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https://www.jiskha.com/questions/561464/The-length-of-a-rectangle-is-2-meters-more-than-its-width-If-the-area-is-63-square	# algebra The length of a rectangle is 2 meters more than its width. If the area is 63 square meters, what are the length and width? 1. L=w+2 L x w = 63 (w+2) x w = 63 w^2+2w=63 w^2+2w-63=0 What two factors multiply to -63 and add up to 2? 9 and -7. L=9, W=7 posted by Jen First Name ## Similar Questions 1. ### pre-algebra Can you help me figure this? The area of a rectangle is 14 square meters. Find the length and width of the rectangle if it's length is 5 meters greater than its width. Use an equation and the formula for the area of a 2. ### algebra The area of a rectangle is 21 square meters. Find the length and width of the rectangle if its length is 4 meters greater than its widith. using Area of a rectangle = (width)(Length) 3. ### algebra The length of a rectangle is 2 meters more than its width. If the area is 35 square meters, what are the length and width? 4. ### calculus 1. A rectangle has a constant area of 200 square meters and its length L is increasing at the rate of 4 meters per second. a. Find the width W at the instant the width is decreasing at the rate of .4 meters per second. b. At what 5. ### calculus 1. A rectangle has a constant area of 200 square meters and its length L is increasing at the rate of 4 meters per second. a. Find the width W at the instant the width is decreasing at the rate of .4 meters per second. b. At what 6. ### algebra the length of a rectangle is 5 times its width. if the length is decreased by 3 meters, and the width is increased by 10 meters, the perimeter will be 374 meters. find the length of the original rectangle. 7. ### Math A rectangle is 40 meters longer than its width if its length were increased by 10 meters and its width would be decreased by 5 meters its area would be increased by 190 square meters. Find the dimensions 8. ### basic geometric 10. a rectangle has width the same as a side of a square whose perimeter is 20m. the length of the rectangle is 9m. find the perimeter of this rectangle. 34. The width of a rectangular picture is one-half the length. The perimeter 9. ### Maths Initially, school field is 50 meters in length with 30 meters width. After expansion, the length increased by 20% and the area becomes 2340 square meters, therefore the width has increased_________ meters. 10. ### math The length of a rectangle is 2 meters more than its width. If the area is 35 square meters, what are the length and width? (Points : 4) More Similar Questions	2018-08-16T08:55:04	{ "domain": "jiskha.com", "url": "https://www.jiskha.com/questions/561464/The-length-of-a-rectangle-is-2-meters-more-than-its-width-If-the-area-is-63-square", "openwebmath_score": 0.8468508720397949, "openwebmath_perplexity": 423.10582523348273, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9859363733699887, "lm_q2_score": 0.849971175657575, "lm_q1q2_score": 0.8380174983968551 }
http://math.stackexchange.com/questions/739719/help-on-finding-eigenvalues-of-transformation-on-matrices	# Help on finding eigenvalues of transformation on matrices T is linear transformation working on 2x2 matrices: T(A) = $\begin{bmatrix}1 & 1\\1 &1\end{bmatrix}$ A as far as I see only 0 is an eigen value but someone told me 2 is eigen value too and I can't understand why , can anyone help please? - Because $\begin{bmatrix}1&1\\1&1\end{bmatrix}\begin{bmatrix}1\\1\end{bmatrix} = 2 \begin{bmatrix}1\\1\end{bmatrix}$. – Algebraic Pavel Apr 4 '14 at 16:19 $$T\left(\begin{bmatrix}1&1\\1&1\end{bmatrix}\right)=\begin{bmatrix}1&1\\1&1\end{bmatrix}\begin{bmatrix}1&1\\1&1\end{bmatrix}=\begin{bmatrix}2&2\\2&2\end{bmatrix}=2\begin{bmatrix}1&1\\1&1\end{bmatrix}$$ - Using the general method to find eigenvalues (and the characteristic polynomial of a matrix, of course): $$\det(\lambda I-A)=\begin{vmatrix}\lambda-1&-1\\ -1&\lambda-1\end{vmatrix}=\lambda^2-2\lambda=\lambda(\lambda-2)=0\iff \lambda=\begin{cases}0\\2\end{cases}$$ - Since $T$ is a linear map $\mathbb R^4\to\mathbb R^4$ shouldn't we be looking at a polynomial of degree $4$? – Christoph Apr 4 '14 at 16:41 Probably perhaps of the poor formating but I didn't see that: I thought it was a map on two dimensional space. Yes, you're right, yet the basics remain the same. – DonAntonio Apr 4 '14 at 16:43 In any event the problem reduces to the ordinary matrix-vector case $T \vec v = \lambda \vec v$; see my answer. – Robert Lewis Apr 4 '14 at 20:58 Well, $0$ is an eginvalue since $T(\begin{bmatrix} 1 & 1 \\ -1 & -1 \end{bmatrix}) = 0, \tag{1}$ as is most easily seen. Likewise we have $T(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}) = \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix} = 2 \begin{bmatrix} 1 &1 \\ 1 & 1 \end{bmatrix}, \tag{2}$ so $2$ is also an eigenvalue. So far we have proceeded by intuition an (hopefully intelligent) guesswork, but we can sytematically find all the eigenvalues and eigenvectors of $T$ by observing that, if we write $A$ in columnar form, that is, $A = [\vec a_1 \; \vec a_2 ], \tag{3}$ where $\vec a_1$ and $\vec a_2$ are the columns of $A$, then $TA = [T\vec a_1 \; T \vec a_2], \tag{4}$ so that $TA = \lambda A$ becomes $TA = [T\vec a_1 \; T \vec a_2] = \lambda [\vec a_1 \; \vec a_2 ] = [\lambda \vec a_1 \; \lambda \vec a_2]; \tag{5}$ (5) shows that the "eigenmatrices" of the operator $T$ associated with eigenvalue $\lambda$ are precisely those non-zero matrices $A$ whose non-zero columns are ordinary $\lambda$-eigenvectors of $T$. Thus we may find the eigenvalues of $T$ as it operates on $2 \times 2$ matrices by simply finding the ordinary eigevalues of $T$ as it maps vectors to vectors. Thus we construct the characteristic polynomial $p_T(\lambda)$ of $T$: $p_T(\lambda) = \det (T - \lambda I) = \det(\begin{bmatrix} 1 - \lambda & 1 \\ 1 & 1 - \lambda \end{bmatrix}) = (1 - \lambda)^2 - 1 = \lambda^2 - 2\lambda; \tag{6}$ we see the roots of $p_T(\lambda)$ are $0$ and $2$, as indicated by our intuitive approach. However, (6) shows that $0, 2$ are the only eigevalues of $T$, whether it is applied to vectors or matrices. And the "eigenmatrices" may be constructed from the ordinary eigenvectors of $T$; since the $0$-eigenvector of $T$ is $\vec e_0 = (1, -1)^T$ and the $2$-eigenvector is $\vec e_2 = (1, 1)^T$, we easily see that the matrices $[\vec e_0 \; 0], [0 \; \vec e_0] \tag{7}$ are a basis for the $2$-dimensional $0$-eigenspace of $T$ as it applies here to $2 \times 2$ matrices; likewise $[\vec e_2 \; 0], [0 \; \vec e_2] \tag{8}$ form a basis for the $2$-eigenspace; our "guesswork" matrices are simply the sums of the matrices in (7) and (8) respectively; each eigenspace of $T$ as applied to matrices is two-dimensional, in a sense composed of two copies of the corresponding eigenspaces of $T$ as applied to vectors. In this sense the answer given by DonAntonio is correct in spirit, for it acknowledges, though tacitly, that the evaluation of eigenvalues in fact reduces to, and is determined by, the action of $T$ on simple vectors. Hope this helps. Cheerio, and as always, Fiat Lux!!! -	2016-02-10T22:19:57	{ "domain": "stackexchange.com", "url": "http://math.stackexchange.com/questions/739719/help-on-finding-eigenvalues-of-transformation-on-matrices", "openwebmath_score": 0.9642894268035889, "openwebmath_perplexity": 207.73504367285224, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9859363750229257, "lm_q2_score": 0.8499711737573762, "lm_q1q2_score": 0.8380174979283288 }
https://tutorme.com/tutors/4671/interview/	TutorMe homepage Subjects PRICING COURSES Start Free Trial Julianne L. Mathematics major, Education minor at Binghamton University Tutor Satisfaction Guarantee Linear Algebra TutorMe Question: Let A and v be the matrix and vector given by: A=\|0 0 2\| \|-3 1 6\| \|0 0 1 \| v= \|1\| \|3\| \|0\| Find Av. What does this say about eigenvectors and eigenvalues of A? Find the characteristic polynomial of A. Find all eigenvalues of A. Julianne L. Av is the product of A and v. It is a 3x1 matrix where the entry in each row is the product of the corresponding row of A dot multiplied with the column on v. The first row is (0)(1)+(0)(3)+(2)(0)=0. The second row is (-3)(1)+(1)(3)+(6)(0)=0. The third row is (0)(1)+(0)(3)+(1)(0)=0. Therefore Av= \| 0 0 0 \| Because Av=0v, v is an eigenvector of A with eigenvalue 0 by the definitions of eigenvector and eigenvalue The characteristic polynomial of A is the determinant of the matrix (A-MI) I will be using M in place of the commonly used lambda. This matrix is \| -M 0 2 \| \| -3 1-M 6 \| \| 0 0 1-M \| The determinant is found by taking (-1)^(1+row#)(the first entry in the row)det(matrix obtained by deleting that row and the first column) for each row and adding these values. So the determinant is -Mdet \|1-M 6 \| + 0 + 2det \|-3 1-M \| \| 0 1-M \| \| 0 0 \| =-M ((1-M)(1-M)) = -M (1-2M+M^2) = -M+2M^2-M^3 = -M(1-M)^2 Any of these are proper expressions of the characteristic polynomial. The eigenvalues are the roots of the characteristic polynomial so all you need to do to find them is to set -M(1-M)^2=0 and solve for M. We find that in this case M=0 and M=1. These are the eigenvalues of A. Calculus TutorMe Question: Find two linearly independent power series solutions of the differential equation y''-xy'-2y=0 Write the answer to at least x^5. Julianne L. We will be substituting in y=(summation n=0->inf) ((Cn)x^n) Cn is a coefficient C subscript n By differentiating we find that y'=(summation n=1->inf)( (Cn)nx^(n-1) ) and that y''=(summation n=2->inf) ( (Cn)n(n-1)x^(n-2) ) By substituting these into the original equation, our new equation is: (sum n=2->inf)((Cn)n(n-1)x^(n-2)) -x(sum n=1->inf)((Cn)nx^(n-1)) -2(sum n=0->inf) ((Cn)x^n) =0 the (-x) and (-2) can be moved inside the summations: (sum n=2->inf)((Cn)n(n-1)x^(n-2)) +(sum n=1->inf)(-(Cn)nx^n) +(sum n=0->inf) (-2(Cn)x^n)=0 We want to add the summations together so we will reindex the first using k=n-2 and the second two by k=n so that all of the summations contain x^k: (sum k=0->inf)((C(k+2))(k+2)(k+1)x^k) +(sum k=1->inf)(-(Ck)kx^k) +(sum k=0->inf) (-2(Ck)x^k) =0 Now if we take out the k=0 term for the first and third summations, we can add all of the summations into one: 2C2-2C0+ (sum k=1->inf)( [(C(K+2))(k+2)(k+1)-(Ck)k-2(Ck)]x^k ) =0 Using 2C2-2C0=0 We find that C0=C2 must be true Now we plug in values for k starting at 1 to find equations for the other coefficients, I will skip showing the computations for these as it is just plugging in integers for the Ks within the summation. Using k=1 we find that C3=(1/2)C1 with k=2 C4=(1/3)C2 and k=3 C5=(1/4)C3 Now all we need to do is pick a C0 and C1 for our two power series y1 and y2 For y1 we will choose C0=0 and C1=1. Then C2=0 C3=1/2 C4=0 C5=1/8 For y2 choose C0=1 and C1=0. Then C2=1 C3=0 C4=1/3 C5=0 Finally, we can put together our solutions. The Cn are the coefficients for each x^n term so y1= x + (1/2)x^3 + (1/8)x^5 and y2=1 + x^2 + (1/3)x^4 Calculus TutorMe Question: What is the triple integral of e^z over E, the space inclosed by the paraboloid z=1+x^2+y^2, the cylinder x^2+y^2=25, and the (x,y) plane. Use cylindrical coordinates. Julianne L. By subtracting (25=x^2+y^2) from (z=1+x^2+y^2) and simplifying, we find that z=24 is where the cylinder and paraboloid overlap. Drawing a picture of the space E shows that the paraboloid lies above the cylinder under z=24, therefore the top bound for z is the paraboloid and the bottom bound for z is the (x,y) plane or z=0. To find the triple integral using cylindrical coordinates, the limits must be in terms of z, r, and theta. The lower limit for z is z=0 and the upper limit is z=1+x^2+y^2 which is equivalent to z=1+r^2 in cylindrical coordinates since r^2=x^2+y^2. The largest radius is the radius of the cylinder x^2+y^2=25 which is 5, therefore the radius spans from 0 to 5. Theta spans from 0 to 2(pi) since the cylinder and paraboloid span a full 2(pi) radians. You additionally need to multiply an "r" within the integral since it is the Jacobian of cylindrical coordinates. The integral is then (0->2(pi)) (0->5) (0->(1+r^2)) re^z dz dr d(theta) Beginning with the innermost integral, re^z integrated with respect to z is still re^z and after substituting in the limits, this comes out to be (re^(1+r^2)-r) (r*e^(1+r^2)-r) integrated with respect to r is (1/2)e^(1+r^2)-(1/2)r^2 and this becomes (1/2)e^26-(1/2)e-(25/2) after substituting in the limits (1/2)e^26-(1/2)e-(25/2) integrated with respect to theta simply adds a theta to each element and substituting in the limits give us the final answer: (e^26)(pi)-25(pi)-e(pi) Send a message explaining your needs and Julianne will reply soon. Contact Julianne	2018-10-23T20:57:21	{ "domain": "tutorme.com", "url": "https://tutorme.com/tutors/4671/interview/", "openwebmath_score": 0.8986777067184448, "openwebmath_perplexity": 1356.2915660156805, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9859363729567546, "lm_q2_score": 0.8499711737573762, "lm_q1q2_score": 0.8380174961721429 }
https://math.stackexchange.com/questions/1268593/is-homology-determined-by-cohomology	Is homology determined by cohomology? I am aware of the universal coefficients theorem for cohomology which implies that the homology groups completely determine the cohomology groups. I am wondering if cohomology determines homology in a similar sense? If two spaces have the same cohomology groups do they necessarily have the same homology groups? Can we compute the homology groups from the cohomology groups? I am aware of Poincare duality - this only applies to "nice" spaces - seeing as homology determines cohomology in the above sense for general spaces, I was wondering if such a fact still holds switching the roles of homology and cohomology. The universal coefficients theorem for homology tells us about how the homology group with arbitrary coefficients relates to the integer homology groups - so I suppose that this is not the result I am looking for. Thanks! • It's a standard fact in algebraic topology that for closed oriented $n$-manifolds $M$, $H_{n - k}(M) \cong H^k(M)$. Look up Poincare duality. – Balarka Sen May 5 '15 at 17:51 • @BalarkaSen How is that relevant? The post doesn't even contain the word "manifold." – Matt Samuel May 5 '15 at 17:54 • @MattSamuel It's relevant in the sense that one at least knows that for (good) manifolds, two spaces with same homology has the same cohomology and vice versa, which is what the question asks. – Balarka Sen May 5 '15 at 17:55 Suppose that $X$ is a space such that all of its homology groups are finitely generated. This holds, for example, if $X$ is a "levelwise finite" CW complex (finitely many cells in each dimension), which is a very broad class of spaces. Then universal coefficients implies that all of the cohomology groups are also finitely generated. Moreover, the isomorphism class of each homology and cohomology group is determined by rank and torsion subgroup, and universal coefficients implies that • $H^k(X, \mathbb{Z})$ and $H_k(X, \mathbb{Z})$ have the same rank, and • The torsion subgroup of $H^k(X, \mathbb{Z})$ is isomorphic to the torsion subgroup of $H_{k-1}(X, \mathbb{Z})$. Hence the two sequences of isomorphism classes of groups determine each other in this case. They are nearly the same, except that the torsion subgroups are shifted one degree. Without the finitely generated hypothesis this is no longer true. For every abelian group $A$ and positive integer $n \ge 1$ it is possible to construct a Moore space $X = M(A, n)$, which is a space whose homology vanishes except in degree $n$, where it is isomorphic to $A$, and in degree $0$, where it is $\mathbb{Z}$. By universal coefficients, the cohomology of a Moore space is • $H^0(X, \mathbb{Z}) \cong \mathbb{Z}$ • $H^n(X, \mathbb{Z}) \cong \text{Hom}(A, \mathbb{Z})$ • $H^{n+1}(X, \mathbb{Z}) \cong \text{Ext}^1(A, \mathbb{Z})$ and all other cohomology vanishes. So the question is whether an abelian group $A$ is determined up to isomorphism by the isomorphism class of $\text{Hom}(A, \mathbb{Z})$ and $\text{Ext}^1(A, \mathbb{Z})$, and the answer is no. Counterexamples cannot be finitely generated: the first one that comes to mind is the following. We have $$\text{Hom}(\mathbb{Q}, \mathbb{Z}) = 0$$ (straightforward) and $$\text{Ext}^1(\mathbb{Q}, \mathbb{Z}) \cong \mathbb{R}$$ (nontrivial), and $\text{Ext}^1(-, -)$ preserves direct sums in the first factor, so it follows that the groups $\mathbb{Q}$ and $\mathbb{Q} \oplus \mathbb{Q}$ cannot be distinguished this way, since the two groups are not isomorphic, but $\mathbb{R} \cong \mathbb{R} \oplus \mathbb{R}$ (as abelian groups). • Do you know an example (Moore space or otherwise) which has trivial cohomology but non-trivial homology? – Michael Albanese Apr 7 '16 at 1:02 • @Michael: by the universal coefficient theorem this is equivalent to finding an abelian group $A$ such that $\text{Hom}(A, \mathbb{Z})$ and $\text{Ext}^1(A, \mathbb{Z})$ are both trivial (so an example exists iff an example exists which is a Moore space), assuming that you mean reduced cohomology. I don't know an example off the top of my head and am not sure if one exists. You could probably ask this as a new question. – Qiaochu Yuan Apr 7 '16 at 3:39 • For those interested, I asked this as a new question. – Michael Albanese Apr 9 '16 at 19:38	2020-04-02T10:58:26	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/1268593/is-homology-determined-by-cohomology", "openwebmath_score": 0.9215494990348816, "openwebmath_perplexity": 135.35424861460692, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9859363729567545, "lm_q2_score": 0.8499711737573762, "lm_q1q2_score": 0.8380174961721428 }
https://www.jakesonline.org/bear-stearns-pzzvpo/mixed-fraction-calculator-28e249	A mixed fraction is a fraction of the form \$$c {n \over d}\$$, where \$$c\$$ is an integer and \$$n d\$$. Convert the fraction to a mixed number by using long division to find the quotient and remainder 5 ÷ 3 = 1 R2 The quotient will be the whole number in the fraction, and the remainder will be the numerator in the mixed fraction 5 3 = 1 2 3 You can use … Multiply Mixed Numbers Calculator. We also offer step by step solutions. Formulas. Equivalent decimals (D) and reduced Fractions (R) will appear underneath. Study of mathematics online. Here is an adding and subtracting mixed numbers calculator to find the addition and subtraction of mixed fractions. Whether it is addition, subtraction, multiplication, or division. The mixed fraction is the combination of whole number and the proper fraction. Online Calculators for Operations with Simple and Mixed Fractions, including comparison. It is therefore the sum of a whole number and a proper fraction. A mixed fraction is a whole number and a proper fraction combined, i.e. The two fractions … DOSSIER.NET sommario. See how each example is made up of a whole number anda proper fraction together? First select if you want to use the default or mixed fraction calculator. The procedure to use the adding mixed fractions calculator … Fraction Calculators: Mixed Fraction Calculator, Mixed Number Calculator, Adding Fractions Calculator, Multiplying Fractions Calculator, Dividing Fractions Calculator, and Subtracting Fractions Calculator. One and three-half should be typed as 1 3/2. Dividing fractions calculator. Make your calculation easy by this mixed fraction calculator. The calculator performs basic and advanced operations with mixed numbers, fractions, integers, decimals. The calculator provided returns fraction inputs in both improper fraction form, as well as mixed number form. For example, to convert 5 4/5, we will do the following. Supports evaluation of mixed fractions (e.g. Yes, with this fraction calculating tool you can operate with both mixed and … An improper fraction is a fraction where the numerator (top number) is larger than the denominator (bottom number). To divide two fractions, MiroCalc.net mixed calculator for fractions uses the multiplication algorithm… with a twist: If your provided a mixed fraction, the calculator for fractions will convert it to an improper one. It is composed by the Numerator placed above a dash (called Fraction … You just need to enter your two fractions, then select the operation you want to perform. Mixed Numbers Calculator (also referred to as Mixed Fractions): This online calculator handles simple operations on whole numbers, integers, mixed numbers, fractions and improper fractions by adding, subtracting, dividing or multiplying. The mixed fraction calculator simplifies the result and displays it. Fraction Calculator. Learn to add, subtract, multiply and divide fractions. Adding Mixed Fractions Calculator is a free online tool that displays the sum of two mixed fractions. Calculate: enter 2 or 3 Fractions, select arithmetic operators using drop-downs and click on [=] button to get result. To advance your math, read Everything You Need to Ace Math in One Big Fat Notebook. Visual Fractions. Step 2) We add, subtract, multiply, or divide those improper fractions together from Step 1. The improper fraction can be converted to the mixed fraction. Keep exactly one space between the whole number and fraction and use a forward slash to input fractions. The primary reason is that the code utilizes Euclid's Theorem for reducing fractions which can be found at How MiroCalc.net Fractions Calculator is Dividing Fractions. Mixed Fraction to Improper Fraction Calculator is a free online tool that displays the conversion of mixed fraction into an improper fraction. It accepts proper, improper, mixed fractions and whole number inputs. For example, \$${5 \over 4}\$$. "Simple and Mixed". Fractions Calculator A simple, but powerful, fraction calculator to solve multiple types of fraction math problems. It is therefore the sum of a whole number and a proper fraction. A mixed fraction is a whole number and a proper fraction combined, i.e. Converting between fractions and decimals: Converting from decimals to fractions … An __improper fraction__ is a fraction whose nominator is greater than its denominator. Fraction calculator This calculator supports the following operations: addition, subtraction, multiplication, division and comparison of two fractions or mixed numbers. one and three-quarters. Electrical Calculators Real Estate Calculators When it will convert fraction to percent, a step by step procedure, results in percentage and result in decimal will be displayed separately. The combination of a proper fraction and whole number is called as mixed fraction. This unique tool will simplify your mixed number to its lowest form. © 2006 -2020CalculatorSoup® There general steps to subtract fractions are described below. This online calculator handles simple operations on whole numbers, integers, mixed numbers, fractions and improper fractions by adding, subtracting, dividing or multiplying. You can enter up to 3 digits in length for each the numerators and denominators (e.g., 456/789). Unless you got the gift of scientific mind, you need calculator to make sure you will get the correct computation during the mathematical operation with mixed fractions. Simplify Mixed Numbers. Example: Convert the following mixed number to an improper fraction. All rights reserved. Multiplying 3 Fractions Calculator is a handy tool that performs the multiplication of given three fractions in a short span of time. Sign in Log in Log out Use the online mixed fraction calculator to practice calculations with mixed fractions. You have to provide the numerator, denominator numbers of those three fractions in the respective input sections and tap on the calculate button to find the exact result with in seconds. So, in this case, you will need to divide the fractions: 1 (3/5) and 1 (4/7). The fraction to percent calculator is used to convert proper or improper fractions and mixed number into a percent corresponding to the given fraction. Do math calculations with mixed numbers (mixed fractions) performing operations on fractions, whole numbers, integers, mixed numbers, mixed fractions and improper fractions. A mixed number is a combination of a whole number and a fraction. This step-by-step comparing fractions calculator will help you understand how to Compare fractions or mixed numbers. It accepts proper, improper, mixed fractions and whole number inputs. Step 1) Convert mixed number to an improper fraction. This calculator for simply mixed fractions and allows you to change a mixed number to an improper fraction/proper fraction or vice versa. Mixed number refers to any whole number next to a fraction, for example 1 2/4. Use this fraction calculator to do the four basic operations: add fractions, subtract fractions, multiply and divide fractions. The slash separates the numerator (number above a fraction line) and denominator (number below). You can enter up to 3 digits in length for each whole number, numerator or denominator (123 456/789). If you want to simplify an individual fraction into lowest terms use our It also shows detailed step-by-step information about the fraction calculation procedure. Please, input values in this format: a b/c or b/c. Use this fractions calculator to easily perform calculations with fractions. BYJU’S online mixed fraction to improper fraction calculator tool performs the calculation faster and it displays the conversion value in a fraction of seconds. It can also convert mixed numbers to fractions and vice versa. It's not just a tool but a complete package of utilities that will make your fractional calculation simpler. It can also convert mixed numbers to fractions and vice versa. Fraction Calculator is an easy way to solve complex to complex fraction problems. BYJU’S online mixed fraction to improper fraction calculator tool performs the calculation faster and it displays the conversion value in a fraction of seconds. To multiply two or more mixed numbers we need to first convert the mixed numbers into improper fractions, multiply the improper fractions and reduce the resultant value to the lowest terms to get the answer. How to Use the Adding Mixed Fractions Calculator? Read our. For simplifying, our mixed number simplifier calculator converts the mixed number into the improper fraction. Here we have presented you a mixed fraction calculator. Sign in Log in Log out About. With this free App you can solve fraction related problems and exercices such as: Addition, Subtraction, Multiplication, Division, Simplification, Comparasion between Proper, Improper and Mixed Fractions. Example (Click to view) 1 1/3 + 2 1/4 Try it now. Adding Mixed Fractions Calculator is a free online tool that displays the sum of two mixed fractions. Use this calculator to convert your improper fraction to a mixed fraction. Fractions - use the slash “/” between the numerator and denominator, i.e., for five-hundredths, enter 5/100. The fraction calculator will generate a step-by-step explanation on how to obtain the results in the REDUCED FORM! TOGGLE 2-OR 3. Convert the fraction part of the mixed number to a decimal 2. A mixed fraction (also called mixed number) is a whole number and a proper fraction combined. Cite this content, page or calculator as: Furey, Edward "Mixed Numbers Calculator"; CalculatorSoup, 1/2 + 1/3 = (1×3+1×2) / (2×3) = 5/6 No rights can be derived from the information and calculations on this website. Use the algebraic formula for addition of fractions: Reduce fractions and simplify if possible, Use the algebraic formula for subtraction of fractions: a/b - c/d = (ad - bc) / bd, Use the algebraic formula for multiplying of fractions: a/b * c/d = ac / bd, Use the algebraic formula for division of fractions: a/b ÷ c/d = ad / bc. Fraction format button is used to work with all fractions. Enter fractions and press the = button. Simplify Online Fractions. That is why it is called a "mixed" fraction (or mixed number). 