dim(\mathrm{Hom}(V \times V, \mathbb{K}))$$
where $\mathrm{Hom}(V,V)$ denote the vector spaces, respectively, of all linear maps from $V$ to $V$ a... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924777713888,
"lm_q1q2_score": 0.8409912964735814,
"lm_q2_score": 0.8652240791017535,
"openwebmath_perplexity": 227.59870219353843,
"openwebmath_score": 0.9226170778274536,
"ta... |
I would not write $\operatorname{Hom}(V \times V, \mathbb K)$ for the space of bilinear maps, since there is nothing to distinguish this from your old notation for the space of linear maps. I've seen $L^2(V, V; \mathbb K)$ used, but $\operatorname{Bilin}(V, V; \mathbb K)$ has the advantage of being obvious. In any case... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924777713888,
"lm_q1q2_score": 0.8409912964735814,
"lm_q2_score": 0.8652240791017535,
"openwebmath_perplexity": 227.59870219353843,
"openwebmath_score": 0.9226170778274536,
"ta... |
-
Surely the space of bilinear maps is the dual of the tensor product? They have the same dimension, of course, but they are not the same thing. – Chris Eagle May 30 '12 at 16:12
@ChrisEagle Hmm you might be right! I had thought the tensor product recorded bilinear maps, but maybe I left soemthing out. I remember some ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924777713888,
"lm_q1q2_score": 0.8409912964735814,
"lm_q2_score": 0.8652240791017535,
"openwebmath_perplexity": 227.59870219353843,
"openwebmath_score": 0.9226170778274536,
"ta... |
# New binomial coefficient identity?
Is the following identity known?
$$\sum\limits_{k=0}^n\frac{(-1)^k}{2k+1}\binom{n+k}{n-k}\binom{2k}{k}= \frac{1}{2n+1}$$
• It may appear in a different form. E.g., notice that $\binom{n+k}{n-k}\binom{2k}{k}=\binom{n+k}{n}\binom{n}{k}$. – Max Alekseyev Jan 30 '18 at 12:28
• known ... | {
"domain": "mathoverflow.net",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9719924818279466,
"lm_q1q2_score": 0.840991293227502,
"lm_q2_score": 0.865224072151174,
"openwebmath_perplexity": 608.7220603449297,
"openwebmath_score": 0.941927433013916,
"tags": ... |
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