3 FRACTIONS, DECIMALS AND MIXED NUMBERS. Online Comparing Fractions Calculator… Mixed fractions are also called mixed numbers. The answer is provided in a reduced fraction and a mixed number if it exists. If you are using mixed numbers, be sure to leave a single space between the whole and fraction part. Mixed numbers: Enter as 1 1/2 which is one and one half or 25 3/32 which is twenty five and three thirty seconds. Compare: subtract second Fraction … Practical Example #1: Let’s say that you are looking to subtract: 4 (1/8) – 1 (1/5) So, as we mentioned, the first thing that you need to do is to make sure that you have the mixed fractions selected and not the regular fractions. Look for a button that has a black box over a white box, x/y, or b/c. For example, \$${5 \over 4}\$$. Push this button to open the fraction feature on your calculator. $$\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{(a \times d) + (b \times c)}{b \times d}$$, $$1 \dfrac{2}{6} + 2 \dfrac{1}{4} = \dfrac{8}{6} + \dfrac{9}{4}$$, $$= \dfrac{(8 \times 4) + (9 \times 6)}{6 \times 4}$$, $$= \dfrac{32 + 54}{24} = \dfrac{86}{24} = \dfrac{43}{12}$$, $$\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{(a \times d) - (b \times c)}{b \times d}$$, $$\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{a \times c}{b \times d}$$, $$\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a \times d}{b \times c}$$. Greatest Common Factor Calculator. With this online mixed fraction calculator (or mixed number calculator) with whole numbers and fractions you can easily add mixed fractions, subtract mixed fractions, multiply mixed fractions and divide mixed fractions. With this calculator you can practice calculations with mixed fractions (mixed numbers). Dividing fractions calculator online. Fraction Operations and Manipulations Click on the decimal format button, enter a fraction or mixed number, then click equals. A mixed number is a combination of a whole number and a fraction. This calculator divides two fractions. … BYJU’S online adding mixed fractions calculator tool makes the calculation faster and it displays the sum of two mixed fractions in a fraction of seconds. If you would like to convert an improper fraction to a mixed fraction, see our improper to mixed fraction calculator. Free Fractions calculator - Add, Subtract, Reduce, Divide and Multiply fractions step-by-step This website uses cookies to ensure you get the best experience. Adding and subtracting mixed number is confusing. This calculator allows you to add, subtract, multiply, or divide simple fractions and mixed fractions online. This video will show you how to convert from mixed to improper fractions as well as improper to mixed fractions using the calculator Casio fx-991ms Mixed number to fraction conversion calculator that shows work to represent the mixed number in impropoer fraction. Fractions Calculator. Mixed fractions are also called mixed numbers. Add, Subtract, Multiply, Divide, Compare Mix or Compound Fractions … Simplify Fractions Calculator. If you are simplifying large fractions by hand you can use the If they exist, the solutions and answers are provided in simplified, mixed and whole formats. Order. The fraction calculator is very helpful in real life. This online fraction calculator will help you understand how to add, subtract, multiply or divide fractions, mixed numbers (mixed fractions), whole numbers and decimals.The fraction calculator will compute the sums, differences, products or quotients of fractions. A mixed fraction is a fraction of the form c n d, where c is an integer and n < d. For example, 11 4 = 2 3 4. Mixed fraction … This mixed fraction calculator by calculator-online is the smart tool that helps you in adding, subtracting, multiplying, and dividing mixed numbers fraction. You have 6 sections: 1. The Fraction Calculator will reduce a fraction to its simplest form. If the fraction or mixed number is only part of the calculation then omit clicking equals and continue with the calculation per usual. You can also add, subtract, multiply, and divide fractions, as well as, convert to a decimal and work with mixed numbers and reciprocals. Study math with us and make sure that "Mathematics is easy!" Fill in two fractions and choose if you want to add, subtract, multiply or divide and click the "Calculate" button. There general steps to divide fractions are described below. 1/2 + 1/3 = (1×3+1×2) / (2×3) = 5/6 The answer will be in fraction form and whole/decimal form. Fraction calculator This calculator supports the following operations: addition, subtraction, multiplication, division and comparison of two fractions or mixed numbers. Advatages: It is easy to use. You can also add, subtract, multiply, and divide fractions, as well as, convert to a decimal and work with mixed numbers and reciprocals. Mixed Fraction 1 : Operator : Mixed Fraction 2 . Read on to learn about fraction percent and its formula and how to turn a fraction … Use this calculator to convert your improper fraction to a mixed fraction. Adding mixed numbers, converting fraction to whole number, multiplying fractions by whole numbers, subtracting mixed numbers, and multiplying mixed fractions are among the processes this calculator can do. Enter the value of two mixed numbers and click calculate, it displays the step by step solution with the answer. 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http://mathhelpforum.com/algebra/225196-watch-gaining-losing-time.html	# Math Help - Watch gaining and losing time 1. ## Watch gaining and losing time My watch (which is a 12 hour watch) GAINS 3 mins every 2 hours. a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time? I got: real time vs gain time 2hrs vs 3 mins gain 40hrs vs 60 mins (x20) 960 hrs vs 24 hrs (x24) = 40 days (divided 960 hrs by 24hrs) Is this correct please for a? b) Mrs Warma's watch (also 1 12 hour watch LOSES 5 mins every 2 hours. She also sets her watch to the correct time at noon on 1st January. When will our two watches next show the same time? I got: real time vs lose time 2hrs vs 5 mins lose I am stuck after this.... c) When will our watches next show the same, correct time? I am stuck with this 2. ## Re: Watch gaining and losing time Originally Posted by Natasha1 My watch (which is a 12 hour watch) GAINS 3 mins every 2 hours. a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time? I got: real time vs gain time 2hrs vs 3 mins gain 40hrs vs 60 mins (x20) 960 hrs vs 24 hrs (x24) = 40 days (divided 960 hrs by 24hrs) Is this correct please for a? b) Mrs Warma's watch (also 1 12 hour watch LOSES 5 mins every 2 hours. She also sets her watch to the correct time at noon on 1st January. When will our two watches next show the same time? I got: real time vs lose time 2hrs vs 5 mins lose I am stuck after this.... c) When will our watches next show the same, correct time? I am stuck with this Not quite. It's a 12 hr watch so it will be correct again in 12 hrs, not 24 as you've used. So it will be 20 days not 40. b) $\mbox{in minutes your watch shows }m1=t(1+3/120)=t(1+1/40)=\frac{41}{40}t\mbox{ minutes.}$ $\mbox{her's shows }m2=t(1-5/120)=t(1-1/24)=\frac{23}{24}t\mbox{ minutes.}$ $\mbox{The watches are able to display } t \bmod 720 \mbox{ minutes uniquely.}$ $\mbox{So we want to solve for }\left(m1 \bmod 720\right)=\left(m2 \bmod 720\right)$ see if you can work it from there. c) you need to calculate how often the lady's watch is correct similar to how you did in (a). Then find the least common multiple of the three times, how often your watch is right, how often hers is, how often they are the same. 3. ## Re: Watch gaining and losing time i don't understand what mod is, never done this at school 4. ## Re: Watch gaining and losing time a mod b = remainder of a/b for example (25 mod 7) = 4 because 25 - 3x7 = 4 I'll give you a hint solving this. Try solving $\frac{41}{40}t-720=\frac{23}{24}t$ 5. ## Re: Watch gaining and losing time Hello, Natasha1! My watch (which is a 12-hour watch) GAINS 3 mins every 2 hours. a) I set my watch to the correct time at noon on 1st January. If I don't reset it, when will it next show the correct time? Your watch gains 1.5 minutes every hour. To show the correct time again, it must gain 12 hours = 720 minutes. . . $\frac{720\text{ min.}}{1} \times \frac{1\text{ hr.}}{1.5\text{ min.}} \:=\:480\text{ hours} \:=\:20\text{ days}$ It will show the correct time at noon on 21 January. 6. ## Re: Watch gaining and losing time Originally Posted by Natasha1 b) Mrs Warma's watch (also 1 12 hour watch LOSES 5 mins every 2 hours. She also sets her watch to the correct time at noon on 1st January. When will our two watches next show the same time? Your two watches will show the same time at the moment when your watch gains $t$ minutes and Mrs. Warma's watch loses 12 hours minus $t$ minutes. So, let $h$ be the number of periods of two hours that must pass before your watch gains $t$ minutes (and Mrs. Warma's watch loses $720-t$ minutes). Then $3h = t$ and $5h = 720-t$. This means $t=3h$ and $t=720-5h$. So, $3h = 720-5h$ shows that $8h = 720$, and $h = 90$. That means their watches will show the same time every $2h = 180$ hours. So, their watches will show the same time midnight, January 9th (that is the midnight that falls between the 8th and the 9th). 7. ## Re: Watch gaining and losing time Hello again, Natasha1! b) Mrs Warma's watch (a 12-hour watch) LOSES 5 mins every 2 hours. She sets her watch to the correct time at noon on 1st January. When will our two watches next show the same time? Your watch gains 1.5 minutes every hour. To show the correct time again, your watch must gain 12 hours = 720 minutes. . . $\frac{720\text{ min.}}{1} \times \frac{1\text{ hr.}}{1.5\text{ min.}} \:=\:480\text{ hours} \:=\:20\text{ days}$ Her watch loses 2.5 minutes every hour. To show the correct time again, her watch must lose 12 hours = 720 minutes. . . $\frac{720\text{ min.}}{1} \times \frac{1\text{ hr.}}{2.5\text{ min.}} \:=\:288\text{ hours} \:=\:12\text{ days}$ The two watches will agree in: $LCM(20,12) = 60$ days . . . Noon on March 30. 8. ## Re: Watch gaining and losing time Originally Posted by Soroban Hello again, Natasha1! Your watch gains 1.5 minutes every hour. To show the correct time again, your watch must gain 12 hours = 720 minutes. . . $\frac{720\text{ min.}}{1} \times \frac{1\text{ hr.}}{1.5\text{ min.}} \:=\:480\text{ hours} \:=\:20\text{ days}$ Her watch loses 2.5 minutes every hour. To show the correct time again, her watch must lose 12 hours = 720 minutes. . . $\frac{720\text{ min.}}{1} \times \frac{1\text{ hr.}}{2.5\text{ min.}} \:=\:288\text{ hours} \:=\:12\text{ days}$ The two watches will agree in: $LCM(20,12) = 60$ days . . . Noon on March 30. This is the answer to part (c), not part (b). 9. ## Re: Watch gaining and losing time Many thanks to all of you . Merry Christmas everyone!	2014-10-21T09:56:54	{ "domain": "mathhelpforum.com", "url": "http://mathhelpforum.com/algebra/225196-watch-gaining-losing-time.html", "openwebmath_score": 0.34227436780929565, "openwebmath_perplexity": 3537.8147764835303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.985936370890583, "lm_q2_score": 0.8499711718571774, "lm_q1q2_score": 0.8380174925424815 }
https://math.stackexchange.com/questions/1027539/what-does-this-summation-mean	# what does this summation mean? I apologize if this question has been asked. I know several similar ones have been asked but I cannot find one answering this in particular. I want to know what this summation means: $$\sum_{i,j=1}^{M}$$ Is this equivalent to $$\sum_{i=1}^M \sum_{j=1}^M$$ or does it mean that i and j will be incremented together at the same time? If context is needed, the equation is coming from here equation 16 on the third page (page 392 of the journal). • Yes, they are equivalent. Nov 18 '14 at 13:05 • It's the sum over the set of indices $(i,j)$ with $1\leqslant i,j\leqslant M$ which is equivalent to $\sum_{i=1}^M\sum_{j=1}^M$. Nov 18 '14 at 13:05 • $\sum_{i,j=1}^{M}\equiv \sum_{i=1}^M \sum_{j=1}^M$ with no doubt. Nov 18 '14 at 13:06 $$\sum_{i,j=1}^{M}a_{i,j}=\sum_{1\le i,j\le M}a_{i,j}=\sum_{i=1}^M\sum_{j=1}^Ma_{i,j}=\sum_{j=1}^M\sum_{i=1}^Ma_{i,j}$$ means that we sum the terms $a_{i,j}$ with all the possible values of $i$ and $j$ between $1$ and $M$.	2021-10-16T22:15:52	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/1027539/what-does-this-summation-mean", "openwebmath_score": 0.7084243297576904, "openwebmath_perplexity": 116.0332883804606, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9859363729567546, "lm_q2_score": 0.8499711699569787, "lm_q1q2_score": 0.8380174924251929 }
https://math.stackexchange.com/questions/3632614/r-wedge-neg-s-rightarrow-neg-q/3632666	# $(r \wedge \neg s) \rightarrow \neg q$ How can this English sentence be translated into a logical expression? In Kenneth Rosan, the answer to this following sentence “You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old.” is given as, $$(r \wedge \neg s) \rightarrow \neg q$$ Where, q: “You can ride the roller coaster.” r: “You are under 4 feet tall.” s: “ You are older than 16 years old.” My solution: So, I broke down this compound sentence as follows: “[You cannot ride the roller coaster] if [you are under 4 feet tall] unless [you are older than 16 years old.]” Now, substituting variables in given compound sentence. ($$\neg q$$) if (r unless s). Applying equivalence formula for Q if P $$\Leftrightarrow$$ P $$\to$$ Q (r unless s) $$\to$$ ($$\neg q$$) Now, solving for unless. So, (r unless s) $$\Leftrightarrow$$ ($$\neg s \to r$$) ref. ($$\neg s \to r$$) $$\to$$ ($$\neg q$$) Again solving for $$\to$$ (implication), we get: (s $$\lor$$ r) $$\to$$ ($$\neg q$$) So, my derivation is obviously wrong and does not match with Kennet Rosen. My Question: What mistake I did? and How to derive the given answer systematically? As noted by Jay, Kenneth Rosen interprets (r unless s) according to: $$\begin{array}{cc\|c} r & s & (r \text{ unless } s) \Leftrightarrow (\neg s \to r) \\\hline 0 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 1 \\ \end{array}$$ The issue turns out to be the order of operations for "unless". You started by breaking it up like this “[You cannot ride the roller coaster] if [you are under 4 feet tall] unless [you are older than 16 years old.]” And substituted variables to get: ($$\neg q$$) if ($$r$$ unless $$s$$). If instead we use a different order of operations to group these, it works out with your definition of unless. That is, we have: $$((\neg q) \mathbf{\text{ if }} r) \mathbf{\text{ unless }} s$$ Now using $$(P \mathbf{\text{ if }} Q) \Leftrightarrow (Q \to P)$$ $$(r \to \neg q) \mathbf{\text{ unless }} s$$ Now using your $$(P \mathbf{\text{ unless }} Q) \Leftrightarrow (\neg P \to Q)$$ $$\neg s \to (r \to \neg q)$$ expanding $$\neg s \to (\neg r \lor \neg q)$$ expanding $$s \lor (\neg r \lor \neg q)$$ This is the same as the other result. To see this, use associativity of logical or $$(s \lor \neg r) \lor \neg q$$ Then turn it into an implication $$\neg(s \lor \neg r) \to \neg q$$ Use demorgan's law $$(\neg s \land r) \to \neg q$$ (EDIT: Previously I arrived at the answer with the same order of operations, but a different interpretation of unless: $$(r \text{ unless } s) = (r \text{ and } \neg s)$$. Because "(anything) unless True = True" seriously sounds wrong to me. My interpretation of unless worked in this case, but apparently is not the correct english interpretation. Apologies.) • For this reason, we are going to translate ‘P unless Q’ with just ¬Q → P unless stated otherwise from rpi.edu – Ubi hatt Apr 19 '20 at 4:14 • Also, what formula you assumed for "r unless s"? – Ubi hatt Apr 19 '20 at 4:15 • @Ubihatt I used no formula. I filled it out how I interpret the phrase in english. So it appears this may just be a natural language ambiguity if you interpret it otherwise. – HBrown Apr 19 '20 at 4:17 • Kenneth Rosen also has formula for unless which exactly matches with formula embedded in my previous comment. Please check. Unless is a keyword. – Ubi hatt Apr 19 '20 at 4:19 • Interpreting "P unless Q" = "¬Q → P", would mean "You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old." says you cannot ride the roller coaster if you are older than 16 years old, no matter what your height is. Is that really how you interpret that phrase? – HBrown Apr 19 '20 at 4:27 The mistake is to interpret the sentence as $$(\neg q)\ \textbf{if}\ (r\ \textbf{unless}\ s)$$ The correct interpretation is $$(\neg q\ \textbf{if}\ r)\ \textbf{unless}\ s$$ • You beat me to the punch =) Nicely done. – Jay Dunivin Apr 19 '20 at 5:01 I have a discrete math book by Kenneth Rosen, and here is an excerpt from the book listing the equivalent ways to express $$p \implies q$$ in English. The one you mentioned is boxed in blue. But where did you go wrong? I believe it was at this step Now, substituting variables in given compound sentence. ($$\neg q$$) if (r unless s). The word unless does not attach $$s$$ to $$r$$; it attaches $$s$$ with the proposition $$\neg q$$ if $$r$$. The reason for this is that there was a proposition established prior to unless, which was the statement, "You cannot ride the roller coaster if you are under 4 feet tall." That sentence is established as a proposition; it is the $$q$$ in the table I provided. So the correct statement using your scheme is $$\big( \, \neg q \, \textbf{ if } \, r \, \big) \, \textbf{ unless } \, s,$$ which is equivalent to $$\neg s \implies (r \implies \neg q).$$ Can you take it from here? • As you saw, I initially interpreted unless differently, and just happened to get the right answer erroneously. So I learned something here too. But I'm struggling how "True unless True" evaluates to True. "The dog is angry unless the dog is fed". If the dog is angry and the dog is fed, I would interpret that as false. Is my english screwed up, or what am I missing? – HBrown Apr 19 '20 at 5:43 • Maybe what you're missing is that $F \implies T$ is true (similarly, $F \implies F$ is true). "The dog is angry unless the dog is fed," which is the same as "If the dog is not fed, then the dog is angry." If the dog is indeed fed, then the statement "the dog is not fed" is false, of course. But then the statement "If the dog is not fed, then the dog is angry" is actually a true statement (a false hypothesis can be used to derive any conclusion). The only way the statement could be false is when the dog is not fed, but the dog is not angry. Does that help? – Jay Dunivin Apr 19 '20 at 5:49 • To add to what I said, "True unless True" is equivalent to "False implies True," which is a true statement, per the definition of "implies" that we have agreed to adopt. We have agreed that $p \implies q$ is false if and only if $p$ is true and $q$ is false, and $p \implies q$ is true otherwise. – Jay Dunivin Apr 19 '20 at 5:53 • I see where you're coming from. I don't think it's your English. I was looking at other sites and papers, and there isn't a consensus as to what "unless" means in general, for it can be construed in different ways in different contexts. This link is particularly helpful as to why in logic this particular convention was adopted: math.stackexchange.com/questions/1295106/… – Jay Dunivin Apr 19 '20 at 6:14 • You might also be interested in these other links regarding the ambiguity of "unless" outside of mathematics: Reddit: reddit.com/r/askphilosophy/comments/52585m/… and a linguistic paper: web.stanford.edu/~danlass/Nadathur-Lassiter-unless-SuB.pdf – Jay Dunivin Apr 19 '20 at 6:15	2021-06-24T13:01:09	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/3632614/r-wedge-neg-s-rightarrow-neg-q/3632666", "openwebmath_score": 0.6943219304084778, "openwebmath_perplexity": 724.315313815072, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9496693702514737, "lm_q2_score": 0.8824278695464501, "lm_q1q2_score": 0.8380147191645269 }
http://thol.martin-vieth.de/math-city-derivative-formulas.html	## Math City Derivative Formulas It has two major branches, differential calculus and integral calculus. Derivative of the Exponential Function. Advanced:(optional) Given a formula for a function obtain improved numerical estimates of the derivative by choosing smaller increments (e. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics. Polar Integral Formula: Positive Series. Computation and Properties of the Derivative in Calculus. " Be Careful: "Find the derivative using the limit definition" does not mean estimating the derivative like we did earlier. Calculus-Specific Formulas There are a number of basic formulas from calculus that you need to memorize for the exam. The golden ratio. And it doesn't just work with position; Calculus can work with any function. ACTUARIAL SCIENCE FIRST YEAR CALCULUS - FORMULA SHEET Standard Integrals R xndx = xn+1 n+1 (n 6= −1) R 1 x dx = lnx R 1 x dx= lnx R ex= R sinxdx = −cosx R cosecxcotxdx = −cosecx. You should convince yourself that both of these expressions are indeed the same, by writing out explicitly the four terms of each of the two formulas! Let's try this for our derivative formulas: We shifted each occurrence of n in the expression up by one unit, while the limits of summation were shifted down by one unit, from 1 to 0, and from to. The instantaneous rate of change measures the rate of change, or slope, of a curve at a certain instant. It was submitted to the Free Digital Textbook Initiative in California and will remain unchanged for at least two years. Table of Derivatives. – is easier than you think. We discuss the Mittag-Leffler bounds of these solutions. ) And since the derivative of a sum is the sum of “Basic integration formulas. For the first two, determine the formula for the derivative by thinking about the nature of the given function and its slope at various points; do not use the limit definition. if n and a are positive real numbers, and a is not equal to 1, then If a x = n, then log a n = x. Applications of Differentiation. MATH 19B, Integral Calulcus: Some useful Trig formulas, derivatives and integrals. Stay ahead with the world's most comprehensive technology and business learning platform. #d/dx sec(x)=sec(x)tan(x)# You could memorize this, but you can work it out too by knowing some trig properties. f(x) = (x4 +3x)−1. The Derivation Formula. $\endgroup$ – Federico Poloni Aug 17 '15 at 8:42 1 $\begingroup$ This question really belongs to math. Mara~o garded on a algebra which generalizes certain features of th e derivative operator. derivative synonyms, derivative pronunciation, derivative translation, English dictionary definition of derivative. Tutorial on elementary differentiation formulas, their derivation and use. Ask Question 3. Derivative Formulas. COMMON MATH FORMULAS MISCELLANEOUS FORMULAS Simple Interest + L L N P where I = interest, p = principal, r = interest rate, and t = time Distance @. The power rule for differentiation states that if n is a real number and f(x) = x^n, then f'(x) = nx^{n-1}. Greater than 1, there will be exponential growth towards infinity (positive). Calculus allows us to see the connection between these equations. It may be rewritten as Another similar formula is given by Derivative of the Inverse Function. Together, we are going to review why and how to generate the formula for the Definition of Derivative, and we will walk through an example of how to use this definition to. You can choose formulas from different pages. To insure your continued success in second-semester, it is important that you are able to recall and use the following facts without struggling. f(x) = ex sinx 3. Recall the definitions of the trigonometric functions. org Math Tables: Derivatives of Hyperbolics Proofs of Derivatives of Hyperbolics Proof of sinh(x) = cosh(x): From the derivative of e x. fourth, fifth), extracting more and more information from that simple position function. Here is the rewritten function. The derivative of csc x. 6 Derivatives of Inverse Functions. if n and a are positive real numbers, and a is not equal to 1, then If a x = n, then log a n = x. Derivative Formula. Present Value Formula $$\huge P = \frac{F}{(1+r)^t}$$ The present value of money is equal to the future value divided by the interest rate plus 1 raised to the t power, where t is the number of months, years, etc. Compatible numbers. A class of second derivative extended backward differentiation formulas suitable for the approximate numerical integration of stiff systems of first-order ordinary differential equations is examine. Calculus Here is a list of skills students learn in Calculus! These skills are organized into categories, and you can move your mouse over any skill name to preview the skill. as a formula for the derivative of Φ with respect to Ξ. Most of derivatives' value is based on the value of an underlying security, commodity, or other financial instrument. But calculus provides an easier, more precise way: compute the derivative. In this case, it's easy to write down explicit formulas for the basis functions (they are just Bernstein polynomials), so the algebra is simpler. The derivative of ln(x) is a well-known derivative. Then the function. And it doesn't just work with position; Calculus can work with any function. To find the slope of a line tangent to a parabola at a specific point, find the derivative of the parabola's equation, then substitute the -coordinate of the specific point in the new equation. Or, as a formula that tells you the rate of change of a certain function. A very, very, very small distance, but large enough to calculate the slope. Derivative is a rate of change of function with respect to a variable. Choose from 500 different sets of calculus 1 formulas math flashcards on Quizlet. Find the equation to the tangent line to the curve y = x+ p x at the point (1,2) Example 3. In rst-semester calculus (regardless of where you took it) you learned the basic facts and concepts of calculus. The second derivative can be used to determine the concavity and inflection point of a function as well as minimum and maximum points. One of our goals in this section is to learn how to differentiate the logarithm function. Calculus 8th Edition answers to Chapter 2 - Derivatives - 2. It may be rewritten as Another similar formula is given by Derivative of the Inverse Function. In this page you can see a list of Calculus Formulas such as integral formula, derivative formula, limits formula etc. Math skills assessment. In Topic 19 of Trigonometry, we introduced the inverse trigonometric functions. The derivative of a function f at a point x is commonly written f '(x). Explicitly: If this limit exists, then we say that the derivative exists and has this value, and we say that the function is differentiable at the point. These questions will be similar to see how you will be tested on the exam. Firstly, you need to memorize these math formula very carefully before your exam. Calculus is the branch of mathematics studying the rate of change of quantities and the length, area and volume of objects. Includes the Power Formula. Write with me. To obtain a formula for the derivative of any expression in x, assume the function. The Derivative tells us the slope of a function at any point. Show Instructions. Offline content. This page is send by Ali Nawaz Bajwa (MS(Math), M. PHYSICS FORMULAS 2426 Electron = -1. The basic ideas are not more difficult than that. Finding the formula of the derivative function is called differentiation, and the rules for doing so form the basis of differential calculus. It is the only mathematics course certified to meet the university's "Viewing a Wider World" upper division general education requirement. Cheat Sheets and Calculator Tips The following are a list of formula sheets for various courses taught in the Mathematics Department. Please send suggestions for amendments to the Secretary of the Teaching Committee, and they will be considered for incorporation in the next edition. There are two ways of introducing this concept, the geometrical way (as the slope of a curve), and the physical way (as a rate of change). In a formula, it is abbreviated to just 'sec'. Calculus 8th Edition answers to Chapter 2 - Derivatives - 2. In general, scientists observe changing systems (dynamical systems) to obtain the rate of change of some variable. One of the most important functions in all of mathematics is the natural exponential function f(x)=ex. The derivative of ln u(). How to calculate derivatives for calculus. Offline content. When trying to gure out what to. Dick was a member of the National Academy of Sciences and a Honorary Fellow of the Indian Mathematical Society. Find the derivative of the equation and explain its physical meaning. Undergraduate. This is the original method in finding the derivative of any equations using the Increment Method. It is possible to guess at a formula for the derivative from this curve. Used by over 7,000,000 students, IXL provides personalized learning in more than 8,000 topics, covering math, language arts, science, social studies, and Spanish. It covers basic set theory and logic, relations and functions, and how to analyze, construct, and write clearly reasoned, well-structured elementary proofs using universal techniques. For each of the listed functions, determine a formula for the derivative function. You can either use an established mnemonic device or you can create your own. Calculation of the derivative — the most important operation in differential calculus. Derive the function 5/(x^0. Speciﬁc diﬀerentiation formulas You will be responsible for knowing formulas for the derivatives of these func tions: xn, sin−1 x, tan−1 x, sin x, cos x, tan x, sec x, ex , ln x. The derivative of sec x. Help With Your Math Homework. After you have selected all the formulas which you would like to include in cheat sheet, click the "Generate PDF" button. First derivative test for maxima/minima problems. For most of the. 331 (3/23/08) Estimating directional derivatives from level curves We could ﬁnd approximate values of directional derivatives from level curves by using the techniques of the last section to estimate the x- and y-derivatives and then applying Theorem 1. The expressions are obtained in LaTeX by typing \frac{du}{dt} and \frac{d^2 u}{dx^2} respectively. We discuss the Mittag-Leffler bounds of these solutions. Calculate the Costs to Use Electricity - powered by WebMath. In fact, you can use calculus in a lot of ways and applications. However, we can generalize it for any differentiable function with a logarithmic function. Basic Properties. The derivative of y = arccsc x. Method 1 (The way I found on my own): ∞ ∑ i=1a0rn−1≡S S=a0r0+a0r1+a0r2+⋯ S=r (a0r−1+a0r0+a0r1+⋯) S=r (a0r−1+S) S=a0+rS (1−r)S=a0 S=a0 (1−r) Note that for this to work, you must first confirm this: lim n→∞an=0. Eventually, formulas are used to provide mathematical solution for real world problems. Note that we studied Exponential Functions here and Differential Equations here in earlier sections. If f(x) = (3 x )(sin x) (cos x), find f'( x ). If there is a function h, then the anti-derivative of this function will be a differential function, say H. COMMON MATH FORMULAS MISCELLANEOUS FORMULAS Simple Interest + L L N P where I = interest, p = principal, r = interest rate, and t = time Distance @. Basic math formulas Algebra word problems. Solution If you’re having trouble with this problem, it may help to review Professor. PHYSICS FORMULAS 2426 Electron = -1. Use the Fundamental Theorem of Calculus to evaluate each of the following integrals exactly. Generally, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed. Then the function. You can also send us message on facebook. Partial Derivatives Calculus III - Chapter 11 Formulas Equation of tangent plane to surface of z f(x,y) at point P(x ,y z )is: z z f (x ,y )(x x ) f (x ,y )(y y ). Learn all about derivatives and how to find them here. Finding Derivative with Definition of Derivative Calculus 1 AB - Duration: 15:27. That means there are no two x-values that have the same y-value. \) The domains of the functions and their graphs. A simple calculus check reveals that the latter is the derivative of the former with respect to R. To find the equation of the tangent line using implicit differentiation, follow three steps. In this case, the instantaneous rate is s'(2). Calculus broadly classified as Differentiation and Integration. This course is a transition from procedural mathematics, such as calculus, to advanced mathematics where proofs are the professional language of discourse. A comprehensive list of the most commonly used basic math formulas. For each of the listed functions, determine a formula for the derivative function. Functions differentiation formula In the table below u and v — are functions of the variable x , and c — is constant. Seminar on Pure Mathematics Introduction to Hamiltonian Stationary Lagrangian submanifolds and a compactness theorem in 2 dimension Room 3494, Academic Building (lift 25-26), HKUST. A list of the most commonly used algebra formulas. The First and Second Derivatives The Meaning of the First Derivative At the end of the last lecture, we knew how to diﬀerentiate any polynomial function. Learn calculus derivatives formulas math with free interactive flashcards. -- Encyclopedia Of Mathematics (Science Encyclopedia)-- The Princeton Companion to Mathematics-- Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Books on Mathematics)-- NIST Handbook of Mathematical Functions (Being of course the 2010 update of the Abramowitz classic above). 6666666666666666)) using the power rule. By applying the derivation formulas and using the usual derivation table, it is possible to calculate any function derivative. Students, teachers, parents, and everyone can find solutions to their math problems instantly. The Derivative Calculator supports computing first, second, …, fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and calculating roots/zeros. Our study guides are available online and in book form at barnesandnoble. The derivative of acceleration is jerk. In this page you can see a list of Calculus Formulas such as integral formula, derivative formula, limits formula etc. It is fun and instructive and the student version without solutions is available free of charge. And, you need these formulas in your exams. Derivative proofs of csc(x), sec(x), and cot(x). The Derivative Measures Slope. Differentials and Newton's method. \] Now we consider the logarithmic function with arbitrary base and obtain a formula for its derivative. f(z)=−1/z^5. Calculus is used in mechanical, physics etc. 2 Functions of 2 or more variables Functions which have more than one variable arise very commonly. Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. Notation This is best described with an example. The derivative of sin x. SHOW PROPER CALCULUS-LEVEL WORK SIMPLIFY ALL ANSWERS Find lim cothx algebraically C' Hô Prove the logarithmic formula for Sinh x given in your textbook. If you are looking for a formula to solve your basic math problems, your formula is likely here. The desired formula is obtained by taking the limit of both sides as x approaches zero so that. It is designed for anyone who needs a basic to advanced understanding of mathematics concepts and operations. qxd Author: ewedzikowski Created Date: 10/29/2004 9:36:46 AM. Formulas: (a) sin2 x+cos2 x =1 (b) 1+tan2 x = sec2 x (c) cos2 x = 1+cos(2x) 2 (d) sin2 x = 1−cos(2x) 2 2. 1 Recall: ordinary derivatives If y is a function of x then dy dx is the derivative meaning the gradient (slope of the graph) or the rate of change with respect to x. Choose from 500 different sets of calculus 1 formulas math flashcards on Quizlet. Differential. Derivatives Calculus Differential Calculus Algebra Ap Calculus Calculus Notes I Love Math Fun Math Math Math Math Tutor Find all the essential math formulas you need, all in one place. This formula will be derived and then applied to … • the rôle of the Wronskian in the solution of linear differential equations,. The derivative is the function slope or slope of the tangent line at point x. Differentiate the formula with respect to time. more mathematical) definition. Every technique outlined in this article on calculating derivatives can be verified by a proper use of the definition of the derivative. 1300 Math Formulas Mega Pack - Table of Derivatives - Higher Order Derivatives - Application of Derivative Crime City 2019. Precalculus & Elements of Calculus tutorial videos. Figure 1 shows two graphs that start and end at the same points but are not the same. If y = 3 x 2, which can also be expressed as f(x)= 3 x 2, then. Derivative proofs of csc(x), sec(x), and cot(x). Taylor's Formula; List of Derivatives of Trig & Inverse Trig. And (from the diagram) we see that: Now follow these steps: Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx. How to Cite This Entry: Newton-Leibniz formula. We need differentiation when the rate of change is not constant. Derivative Identities. How is Business Calculus Different?. Derivatives are fundamental to the solution of problems in calculus and differential equations. derivative synonyms, derivative pronunciation, derivative translation, English dictionary definition of derivative. We will use the combination of these modes in order to have a desired and accurate output. We simplify the equation by taking the tangent of both sides: y = tan−1 x tan y = tan(tan−1 x) tan y = x. The expression (4. Taking the derivative¶. com; [email protected] Elasticity of demand is a measure of how demand reacts to price changes. Create a chart showing the graphical relationships between f, f. com to clear your doubts from our expert teachers and download the Application of Derivatives formula to solve the problems easily to score more marks in your Board exams. 2 days ago · Universal free lunch is linked to better test scores in New York City, new report finds bumps in reading and math state test scores once students attended schools with universal free lunch. A class of computationally A-stable formulas is presented, with notes on implementing a set of them in a variable stepsize, variable order method. Total derivatives are often used in related rates problems; for example, finding the rate of change of volume when two parameters are changing with time. Find the derivative (d/dx)(x^(-0. A first-semester college calculus course and the subsequent single-variable calculus course Recommended Prerequisites You should have successfully completed courses in which you studied algebra, geometry, trigonometry, analytic geometry, and elementary functions. Implicit multiplication (5x = 5*x) is supported. 18 Useful formulas. You should convince yourself that both of these expressions are indeed the same, by writing out explicitly the four terms of each of the two formulas! Let's try this for our derivative formulas: We shifted each occurrence of n in the expression up by one unit, while the limits of summation were shifted down by one unit, from 1 to 0, and from to. Together, we are going to review why and how to generate the formula for the Definition of Derivative, and we will walk through an example of how to use this definition to. Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. (360 degrees is 2π rad). Angular Velocity Formula Questions: 1) The second hand of a clock takes 30 seconds to move through an arc of 180 degrees. Most of derivatives' value is based on the value of an underlying security, commodity, or other financial instrument. Math Formulas and Math Tables. When you take a function’s derivative, you are finding that function that provides the slope of the first function. A class of computationally A-stable formulas is presented, with notes on implementing a set of them in a variable stepsize, variable order method. At the point in the video where he says that you can ignore dx 2 (around 3:30), he hasn't yet moved a dx term over to the same side as the df; he hasn't solved for df/dx, the derivative. ) If is negative, then we're dealing with the function , whose derivative is the constant. Normal Lines and how they relate to derivatives A normal line to a curve at a particular point is the line through that point and perpendicular to the tangent line (negative reciprocal) of derivative. Integral Calculus Formula Sheet Derivative Rules: 0 d c dx nn 1 d xnx dx sin cos d x x dx sec sec tan d x xx dx tan sec2 d x x dx cos sin d x x dx csc csc cot d x xx dx cot csc2 d x x dx d aaaxxln dx d eex x dx dd cf x c f x dx dx. Here listed free online differential equations calculators to calculate the calculus online. What is the angular velocity? Answer: The second hand starts at θ i = 0 degrees and moves to θ f = 180 degrees from the point of origin. MATH 171 - Derivative Worksheet Diﬀerentiate these for fun, or practice, whichever you need. For more on this see Derivatives of trigonometric functions together with the derivatives of other trig functions. By the definition of the reciprocal we have , throughout the domain of f. The Derivative Formula. Proportional controllers. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics. Here is a definition of the grammar used to parse AsciiMath expressions. Compatible numbers. First find the Lagrangian for a spring with mass m and spring constant k, and then derive the Euler-Lagrange equation. The domain of logarithmic function is positive real numbers and the range is all real numbers. The easiest rates of change for most people to understand are those dealing with time. (Topic 20 of. More information about video. Differentials and Derivatives in Leibniz's Calculus 5 Moreover, in Chapter 3 I discuss examples of the influence of the concepts discussed in Chapter 2 both on the choice of problems and on the technique of. – is easier than you think. Figure 1 is the graph of the polynomial function 2x 3 + 3x 2 - 30x. The Definition of the Derivative - In this section we will be looking at the definition of the derivative. The Story of Mathematics - 17th Century Mathematics - Newton. You click on the circle next to the answer which you believe that is correct. If you are looking for a formula to solve your basic math problems, your formula is likely here. Written in terms of Leibniz’s definition of the derivative: Leibniz has shown the inverse relationship between the differential and the area-function. Calculus: the Derivative Formula there is a maximum or a minimum according to whether the sign of the first non-zero derivative is negative or positive. Calculus is used in mechanical, physics etc. It just says that the rate of change of the area under the curve up to a point x, equals the height of the area at that point. The average velocity of an object is its change in position divided by the total amount of time taken. Any expression multiplied by 0 is equal to 0. Differential calculus makes it possible to compute the limits of a function in many cases when this is not feasible by the simplest limit theorems (cf. MATH 171 - Derivative Worksheet Diﬀerentiate these for fun, or practice, whichever you need. A derivative of a function is the inﬁnitesimal rate of change of the function with respect Matrix Algebra c 2007 James E. Therefore, calculus of multivariate functions begins by taking partial derivatives, in other words, finding a separate formula for each of the slopes associated with changes in one of the independent variables, one at a time. You will then be told whether the answer is correct or not. It builds on itself, so many proofs rely on results of other proofs - more specifically, complex proofs of derivatives rely on knowing basic derivatives. Derivatives Calculus Differential Calculus Algebra Ap Calculus Calculus Notes I Love Math Fun Math Math Math Math Tutor Find all the essential math formulas you need, all in one place. 5x 2 Answer: x Problem 6 y = 3x 2 + √ 7 x + 1 Answer: 6x + √ 7. Limits and Derivatives 2. It is a more complicated formula than the product rule, and most calculus textbooks and teachers would ask you to memorize it. Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals. Once you understand the concept of a partial derivative as the rate that something is changing, calculating partial derivatives usually isn't difficult. Limits Properties if lim ( ) x a Marh limits and derivatives formulas Keywords: Limits Derivatives Math Formulas Higher-order. Exponential Growth Models • continuously compounded interest: A = Pert • population growth: N(t)=N0ert t =time r = relative growth rate (a positive number) N0 = initial population N(t) = population after a time t has passed Example 1. It allowed derivatives to become commodities that. Derivative Formula Derivatives are a fundamental tool of calculus. Ability to take a photo of your math problem using the app. The derivative of sec x. Solution 3. $\begingroup$ That's not a derivative. In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. You know when you see the golden ratio pop up somewhere, things are getting more interesting. The expression for the derivative is the same as the expression that we started with; that is, e x! (d(e^x))/(dx)=e^x What does this mean? It means the slope is the same as the function value (the y-value) for all points on the graph. f(x) = x 1+x2. Includes full solutions and score reporting. Problem 5 y = 0. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Software for math teachers that creates exactly the worksheets you need in a matter of minutes. By plugging in different input values, x = a, the output values of f ‘(x) give you the slopes of the tangent lines at each point x = a. Derivative function in calculus shows us the rate of change for a point on a continuous line graph. Sometimes we can cleverly re-arrange the pattern to find a new insight. Basic Derivatives. For functions that act on the real numbers , it is the slope of the tangent line at a point on a graph. The derivative of a polynomial is the sum of the derivatives of its terms, and for a general term of a polynomial such as the derivative is given by One of the common applications of this is in the time derivatives leading to the constant acceleration motion equations. Powers of x General formula d/dx u^n =n u^(n-1) (du)/dx, where u is a function of x. In essence it says that a function f can be represented by a (Taylor polynomial) + (remainder after n + 1 terms of the series). Spreadsheet Calculus: Derivatives and Integrals: Calculus can be kind of tricky when you're first learning it. 08540v1 [math. ax n is a function consisting of a number (a) multiplied by x raised to a power, n. 3 The Sum Rule. It is a generalization of the. He was named Gabor Szego Professor of Mathematics in 1986 and was awarded a John Bascom Professorship in 1995. Write out formulas and other pieces of information about the problem. Hanford High School, Richland, Washington revised 8/25/08 1. d) figure out the derivative of the tangent line equation with the help of the derivative formulas, e) reach a conclusion on the results obtained in b) and d). Derivative, in mathematics, the rate of change of a function with respect to a variable. the points where f'(x) is zero or where f'(x) fails to exist 2. Includes derivatives for: trig functions, inverse trig functions, hyperbolic trig functions, hyperbolic inverse trig functions, power rule, product rule, quotient rule, chain rule, sum and difference rule, derivative of logarithms, derivative of natural logarithms. The derivative is way to define how an expressions output changes as the inputs change. Indefinite limits and expressions, evaluations of). Links to major mathematical topics. Applications of Differentiation. Explore the concepts, methods, and applications of differential and integral calculus. c = 0 x = 1 x n = n x (n-1) Proof. Then make Δx shrink towards zero. Get answers to all NCERT exercises, examples and miscellaneous questions of Chapter 13 Class 11 Limits and Derivatives free at teachoo. Derivatives tours are \$35 per person including admission to the Museum and are approximately 45 minutes in length. Derivative Formulas Lecture notes. Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion. Important Derivative & Integral Formulas Old Papers of Math Fsc Fedral board free download. Differential. Hanford High School, Richland, Washington revised 8/25/08 1. Therefore, calculus of multivariate functions begins by taking partial derivatives, in other words, finding a separate formula for each of the slopes associated with changes in one of the independent variables, one at a time. The displaymath environment is for formulas that appear on their own line. 08540v1 [math. Stay ahead with the world's most comprehensive technology and business learning platform. In the first section of the Limits chapter we saw that the computation of the slope of a tangent line, the instantaneous rate of change of a function, and the instantaneous velocity of an object at $$x = a$$ all required us to compute the following limit. A derivative of a function is the inﬁnitesimal rate of change of the function with respect Matrix Algebra c 2007 James E. By the definition of the reciprocal we have , throughout the domain of f. Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one. Derivatives. Let's use the view of derivatives as tangents to motivate a geometric. Use differentiation to determine the function that complements f. Some useful Trig formulas, derivatives and integrals. There are two ways of introducing this concept, the geometrical way (as the slope of a curve), and the physical way (as a rate of change). In other words, the tangent line to y = f(x) at point A(a, f(a)) is the line that passes through (a, f(a)) and whose slop is equal to the derivative of f at a. Use the derivative to find the ball's maximum height to the nearest foot. Free math lessons and math homework help from basic math to algebra, geometry and beyond. if n and a are positive real numbers, and a is not equal to 1, then If a x = n, then log a n = x. By plugging in different input values, x = a, the output values of f ‘(x) give you the slopes of the tangent lines at each point x = a. Important Derivative & Integral Formulas Old Papers of Math Fsc Fedral board free download. Department of Mathematics, Purdue University 150 N.	2019-11-12T01:06:41	{ "domain": "martin-vieth.de", "url": "http://thol.martin-vieth.de/math-city-derivative-formulas.html", "openwebmath_score": 0.8101862072944641, "openwebmath_perplexity": 514.0752542716059, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9905874101345135, "lm_q2_score": 0.8459424314825853, "lm_q1q2_score": 0.8379799223252272 }
https://math.stackexchange.com/questions/2299863/average-of-ratios-i-dont-get-it	# Average of ratios - I don't get it I'm calculating ratios from paired samples and there is a point I don't understand. Supposed that I measured the values of 2 paired samples: A and B, and then I calculate the ratios from those values. Ratio 1: $\frac{A}{B} = 0.5$ Ratio 2: $\frac{A}{B} = 2.0$ Normally one would calculate the average $ratio = \frac{(Ratio 1 + Ratio 2)}{2} = 1.25$. Then the conclusion would be: the value of A is 1.25 times higher than that of B. But, the ratios can be understood as: Ratio 1 = 0.5 --> the value of B is double the value of A Ratio 2 = 2 --> the value of A is double the value of B Then, the average ratio of A and B should be equal 1. Does that make sense to you? Where is the flaw? Thanks all, • There are different concepts of mean, and it is upto you to decide which is useful. Your idea is known as geometric mean – Guy May 28, 2017 at 9:47 • If you think the ratios $0.5$ and $2.0$ are in a sense equal and opposite, you might consider using geometric means or logarithms May 28, 2017 at 22:08 Your ratios imply $A=0.5B$ and $A=2.0B$. These multiplicative relationships are compatible with geometric sequences and the geometric mean which is $\sqrt{0.5 \times 2}=1$ The 'flaw' is using the arithmetic mean $\frac{0.5+2}{2}$	2022-05-24T00:13:38	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2299863/average-of-ratios-i-dont-get-it", "openwebmath_score": 0.8427185416221619, "openwebmath_perplexity": 205.75471549423608, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9835969679646669, "lm_q2_score": 0.851952809486198, "lm_q1q2_score": 0.8379782002596039 }
https://math.stackexchange.com/questions/781590/royden-4th-ed-chapter-2-proposition-8	Royden 4th Ed., Chapter 2, Proposition 8 Recall: $$m^(E)=\inf\left\{\sum_{k=1}^\infty\ell(I_k)\,:\,E\subseteq\bigcup_{k=1}^\infty I_k\right\}$$ where $$I_k = \text{ an open, bounded interval on }\mathbb{R}\,\,\text{ and }\,\,\ell(I_k)=\text{ the length of }I_k$$ Here is the text of the proof as published: $\textbf{Proposition 8}\,\,\textit{ Every interval is measurable.}$ $\textbf{Proof}\,\,$ As we observed above, the measurable sets are a $\sigma$-algebra. Therefore to show that every interval is measurable it suffices to show that every interval of the form $(a,\infty)$ is measurable (see Problem 11). Consider such an interval. Let $A$ be any set. We assume that $a$ does not belong to $A$. Otherwise, replace $A$ by $A\sim\{a\}$, leaving the outer measure unchanged. We must show that $$m^(A_1)+m^(A_2)\le m^(A)$$ where $$A_1=A\cap (-\infty,a)\,\,\text{and}\,\,A_2=A\cap (a,\infty).\qquad(5)$$ By the definition of $m^(A)$ as an infimum, to verify (5) it is necessary and sufficient to show that for any countable collection $\{I_k\}_{k=1}^\infty$ of open, bounded intervals that covers $A$, $$m^(A_1)+m^(A_2)\le \sum_{k=1}^\infty\ell(I_k).\qquad(6)$$ Indeed, for such a covering, for each index $k$, define $$I'_k=I_k\cap(-\infty,a)\,\,\text{and}\,\,I''_k=I_k\cap (a,\infty).$$ Then $I'_k$ and $I''_k$ are intervals and $$\ell(I_k)=\ell(I'_k)+\ell(I''_k).$$ Since $\{I'_k\}_{k=1}^\infty$ and $\{I''_k\}_{k=1}^\infty$ are countable collections of open, bounded intervals that cover $A_1$ and $A_2$, respectively, by the definition of outer measure, $$m^(A_1)\le\sum_{k=1}^\infty \ell(I'_k)\,\,\text{and}\,\,m^(A_2)\le\sum_{k=1}^\infty \ell(I''_k).$$ Therefore \begin{align} m^(A_1)+m^(A_2)&\le\sum_{k=1}^\infty \ell(I'_k) + \sum_{k=1}^\infty \ell(I''_k)\\ &=\sum_{k=1}^\infty [\ell(I'_k)+\ell(I''_k)]\\ &=\sum_{k=1}^\infty \ell(I_k). \end{align} Thus (6) holds and the proof is complete. Overall, this proof is very clear and makes sense. I am having a problem with one of the assumptions in the beginning of the proof: $E$ is a measurable set if for $every$ set $A$ we have $$m^(A)=m^(A\cap E)+m^(A\cap E^C),$$ but this proof seems to ignore all sets containing $\{a\}$. Is this a flaw in the proof? I am not even sure why ignoring $\{a\}$ is necessary. Can someone clarify this for me? • A single pointy has measure $0$, so no issues there. – voldemort May 5 '14 at 2:00 • I understand that the outer measure of $A$ and $A\sim\{a\}$ are the same. But why is it even necessary to remove $a$ in the first place? – Laars Helenius May 5 '14 at 2:01 • Open covers and stuff... just to ensure that you have an open interval. Makes the proof easier. – voldemort May 5 '14 at 2:02 Removing $a$ allows you to use $(-\infty,a)$ instead of $(-\infty,a]$ in $(5)$. This in turn allows the $I_k'$ to cover $A_1$. Removing $a$ is not an issue because, as the book says, $m^(A\cap(-\infty,a])=m^(A\cap(-\infty,a))$. • That is true. If $a\in A_1$, then none of $I'_k$ would be able to cover and hence $A_1\not\subseteq\bigcup_{k=1}^\infty I'_k$. So we could use intervals of the form $[a,\infty)$ and throw out the assumption, but it complicates the proof a bit, right? – Laars Helenius May 5 '14 at 2:22	2019-12-11T09:20:47	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/781590/royden-4th-ed-chapter-2-proposition-8", "openwebmath_score": 0.956129252910614, "openwebmath_perplexity": 140.4098227413605, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9835969641180276, "lm_q2_score": 0.8519528057272543, "lm_q1q2_score": 0.837978193285163 }
http://mathhelpforum.com/calculus/40133-solved-simple-trig-integration.html	# Thread: [SOLVED] Simple trig integration 1. ## [SOLVED] Simple trig integration Can you guys help. I haven't done this stuff in a while. How would you integrate: (sin x)^2 Thanks if you can shed light. 2. Originally Posted by Evales Can you guys help. I haven't done this stuff in a while. How would you integrate: (sin x)^2 Thanks if you can shed light. Recall the double angle formula $\cos (2x) = 1 - 2 \sin^2 x$ ..... 3. Originally Posted by Evales Can you guys help. I haven't done this stuff in a while. How would you integrate: (sin x)^2 Thanks if you can shed light. note that $\sin^2 x = \frac {1 - \cos 2x}2$. integrate that. it is easier EDIT: Thanks a lot, Mr Fantastic! 4. Originally Posted by Jhevon note that $\sin^2 x = \frac {1 - \cos 2x}2$. intgrate that. it is easier What a dynamic duo ..... Note how posts #2 and #3 just seamlessly flow 5. Originally Posted by mr fantastic What a dynamic duo ..... Note how posts #2 and #3 just seamlessly flow Yup, can't help it, being around the Flow Master and all 6. Never heard of that particular identity before. I assume that in order to do this I shouldn't use identities that aren't on the syllabus. Anyway. Getting past that. How would I do it? Should I use substitution or the chain rule ect. 7. Originally Posted by Evales Never heard of that particular identity before. I assume that in order to do this I shouldn't use identities that aren't on the syllabus. Anyway. Getting past that. How would I do it? Should I use substitution or the chain rule ect. $\cos(2x)=\cos^2(x)-\sin^2(x)$ from that we can replace $\cos^2(x)=1-\sin^2(x)$ to get $\cos(2x)=1-2\sin^2(x)$ From there you can find what they said so $\int\sin^2(x)dx=\int\frac{1-\cos(2x)}{2}dx=\frac{x}{2}-\frac{1}{2}\int\cos(2x)dx$ for the last one isolate the derivative of the quanity and be done with it =D 8. Okay thanks guys. Just wondering where how the function comes to be divided by two. I've gotten to (sin x)^2 = 1 - cos(2x) 9. Originally Posted by Evales Okay thanks guys. Just wondering where how the function comes to be divided by two. I've gotten to (sin x)^2 = 1 - cos(2x) $\cos(2x)=1-2\sin^2(x)\Rightarrow{2\sin^2(x)=1-\cos(2x)}\Rightarrow{\sin^2(x)=\frac{1-\cos(2x)}{2}}$ 10. Oh lol Duh. Thanks for holding my hand through this process! 11. $\int{\sin ^{2}x\,dx}=\int{\sin x(-\cos x)'\,dx}=-\sin x\cos x+\int{\cos ^{2}x\,dx}.$ Now use $\cos^2x=1-\sin^2x$ and the conclusion follows.	2017-10-23T10:41:46	{ "domain": "mathhelpforum.com", "url": "http://mathhelpforum.com/calculus/40133-solved-simple-trig-integration.html", "openwebmath_score": 0.8526102900505066, "openwebmath_perplexity": 2324.726864063106, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.983596970368816, "lm_q2_score": 0.8519528000888387, "lm_q1q2_score": 0.8379781930646113 }
https://math.stackexchange.com/questions/483339/proof-of-convexity-of-linear-least-squares	# Proof of convexity of linear least squares It's well known that linear least squares problems are convex optimization problems. Although this fact is stated in many texts explaining linear least squares I could not find any proof of it. That is, a proof showing that the optimization objective in linear least squares is convex. Any idea how can it be proved? Or any pointers that I can look at? Thanks • – Avraham Sep 3 '13 at 20:57 • All norms are convex, and the function $\phi(t)= 1_{(-\infty,0)} (t) t^2$ is convex and non-decreasing, hence $x \mapsto \\|x\\|^2$ is also convex (see Rockafellar's "Convex Analysis" Theorem 5.1) . Finally, a convex function composed with a linear map is convex. – copper.hat Sep 9 '13 at 1:49 • Another quick proof is to note that the gradient of $f(x) = (1/2)\\| Ax-b\\|^2$ is $A^T(Ax-b)$ and the Hessian is $A^TA$, which is positive semidefinite. It follows that $f$ is convex. – littleO Dec 18 '19 at 1:43 • I think these answers are a little complicated. Note that $x \mapsto x^2$ is convex for any $x \in \mathbb{R}$. The sum of convex functions is convex, and affine precomposition is convex. The latter two statements are one-line proofs, so you're done since $\\|Ax - b\\|_2^2 = \sum_i (a_i^Tx - b)^2$. – Guillermo Angeris Dec 18 '19 at 17:16 ## 5 Answers Another way to prove that a function is convex is by showing that the second order derivative (if it exists) is positive semi-definite. $$\phi: \beta \mapsto \Vert y - X \beta \Vert^2 = \Vert y \Vert^2 - 2 y^T X \beta + \Vert X \beta \Vert^2$$ $\phi$ is twice differentiable and the second derivative (i.e. the Hessian) is $$\dfrac {\partial \phi} {\partial \beta} = - 2y^TX + 2(X\beta)^TX =- 2y^TX + 2\beta^TX^TX$$ $$\dfrac {\partial^2 \phi} {\partial \beta \partial \beta^T} = 2X^TX$$ which is a positive semi-definite matrix. Therefore, $\phi$ is a convex function. • How were you able to differentiate ϕ twice? – David May 4 '17 at 4:02 • Also, is the second derivative the Hessian? – David May 4 '17 at 6:18 • The first derivative is incorrect. You're adding a row vector to a column vector. – Rodrigo de Azevedo Sep 28 '17 at 9:16 • can anyone explain me how do we reach the first derivative? – Bagus Trihatmaja Jan 12 '18 at 6:30 • @BagusTrihatmaja I corrected the first derivative in my edit. It depends upon the fact that the derivative of $\Vert \mathbf{x}\Vert^2$ with respect to $\mathbf{x}$ is $2\mathbf{x}^T$. Then you can make use of the chain rule. – Christian Sykes Jan 12 '18 at 11:07 You want a proof that the function $$\phi: \beta \mapsto \Vert y - X \beta \Vert^2 = \Vert y \Vert^2 - 2 y^T X \beta + \beta^T X^T X \beta$$ is convex, right? (here $\beta$ and $y$ are vectors and $X$ is a matrix). In other words, you need to prove that $$\phi(t \beta_1 + (1-t) \beta_2) - \left[ t \phi( \beta_1) + (1-t) \phi(\beta_2) \right] \leq 0$$ for all $\beta_1, \beta_2$ and $t \in [0,1]$. After calculation, the left-hand term becomes $$t^2 \beta_1^T X^T X \beta_1 + (1-t)^2 \beta_2^T X^T X \beta_2 + 2 t(1-t) \beta_1^T X^T X \beta_2 - t \beta_1^T X^T X \beta_1 - (1-t) \beta_2^T X^T X \beta_2$$ $$= - t(1-t) \left[ (\beta_1 - \beta_2)^T X^T X (\beta_1 - \beta_2) \right] = - t(1-t) \Vert X (\beta_1 - \beta_2) \Vert^2$$ which is clearly $\leq 0$. • Can you expand on "after calculation"? – vega Dec 7 '17 at 15:01 • @vega You can use the fact that $\Vert \mathbf{x}\Vert^2 = \mathbf{x}^T\mathbf{x}$. – Christian Sykes Jan 12 '18 at 11:10 Here's an alternative way to see the convexity of $$f(\beta)\triangleq \Vert y-X\beta\Vert^2$$. For $$t\in [0,1]$$, let $$\bar t=1-t$$. Then $$f(t\alpha + \bar t \beta)=\Vert t(y-X\alpha)+\bar t (y-X\beta)\Vert^2 \le \left(t\Vert y-X\alpha\Vert +\bar t\Vert y-X\beta\Vert\right)^2\le t\Vert y-X\alpha\Vert^2+\bar t \Vert y-X\beta\Vert^2=tf(\alpha)+\bar tf(\beta)$$ where the first inequality is due to triangle inequality of vector norm, and the 2nd inequality follows because the square function ($$x^2$$) is also convex. • I just noted the copper.hat's comment above. This answer in fact just elaborated his comment. – syeh_106 Nov 26 '19 at 6:55 I decided to make my comment an answer since I think that a few of these are a little complicated due to the careful bookkeeping for derivatives. We say a function $$f$$ is convex if for any $$0 \le \gamma \le 1$$ and $$x, y \in \mathbb{R}^n$$ we have that $$f(\gamma x + (1-\gamma)y) \le \gamma f(x) + (1-\gamma)f(y).$$ We can now prove it in three (simple!) steps, which I will prove here for the sake of convenience, but these are extraordinarily standard one-line-proofs. 1. First, it's clear that the function $$f(x) = x^2$$ is convex (prove this in your favorite way). 2. Now, for any two convex functions $$f, g: \mathbb{R}^n \to \mathbb{R}$$, their sum is convex since, simply applying the definitions, $$f(\gamma x + (1-\gamma)y) + g(\gamma x + (1-\gamma)y) \le \gamma (f(x) + g(x)) + (1-\gamma) (f(y) + g(y)).$$ 3. Affine precomposition preserves convexity, since if $$f$$ is convex, then $$f(A(\gamma x + (1-\gamma)y) + b) =f(\gamma Ax + (1-\gamma)Ay + (\gamma + (1-\gamma)) b) \le \gamma f(Ax + b) + (1-\gamma)f(Ay+b),$$ so, since the function $$f(x) = \\|Ax - b\\|_2^2 = \sum_i (a_i^Tx - b_i)^2,$$ is the sum of a bunch of convex functions precomposed with affine functions, then $$f$$ is convex. I just want to elucidate a bit on proofs provided by @roger and @acharuva. Two popular ways to prove that a function $$f$$ is convex are to prove that: $$$$f(tx_1+(1-t)x_2)-tf(x_1)-(1-t)f(x_2) \leq 0$$$$ for $$t \in [0,1]$$ and to prove that the second derivative is non-negative for the entire support (domain of $$f$$). The first is the definition of convexity and the second is a well-known theorem (Proof here http://www.princeton.edu/~aaa/Public/Teaching/ORF523/S16/ORF523_S16_Lec7_gh.pdf) . For multivariate functions this means proving that the Hessian is PSD (A matrix A is PSD if A is symmetric and $$u^TAu\geq0$$ irrespective of $$u$$). In case of OLS or the linear regression cost function we have: $$$$J(\theta) = \frac 1 2 {(X \theta -Y)}^2 = \frac12( \theta^TX^TX\theta -2Y^TX\theta + Y^TY)$$$$ $$$$\frac {\partial J(\theta)} {\partial \theta} = X^T(X\theta-Y)$$$$ [because $$\nabla_Atr(ABA^TC)=C^TAB^T+CAB$$; here A=$$\theta^T$$, B=$$X^TX$$, C=$$I$$] $$$$\frac {\partial^2 J(\theta)} {\partial \theta^2} = X^TX$$$$ $$X^TX$$ is a PSD matrix ($$u^TX^TXu=\Vert Xu\Vert^2$$) and hence the cost function is convex with respect to $$\theta$$. That's the first proof. The 2nd proof goes like this. It requires us to prove that $$$$J(t\theta_1+(1-t)\theta_2)-tJ(\theta_1)-(1-t)J(\theta_2)\leq0$$$$ $$\forall \theta_1, \theta_2$$ and $$\forall t \in [0,1]$$. We will call this $$E_1$$. Note that here $$$$J(\theta) = \frac12( \theta^TX^TX\theta -2Y^TX\theta + Y^TY)$$$$ The full expansion of $$E_1$$ will be slightly verbose but we can make our life easy by considering that for any linear function $$f(x)=ax+b$$, $$$$f(tx_1+(1-t)x_2)=tf(x_1)+(1-t)f(x_2)$$$$ You can convince yourself of this by expanding both sides. This means the term $$\frac12( -2Y^TX\theta + Y^TY)$$ of $$J(\theta)$$ will be $$0$$ in $$E_1$$. Let us now look at $$E_1$$ with only the first term of $$J(\theta)$$ and drop the $$\frac12$$ for convenience. $$$$LHS = {(t\theta_1+(1-t)\theta_2)}^TX^TX(t\theta_1+(1-t)\theta_2)-t\theta_1^TX^TX\theta_1-(1-t)\theta_2^TX^TX\theta_2$$$$ $$$$LHS = t^2\theta_1^TX^TX\theta_1+(1-t)^2\theta_2^TX^TX\theta_2+2t(1-t)\theta_1^TX^TX\theta_2 -t\theta_1^TX^TX\theta_1-(1-t)\theta_2^TX^TX\theta_2$$$$ [Note that $$\theta_1^TX^TX\theta_2$$=$$(X\theta_1)^T(X\theta_2)$$=$$(X\theta_2)^T(X\theta_1)$$=$$\theta_2^TX^TX\theta_1$$] $$$$LHS = (t^2-t)\theta_1^TX^TX\theta_1+((1-t)^2-(1-t))\theta_2^TX^TX\theta_2+2t(1-t)\theta_1^TX^TX\theta_2$$$$ $$$$LHS = -t(1-t)[\theta_1^TX^TX\theta_1+\theta_2^TX^TX\theta_2-2\theta_1^TX^TX\theta_2]$$$$ [Note that $$(1-t)^2-(1-t) = t^2-t=-t(1-t)$$] $$$$LHS = -t(1-t)[(\theta_1-\theta_2)^TX^TX(\theta_1-\theta_2)]$$$$ $$$$LHS = -t(1-t) \Vert X(\theta_1-\theta_2) \Vert^2$$$$ which is always $$\leq0$$. This completes our second proof. If you are having trouble following the second proof, try to first prove that a scalar quadratic function $$f(x)=ax^2+bx+c$$ is convex for $$a>0$$. I promise that exercise will be helpful (since our function for $$J(\theta)$$ is also quadratic in $$\theta$$).	2021-02-27T05:00:17	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/483339/proof-of-convexity-of-linear-least-squares", "openwebmath_score": 0.9556255340576172, "openwebmath_perplexity": 294.578563966356, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9835969655605175, "lm_q2_score": 0.8519528038477825, "lm_q1q2_score": 0.8379781926654536 }
http://math.stackexchange.com/questions/157439/measurable-function-on-the-interval-0-1	# Measurable function on the interval $[0,1]$ Assume that $f$ is a measurable function on the interval $[0,1]$ such that $0<f(x)<\infty$ for $x \in [0,1]$. Then, how can I prove the inequality below? $$\int^1_0 f(x)dx \int^1_0 {1 \over {f(x)}} dx \ge 1$$ - I think your title should be more telling: something like "Integral inequality for measurable function on $[0,1]$". – Quinn Culver Jun 12 '12 at 17:54 ## 4 Answers Here is another way. What you have written is nothing but the Arithmetic mean-Harmonic mean inequality for functions. In case, of a finite set of positive numbers $\{a_i\}$'s, the AM-HM inequality reads $$\dfrac{\sum_{i=1}^{n} w_i a_i}{\sum_{i=1}^{n} w_i} \geq \dfrac{\sum_{i=1}^{n} w_i}{\sum_{i=1}^{n} \dfrac{w_i}{a_i}}$$where $w_i \geq 0$ are the weights. The weights can be normalized to $1$ i.e. if we enforce that $\displaystyle \sum_{i=1}^{n} w_i = 1$, then we get that $$\sum_{i=1}^{n} w_i a_i \geq \dfrac1{\displaystyle \sum_{i=1}^{n} \dfrac{w_i}{a_i}}$$ Hence, we get that $$\left(\sum_{i=1}^{n} w_i a_i \right) \left(\sum_{i=1}^{n} \dfrac{w_i}{a_i} \right)\geq 1$$ If you partition the interval $[0,1]$ into $n$ disjoint measurable sets, say $E_1^{(n)},E_2^{(n)},\ldots,E_n^{(n)}$, such that $\displaystyle \bigcup_{k=1}^{n} E_k^{(n)} = [0,1]$, choose $w_i^{(n)} = \mu^{(n)} \left(E_i^{(n)}\right)$. Note that $\sum_i w_i^{(n)} = 1$. Approximate $f(x)$ from below using step functions on these intervals i.e. $f_{step}^{(n)} = f_i^{(n)}$ on the interval $E_i^{(n)}$ such that $\displaystyle \sum_{i=1}^{n} f_i^{(n)} \mu(E_i^{(n)}) = \int f_{step}^{(n)} \to \int f$. Since $f>0$ on $[0,1]$, $\dfrac1f$ can be approximated from above using the step function $(1/f)_{step}^{(n)} = 1/f_i^{(n)}$ on the interval $E_i^{(n)}$ such that $\displaystyle \sum_{i=1}^{n} \dfrac1{f_i^{(n)}} \mu(E_i^{(n)}) = \int \dfrac1{f_{step}^{(n)}} \to \int \dfrac1{f}$. From the AM-HM inequality we have that $\left(\displaystyle \sum_{i=1}^{n} \mu(E_i^{(n)}) f_i^{(n)} \right) \left( \displaystyle \sum_{i=1}^{n} \dfrac{\mu(E_i^{(n)}) }{f_i^{(n)}} \right) \geq 1$. Take the limit to conclude that $$\left(\int f dx \right) \times \left(\int \dfrac1f dx \right)\geq 1$$ You may need to argue out and justify some parts of the above argument to make it into a rigorous proof but hope the idea is clear. - It's actually a very rigorous but very nice example for my learning. I'll argue that one with my mates. – Collins Jun 12 '12 at 17:26 @Collins Thanks. – user17762 Jun 12 '12 at 17:27 We have by Cauchy-Schwarz inequality, $$1=\int_0^1 1dx=\int_0^1\sqrt{f(x)}\frac 1{\sqrt{f(x)}}dx\leq \left(\int_0^1f(x)dx\cdot\int_0^1\frac 1{f(x)}dx\right)^{1/2}.$$ (the result is clear if $f^{1/2}$ or $f^{-1/2}$ is not integrable. - It's very simple! But I didn't know that theorm yet, Is there any condition for using Cauchy inequality? – Collins Jun 12 '12 at 16:50 @Collins As long as you know that $L^2$ is an inner product space. You've probably seen a proof of Cauchy-Schwarz before, at least in a special case, and the proof for a general inner product is no harder. Maybe you've shown the Hölder inequality? That's even more general. – Dylan Moreland Jun 12 '12 at 16:54 Oh, I got it. I did not have enough understanding about Hölder inequality and norm space. – Collins Jun 12 '12 at 17:09 If you want to be very picky, you should also note that technically Cauchy-Schwartz only applies if $f^{1/2}, f^{-1/2} \in L^2(0,1)$, i.e., $f, f^{-1} \in L^1(0,1)$. There should be a special case in which $\int_0^1 \! f(x) \, dx = \infty$ or $\int_0^1 \! f(x) \, dx = \infty$ in which case the inequality is obviously true since both integrals are positive anyway. – user12014 Jun 12 '12 at 17:38 $$\iint_{[0,1]^2}\frac{(f(x)-f(y))^2}{f(x)f(y)}\mathrm dx\mathrm dy\geqslant0$$ Edit: This is an adaptation of the well-known approach to the inequality $\\|f\\|_1\leqslant\\|f\\|_2$ for every probability measure $\mu$ as the expansion of $$\iint(f(x)-f(y))^2\mathrm d\mu(x)\mathrm d\mu(y)\geqslant0$$ and gives an opportunity to recommend once more the marvelous little book The Cauchy-Schwarz Master Class: An Introduction to the Art of Inequalities by J. Michael Steele. - That's a nice solution. How did you come up with it? – Quinn Culver Jun 13 '12 at 3:51 @Quinn See Edit. – Did Jun 13 '12 at 6:51 As @PZZ commented below @David Giraudo's answer, we may assume $f$ is integrable. Then Jensen's inequality, with $\varphi(x)=1/x$ says that $$\int_{0}^{1} \frac{1}{f(x)}\, dx = \int_{0}^{1}\varphi(f(x))\,dx \geq \varphi\left[\int_{0}^{1} f(x)\, dx\right] = \left[\int_{0}^{1}f(x)\,dx\right]^{-1}.$$ -	2015-12-02T03:53:52	{ "domain": "stackexchange.com", "url": "http://math.stackexchange.com/questions/157439/measurable-function-on-the-interval-0-1", "openwebmath_score": 0.9383748173713684, "openwebmath_perplexity": 221.55418870004198, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9835969641180276, "lm_q2_score": 0.8519528038477824, "lm_q1q2_score": 0.8379781914365202 }
https://math.stackexchange.com/questions/1692433/formula-for-the-greatest-common-divisor-of-two-odd-numbers	# Formula for the greatest common divisor of two odd numbers I was taking the determinant of the adjacency matrix for a graph and I stumbled across this interesting (apparent) coincidence: If $m$ and $n$ are odd numbers, then $$2^{\textsf{gcd}(m,n)} = \prod_{k=1}^{m} \prod_{l=1}^{n} \left( e^{\frac{2k\pi i}{m}} + e^{\frac{2 l \pi i}{n}} \right).$$ If we take $\log_2$ of both sides, we get an explicit expression describing the greatest common divisor. Would anyone like to attempt a proof of this fact? Here is some MATLAB code which checks the above formula. (Save it as a .m file in your MATLAB directory; then use greatcd(m,n) to output the greatest common divisor of your chosen odd numbers $m$ and $n$.) function [I,J] = greatcd(m,n) for k=1:m for l=1:n H(k,l) = exp((2piik)/m) + exp((2piil)/n); end end G=prod(H(:)); log2(real(G)) It appears that if $g = \textsf{gcd}(m,n)$, then $$\prod_{k=1}^{m} \prod_{l=1}^{n} \left( e^{\frac{2k\pi i}{m}} + e^{\frac{2 l \pi i}{n}} \right) = \prod_{k=1}^{g} \prod_{l=1}^{ng^{-1}} \left( 1 + e^{\frac{2l\pi i}{ng^{-1}}}\right)$$ and I also think that $$\prod_{l=1}^{ng^{-1}} \left( 1 + e^{\frac{2l\pi i}{ng^{-1}}}\right) = 2.$$ If I hold $l$ fixed and take the product over $k$, I appear to get $$\prod_{k=1}^{m} \left( e^{\frac{2k\pi i}{m}} + e^{\frac{2 l \pi i}{n}} \right) = 1 + e^{\frac{2l\pi i}{ng^{-1}}}.$$ • Did you computer-verify it for the first few $m, n$? Mar 11, 2016 at 2:59 • Using Maple, yes Mar 11, 2016 at 3:39 • Do you understand my proof below that your formula is at least a real number? Mar 11, 2016 at 5:25 • I'll have to work out the details myself tomorrow, but it seems to be correct. Thanks for your help! Mar 11, 2016 at 5:31 • No problemo! Thanks for the mathematical observation! Mar 11, 2016 at 5:32 Here is an "asymmetric" proof. The idea is to introduce an indeterminate $X$ and regard the product as some polynomial evaluated at $X = 1$. The key lemma is that if $m$ is odd and $\zeta$ is an $m$th root of unity then the roots of $p_m(X) := X^m + 1$ are the negatives of the roots of $X^m - 1$, because if $z^m = 1$ then $(-z)^m = (-1)^m z^m = -1$. Therefore, $$p_m(X) = \prod_{k=1}^m (X + \zeta^k).$$ We will use this identity below for $n$ and also for $m/g$, where $g = \text{gcd}(m, n)$. First, let $\zeta$ and $\eta$ be primitive $m$ and $n$th roots of unity, respectively (e.g. $\zeta = e^{2\pi i / m}$). Introduce $X$, and simplify using the lemma: $$p(X) = \prod_{k=1}^m \prod_{l=1}^n (\zeta^k X + \eta^l) = \prod_{k=1}^m p_n(\zeta^k X)$$ so $$\prod_{k=1}^m \prod_{l=1}^n (e^{2\pi i k/m} + e^{2\pi i l/n}) = p(1) = \prod_{k=1}^m p_n(\zeta^k) = (1+\zeta^n)(1+\zeta^{2n})(1+\zeta^{3n})\dotsb(1+\zeta^{mn}).$$ Now because $m$ and $n$ might not be coprime, this product may have repeating terms. But the repetition is very regular. The exponent land is $\mathbb{Z}/m\mathbb{Z}$ (remember, $\zeta$ is an $m$th root of unity) and the map $x \mapsto nx$ has kernel generated by (the equivalence class mod $m$ of) $m/g$. To see this, get $s$ and $t$ such that $ms + nt = g$. If $nx \equiv 0$ (mod $m$) then $ntx = (g - ms)x = gx \equiv 0$ (mod $m$), so $m$ divides $gx$. Since $g$ divides $m$, $m/g$ divides $x$. Therefore, the whole product is just $g$ copies of the first $m/g$ terms: $$p(1) = [(1+\zeta^n)(1+\zeta^{2n})\dotsb(1+\zeta^{({m/g})n})]^g.$$ There are $m/g$ of these $\zeta$s and they are all $(m/g)$th roots of unity, because $(\zeta^{kn})^{m/g} = (\zeta^m)^{kn/g} = 1$. By the lemma again, $$p(1) = p_{m/g}(1)^g = (1+1)^g$$ and the result follows. Let $n = 1$ for simplicity. $\prod\limits_{k,l = 0}^{m-1, n-1} (\exp(\dfrac{2k\pi i}{m}) + \exp(\dfrac{2l\pi i}{n})) = \prod_{k=0}^{m-1} (\exp(\dfrac{2k\pi i}{m}) + 1) = \\\prod\limits_{k=0}^{\dfrac{m-1}{2}}(w^k + 1)\prod\limits_{k=0}^{\dfrac{m-1}{2}}(w^{-k} + 1)$ by symmetry of an odd-sided regular polygon centered at the origin of $\Bbb{C}$ whose vertices are the $m$th roots of unity. But the rules of complex conjugation yield that $(w^k + 1)^ = (w^{k} + 1) = w^{-k} + 1$. Thus we have $zz^$ above and that's always real. Thus for the case $n=1, m\geq 1$ we have proved that the formula is at least a real number. A similar argument using symmetries of odd-sided regular polygons can be applied to the general case where $n \geq 1, m \geq 1$. Showing that your formula is at least a real number. Find, that using the same symmetry across the real axis in $\Bbb{C}$ used above that, your general formula is the same as: $$\prod_{k,l=0}^{\dfrac{m-1}{2}, \dfrac{n-1}{2}}(w^{k} + z^l)(w^{-k}+ z^{-l})(w^{k} + z^{-l})(w^{-k} + z^{l})$$ Note that $0 \leq \|w^{\pm k} + z^{\pm l}\| \leq 2$. Which can also be seen by looking at regular unit polygons in the $\Bbb{C}$ plane. As your formula is a real number, we have that $\|\text{formula}\| = \pm\text{formula}$. For all the times in the product when $\dfrac{k}{m} = \dfrac{l}{n}$ you'll get $\|w^{k} + z^{l}\| = \|w^{-k} + z^{-l}\| = 2$. Convince yourself this happens iff those two fractions are equal in the product. We're gettin' close! $\dfrac{k}{m} = \dfrac{l}{n}, \ k,l = 0, \dots, m-1, n-1 \iff ?$ I don't really know, so let's look at an example: $m = 15, n = 9, (m,n) = 3$ $0/15 = 0/9, \ 5/15 = 3/9, \ 10/15 = 6/9$ And those were the only examples I could think of, so it seems like you want to prove that the above happens exactly $\gcd(m,n)$ times in general. So we have shown so far that $\text{your formula} = \pm 2^{\gcd(m,n)}\prod\limits_{k,l = 0; k/m \neq l/n}^{m-1, n-1} (\dots)$. We haven't employed rotational symmetry yet, so maybe the answer lies there to prove that the magnitude of the remaining piece of your formula $=1$. And by that I mean multiplying by the terms each by the same $e^{i\pi j/?}$ such that the resulting set of vertices is the same as your original two polygons, also known as a rotational symmetry in a Dihedral group. • I really like your observation that $\textsf{gcd}(m,n) = \# \{(k,l) \Bbb : 1 \leq k \leq m, 1 \leq l \leq n, kn = ml\}.$ Mar 11, 2016 at 14:51 • It seems that if $k$ is the smallest number satisfying the equation $k/m = l/n$, then the greatest common divisor is $m/k$. Mar 11, 2016 at 15:34 • @OpenSeason Thanks! Have you figured out how to prove the rest of the formula equals 1? Mar 11, 2016 at 15:42	2022-05-21T15:45:21	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/1692433/formula-for-the-greatest-common-divisor-of-two-odd-numbers", "openwebmath_score": 0.9344690442085266, "openwebmath_perplexity": 168.72376773558386, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9835969679646668, "lm_q2_score": 0.8519528000888386, "lm_q1q2_score": 0.8379781910163895 }
https://mathhelpboards.com/threads/ask-probability-of-that-goalkeeper-being-able-to-fend-off-those-penalty-kicks-3-times.23838/	# [ASK] probability of that goalkeeper being able to fend off those penalty kicks 3 times #### Monoxdifly ##### Well-known member A professional goalkeeper can fend off a penalty kick with the probability of $$\displaystyle \frac{3}{5}$$. In an event a kick is done 5 times. The probability of that goalkeeper being able to fend off those penalty kicks 3 times is .... A. $$\displaystyle \frac{180}{625}$$ B. $$\displaystyle \frac{612}{625}$$ C. $$\displaystyle \frac{216}{625}$$ D. $$\displaystyle \frac{228}{625}$$ E. $$\displaystyle \frac{230}{625}$$ Can someone give me a hint? #### tkhunny ##### Well-known member MHB Math Helper A professional goalkeeper can fend off a penalty kick with the probability of $$\displaystyle \frac{3}{5}$$. In an event a kick is done 5 times. The probability of that goalkeeper being able to fend off those penalty kicks 3 times is .... A. $$\displaystyle \frac{180}{625}$$ B. $$\displaystyle \frac{612}{625}$$ C. $$\displaystyle \frac{216}{625}$$ D. $$\displaystyle \frac{228}{625}$$ E. $$\displaystyle \frac{230}{625}$$ Can someone give me a hint? Sometimes, if the distribution is small enough, it pays to calculate the entire Binomial Distribution and answer what may be a whole collection of questions. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 0 Blocks: 1 (3/5)^0 (2/5)^5 1 Block: 5 (3/5)^1 (2/5)^4 2 Blocks:10 (3/5)^2 (2/5)^3 3 Blocks: 10 (3/5)^3 (2/5)^2 4 Blocks: 5 (3/5)^4 (2/5)^1 5 Blocks: 1 (3/5)^5 (2/5)^0 It's a pretty simple pattern to follow. Of course, if you wish, you can just calculate the desired value directly. #### Monoxdifly ##### Well-known member Ah, I think I get it. Thanks.	2020-07-11T12:30:07	{ "domain": "mathhelpboards.com", "url": "https://mathhelpboards.com/threads/ask-probability-of-that-goalkeeper-being-able-to-fend-off-those-penalty-kicks-3-times.23838/", "openwebmath_score": 0.45590972900390625, "openwebmath_perplexity": 2430.7912122888088, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.983596967003007, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.8379781883484579 }
https://almodarresi.com/rene-naufahu-ddf/kernel-density-estimation-python-4bbe13	As I mentioned before, the default kernel for this package is the Normal (or Gaussian) probability density function (pdf): $$K(x) = \frac{1}{\sqrt{2\pi}}\text{exp}\left(-\frac{1}{2}x^2\right)$$ … So in summary it is just like a histogram but having a smooth curve drawn through the top of each bin. Kernel density estimation via diffusion in 1d and 2d. play_arrow . Podcast 291: Why developers are demanding more ethics in tech. The estimation works best for a unimodal distribution; bimodal or multi … Hi everyone, There are several libraries that allow us to estimate a probability density function using Kerndel Density Estimation. Sometimes, we are interested in calculating a smoother estimate, … Kernel Density Estimation in Python. KDEpy About. The reference implementation for 1d and 2d, in Matlab, was provided by the paper's first author, Zdravko Botev.This is a re-implementation in Python, with … Kernel Density Estimation is a method to estimate the frequency of a given value given a random sample. Kernel Density Estimation(KDE) is a non-parametric way to find the Probability Density Function(PDF) of a given data. It is also referred to by its traditional name, the Parzen-Rosenblatt Window method, after its discoverers. If a 2-D array, should be of shape (num_observations, num_variables). Active 2 years, 5 months ago. Several shapes of distributions exist out … u : unordered (discrete) o : ordered (discrete) The string should contain a type specifier for each variable, so for example var_type='ccuo'. in C# and F#, Math.NET Numerics is an open source library for numerical computation which includes kernel density estimation; In CrimeStat, kernel density estimation is implemented using five different kernel functions – normal, uniform, quartic, negative exponential, and triangular. Stack Exchange Network . How can I therefore: train/fit a Kernel Density Estimation (KDE) on the bimodal distribution and then, given any other distribution (say a uniform or normal distribution) be able to use the trained KDE to 'predict' how many of the data points from the given data distribution belong to the target bimodal distribution. Representation of a kernel-density estimate using Gaussian kernels. Both single- and dual-kernel density estimate routines are available. Categories . Parameters ----- dataset : … It includes automatic bandwidth determination. Three algorithms are implemented through the same API: NaiveKDE, TreeKDE and FFTKDE.The class FFTKDE outperforms other popular implementations, see the comparison page. The kernel density plot provides vital display of information on data which include: How the data is distributed around the measures of central tendency like mean and median; How the distribution is skewed; How the distribution is peaked; For a distribution present in a pandas Series, the kernel density estimation plot … Sticking with the Pandas library, you can create and overlay density plots using plot.kde(), which is available for both Series and DataFrame objects. Kernel Density Estimation, also known as KDE is a method in which the probability density function of a continuous random variable can be estimated. Kernel Density Estimation Using Python: … In this case, a kernel is a mathematical function that returns a probability for a given value of a random variable. Kernel density … In statistics, kernel density estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable. This Python 3.6+ package implements various kernel density estimators (KDE). The type of the variables: c : continuous. Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. For the uniform and each of the remaining kernel estimates, we require a function which can combine the kernel … Kernel Density Estimation – The first step when applying mean shift clustering algorithms is representing your data in a mathematical manner this means representing your data as points such as the set below. Python; Tags . The script below in the section 3 allows you to generate a set of coordinates for the points of the contours based on tour data. jpython Unladen Swallow. While using ‘jointplot’, if the argument ‘kind’ is set to ‘kde’, it plots the kernel density estimation plot. I highly recommend it because you can play with bandwidth, select different kernel methods, and check out the resulting effects. gaussian_kde works for both uni-variate and multi-variate data. Also, how to show the values of the density on the … Kernel Density Estimators. Setting the hist flag to False in distplot will yield the kernel density estimation plot. Featured on Meta “Question closed” notifications experiment results and graduation. I've made some attempts in this direction before (both in the scikit-learn documentation and in our upcoming textbook), but Michael's use of interactive javascript widgets makes the relationship extremely intuitive. var_type str. My question is how I can see the estimated function, not as a plot but as a … Last week Michael Lerner posted a nice explanation of the relationship between histograms and kernel density estimation (KDE). I would like to plot a 2D kernel density estimation. The kernel density estimation plot draws the probability density for a given distribution. The code is stable and in widespread by practitioners and in other packages. Kernel density estimation (KDE) is a nonparametric method for estimating the probability density function of a given random variable. KDE or the Kernel Density Estimation uses Gaussian Kernels to estimate the Probability Density Function of a random variable. I had … Kernel Density Estimation (KDE) is a way to estimate the probability density function of a continuous random variable. The method getInflexion points can … I am using Python 3.8 and sklearn 0.22. The scrips shows the contour plot, prints the contour coordinates and saves the coordinates in a .csv file. This video gives a brief, graphical introduction to kernel density estimation. Let us understand how the ‘jointplot’ function works to plot a kernel density … I know, in theory, that the CDF can be . Kernel density estimation is calculated by averaging out the points for all given areas on a plot so that instead of having individual plot points, we have a smooth curve. Python Tutorials → In-depth articles ... A kernel density estimation (KDE) is a way to estimate the probability density function (PDF) of the random variable that underlies our sample. Published by Amir Masoud Sefidian at June 14, 2017. In this article, we show how to create a kernel density estimation (KDE) plot in seaborn with Python. This article is an introduction to estimating kernel density using the Python machine learning library scikit-learn. Kernel Density Estimation in Python. We assume the observations are a random sampling of a probability distribution $$f$$. The density plots are not affected by the number of bins which is a major parameter when histograms are to be considered, hence allows us to better visualize the distribution of our data. While a histogram counts the number of data points in somewhat arbitrary regions, a kernel density estimate is a function defined as the sum of a kernel function on every data point. link brightness_4 code # for 'tip' attribute # using plot.kde() data.tip.plot.kde(color='green') plt.title('KDE-Density plot for Tip') plt.show() … Python and R interfaces available. Implementation of 1-D and 2-D Kernel Density Estimation Methods in Python using Numpy and Matplotlib Only. Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. Please use a supported browser. This article is an introduction to kernel density estimation using Python's machine learning library scikit-learn. This PDF was estimated from Kernel Density Estimation (with a Gaussian kernel using a 0.6 width window). Below is the implementation of plotting the density plot using kde() for the dataset ‘tips’. There is a great interactive introduction to kernel density estimation here. 8. Kernel Density Estimation: Nonparametric method for using a dataset to estimating probabilities for new points. The kernel density estimate of the input will be returned, and when combined with the kernel density estimators for all other points in the dataset of interest, we obtain a rough estimate of the distribution’s underlying density. With the correct choice of bandwidth, important features of the distribution can be seen, while an incorrect choice results in … Visit Stack Exchange. Kernel density estimation (KDE) is a non-parametric method for estimating the probability density function of a given random variable. Kernel density estimate allows smoother distributions by smoothing out the noise. Posts: 4 Threads: 2 Joined: Dec 2019 Reputation: 0 Likes received: 0 #1. The kernel function typically exhibits the following properties: Symmetry such that $$K(u) = K( … Dec-04-2019, 11:02 PM . We first consider the kernel estimator: $\hat{f}(x) = \frac{1}{Wnh} \sum_{i=1}^n \frac{w_i}{\lambda_i} K\left(\frac{x_i - x}{h\lambda_i}\right)$ Where: \(K: \R^p\rightarrow … gaussian_kde works for both uni-variate and multi-variate data. Python3. Mean-shift builds upon the concept of kernel density estimation is sort KDE. edit close. The kernel effectively smooths or interpolates the probabilities across the range of outcomes for a random variable such that the sum of probabilities equals one, a … Python has the ability to calculate and visualize contours. Loading… This can be done by identifying the points where the first derivative changes the sign. Drawing a Kernel Density … Browse other questions tagged scikit-learn python-3.x kernel density-estimation or ask your own question. The training data for the Kernel Density Estimation, used to determine the bandwidth(s). Once we have an estimation of the kernel density funtction we can determine if the distribution is multimodal and identify the maximum values or peaks corresponding to the modes. The Kernel Density Estimation function has a smoothing parameter or bandwidth ‘h’ based on which the resulting PDF is either a close-fit or an under-fit or an over-fit. filter_none. JED is a powerful … share \| cite \| … Plotting 2D Kernel Density Estimation with Python. However, after searching for a long time, I couldn't figure out how to make the y-axis and x-axis non-transparent. Kernel density estimation is the process of estimating an unknown probability density function using a kernel function \(K(u)$$. It includes automatic bandwidth determination. Kernel density estimation is a way of smoothing out plotting points in a graph in order to get an estimation of the plotting points. Provides the fast, adaptive kernel density estimator based on linear diffusion processes for one-dimensional and two-dimensional input data as outlined in the 2010 paper by Botev et al. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Kernel density estimation is a method of estimating the probability distribution of a random variable based on a random sample. It includes automatic bandwidth determination. kernel-density-estimation statistical-pattern-recognition histogram-density-estimation Updated Apr 27, 2018; Python; charlesdavid / JED Star 1 Code Issues Pull requests JED is a program for performing Essential Dynamics of protein trajectories written in Java. It is also referred to by its traditional name, the Parzen-Rosenblatt window method, according to its discoverers. Pygator on Sept. 16, 2019. what … Porting popular R library KernSmooth to python. MAINTENANCE WARNING: Possible downtime early morning Dec … Get underlying function from Kernel Density Estimation. More info Kernel Density Estimation can be applied regardless of the underlying distribution of the dataset. The following python package https: ... Identify and plot local maximum values of the KDE. The estimation works best for a unimodal distribution; bimodal or multi … The estimation works best for a unimodal distribution; bimodal or multi-modal distributions tend to be oversmoothed. … This site may not work in your browser. In contrast to a histogram, kernel density estimation produces a smooth estimate.The smoothness can be tuned via the kernel’s bandwidth parameter. KDE is a means of data smoothing. gaussian_kde works for both uni-variate and multi-variate data. Thanks! Representation of a kernel-density estimate using Gaussian kernels. python density-estimation. Given a sample of independent and … KDE is a method to estimate the … It is used for non-parametric analysis. This method is used for the analysis of the non-parametric values. Transformed R and Fortran functions into Python(2,3) code. If a list, each list element is a separate observation. However we choose the interval length, a histogram will always look wiggly, because it is a stack of rectangles (think bricks again). Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. It is possible to estimate the distribution density in two … Python; Kernel Density Estimation. Functions for Kernel Smoothing and Density Estimation. A density estimate or density estimator is just a fancy word for a guess: We are trying to guess the density function f that describes well the randomness of the data. The Overflow Blog Does your organization need a developer evangelist? Viewed 26k times 21. Given a set of observations $$(x_i)_{1\leq i \leq n}$$. Ask Question Asked 5 years, 8 months ago. Imagine that the above data was sampled from a probability distribution. I find the seaborn package very useful here. Kernel Density Estimation in Python Sun 01 December 2013. Kernel density estimation is a fundamental data smoothing problem where inferences about the population … So in … Peg Perego Gator Motor Upgrade, 36 Lessons Of Vivec Writer, Klipsch R-10sw Connection, Odisha Pg Entrance 2020 Syllabus, Realized Yield Calculator, What Duck Walks On Two Legs Answer, John Deere 5115m Manual, Aerosoft A330 Checklist,	2021-03-07T02:10:38	{ "domain": "almodarresi.com", "url": "https://almodarresi.com/rene-naufahu-ddf/kernel-density-estimation-python-4bbe13", "openwebmath_score": 0.4124547243118286, "openwebmath_perplexity": 834.4257758144151, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9835969641180276, "lm_q2_score": 0.8519527982093666, "lm_q1q2_score": 0.8379781858905916 }
https://math.stackexchange.com/questions/2908939/linear-and-algebraic-independence-of-exponentials	# Linear and algebraic independence of exponentials Original question Let $$P_1,\ldots, P_n$$ be distinct polynomials of a complex variable. We suppose that they are without constant term ($$P_i(0)=0$$). Is it true that the functions $$z\mapsto e^{P_i(z)}$$ are linearly independant (over $$\mathbb{C}$$) ? Late edit: The proof given in the selected answer proves the following Theorem Let $$(P_i)_{i\in I}$$ a family of complex univarariate polynomials without constant term. Then if $$i\mapsto P_i$$ is injective, $$(e^{P_i})_{i\in I}$$ is a family of (entire) functions linearly independent over $$\mathbb{C}[z]$$. One then has the Corollary Under the same conditions (no constant term), if the family $$(P_i)_{i\in I}$$ is $$\mathbb{Z}$$-independant, then $$(e^{P_i})_{i\in I}$$ is algebraically independant with respect to $$\mathbb{C}[z]$$. Proof Call $$G$$ the family of exponentials ($$G=(e^{P_i})_{i\in I}$$). For every multiindex $$\alpha \in \mathbb{N}^{(I)}$$, one has $$G^\alpha=\prod_{i\in I} (e^{P_i})^{\alpha(i)}= \prod_{i\in I} (e^{\alpha(i)\,P_i})= e^{\sum_{i\in I}\alpha(i)\,P_i}$$ but the fact that $$(P_i)_{i\in I}$$ is $$\mathbb{Z}$$-independant (linearly) implies (and is indeed equivalent to) $$\alpha \mapsto \sum_{i\in I}\alpha(i)\,P_i$$ is into. One has, from the theorem, that $$(G^\alpha)_{\alpha \in \mathbb{N}^{(I)}}$$ is $$\mathbb{C}[z]$$-linearly independant which amounts to say that $$G$$ is algebraically independant over $$\mathbb{C}[z]$$. • Indeed it is. This can be shown by realizing that the sum of terms in the polynomial becomes a product of exponential terms, each of which is independent/ – Don Thousand Sep 7 '18 at 18:34 • Could you elaborate this technically ? In particular you do not use the fact that they are without constant term and the conclusion does not hold when you take polynomials with constant terms. – Duchamp Gérard H. E. Sep 7 '18 at 20:16 • @RushabhMehta You should write an official answer. – Paul Frost Sep 7 '18 at 21:58 • This question has been satisfactorily answered in MO there. I'll put details later – Duchamp Gérard H. E. Sep 12 '18 at 7:40 • @DuchampGérardH.E.: Any progress on this one? – Martin R Oct 6 '18 at 14:35 As mentioned in the comments, this has been answered affirmative in linear independence of exponentials on Math Overflow for the more general case of exponentials of entire functions of several complex variables. In the case of exponentials of polynomials of a single complex variable it can be proven with more elementary methods. Here is something I came up with. The idea is to consider slightly more general linear combinations with rational coefficients, in order to make a proof by induction work. Claim: Let $$P_1, \ldots, P_n$$ be distinct polynomials without constant term, and $$R_1, \ldots, R_n$$ be rational functions, such that $$R_1 e^{P_1} + \ldots + R_n e^{P_n} = 0 \, .$$ Then $$R_1 = \ldots = R_n = 0$$. In particular, $$e^{P_1}, \ldots, e^{P_n}$$ are linearly independent over $$\Bbb C$$. Proof by induction: The case $$n=1$$ is trivial, $$R_1 e^{P_1} = 0$$ clearly implies $$R_1 = 0$$. Now let $$n \ge 2$$ be arbitrary and assume that the claim is true for all smaller values of $$n$$. Let $$P_1, \ldots, P_n$$ be distinct polynomials without constant term, and $$R_1, \ldots, R_n$$ be rational functions, with $$R_1 e^{P_1} + \ldots + R_n e^{P_n} = 0 \, .$$ If all $$R_k$$ are zero then we are done. Otherwise (without loss of generality) $$R_n \ne 0$$, and it follows that $$\frac{R_1}{R_n}e^{P_1 - P_n} + \ldots + \frac{R_{n-1}}{R_n}e^{P_{n-1} - P_n} + 1 = 0 \, .$$ Differentiating this identity gives $$\left( \left(\frac{R_1}{R_n}\right)' + \frac{R_1}{R_n}(P_1'-P_n')\right)e^{P_1 - P_n} + \ldots + \left( \left(\frac{R_{n-1}}{R_n}\right)' + \frac{R_{n-1}}{R_n}(P_{n-1}'-P_n')\right)e^{P_{n-1} - P_n} = 0\, .$$ Now we can apply the induction hypotheses, since the $$P_k - P_n$$ ($$k=1,\ldots, n-1$$) are distinct polynomials without constant term. It follows that $$\left(\frac{R_k}{R_n}\right)' + \frac{R_k}{R_n}(P_k'-P_n') = 0$$ for $$k=1,\ldots, n-1$$, and consequently that the functions $$\frac{R_k}{R_n}e^{P_k - P_n}$$ are constant: $$\frac{R_k}{R_n}e^{P_k - P_n} = C_k \in \Bbb C \quad (k = 1, \ldots, n-1) \, .$$ If $$C_k \ne 0$$ for some $$k$$ then $$\frac{R_k}{R_n}$$ is a rational function without zeros and poles and therefore constant. It follows that $$e^{P_k - P_n}$$ and consequently $$P_k - P_n$$ is constant, which is a contradiction to the assumption that the polynomials are distinct without constant term. Therefore $$C_k=0$$ for $$k=1,\ldots, n-1$$, so that $$R_k = 0$$ for $$k=1,\ldots, n-1$$, which also implies that $$R_n = 0$$. Remark: Instead of $$P_k(0)=0$$ for all $$k$$ it suffices to assume that no difference $$P_k - P_j$$ for $$k \ne j$$ is constant. The proof remains the same. • Great, thanks a lot. You're right, generalizing the coefficients makes the recurrence possible. Of course, your computations live within the space of germs with respect to Fréchet filter (complements of finite sets). I adopt your answer. – Duchamp Gérard H. E. Oct 26 '18 at 2:30 • Moreover, I think that your proof proves that $\{1,e^{P_1},\cdots ,e^{P_n}\}$ are linearly independant over polynomials, which would imply that $\{e^{P_1},\cdots ,e^{P_n}\}$ are algebraically independant. I must think of this ... – Duchamp Gérard H. E. Oct 26 '18 at 19:15 • I think, I have a stable statement: if the polynomials $P_i$ are $\mathbb{Z}$-linearly independent, then their exponential s are algebraically independent I'll add this in the question. It rests on your proof. – Duchamp Gérard H. E. Oct 27 '18 at 9:55 • @DuchampGérardH.E.: $P_1(z) = z$ and $P_2(z) = z + i$ are $\Bbb Z$-linearly independent, but their exponentials are linearly dependent. – Martin R Oct 27 '18 at 13:00 • Of course ! when I say "if the polynomials $P_i$" it is understood "under the same condition". I put it explicitely in my late edit. – Duchamp Gérard H. E. Oct 27 '18 at 13:38	2019-09-18T12:12:23	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2908939/linear-and-algebraic-independence-of-exponentials", "openwebmath_score": 0.9498687386512756, "openwebmath_perplexity": 203.33616547621241, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9835969645988575, "lm_q2_score": 0.8519527963298947, "lm_q1q2_score": 0.8379781844515931 }
https://dralb.com/2019/06/26/properties-of-matrix-operations/	Properties of Matrix Operations Now that we have defined the matrix operations we will be using in the last post, we can now look at some properties of these operations. When looking at these operations, we often want to compare them to properties held by the real numbers. While not all of the same, many are. Here we will look at two properties of real numbers and prove that the do or do not hold for matrices. Theorem: The associative property holds for matrix addition. We will prove the given theorem the matrix addition is associative. However, before we do that, we should recall what associativity is. In this case, we assume that $$A$$, $$B$$ and $$C$$ are matrices of the same size. We then want to show that \begin{align} (A+B)+C=A+(B+C). \end{align} That is, we want to show that the order of the parenthesis don’t matter when adding three matrices. As a further note, matrix addition is only defined for matrices of the same size, this is why we are restricted our matrices to that condition. Now, to show our theorem, we are really trying to show that two things are equal. We, therefore, need to start with one side of the equation and simplify it step by step until we arrive at the other side. As we try to do this, we really will just work step by step until we can make some connections. The first thing we will do is try to use the notation we have. As such, we will denote $$A=[a_{ij}]$$, $$B=[b_{ij}]$$ and $$C=[c_{ij}]$$. That is, we will express the matrices as arbitrary entries in the matrix. We now have \begin{align} (A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]. \end{align} Now, since addition of matrices is defined component wise, we know that the elements of $$A+B$$ will just be the sum of the entries in $$A$$ and $$B$$, this is \begin{align} (A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\ &=[a_{ij}+b_{ij}]+[c_{ij}]. \end{align} That is, we wrote $$A+B$$ in terms of its entries. We can do this again, combing $$C$$ and we get \begin{align} (A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\ &=[(a_{ij}+b_{ij})+c_{ij}]. \end{align} Now that we’ve combined these into one matrix, we can look at the entries of this matrix. The entry in the ith row and jth column is now $$(a_{ij}+b_{ij})+c_{ij}$$. However, these are just real numbers. Since addition for real numbers is associative, we can rearrange these parenthesis and get \begin{align} (A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\ &=[(a_{ij}+b_{ij})+c_{ij}]\\ &=[a_{ij}+(b_{ij}+c_{ij})]. \end{align} We can now work backwards from before and break one matrix into two. Here we have, \begin{align} (A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\ &=[(a_{ij}+b_{ij})+c_{ij}]\\ &=[a_{ij}]+[b_{ij}+c_{ij}]. \end{align} Using this same step again, we finally arrive at \begin{align} (A + B ) + C & = ([a_{ij}]+[b_{ij}])+[c_{ij}]\\ &=[(a_{ij}+b_{ij})+c_{ij}]\\ &=[a_{ij}]+[b_{ij}+c_{ij}] \\ &=[a_{ij}]+([b_{ij}]+[c_{ij}]) \\ &=A+(B+C). \end{align} This is precisely what we wanted to show. If we were to give a formal proof, we would like to be more concise with the writing we used here. However, this would be a great way to work through the proof the first time you are working on it. As an additional note, we assumed here that we were working on matrices with real entries. Later in this class, we will consider matrices with entries from other sets than just the real numbers. For this reason I will point out the associativity of matrix multiplication didn’t depend on the use of real numbers, but rather just that the set used for entries had associative addition. That is, with slight modification, our proof would work over any set of numbers with associative addition. Commutativity of Matrix Multiplication Theorem: Matrix multiplication is not commutative. Note that we are stating that, unlike the real numbers, we will not always get the same answer if we switch the order that we multiply matrices. In order to provide a proof of this, we simply need to find an example where switching the order gives a different answer. First Example Let $A=\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$. In this case, we note that $AB=\begin{bmatrix} 2 \\ 2 \end{bmatrix},$ whereas $$BA$$ is not defined. Therefore, the changing the order can change whether or not the multiplication is even defined. Second Example While our last example was enough to show that commutativity does not hold, we can also give an example where the multiplication is defined both times, but still not equal. For this example, we’ll let $A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B=\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$. We now get that $AB=\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix},$ whereas $BA=\begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}.$ Therefore, again, we cannot change to order of multiplication. Other properties We won’t prove all of these here, but it is important that we remember the properties that the matrix operations have. On the one hand, using properties that are correct can save us time and work. On the other hand, if we try to use properties that are not true, we will end up getting things that don’t make sense. Below, we have a list of some of the properties you will be using. In each case, however, you do need to be careful as each has implication about the size of the respective matrices. • is associative, • is commutative, • has an identity, • has inverses. • Matrix multiplication • is associative, • has an identity. Conclusion Going through the proof and disproof of the given properties has hopefully helped you to better understand why some things work for matrices and other things don’t. The deeper understanding this gives will help you as you work through any number of problems that will arise with matrices. If you learned something from this post, make sure to share it social media so that other people can also get help. Furthermore, make sure to subscribe to our YouTube channel so that you can watch the associated videos for our posts. This site uses Akismet to reduce spam. Learn how your comment data is processed.	2021-10-27T12:32:48	{ "domain": "dralb.com", "url": "https://dralb.com/2019/06/26/properties-of-matrix-operations/", "openwebmath_score": 0.8865190744400024, "openwebmath_perplexity": 204.22912736636664, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9929882052410105, "lm_q2_score": 0.8438950986284991, "lm_q1q2_score": 0.8379778793987989 }
https://math.stackexchange.com/questions/714131/derivatives-of-sine-and-cosine-at-x-0-give-all-values-of-fracddx-sin-x	# Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$? In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left\|\right._{x=0}=\lim\limits_{\Delta x\to0}\dfrac{\cos\Delta x-1}{\Delta x}=0$ $\dfrac{d}{dx}\sin x\left\|\right._{x=0}=\lim\limits_{\Delta x\to0}\dfrac{\sin\Delta x}{\Delta x}=1$ Okay I understand this, but then he says: Derivatives of sine and cosine at %x=0% give all values of $\dfrac{d}{dx}\sin x$ and $\dfrac{d}{dx}\cos x$. What?? What does he mean? mookid's answer is fine. Or try this. Suppose you want to compute the derivative of $\cos$ at a point $a$. Use the identity $$\cos(a+x)=\cos(a)\cos(x)-\sin(a)\sin(x) .$$ Differentiate that with respect to $x$, $$\cos'(a+x)=\cos(a)\cos'(x)-\sin(a)\sin'(x) ,$$ then plug in $x=0$ $$\cos'(a)=\cos(a)\cos'(0)-\sin(a)\sin'(0) .$$ Now if you know $\sin'(0)=1$ and $\cos'(0)=0$, you get $$\cos'(a) = -\sin(a) .$$ • You do not prove the existence of the derivative. – mookid Mar 16 '14 at 13:29 • His answer is more clear! – user135767 Mar 16 '14 at 13:33 • but it is not true. anyway. – mookid Mar 16 '14 at 16:06 He makes use of $$\cos(x+h) - \cos(x) = -2\sin (x+h/2)\sin(h/2)\\ \sin(x+h) - \sin(x) = 2\cos (x+h/2)\sin(h/2)\\$$ then $$\lim_{h\to 0} \frac 1h [\cos(x+h) - \cos(x)] = -\sin (x) \lim_{h\to 0} \frac 2h\sin(h/2) = -\sin (x)\frac d{dx}\sin(x)\|_{x=0} \\ \lim_{h\to 0} \frac 1h [\sin(x+h) - \sin(x)] = \cos (x)\lim_{h\to 0} \frac 2h\sin(h/2) =\cos(x) \frac d{dx}\sin(x)\|_{x=0}$$ and you know that • That's not what I said. I said that Idon't understand the statement that he said. – user135767 Mar 16 '14 at 11:58 • please re read my question – user135767 Mar 16 '14 at 11:59 • I completed my answer. I you know the derivative for x=0, then you know it everywhere. – mookid Mar 16 '14 at 11:59 • why? why if we know the derivative at x=0 then we know it everywhere? – user135767 Mar 16 '14 at 12:02 • because of my argument. – mookid Mar 16 '14 at 12:50 I didn't watch the video, so this might not be on point. But just compute the derivative of $\sin x$ at the point $x = a$: $$\left.{d\over dx}\sin{x}\right\|_{x=a} = \ \lim_{h\rightarrow 0} \frac{\sin{\!(a + h)}-\sin{a}}{h}\,.$$ Using the identity $\sin{\!(a + h)} = \sin{a}\cos{h} + \cos{a}\sin{h}$, $$\left.{d\over dx}\sin{x}\right\|_{x=a} = \ \lim_{h\rightarrow 0} \frac{\sin{a}\cos{h} + \cos{a}\sin{h}-\sin{a}}{h}$$ $$= \cos{a}\left(\lim_{h\rightarrow 0} \frac{\sin{h}}{h}\right) + \sin{a}\left(\lim_{h\rightarrow 0} \frac{\cos{h} - 1}{h}\right)\,.$$ Perhaps now you can see that if we know the two principal limits at $0$, we obtain $(\sin{x})^\prime$ at any value of $x$. Was that your question? You might try this with $(\cos{x})^\prime$ as an exercise. Jason Zimba's answer actually is the one that explains the most about the particular subtlety of the statement in the video. In the video before the moment in question, the lecturer derives formulae of $sin'(x)$ and $cos'(x)$ based upon the (yet unproven) facts which he denotes as $A$ and $B$: \begin{align} \lim_{\Delta x \to 0} {\frac{\cos \Delta x - 1}{\Delta x}} = 0 \tag{A} \\ \lim_{\Delta x \to 0} {\frac{\sin \Delta x}{\Delta x}} = 1 \tag{B} \end{align} Then he says that we surely need to prove $A$ and $B$ in order for these derivations to be valid. In other words, he says that $$A, B \implies \text{proofs of the derivations of sin' and cos'}$$ (i.e. if we prove A, B then the proofs are sound) Then, in the remark in question, the lecturer is considering the derivatives of the sine and cosine at $x = 0$ using the definition of a derivative and derives that: \begin{align} \dfrac{d}{dx} \cos x \|_{x=0} &= \lim_{\Delta x \to 0} {\frac{\cos \Delta x - 1}{\Delta x}} \\ \dfrac{d}{dx} \sin x \|_{x=0} &= \lim_{\Delta x \to 0} {\frac{\sin \Delta x}{\Delta x}} \end{align} From which we can see that the derivatives at $0$ equal to the left hand sides of the equations $A$ and $B$ that we need to prove. Hence, if we'd be able to prove that \begin{align} \dfrac{d}{dx} \cos x \|_{x=0} &= 0 \\ \dfrac{d}{dx} \sin x \|_{x=0} &= 1, \end{align} by transitivity the statements $A$ and $B$ will be proven. I think that in the lecture the last idea was formulated a bit unclear, particularly the lecturer said that "the derivatives at $0$ give us all the values of the derivatives" while effectively he meant that the derivatives at $0$ give us the values of l.h.s. terms in A and B which in turn give us the formulae for $sin'(x)$ and $cos'(x)$ Basically, David Jarison has just used the wording "all the values of the derivatives" in place of "analytical formulae for $sin(x)$ and $cos(x)$".	2019-08-22T15:47:49	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/714131/derivatives-of-sine-and-cosine-at-x-0-give-all-values-of-fracddx-sin-x", "openwebmath_score": 0.9976234436035156, "openwebmath_perplexity": 428.60029163775954, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9664104972521579, "lm_q2_score": 0.8670357666736772, "lm_q1q2_score": 0.8379124664065143 }
https://gateoverflow.in/41/gate-cse-2012-question-9	8,231 views Consider the function $f(x) = \sin(x)$ in the interval $x =\left[\frac{\pi}{4},\frac{7\pi}{4}\right]$. The number and location(s) of the local minima of this function are 1. One, at $\dfrac{\pi}{2}$ 2. One, at $\dfrac{3\pi}{2}$ 3. Two, at $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2}$ 4. Two, at $\dfrac{\pi}{4}$ and $\dfrac{3\pi}{2}$ Local Minima :- $f(x)$ will have local minima at $x=a$ iff $f(a-h)>f(a)<f(a+h)$where $h->0$ Local Maxima :- $f(x)$ will have local maxima at $x=a$ iff $f(a-h)<f(a)>f(a+h)$ where $h->0$ Global Minima :- $f(x)$ will have global minima at $x=a$ iff $f(x)≥f(a)$ ∀$x$∈ Domain of function. Global Maxima :-$f(x)$ will have global maxima at $x=a$ iff $f(x)≤f(a)$ ∀$x$∈ Domain of function. So overall we can say , when we talk about local minima/maxima we compare it with nearest points and then declare nature of function at that point but when we talk about Global minima/maxima we compare it with whole domain. Global minima $⊆$ Local minima and similarly Global maxima $⊆$ Local maxima The answer should be B. The point at Pi/4 is not a local minima. Here's the definition from NCERT book, it should be clear that D does not satisfy the definition. I think the question was wrong and remain unchallenged, or perhaps I am wrong in which case I would love to be corrected :) @Divy Kala My approach is end points can be global maximum/minimum Also Global minima ⊆ Local minima So if end point is can be global minima , it can also be local minima . So the concept of end point can be local minima is justified. @Arjun Sir plz tell if I 'm thinking correctly ? @Apoorva Jain i guess it's a matter of convention then. the link you provide has a different definition than the ncert one i gave above. @Arjunsir please guide on this matter edit: apparently @Arjunsir has already commented below, the answer is D according to him as well as the official key. This means NCERT definition is to be followed. ### Subscribe to GO Classes for GATE CSE 2022 Answer is $(D)$ $f '(s) = \cos x =0$ gives root $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ which lie between the given domain in question $\left[\frac{\pi}{4},\frac{7\pi}{4}\right]$ $f ''(x)= - \sin x$ at $\frac{\pi}{2}$ gives $-1<0$ which means it is local maxima and at $\frac{3\pi}{2}$ it gives $1>0$ which is local minima. Since, at $\frac{\pi}{2}$ it is local maxima so, before it, graph is strictly increasing, so $\frac{\pi}{4}$ is also local minima. So, there are two local minima $\frac{\pi}{4}$ and $\frac{3\pi}{2}.$ edited +1 (y) @Praveen Saini@Arjun @Bikram sir I understand that if the endpoint is to the left of local maxima , then the endpoint is local minima and if the endpoint is to the left of local minima then the endpoint should be local maxima ... now what if the stationary point next to endpoint is an inflection point ??? Now how to find out whether endpoint is local maxima or minima ??? Why pi/4 is local minima? how is pi/4 even a local mimima???? okay finally got it.... @Sushmita, Can you please explain in detail? Can we say that $\frac{\pi}{4}$ is a local minima because we have for all x in an interval $(\frac{\pi}{4},\frac{\pi}{2})$ such that $f(x)>f(\frac{\pi}{4})$ ? See the pic Here it is not asked for absolute minima but it asked for local minima local minima is the point, where the graph starts strictly increasing Here the graph starts at $\pi /4$ and then strictly increases So, it is one local minima Sine function increases till $\frac{\pi}{2}$ and so for the considered interval $\frac{\pi}{4}$ would be a local minimum. From $\frac{\pi}{2}$, value of sine keeps on decresing till $\frac{3\pi}{2}$ and hence $\frac{3\pi}{2}$ would be another local minima. So, (D) is the correct answer here. by Answer is D only. 2012 onward official keys are available. http://gatecse.in/w/images/b/b5/Key_CS_2012.pdf This is trick question. Local word matters ! Why are we considering pi/4 here. They have asked for local minima and not absoulte minina, so shouldnt we consider the open interval only? what has "local" to do with interval being closed or open? [ ] means it is closed interval and ( ) means it is open interval. Sir, accroding to me there should be only one minima at 3pi/2. I am under the impression that "Local" means we should consider the open interval (pi/4,7pi/4). Please correct me if I am wrong. Definition : http://tutorial.math.lamar.edu/Classes/CalcI/MinMaxValues.aspx @Arjun SIR f(x) = sin x f'(x) = cos x solve for cos x = 0 x= n (pi) / 2 f''(x) = - sin x at x= n (pi) / 2 f''(x) <0 implies local maxima How did u find local minima ? @arjun sir please explain why we have consider $\prod /4$ here ?? Simply visualize the graph of sinx between those interval. why is pi/4 considered? Simplest way possible! by plz rotate the image The signs are wrong. In fact, they should be opposite to what is given by you. signs are wrong. opposite is even not right. they should be respectively : + + – + + signs here represents whether is function is increasing (if f’(x) > 0 ) or decreasing (if f’(x) < 0 ). so whenever we go from – to + , that is a poing of minima.( i.e 3pi/2) and check explicitly at end points. ( reverse ). P.S: simplest way to do this ques is : by making graph and locate pi/4 and 7pi/4. NCERT 12th class maths book pg no:228 (chapter 6): I don't think there is any doubt regarding local minima at point $c=\frac{\pi}{4}$. by	2021-09-27T01:37:18	{ "domain": "gateoverflow.in", "url": "https://gateoverflow.in/41/gate-cse-2012-question-9", "openwebmath_score": 0.8030589818954468, "openwebmath_perplexity": 1072.4184923047064, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9664104972521579, "lm_q2_score": 0.8670357666736772, "lm_q1q2_score": 0.8379124664065143 }
http://www.dccapco.com/company-location-yflc/1978cd-cardinal-number-example	They are sometimes called counting numbers.. Cardinal numbers (or cardinals) are numbers that say how many of something there are, for example: one, two, three, four, five, six. They answer the question, How many? We use these numbers in different ways. Not sure what college you want to attend yet? Cardinal numbers are used to tell how many marks you obtained; I got eighty marks out of hundred etc. Cardinal numbers, known as the “counting numbers,” indicate quantity. For example, the set {1, 2, 3} has three distinct elements, so its cardinal number is 3. Ordinal numbers tell the position of something on a list. In business writing and technical writing, figures are used in nearly all cases. Example 1 : Find the cardinal number of the following set A = { -1, 0, 1, 2, 3, 4, 5, 6} Solution : Number of elements in the given set is 7. In Latin, most cardinal numerals behave as indeclinable adjectives. 43. The answer to this question is that there are 6 baseballs in the picture. Example 2 : Find the cardinal … 1st is an Ordinal Number (it tells position) 3. Mathematicians manipulate cardinal numbers in many of their equations. Cardinal numbers represent ‘quantity’. Ordinal numbers are used for putting thing in order. The numbers which give us the exact quantity of an object are called cardinal numbers. In this photo there are 6 cars. The definition of a Cardinal number of a set as follows: The Number of elements present or contains in any given set is called as cardinal number of a set. Let Q \space c \space \mathbb{R} be the set of all rational numbers. Regardless of which rule you choose to follow, exceptions are made for dates, decimals, fractions, percentages, scores, exact sums of money, and pages--all of which are generally written in figures. let a, b , and c be sets in a universal set u . Hence, 4 or ‘four’ here is a cardinal number. Knowing all three types of numbers is important for a situation where all three types of numbers could be used like this: How many 99s are in the third row? flashcard set{{course.flashcardSetCoun > 1 ? If the given set is D then Cardinal number of a set is represented by n(D). An alternative rule is to spell out numbers of one or two words (such as two and two million), and use figures for numbers that require more than two words to spell out (such as 214 and 1,412). Infinite cardinals only occur in higher-level mathematics and logic. The cardinality of a set is the cardinal number that tells us, roughly speaking, the size of the set.. There are eleven birds on the tree. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons A number is a cardinal number if it describes a quantity related to objects or events, for example. This lesson will explain what they are and also give some examples. lessons in math, English, science, history, and more. Try refreshing the page, or contact customer support. Examples of cardinal number in a Sentence Recent Examples on the Web With the hypothesis unresolved, many other properties of cardinal numbers and infinity remain uncertain too. And so ostensibly are the greatest cardinal number and the abominable snowman. {{courseNav.course.topics.length}} chapters \| They do not show quantity or rank. The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and 13, satisfying the conditions (1) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in common, and then by defining by reference to a and (3 two other suitable classes whose cardinal numbers are defined to be … An example of using nominal numbers can be shown with this picture: What type of ball is in this picture? Services. They are usually associated with a noun that is counted, but do not change their endings to agree grammatically with that noun. Ordinal numbers do not show quantity. Though not all style guides agree, a common rule is that cardinal numbers one through nine are spelled out in an essay or article, while numbers 10 and above are written in figures. How are cardinal numbers formed? Examples: Ram came first in class; Sarita was the third girl standing in a row First, I think of the famous fierce red bird. 3. succeed. What does cardinal mean? COBUILD Advanced English Dictionary . I have two sisters. Cardinal numbers tell us "how many." All other trademarks and copyrights are the property of their respective owners. \{x\|x= k^2 , k=1,2,3, \cdots 50 \}, Find the cardinal number of each of the following sets. Ordinal numbers are numbers that represent place or position. By using ThoughtCo, you accept our, The Difference Between Cardinal Numbers and Ordinal Numbers, Learn the Months, Seasons, Days, and Dates in German. By further studying cardinal numbers, you will see we can include the number zero, because it can identify the quanity of something. Find the cardinal number of each of the following sets. The number of elements in a set is called the cardinal number of the set. 27 indicated they would participate in sports and student government, 24 in sports an, Find the cardinal number of the set. To know “how many” there should be something. Synonym: Cardinalis cardinalis, Richmondena Cardinalis, cardinal grosbeak, cardinal number, carmine, central, fundamental, key, primal, redbird. Looking at this picture, a baseball is third in line. The English words one, two, three, four, etc. What Are French Ordinal Numbers and Fractions? in Educational Leadership. Cardinal sentence examples. we are given n(u) = 67 , n(a) = 27 , n(b) = 28, n(c) = 27 , n(a cap b) = 16, n(a cap c) = 10, n(b cap c) = 9 , and n(a cap bcap c) = 12 . A cardinal number answers the question "How many?" Create your account. When we have a number of items and we need to tell their quantity we use cardinal numbers for example one, two, three, four, five, and so on. The digits 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 are used to form several other cardinal numbers. To unlock this lesson you must be a Study.com Member. 16 sentence examples: 1. The counting numbers are exactly what can be defined formally as the finite cardinal numbers. He has a master's degree in Educational Administration and is working toward an Ed.D. - Definition & Examples, What Are Opposite Numbers? Log in or sign up to add this lesson to a Custom Course. Create an account to start this course today. There are a couple of other types of numbers that can be confused for cardinal numbers, and they are ordinal numbers and nominal numbers. \{x\|x= k^2 , k=1,2,3, \cdots 50 \} Plus, get practice tests, quizzes, and personalized coaching to help you 47. The ordinal number tells the position of Tom's car in the race. - Definition & Examples, What Are Rectangular Numbers? The exceptions are ūnus (“one”), duo (“two”), trēs (“three”), and multiples … If the number of objects/persons are specified in a list: the position of the objects/persons is defined by ordinals. How many rolls satisfy the following condition? Show that the set \begin{Bmatrix} sinx + cos x + 1: \space x \space \epsilon \space Q \end{Bmatrix} has Lebesgue measure zero. Since 0 means nothing; it is not a cardinal number. credit-by-exam regardless of age or education level. Nominal numbers name or identify something (e.g., a zip code or a player on a team.) 6 is a Cardinal Number (it tells how many) 2. (adjective) A cardinal rule; cardinal sins. Italian Ordinal Numbers and Numerical Rank, Proofreaders' and Teachers' Correction Marks, Counting: The Cardinal Numbers of Spanish, Counting and Calculating in German from 0 to the Trillions, How to Say and Write Fractions in Spanish, Ph.D., Rhetoric and English, University of Georgia, M.A., Modern English and American Literature, University of Leicester, B.A., English, State University of New York, When using number words, it is important to keep the difference between. Here are few examples of cardinal numbers: There are six coins. 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Cardinal numbers are numbers that tell how many of something there are and answers the question, How many? - Definition & Examples, Biological and Biomedical imaginable degree, area of Have you wondered what that means? credit by exam that is accepted by over 1,500 colleges and universities. \| Common Core Math & ELA Standards, TCI History Alive World Connections: Online Textbook Help, AEPA Physics (NT308): Practice & Study Guide, ScienceFusion Intro to Science & Technology: Online Textbook Help, Praxis Middle School Science (5440): Practice & Study Guide, ORELA Middle Grades Math: Simplifying Whole Number Expressions, Quiz & Worksheet - History of INTERPOL Washington, Quiz & Worksheet - Using Personality Career Assessments to Choose a Career, Quiz & Worksheet - Multicell & Supercell Thunderstorms, Quiz & Worksheet - Bottom-Up Processing in Psychology, Arctic Oscillation & North Atlantic Oscillation, How To Get a Copy of Your High School Diploma, Mechanical Engineering Scholarships for High School Seniors, Tennessee Science Standards for 8th Grade, National Science Standards for High School, Tech and Engineering - Questions & Answers, Health and Medicine - Questions & Answers, What is the difference between "ordinal" and "cardinal" ranking of utility? How Do I Use Study.com's Assign Lesson Feature? {110, 112, 114, 118, 310} Find the cardinal number of the set. They may be identified with the natural numbers beginning with 0. Do you know, equivalent sets are described or defined by the cardinal number only. Log in here for access. Compare ordinal number. 95. The adjective terms which are used to denote the order of something are 1st, 2nd, 3rd, 4th, 5th, 6th, and so on. For each of the following ranking systems, explain whether an ordinal or cardinal ranking is being used: (a) military or ac. The ordinal number is third because the question asks for the number in the third row, and the nominal number is 99 because it names what is being asked for. - Definition & Examples, What is Subtraction in Math? Numbers not affiliated with their bird namesakes. What are cardinal numbers? 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Take the familiar mathematical example of the greatest cardinal number. Here are some examples, Cardinal numbers can be used to describe the quantity of shirts sold by an apparel retailer. Cardinal numbers are numbers that represent how many of something someone has. b) Which indices are cardinally equivalent? That is n (A) = 7. Car Number \"99\" (with the yellow roof) is in 1st position: 1. Cardinal Numbers (Numbers for Counting) Cardinal numbers are the numbers we count with and probably what you think of first when you hear the word “number.” For example, “one,” “two,” and “three” are all cardinal numbers. Sciences, Culinary Arts and Personal In this blog, we explain the concept of cardinal numbers in a simple manner, devoid of technical jargon. For example, words like first, second, and third are ordinal numbers. 's' : ''}}. A cardinal number is a number such as 1, 3, or 10 that tells you how many things there are in a group but not what order they are in. On this page you will find a list of ordinal numbers in English with the correct English spelling. We can write cardinal numbers in numerals as 1, 2, 3, 4, and so on as well as in words like one, two, … and career path that can help you find the school that's right for you. Meaning: ['kɑːdɪnl] n. 1. Examples of cardinal numbers are 1, 7, 9, and 123. Diary of an OCW Music Student, Week 4: Circular Pitch Systems and the Triad, Types of Cancer Doctors: Career Overview by Specialization, Master of Arts MA Guidance Counseling Degree Overview, Bachelor of Science in Secondary Education Major in Biological Science Programs, Bachelor of Science BS Administration of Justice Degree Overview, PhD in Telecommunications Degree Program Overview, Working with Numbers for Elementary School, What Are Cardinal Numbers? Let us look into some examples based on the above concept. Here are some examples of cardinal numbers. A cardinal number answers the question "How many?" Could cardinal numbers be referring to their population? 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In the above example, three (3) is the cardinal number. The average is found by adding up all the children and dividing by the number of households. courses that prepare you to earn These are numbers like first, second, third, and fourth. He became dean of Toledo early, and was made cardinal on the 5th of August 1669. Anyone can earn Also called a counting number or a cardinal numeral. Cardinal refers to a basic or primary value. Cardinal nos – Illustrative Example 1: 7 mangoes in a basket In the first example, the ordinal number first precedes the cardinal number two. To learn more, visit our Earning Credit Page. cardinal. Visit the Math for Kids page to learn more. sentence examples. Also, learn here cardinal numbers. \| {{course.flashcardSetCount}} Contents 1 Introduction 2 ... (which underlies, for example, our intuitive understanding of the natural numbers as well as the syntax of logic) towards a formalistic point of view about mathematics. Rome as cardinal protector of the Spanish nation. Earn Transferable Credit & Get your Degree, What is an Ordinal Number? For example, a bag of 10 apples, the number 10 indicating the … Therefore, the cardinal number is 6. Similar words: boarding, rewarding, regarding, ordinary, ordinance, subordinate, coordinate, coordinator. Both first and two are determiners. There were three cars in the race. So, cardinal number of set A is 7. 33. received the cardinal archbishop of Bordeaux and determined to support the cardinal s … one (1) - Definition & Examples, Complement of a Set in Math: Definition & Examples, What is a Limiting Adjective? A cardinal rule is a rule that is basic or essential. One important thing to remember is that cardinal numbers do not include decimals or fractions; they are only used for simple counting and show quantity. Did you know… We have over 220 college Have you ever heard numbers called cardinal numbers? Get the unbiased info you need to find the right school. You may have heard something like "the average household has 1.8 children". What is the Difference Between Blended Learning & Distance Learning? Ordinal numbers indicate the order or rank of things in a set (e.g., sixth in line; fourth place). (This is not true for the ordinal numbers.) They do not include fractions or decimals because they are only used for simple counting. Fin, Of the 196 students in the entering class at Lakeside Community College, 100 of them plan to participate in sports, 85 in student government, and 55 in the theater arts. Study.com has thousands of articles about every No, zero (0) is not a cardinal number. just create an account. \"99\" is a Nominal Number (it is basically just a name for the car) If you roll a pair of six sided dice, one red and one black, there are 36 possible rolls, e.g. Obviously, every family has a whole number of children. Cardinal numbers are numbers that describe quantity, like one pot, two kettles, or three pans. Note: The tables below represent four inequality scales in percentages: a) Which indices are ordinally equivalent? the red die comes up 3 and the black die comes up 5. An example of a question using ordinal numbers is: What type of ball is third in line? The players all wear jerseys with numbers on them; could this be what is meant by cardinal numbers? The term cardinal has two definitions: one in linguistics and one in mathematics. The second three innings were quite dull. The cardinal number of the set A is denoted by n(A). Two Thanksgiving Day Gentlemen: Summary & Theme, Systems of Racial Hierarchy: History & Cultural Influence, Mortgage Brokerage Fees & Broker Liens in Connecticut, Quiz & Worksheet - Rounding Dividends & Divisors to Estimate Quotients, Quiz & Worksheet - The Yellow Wallpaper Literary Devices, Quiz & Worksheet - Modernist Furniture History, Flashcards - Real Estate Marketing Basics, Flashcards - Promotional Marketing in Real Estate, What is Common Core? Copyright © HarperCollins Publishers — Quanta Magazine, "To Settle Infinity Dispute, a New Law of Logic," 26 Nov. 2013 In linguistics or traditional grammar, the term ‘cardinal’ is a part of speech that is used to count or indicate how many. Cardinal numbers are used for counting. first two years of college and save thousands off your degree. The number of distinct elements in a finite set is called its cardinal number. It tells us how many cars were in the race. Last night I counted twenty stars in the sky. See the fact file below for more information on the cardinal numbers or alternatively, you can download our 28-page Cardinal Numbers worksheet pack to utilize within the classroom or home environment. We explain Cardinal Numbers with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. I just have one pencil, the kid said sadly. These are numbers like one, two, three, and four. There are two chairs and one dining table. ThoughtCo uses cookies to provide you with a great user experience. Cardinal numbers can be contrasted with ordinal numbers. A cardinal number, on the other hand, refers to numbers which indicate how many of something there are. Enrolling in a course lets you earn progress by passing quizzes and exams. We use cardinal numbers for the quantity of something. In grammatical terms, a cardinal numeral is a word used to represent such a countable quantity. Cardinal numbers are also called counting numbers. Also called a counting number or a cardinal numeral. For example: How many baseballs are in this picture? The cardinal numbers refer to the size of a group: Dr. Richard Nordquist is professor emeritus of rhetoric and English at Georgia Southern University and the author of several university-level grammar and composition textbooks. For example, 1 (one), 2 (two), 3 (three), etc. A cardinal number is a number used in counting to indicate quantity. 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T.J. is currently a grade 5 teacher and Vice-Principal. {110, 112, 114, 118, 310}, Working Scholars® Bringing Tuition-Free College to the Community. Select a subject to preview related courses: The last type of numbers are nominal numbers, and they are used only as a name or to identify something. - Definition & Examples, Over 83,000 lessons in all major subjects, {{courseNav.course.mDynamicIntFields.lessonCount}}, What Are Figurate Numbers? Looking at the table, there are three 99s, that means the cardinal number is 3 because it tells how many there are. In other words, cardinal numbers answer “How many?” For instance: The given picture shows 4 cars in a parking lot. Like the greatest cardinal number, Russell's class similarly fails t Cardinal and Ordinal Numbers Math 6300 Klaus Kaiser April 9, 2007. "the total is 4". Cardinal numbers are numbers that tell how many of something there are, such as one, two, three, four, or five. © copyright 2003-2021 Study.com. For example: Is zero (0) a cardinal number? - Definition & Examples, How to Read & Write Numbers Up to Six Digits: Lesson for Kids, Indefinite and Definite Articles: Definition and Examples, The Empty Set in Math: Definition & Symbol, English System Of Measurement: Definition, History, Advantages & Disadvantages, What is Place Value? In this situation, the number is identifying what the object is, so it is a nominal number. A Cardinal Number is a number that says how many of something there are, such as … In either case, numbers that begin a sentence should be written out as words. Already registered? Finally, remember don't get cardinal numbers confused with ordinal numbers, which tell the position, or nominal numbers, which name or identify something. The kid ate nine candies. - Definition & Example, Cardinality & Types of Subsets (Infinite, Finite, Equal, Empty), What Are Odd & Even Numbers? Examples of Cardinal Numbers. "When a cardinal number and an ordinal number modify the same noun, the ordinal number always precedes the cardinal number: The first two operations were the most difficult to watch. This is an interesting question because the word cardinal makes me picture a couple different things. In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. All other trademarks and copyrights are the greatest cardinal number of the set on this page you will we! Like the average household has 1.8 children '' car number \ '' 99\ '' ( the! Out as words of something there are '' ( with the natural numbers beginning 0... This picture: What type of ball is in 1st position: 1 99s, means... Think of the set trademarks and copyrights are the property of their respective owners in! Coursenav.Course.Mdynamicintfields.Lessoncount } }, What is the cardinal number two 's car in the sky of a is. The “ counting numbers, ” indicate quantity and four of objects/persons specified! 36 possible rolls, e.g a great user experience a set is called its cardinal number 3 is! To provide you with a great user experience ( e.g., a zip code or a cardinal is... Ostensibly are the property of their equations to a Custom Course baseball is third in line using numbers. ‘ four ’ here is a cardinal number then cardinal number of distinct elements a... Cardinal and ordinal numbers indicate the order or rank of things in a list ordinal! Provide you with a great user experience cardinal number example example of the set 1! Quizzes, and third are ordinal numbers Math 6300 cardinal number example Kaiser April 9,.!, 24 in sports and student government, 24 in sports an, Find the number. They would participate in sports an, Find the cardinal … they may be identified the... Number zero, because it can identify the quanity of something comes up 5 e.g.., 112, 114, 118, 310 }, Find the number! First precedes the cardinal number only order or rank of things in a list of ordinal numbers )... A sentence should be something in higher-level mathematics and logic 's degree in Educational Administration and is working toward Ed.D! Rank of things in a list: the position of the greatest cardinal number of children,! What can be defined formally as the “ counting numbers, '' they... English spelling will Find a list of ordinal numbers indicate the order rank... Out as words Personal Services a Custom Course counting number or a player on a list ordinal... Objects/Persons is defined by ordinals a universal set u true for the quantity of shirts sold an! Must be a Study.com Member the property of their equations test out of the following sets government, in. Assign lesson Feature is defined by the cardinal number }, working Scholars® Bringing college. Mathematics and logic at the table, there are six coins lesson to a Custom Course number,! A number is a cardinal numeral is a cardinal rule is a number! Is the Difference Between Blended Learning & Distance Learning apparel retailer obtained I. Family has a whole number of distinct elements, so it is a cardinal number the... Shown with this picture a finite set is represented by n ( a ) Which are... Get practice tests, quizzes, and personalized coaching to help you succeed Which indices ordinally... Numbers Math 6300 Klaus Kaiser April 9, and c be sets in set... Two, three ( 3 ) is the professional baseball team. associated with a noun that is or... Is zero ( 0 ) is cardinal number example professional baseball team. at this picture: What of. One ), etc black, there are and answers the question as to What cardinal numbers: are... Administration and is working toward an Ed.D note: cardinal numbers. us how many ) 2,. Abominable snowman its cardinal number of each of the following sets the English words one, two, (... Culinary Arts cardinal number example Personal Services that there are 36 possible rolls, e.g this an. We use cardinal numbers be a Study.com Member set { 1, 2 3! List: the position of Tom 's car in the picture ( 0 ) is in this,... College and save thousands off your degree endings to agree grammatically with that noun also give some based... ( 3 ) is in this picture, a baseball is third in line credit-by-exam of! Makes me picture a couple different things you know, equivalent sets described... Obtained ; I got eighty marks out of hundred etc take the mathematical... 50 \ }, Find the cardinal number cardinal … they may be identified with the yellow )... Decimals because they show quantity or defined by the cardinal number is 3 ; fourth place ) ; is! Master 's degree in Educational Administration and is working toward an Ed.D written out as words: Definition Examples... Of a cardinal number example using ordinal numbers. all cases baseballs are in this situation, the kid said sadly to. Kids page to learn more, visit our Earning Credit page 36 possible,... Is basic or essential red bird to this question is that there are and also give Examples. Comes up 3 and the abominable snowman in grammatical terms, a baseball is in. Items, that means the cardinal number of each of the first two years college... Shirts sold by an apparel retailer, ordinance, subordinate, coordinate, coordinator heard like..., known as the “ counting numbers, ” indicate quantity used for putting in... Regarding, ordinary, ordinance, subordinate, coordinate, coordinator interesting question because the word cardinal makes picture... The object is, so it is not a cardinal number of each of the set with numbers on ;! List: the position of something on a team. the first two years of college save. Because they are and answers the question as to What cardinal numbers are exactly can! 2: Find the cardinal number that tells us how many there are three 99s, that the! Custom Course they would participate in sports an, Find the cardinal number that tells us many. ( 0 ) a cardinal rule ; cardinal sins Examples, Complement of a using. Indeclinable adjectives and was made cardinal on the above concept What the object is, so it is cardinal! Rational numbers. in percentages: a ) Which indices are ordinally equivalent was made cardinal the... ) Which indices are ordinally equivalent, Over 83,000 lessons in all major subjects, { courseNav.course.mDynamicIntFields.lessonCount. ; fourth place ) set a is denoted by n ( a.... Number or a cardinal number of children visit the Math for Kids to! 2, 3 ( three ), 2 ( two ), 3 three. Cardinal numbers. the greatest cardinal number of the greatest cardinal number of elements in finite... Lesson you must be a Study.com Member info you need to Find the right school, visit our Earning page!: the position of something there are 6 baseballs in the above example, 1 ( one ),,! To provide you with a great user experience is defined by ordinals can identify the quanity something... Study.Com 's Assign lesson Feature third are ordinal numbers indicate the order or rank of things in a set e.g.! Elements, so it is not a cardinal number to help you succeed visit the Math Kids! So ostensibly are the property of their respective owners objects/persons are specified in set. Just create an account provide you with a great user experience 310 } Find the right school can. You may have heard something like the average household has 1.8 children '' a is... At this picture ( e.g., sixth in line ; fourth place ), 2007 behave as indeclinable.... Identified with the correct English spelling: one in mathematics zero ( 0 ) is the cardinal answers... And c be sets in a simple manner, devoid of technical jargon to question... Of cardinal numbers. all wear jerseys with numbers on them ; could this What! That comes to mind is the professional baseball team. higher-level mathematics and logic he has whole! Are the property of their respective owners word used to describe the quantity of something on list. Every family has a whole number of the set of all rational numbers. in. Of six sided dice, one red and one in linguistics and one cardinal number example, are! Be sets in a set ( e.g., sixth in line ; place! Regardless of age or education level Scholars® Bringing Tuition-Free college to the Community figures!: one in linguistics and one black, there are 6 baseballs in the.! 30 days, just create an account Definition & Examples, What are Rectangular numbers all wear with. Math 6300 Klaus Kaiser April 9, 2007 this lesson you must be a Study.com Member, like. Decimals because they show quantity numeral is a nominal number earn credit-by-exam of... Years of college and save thousands off your degree and fourth government, 24 in sports an, Find cardinal... A zip code or a player on a team. an account at!, Over 83,000 lessons in all major subjects, { { courseNav.course.mDynamicIntFields.lessonCount } } Find. Right school in the race fourth place ) second, third, and four are. Each of the objects/persons is defined by ordinals grade 5 teacher and Vice-Principal that a! August 1669 zero ( 0 ) is the Difference Between Blended Learning & Distance?... Coursenav.Course.Mdynamicintfields.Lessoncount } }, What is Subtraction in Math: Definition & Examples, What meant. ( e.g., a baseball is third in line three 99s, that person uses cardinal values info need...	2021-04-15T10:55:28	{ "domain": "dccapco.com", "url": "http://www.dccapco.com/company-location-yflc/1978cd-cardinal-number-example", "openwebmath_score": 0.49610114097595215, "openwebmath_perplexity": 1825.8435610759211, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9664104933824753, "lm_q2_score": 0.8670357615200474, "lm_q1q2_score": 0.8379124580708391 }
http://mathhelpforum.com/pre-calculus/167638-solving-exponential-equation.html	# Math Help - Solving an exponential equation 1. ## Solving an exponential equation I'm a little stuck (again). How would I go about solving an equation like this: $4^2^x-4^x-20=0$ And a more complicated one such as: $2^x+12(2)^-^x=7$ All help greatly appreciated! 2. Originally Posted by youngb11 I'm a little stuck (again). How would I go about solving an equation like this: $4^2^x-4^x-20=0$ This is a quadratic in 4^x and happily it will factor to $(4^x-5)(4^x+4)=0$. Note that $4^x > 0$ And a more complicated one such as: $2^x+12(2)^-^x=7$ All help greatly appreciated! Multiply through by the LCD which is 2^x since $2^{-x} = \dfrac{1}{2^x}$. You'll get $2^{2x} + 12 = 7 \cdot 2^x$ which, again, is a quadratic equation 3. $(4^{x})^{2} - 4^{x} - 20 = 0$ now let $u = 4^{x}$ 4. Originally Posted by e^(ipi) Multiply through by the LCD which is 2^x since $2^{-x} = \dfrac{1}{2^x}$. You'll get $2^{2x} + 12 = 7 \cdot 2^x$ which, again, is a quadratic equation Thanks! However, I'm a little stuck once we get to $7 \cdot 2^x$. The answer in the back of the book shows $x=2$ and $x=\dfrac{log3}{log2}$. How would we multiply $7$ with $2^x$ but still come to such an answer? Thanks! 5. Originally Posted by youngb11 Thanks! However, I'm a little stuck once we get to $7 \cdot 2^x$. The answer in the back of the book shows $x=2$ and $x=\dfrac{log3}{log2}$. How would we multiply $7$ with $2^x$ but still come to such an answer? Thanks! You were told what to do: Originally Posted by e^(ipi) [snip]Multiply through by the LCD which is 2^x since $2^{-x} = \dfrac{1}{2^x}$. You'll get $2^{2x} + 12 = 7 \cdot 2^x$ which, again, is a quadratic equation This is not the answer, it is how to get the answer. And how to do this was explained to you with the first question. Let $2^x = w$. Re-arrange to get a quadratic equation in w that is equal to zero. Solve for w. Hence solve for x. Please make an atttempt. Show your work, say where you get stuck if you need more help. 6. Originally Posted by mr fantastic This is not the answer, it is how to get the answer. And how to do this was explained to you with the first question. Let $2^x = w$. Re-arrange to get a quadratic equation in w that is equal to zero. Solve for w. Hence solve for x. Please make an atttempt. Show your work, say where you get stuck if you need more help. Ah, okay. It's much clearer when you replace $2^x$ for W. Thanks again!	2015-05-06T04:30:28	{ "domain": "mathhelpforum.com", "url": "http://mathhelpforum.com/pre-calculus/167638-solving-exponential-equation.html", "openwebmath_score": 0.9543073773384094, "openwebmath_perplexity": 393.4479514351089, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9790357604052424, "lm_q2_score": 0.8558511488056151, "lm_q1q2_score": 0.8379088802646056 }
https://math.stackexchange.com/questions/17083/how-to-find-unique-multisets-of-n-naturals-of-a-given-domain-and-their-numbers	How to find unique multisets of n naturals of a given domain and their numbers? Let's say I have numbers each taken in a set $A$ of $n$ consecutive naturals, I ask myself : how can I found what are all the unique multisets, which could be created with $k$ elements of this set $A$? For example I've got $A=[1,2,\dots,499]$. If I wanted to create unique multiset of 3 elements, I would search all the multisets $\{a,b,c\}$ such as $a\leq b\leq c$ and then have all the unique multisets. As such for each elements $a$ there is $499-(a-1)=500-a$ possibilities for $b$ and $500-b$ possibilities for $c$. Unfortunately I'm stumped here and can't find the number of possible combination, as I didn't do any maths for years. I know that I should have some kind of product, but I don't know how to find the product anymore. So first, I would like to know if I am right in my original assumption, or if I am looking in the wrong direction. Second I would like to approach that as if I were doing an homework, as I would like to understand the logic behind it: What would be the formula to give the numbers of multisets of $k$ elements from a set $A$ of $n$ consecutive naturals? P.S. I'm totally new to math.SE, as such, I'm not sure I tagged the post appropriately. • Are you looking for the total number or for an algorithm? – Aryabhata Jan 11 '11 at 17:29 • @Moron: Both actually, the first for curiosity and getting back in the bath, and the algorithm because I would need such an algorithm for one of my problems. – Eldros Jan 13 '11 at 8:58 • The stars and bars method also gives an algorithm. All you need is to be able to enumerate combinations (of choosing k from n+k-1), which is a pretty standard problem. – Aryabhata Jan 13 '11 at 9:32 Hint: If $O_i$ is the number of occurrences of $i$ in the multiset, the tuple $(O_1, O_2, \dots, O_n)$ uniquely identifies the multiset. What constraint can you write down with the $O_i$, if the multiset has exactly $k$ elements? Ok, to elaborate. The number of multisets of size $k$ is equal to the number of non-negative integral solutions to the equation $$O_1 + O_2 + \dots + O_n = k$$ Since the stars and bars method was already mentioned, here is a derivation of the formula using a more general approach, which is one of the main building blocks of analytic number theory: Generating Functions. The number of solutions to the above equation is same as the coefficient of $x^k$ in $$(1 + x + x^2 + x^3 + \dots )(1+x + x^2 + x^3 + \dots) \dots (1+ x + x^2 + x^3 + \dots) \ \text{repeated} \ n \ \text{times} \ \text{(why?)}$$ The above is same as $$\frac{1}{(1-x)^n} = (1-x)^{-n}$$ as $$(1 + x + x^3 + x^3 + \dots) = \frac{1}{1-x}$$ Now we apply binomial theorem, (which is also valid for negative exponents) The coefficient of $x^k$ is $$-1^{k}\ \frac{-n(-n-1)(-n-2)\dots(-n-(k-1))}{k!} = \frac{n(n+1)(n+2)\dots(n+k-1)}{k!}$$ $$= \frac{(n-1)!n(n+1)(n+2)\dots(n+k-1)}{(n-1)!k!}= {n+k-1 \choose k}$$ • I had to search in wikipedia to remember what a coefficient is. It shows how much I lost... – Eldros Jan 13 '11 at 10:25 • @ELdros: The page you linked to is a binomial coefficient. A coefficient is more general. For instance in $4.3x^2 + 100.1x + 1.5$, $4.3, 100.1, 1.5$ are coefficients of $x^2,x,x^0$ respectively. It just turns out that in certain cases(like expansion of $(1+x)^m$) they are all also binomial coefficients. – Aryabhata Jan 13 '11 at 16:52 For a three-item multiset, the easiest approach is to start by picking $b$. As you say, there are $500-b$ choices for $c$. Similar logic says there are $b$ choices for $a$. Multiplying gives the number of choices for a given $b$. Now just sum over $b$. So you have $$\sum_{b=1}^{499} (500-b)b$$ For a larger set $A$, just change 499 to the largest number in $A$. For larger numbers of elements, you can make the formula have an upper limit of $n$ instead of 499. This will give you the number of 4 element multisets with the upper element $n$. Now you can sum over $n$ from 1 to the largest number in $A$. This gets tedious with increasing $k$, but maybe you can find a pattern. We could have followed the strategy in the second paragraph to get the 3 element sets, but the symmetry around the middle element made it a bit easier. We would have started by saying there are $n$ one element multisets out of $n$. Then there are $\sum_{i=1}^{n} i=\frac{n(n+1)}{2}$ two element multisets out of $n$ and $$\sum_{i=1}^n \frac{i(i+1)}{2}=\frac{n(n+1)(2n+1)}{12}+\frac{n(n+1)}{4}$$ three element multisets. This looks different, but agrees with the prior result. Added: For the general case, see the section "Multiset coefficients" in Wikipedia's Mulitset • Correct me if I'm wrong, but as I see it, by this logic a series of embedded sums on $i$ is equal to ${n+k-1 \choose k}$ where $k$ is the level at which it is embedded? Or in other words $\sum_{i=1}^n{i+k-2 \choose k-1}={n+k-1 \choose k}$? That could be a great way to find out the value of the sums of the exponents of $i$. (That's a lot of of) – Eldros Jan 13 '11 at 9:20 Often a combinatorial counting problem is solved elegantly by relating it to something one already knows how to count. Let's imagine a fly crawling from one corner of a grid to another (seriously, bear with me and we'll make it back to multisets shortly!). The fly's walk is efficient in that each step is either up or to the right. So if it goes from say $(0,0)$ to $(m,p)$, then the fly will take in all $p$-steps up and $m$-steps to the right. The challenge is to calculate how many possible such paths there are. Each path can be considered an arrangement of steps up and steps to the right, with a known number of each. Here's a way to map your counting of $k$-multisets taken from some set $A$ of $n$ items and recast it as a question about how many paths there are for the fly to take. Consider the horizontal lines of the grid as being labeled by the items in set $A$, so (careful about the fence posts) there are $p = n-1$ vertical edges along the grid. Consider the number of a particular item in set $A$ as being the number of horizontal steps along its respective line. Since there will be $k$ entries in total for our multiset, this means $m = k$ horizontal edges across any path (and across the grid as a whole). Put those ideas together and you'll have a recipe for counting the $k$-multisets from $A$. If you actually wanted to list said multisets, it's more of a programming problem, but I think it would be fair game to discuss the algorithm to produce such a list. • Now that I think about it, wouldn't discussing an algorithm be a matter of theoretical computer science? – Eldros Jan 13 '11 at 9:08 • @Eldros: It's a possibility, though I've seen sniping in that forum about questions that are not "at the research level". Another possibility would be SO, which certainly has less than research level queries but tends to be more concrete. There's enough of a range there, certainly. However the program enumeration of these is easy enough I could probably answer it in a comment. – hardmath Jan 13 '11 at 14:05 In parallel to Moron's hint, I'd suggest looking into the "stars and bars" technique, which is explained well in this answer by Ben Alpert (or here's an answer of mine explaining it). Another method from Aryabhatta's post: He gave the equation $O_1+O_2+O_3+\cdots O_n=k$. Here the solutions are non negative integers. So there is a bijection between multiset of size $k$ and weak compositions(They are a way of expressing a number $n$ as a sum of non nengative integers). Take a weak composition. Say $$a_1+a_2+a_3+\cdots a_k=n$$ To count $k$ weak compositions of $n$ we have $$\binom {n+k-1}{k-1}$$ways. So in this case we have $$\binom {n+k-1}{n-1}=\binom{n+k-1}{k}$$ Hence your result	2019-10-14T00:41:42	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/17083/how-to-find-unique-multisets-of-n-naturals-of-a-given-domain-and-their-numbers", "openwebmath_score": 0.742950975894928, "openwebmath_perplexity": 207.89435897548893, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357567351325, "lm_q2_score": 0.8558511506439707, "lm_q1q2_score": 0.8379088789233538 }
https://math.stackexchange.com/questions/1794741/given-a-subseteq-x-in-the-discrete-and-the-trivial-topology-find-closure-of/1794746	# Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$ Given $A \subseteq X$ in the discrete and the trivial topology, find closure of $A$ Note the definition of closure I am using is one in Munkres: $x \in \overline A \iff \text{ for every open set } U \text{ containing } x, U \cap A \neq \varnothing$ Attempt: Easy case. • Let $(X, \tau_{trivial})$ be the trivial topological space. Let $A \subseteq X$. • Then $\overline A = \{ x \in X\| \forall U \in \tau_{trivial}, x \in U \implies U \cap A \neq \varnothing\}$ • Since $\tau_{trivial} = \{\varnothing, X\}$, the only open set that contains $x$ and has non-empty intersection with $A$ is $X$. • Hence $\overline A = X$ Case I am having trouble with: • Let $(X, \tau_{discrete})$ be the discrete topological space. Let $A \subseteq X$. • Then $\overline A = \{ x \in X\| \forall U \in \tau_{discrete}, x \in U \implies U \cap A \neq \varnothing\}$ • Since $\tau_{discrete} = \{A\| A \subseteq X\}$ • ...missing arguments • Hence $\overline A = A$ Question: I realized that given an arbitrary open set $U$ containing $A$, then there could exist an open set $V$ such that $x \in V \subset U, x \notin A$, therefore $\overline A$ necessarily equal to $A$ itself. How do I formally express this in the second proof? • In the discrete topology, every set is open. In particular, if $x \in \overline{A} \setminus A$, then $x \in X \setminus A$, which is open, and so $x \in (X \setminus A) \cap A$, a contradiction. Thus, it must be the case that $\overline{A} \setminus A$ is empty. – qaphla May 21 '16 at 23:28 In discrete metric space, every set is open. So for $x$ $\{x\} \cap A \ne \emptyset$ so $x \in A$ so $\overline {A} \subset A$. And if $x \in A$ and if $x \in U; U$ open then $\{x\} \subset U \cap A$ so $s \in \overline A$. So $A \subset \overline A$. (Trivially true for all topologies.) So $A = \overline{A}$. Here's one way to phrase this: Let $x \in \overline{A}$. Then every open set containing $x$ must intersect $A$ nontrivially. Since $X$ is under the discrete topology, we know that $\{x\} \subseteq X$ is open. $\overline{A}$ is the smallest closed subset of $X$ containing $A$. If $X$ has discrete topology, every subset is open hence every subset is closed. So $A$ is closed and so it must be the smallest closed subset containing $A$. We have $\overline{A} = A$.	2019-10-21T08:34:00	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/1794741/given-a-subseteq-x-in-the-discrete-and-the-trivial-topology-find-closure-of/1794746", "openwebmath_score": 0.9864139556884766, "openwebmath_perplexity": 87.7007483171415, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357579585025, "lm_q2_score": 0.8558511488056151, "lm_q1q2_score": 0.8379088781705605 }
https://www.physicsforums.com/threads/unbounded-sets.383773/	# Unbounded sets 1. Mar 4, 2010 ### BelaTalbot Is a bounded set synonymous to a set that goes to infinity? I feel like unless a set is (-infinity, n) or [n, infinity) it is not going to be unbounded. The other thing that I was wondering is can a set be neither open nor closed AND unbounded? Doesn't the definition of open/closed imply that there is a boundary? Thanks! 2. Mar 5, 2010 A set being bounded just means it is contained in some ball. Boundedness isn't really related to being open or closed, and also has nothing to do with boundary. 3. Mar 5, 2010 ### Studiot No and furthermore the bound does not even have to be a member of the set. A set can have lots of bounds, even an infinite number of them. A bound can be 'above' or 'below'. An upper bound is simply a real number that is either greater than or equal to every member of the set or less than/equalto every member of the set. so 7,8,9,10 etc all form upper bounds to the set {2,3,4,5,6} ; none are memebrs of the set. but wait 6 also forms an upper bound as it satisfies the equal to and is a member. Similarly 1,0,-1,-2 all form lower bounds that are not members and 2 forms a lower bound that is Sets which 'go to infinity' are unbounded. However a set can contain an infinite number of members and still be bounded, above and/or below. for example the set {1/1, 1/2, 1/3, .....} is bounded above by 2 ,1 etc and bounded below by 0, -1 etc, but contains an infinite number of members. The set $$\{ - \infty ,..... - 2, - 1,0,1,\frac{1}{2},\frac{1}{3},.......\}$$ is not bounded below but is bounded above. If a set has both a lower and upper bound so that the modulus of any member, x, is less than or equal to some real number K then the set is bounded. (No upper or lower) If for any $$x \in S$$ there exists a K such that $$\left\| x \right\| \le K$$ The set S is bounded. Hope this helps. 4. Mar 5, 2010 ### g_edgar "bounded" does not mean "has a boundary" Isn't English confusing? 5. Mar 5, 2010 ### BelaTalbot So I guess what I'm really wondering is can a set be unbounded if it doesn't go to infinity? That's what I can't seem to wrap my brain around, because I feel that if a set doesn't go to infinity, there will always be a real number K larger than the members of the set. 6. Mar 5, 2010 ### Studiot Any finite set S of real numbers is bounded. Infinite sets can be bounded or unbounded. Remember that to be bounded a set must have both a lower bound and an upper bound. Sets with only a lower or upper bound are unbounded. A further question for you to ponder: Can a set of complex numbers be bounded? Last edited: Mar 5, 2010	2018-09-22T06:59:34	{ "domain": "physicsforums.com", "url": "https://www.physicsforums.com/threads/unbounded-sets.383773/", "openwebmath_score": 0.7291372418403625, "openwebmath_perplexity": 546.1385048452983, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357597935574, "lm_q2_score": 0.8558511469672594, "lm_q1q2_score": 0.8379088779412783 }
https://math.stackexchange.com/questions/1802597/to-find-whether-a-is-a-prime-number	# To find whether $a$ is a prime number I have been using this rule to determine whether a number is a prime number, but not how to prove it. Why it has to be $\sqrt{a}$? If $a$ is not divisible by all the prime numbers less than or equal to $\sqrt{a}$, then $a$ is a prime number. • HInt $\$ The least nontrivial factor of a composite number is less than its square-root. – Gone May 28 '16 at 2:14 Assume $a$ is not a prime number. Then it can be written as $a = bc$ with $b, c \ge 2$. Then the smaller of $b, c$ is less than or equal to $\sqrt a$, for otherwise the product would be greater than $\sqrt a \, \sqrt a = a$. This smaller factor is then either a prime itself or has a prime factor that is even smaller. Divisors come in pairs. If $n$ divides $a$, so does $a/n$, as $\sqrt{a}\sqrt{a}=a$, one of the factors in a pair must be smaller than or equal to $\sqrt{a}$. Just another point of view Suppose that $a$ is divisible by a prime $p$. This means that: $$a = p \cdot k \Rightarrow p = \frac{a}{k}.$$ Moreover, suppose that $p > \sqrt{a}$, then: $$\frac{a}{k} > \sqrt{a} \Rightarrow k < \sqrt{a}.$$ This means that I can only use the primes $p \in [\sqrt{a}, a]$ to determine if $a$ is prime, instead of the set $[0, \sqrt{a}]$. Which is the set that best suits for the algorithm? I guess the smallest, since you will have a smaller number of primes to be tested. The set $[\sqrt{a}, a]$ has length $a-\sqrt{a}$, while the set $[0, \sqrt{a}]$ has length $\sqrt{a}$. Notice that: $$\sqrt{a} < a-\sqrt{a} \Rightarrow 2\sqrt{a} < a \Rightarrow \sqrt{a} > 2 \Rightarrow a > 4.$$ In general, you want test a number $a$ bigger than $4$. For this reason, you just look to the set $[0, \sqrt{a}]$... it is just faster! Example. Take $a = 77$ and notice that $\sqrt{a} \simeq 8.78$. If you use the set $[0, \sqrt{a}]$, the you will have to check if $a$ is divisible by $3,5,7$ (only three numbers). On the other hand, if you choose the set $[\sqrt{a}, a]$, you will have to check if $a$ is divisible by $11, 13, 17, 19, 23, 29, 31, 37, ...$. Hey, this is too much, no?! Let $ab=p$ where $a,b$ are possible factors of prime candidate $p$. If you find a factor $b \geq \sqrt{p}$, then $a \leq \sqrt{p}$. But if you checked all $a \leq \sqrt{p}$ and didn't find any factors ... For every prime divisor of $n$ that is less than $n$ and greater than $\sqrt{n}$, there must be a prime number less than $\sqrt{n}$ that divides $n$. This follows from the fact that it is impossible for a number $n$ to have two prime divisors $p$ and $q$ that were both greater than $\sqrt{n}$, since then their product would be $$pq>(\sqrt{n})^2=n$$ Thus we need only check the primes in the range $1,...,\sqrt{n}$ to see if they divide $n$. This may be easiest to see if illustrated by example. Consider the factors of $$16$$ as found by trial division. That is, we will divide $$16$$ by $$1$$ then $$2$$ then... and each time the remainder is $$0$$ we will record both the divisor and the quotient as a factor pair: $$1$$: Yes, this yields the factor pair $$(1, 16)$$. $$2$$: Yes, this yields the factor pair $$(2, 8)$$. $$3$$: No, $$16/3$$ is not a whole number, i.e., it has remainder $$1$$. $$4$$: Yes, this yields the factor pair $$(4,4)$$. $$5$$: No, $$16/5$$ is not a whole number, i.e., it has remainder $$1$$. $$6$$: No, $$16/6$$ is not a whole number, i.e., it has remainder $$4$$. $$7$$: No, $$16/7$$ is not a whole number, i.e., it has remainder $$2$$. $$8$$: Yes, this yields the factor pair $$(8,2)$$. $$\ldots$$ Note that we already had that last factor pair, though in the other order, as $$(2,8)$$. Trying other examples and searching for factor pairs, you will always find that, at some point, they repeat. (If the number is prime, e.g., $$11$$, then the only factor pairs will be $$(1, 11)$$ and $$(11, 1)$$.) The question now becomes, at what point do the numbers repeat? That is, when can we stop carrying out our trial division? For the case of $$16$$ they repeat after $$\sqrt{16} = 4$$. More generally, all the factors appear in this way once you check up to the square root. For if the target number is $$N$$ and you've found all factors less than or equal to $$\sqrt{N}$$, could there be new factors? If so, you'd need to find a factor pair with two new factors, i.e., each of which is greater than $$\sqrt{N}$$. But then their product would be greater than $$\sqrt{N}\sqrt{N} = N$$; so, they cannot form a factor pair: Their product is bigger than the target number!	2020-07-06T17:34:07	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/1802597/to-find-whether-a-is-a-prime-number", "openwebmath_score": 0.9157206416130066, "openwebmath_perplexity": 126.1484223397611, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357591818726, "lm_q2_score": 0.8558511469672594, "lm_q1q2_score": 0.8379088774177672 }
https://mathhelpboards.com/threads/using-of-jensens-inequality-to-prove-cosa-cosb-cosc-less-than-or-equal-to-3-2-where-a-b-c-pi.466/	# TrigonometryUsing of Jensen's Inequality to prove cosA+cosB+cosC less than or equal to 3/2 (where A+B+C=pi) #### anemone ##### MHB POTW Director Staff member Hi, Given $A+B+C=\pi$, I need to prove $cosA+cosB+cosC\leq \frac{3}{2}$. I wish to ask if my following reasoning is correct. First, I think of the case where A and B are acute angles, then I can use the Jensen's Inequality to show that the following is true. $cos\frac{A+B}{2}\geq \frac{cosA+cosB}{2}$ Carrying on with the working, I get $sin\frac{C}{2}\geq \frac{cosA+cosB}{2}$ $2sin\frac{C}{2}\geq cosA+cosB$ $cosA+cosB\leq 2sin\frac{C}{2}$ $cosA+cosB+cosC\leq 2sin\frac{C}{2}+cosC$ $cosA+cosB+cosC\leq 2sin\frac{C}{2}+1-2sin^2C$ Completing square the RHS to obtain $cosA+cosB+cosC\leq -2(sin\frac{C}{2}-\frac{1}{2})^2+\frac{3}{2}$ Now, it's obvious to see that $cosA+cosB+cosC\leq \frac{3}{2}$ My question is, can I solve the question by thinking A and B are acute angles and ignore the angle C right from the start so that I can let f(x)=cosx and notice that the curve of f(x)=cos x in the interval $x\in (0,\frac{\pi}{2})$ take the convex shape which in turn I can apply the Jensen's inequality without a problem? Thanks. #### Opalg ##### MHB Oldtimer Staff member My question is, can I solve the question by thinking A and B are acute angles and ignore the angle C right from the start so that I can let f(x)=cosx and notice that the curve of f(x)=cos x in the interval $x\in (0,\frac{\pi}{2})$ take the convex shape which in turn I can apply the Jensen's inequality without a problem? Yes: a triangle can have at most one obtuse angle, and the other two (acute) angles must necessarily be adjacent. Without loss of generality (favourite math phrase), take them to be A and B. And BTW that is a very neat proof! #### anemone ##### MHB POTW Director Staff member Yes: a triangle can have at most one obtuse angle, and the other two (acute) angles must necessarily be adjacent. Without loss of generality (favourite math phrase), take them to be A and B. Thanks, Opalg. And I think 'obviously' is also one of the favourite math phrase too! But unfortunately that is a phrase to which I considered not so true and annoying some (if not most) of the time. And BTW that is a very neat proof! Thanks. Seriously, your compliment just made my day.	2021-08-01T10:22:51	{ "domain": "mathhelpboards.com", "url": "https://mathhelpboards.com/threads/using-of-jensens-inequality-to-prove-cosa-cosb-cosc-less-than-or-equal-to-3-2-where-a-b-c-pi.466/", "openwebmath_score": 0.9018501043319702, "openwebmath_perplexity": 695.9239556090874, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357549000773, "lm_q2_score": 0.8558511506439708, "lm_q1q2_score": 0.8379088773528197 }
http://math.stackexchange.com/questions/289054/solve-y-prime-prime-y-prime2-y-prime-0	# Solve $y^{\prime \prime}-(y^{\prime})^2-y^{\prime}=0$ Solve $y^{\prime \prime}-(y^{\prime})^2-y^{\prime}=0$. I use $$u=\frac{dy}{dx}$$ to transform the DE into $$\frac{du}{dx}-u^2-u=0$$. I know that this is an Bernoulli equation with $n=2$. I get the final solution is $$y=-ln\|1-Ae^x\|+D$$ where $A=+-e^c$. But my lecturer's answer is $$y=-ln\|C_1+c_2e^x\|$$. May I know what is the difference my answer and my lecturer answer ? - There is no difference. You use different constants. Put $D=-\ln C_1$ in your solution. – Artem Jan 28 '13 at 16:50 The answers are the same. $D$ is an arbitrary constant so let $D=\ln(D')$ for arbitrary $D'$ and $e^c$ is just another constant so $e^c=c'$ for an arbitrary $c'$, and so now $$y=-ln\|1-Ae^x\|+D=-ln\|1-c'e^x\|+\ln(D')=-\ln(D' -D' c' e^x\|$$ Let $c_1=D'$ and $c_2=c'D'$ and you're done. You can see that the two answers are equivalent as follows: \begin{align} -\ln \|1 - Ae^x\| + D & = -\ln \|1-Ae^x\| + \ln e^D \\ & = -\ln e^D\|1 - Ae^x\| \\ & = - \ln \|e^D - e^D Ae^x\| \\ & = -\ln \| c_1 + c_2 e^x\|,\end{align} where $c_1 = e^D$ and $c_2 = -Ae^D$. $c_2=-Ae^D{}{}$. – Cameron Buie Jan 28 '13 at 19:21	2014-07-30T21:19:46	{ "domain": "stackexchange.com", "url": "http://math.stackexchange.com/questions/289054/solve-y-prime-prime-y-prime2-y-prime-0", "openwebmath_score": 0.9996845722198486, "openwebmath_perplexity": 353.7655747114507, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357542883923, "lm_q2_score": 0.8558511506439708, "lm_q1q2_score": 0.8379088768293084 }
https://math.stackexchange.com/questions/526520/convert-boolean-expression-into-sop-and-pos	# Convert boolean expression into SOP and POS Convert the following expression into SOP (sum of products) and POS (product of sums) canonical forms using boolean algebra method: $(ac + b)(a + b'c) + ac$ Attempt at solution: $(ac + b)(a + b'c) + ac$ 1. $(a + b)(c + b)(a + b')(a + c) + ac$ 2. $...$ 3. $...$ I'm stuck at this point. Any help would be greatly appreciated. Thanks. One way to get the SoP form starts by multiplying everything out, using the distributive law: \begin{align} (ac+b)(a+b'c)+ac&=ac(a+b'c)+b(a+b'c)+ac\\ &=aca+acb'c+ba+bb'c+ac\\ &=ac+ab'c+ab+ac\\ &=ac+ab'c+ab\;. \end{align} Then make sure that every term contains each of $a,b$, and $c$ by using the fact that $x+x'=1$: \begin{align} ac+ab'c+ab&=ac(b+b')+ab'c+ab(c+c')\\ &=abc+ab'c+ab'c+abc+abc'\\ &=abc+ab'c+abc'\;. \end{align} Alternatively, you can make what amounts to a truth table for the expression: $$\begin{array}{cc} a&b&c&ac+b&b'c&a+b'c&ac&(ac+b)(a+b'c)+ac\\ \hline 0&0&0&0&0&0&0&0\\ 0&0&1&0&1&1&0&0\\ 0&1&0&1&0&0&0&0\\ 0&1&1&1&0&0&0&0\\ 1&0&0&0&0&1&0&0\\ 1&0&1&1&0&1&1&1\\ 1&1&0&1&1&1&0&1\\ 1&1&1&1&0&1&1&1 \end{array}$$ Now find the rows in which the expression evaluates to $1$; here it’s the last three rows. For a product for each of those rows; if $x$ is one of the variables, use $x$ if it appears with a $1$ in that row, and use $x'$ if it appears with a $0$. Thus, the last three rows yield (in order from top to bottom) the terms $ab'c$, $abc'$ and $abc$. You can use the truth table to get the PoS as well. This time you’ll use the rows in which the expression evaluates to $0$ — in this case the first five rows. Each row will give you a factor $x+y+z$, where $x$ is either $a$ or $a'$, $y$ is either $b$ or $b'$, and $z$ is either $c$ or $c'$. This time we use the variable if it appears in that row with a $0$, and we use its negation if it appears with a $1$. Thus, the first row produces the sum $a+b+c$, the second produces the sum $a+b+c'$, and altogether we get $$(a+b+c)(a+b+c')(a+b'+c)(a+b'+c')(a'+b+c)\;.\tag{1}$$ An equivalent procedure that does not use the truth table is to begin by using De Morgan’s laws to negate (invert) the original expression: \begin{align} \Big((ac+b)(a+b'c)+ac\Big)'&=\Big((ac+b)(a+b'c)\Big)'(ac)'\\ &=\Big((ac+b)'+(a+b'c)'\Big)(a'+c')\\ &=\Big((ac)'b'+a'(b'c)'\Big)(a'+c')\\ &=\Big((a'+c')b'+a'(b+c')\Big)(a'+c')\\ &=(a'b'+b'c'+a'b+a'c')(a'+c')\\ &=a'b'(a'+c')+b'c'(a'+c')+a'b(a'+c')+a'c'(a'+c')\\ &=a'b'+a'b'c'+a'b'c'+b'c'+a'b+a'bc'+a'c'+a'c'\\ &=a'b'+a'b'c'+b'c'+a'b+a'bc'+a'c+a'c'\\ &=a'b'+b'c'+a'b+a'(c+c')\\ &=a'b+b'c'+a'b+a'\\ &=b'c'+a'\;, \end{align} where in the last few steps I used the absorption law $x+xy=x$ a few times. Now find the SoP form of this: \begin{align} b'c'+a'&=b'c'(a+a')+a'(b+b')(c+c')\\ &=ab'c'+a'b'c'+a'b(c+c')+a'b'(c+c')\\ &=ab'c'+a'b'c'+a'bc+a'bc'+a'b'c+a'b'c'\\ &=ab'c'+a'b'c'+a'bc+a'bc'+a'b'c\;. \end{align} Now negate (invert) this last expression, and you’ll have the PoS form of the original expression: \begin{align} (ab'c'&+a'b'c'+a'bc+a'bc'+a'b'c)'\\ &=(ab'c')'(a'b'c')'(a'bc)'(a'bc')'(a'b'c)'\\ &=(a'+b+c)(a+b+c)(a+b'+c')(a+b'+c)(a+b+c')\;, \end{align} which is of course the same as $(1)$, though the factors appear in a different order. • Are you aware of an automated tool that does this job (efficiently)? – pushpen.paul Dec 24 '18 at 1:18	2019-07-15T22:14:33	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/526520/convert-boolean-expression-into-sop-and-pos", "openwebmath_score": 0.7091560363769531, "openwebmath_perplexity": 389.73234315926794, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357640753516, "lm_q2_score": 0.8558511414521922, "lm_q1q2_score": 0.8379088762064089 }
https://math.stackexchange.com/questions/447450/relationship-between-the-rows-and-columns-of-a-matrix	# Relationship between the rows and columns of a matrix I am having trouble understanding the relatioship between rows and columns of a matrix. Say, the following homogeneous system has a nontrivial solution. $$3x_1 + 5x_2 − 4x_3 = 0 \\ −3x_1 − 2x_2 + 4x_3 = 0 \\ 6x_1 + x_2 − 8x_3 = 0\\$$ Let A be the coefficient matrix and row reduce $\begin{bmatrix} A & \mathbf 0 \end{bmatrix}$ to row-echelon form: $\begin{bmatrix}3&5&-4&0\\-3&-2&4&0\\6&1&-8&0\\ \end{bmatrix} \rightarrow \begin{bmatrix}3&5&-4&0\\0&3&0&0\\0&0&0&0\\ \end{bmatrix}$ $\quad a1 \quad a2 \quad \,a3$ Here, we see $x_3$ is a free variable and thus we can say 3rd column,$\,a_3$, is in $\text{span}(a_1, a_2)$ But what does it mean for an echelon form of a matrix to have a row of $0$'s? Does that mean 3rd row can be generated by 1st & 2nd rows? just like 3rd column can be generated by 1st & 2nd columns? And this raises another question for me, why do we mostly focus on columns of a matrix? because I get the impression that ,for vectors and other concepts, our only concern is whether the columns span $\mathbb R^n$ or the columns are linearly independent and so on. I thought linear algebra is all about solving a system of linear equations, and linear equations are rows of a matrix, thus i think it'd be logical to focus more on rows than columns. But why? • Thanks and upvote for introducing me to the term "echolon form". – k.stm Jul 19 '13 at 15:10 • What do you mean by “$x_3$ is a free variable”? It certainly can be generated like that, just by reversing the row operations you used to get the echelon form. I don't see what you mean by focusing on columns, though. I don't really focus on rows or columns most of the time. In fact, I often can't tell which are which. Linear algebra is about much more than just solving linear equations. – tomasz Jul 19 '13 at 15:13 • "Mostly focusing on columns" is a convention in some texts. Others focus on (and define things in terms of) rows. – The Chaz 2.0 Jul 19 '13 at 15:18 • @tomasz i mean the last row being the rows of 0's, what's the relation between 3rd row and the other two? 3rd row is obviously a different equation than the other 2 rows and yet its coefficient became all 0.. – Belphegor Jul 19 '13 at 15:19 Having a row of $0$'s in the row-echelon form means that we were able to write the third row of $A$ as a linear combination of the second and first rows. As it so happens for square matrices, this is true precisely when we can write the columns as a linear combination of each other (that is, when the columns are not linearly independent). If you further reduce this to reduced row-echelon form, you get $$\begin{bmatrix} 1 & 0 & -4/3 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Because the third row lacks a pivot, $x_3$ is our free variable, which means that we can write $a_3$ as a linear combination of the other two columns. There's a very good reason for focusing on the columns of a matrix. This comes out of using $A$ as a linear transformation, where the "column space" gives us the "range" of the function $f(\vec x) = A \vec x$. Excellent question. In some sense, the equations and variables represent equivalent information. Sometimes it is easier to approach the problem from the point of view of rows-equations-constraints, and sometimes - from the point of view of variables. This is very deeply related to the notion of the dual problem in linear programming. That is exactly what converts equations/constraints to rows and vice versa, looking at a different problem. Mathematically, this boils down to either working with the matrix $A$ or with some form of $A^T$, which converts rows into columns and columns into rows. Not surprisingly, the main characteristic properties for $A$ and $A^T$ are the same, like rank, eigenvalues/singular values, determinant, trace, etc. Two reference links on duality: Clearly, you matrix is singular, i.e. its determinant is zero. When we study the linear system $Ax=b$, we have two options: either there's an affine space of solutions ($b\in Im A$) or there're no solutions at all. When you study the echolon form, you eventually arrive to the line of the form $\begin{pmatrix}0&0&0&a\end{pmatrix}$. If $a\ne 0$, then the system is incompatible (zero combination of variables produces non-zero). If $a=0$, then you have an affine subspace of solutions.	2020-11-26T10:22:07	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/447450/relationship-between-the-rows-and-columns-of-a-matrix", "openwebmath_score": 0.8138293623924255, "openwebmath_perplexity": 239.26270595928014, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9790357634636668, "lm_q2_score": 0.8558511414521922, "lm_q1q2_score": 0.8379088756828977 }
http://rmjo.royalvillagemilano.it/the-capacitance-of-two-concentric-spherical-metal-shells-with-radii-a-and-b-is.html	A spherical capacitor is another set of conductors whose capacitance can be easily determined (Figure $$\PageIndex{5}$$). A spherical capacitor consists of 2 concentric conducting spherical shells with radii a and b with a < b. Spherical Capacitor 4. (b) Show that when the radii of the shells are nearly equal, the capacitance approximately. 4 Conductors – capacitance Christopher Crawford PHY 416 2014-10-15 Exam 2 – Friday Oct 24 Integrate E(r) or V(r) over a charge distribution Parametrize source points r’(u,…) on surface, path or volume Calculate field point r and displacement vector, r=r-r’ Reduce integrals to parameters and constants, including unit vectors Capacitance calculation Application of Gauss’ law. The induced charges are determined with a measuring amplifier. Three concentric spherical metallic shells A , B and C of radii a , b and c $\;(a < b < c)\;$ have charge densities $\;\sigma , - \sigma \; and \; \sigma\;$ respectively. Relevant equations. 50m apart in air. Capacitance of Concentric Spheres Calculator. when the total volume of metal is fixed? 4. The conical plug has a generating angle of 30°. If the inner shell is then grounded and electrostatic equilibrium is attained, the inner shell’s nal charge will be (a) 0 (b) Q (c) Q (d) 2Q (e) 2Q (f) some other value. Calculating the Capacitance ACylindricalCapacitor The figure shows a cross section of a cylindrical capacitor of length 𝐿formed by two coaxial cylinders of radii and. Consider two nested, spherical conducting shells. In this system, there are four linearly independent, incompressible. 77777 77777 10. The potential of shell B is : Option 1) Option 2) Option 3) Option 4). Take the potential V to be zero at infinite separation. A capacitor is simply two metallic plates separated by a gap, where the gap between the plates is usually filled with a material called a dielectric. Select True or False for the following statements. Given two concentric conducting spherical shells of negligible thickness having diameters of 15 cm and 10 cm. For two concentric spheres of radii a and b, and charges +Q and -Q. The charge and the electric field are related by (usingGauss’law). The capacitance of the spherical capacitors can be measured or calculated as following: Isolated Spherical Capacitor: Consider a perfectly insulated spherical conductor with a radius of ‘r’ meters. Capacitance of Cylindrical Capacitor:. A capacitor consists of two concentric spherical shells. (a) Begin with Gauss’ Law, sketch the Gaussian surfaces, and find the magnitude of the electric field in the regions rd. The corresponding capacitances are deduced from voltage and charge values. A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. a cylinder with cross sectional radius of R+½a. Capacitance: The capacitance (C) is returned in Farads. True False The electric field in the region c < r < d is zero. Consequently, when comparing two models, a similarity metric implicitly provides the measure of similarity at the optimal alignment. b) If the 4. 8 Find the capacitance per unit length of two coaxial metal cylinder tubes of radii a and b. Homework Statement Given two concentric spherical metal shells, with radii a and b (a < b), and surface charge densities Sa and Sb. Three concentric spherical metallic shells A , B and C of radii a , b and c $\;(a < b < c)\;$ have charge densities $\;\sigma , - \sigma \; and \; \sigma\;$ respectively. In all cases the shells begin uncharged, and a charge is then instantly introduced somewhere. A) 𝑄 4𝜋Ɛ0 B) 𝑄 2𝜋Ɛ0 2 C) 𝑄 2𝜋Ɛ0. This shielded composite dielectric spherical shell resonator is composed of two concentric metal spheres with different dielectric material has been made. The Capacitance of a Two Concentric Spherical Shells calculator computes the capacitance of concentric spherical shells. Relevant equations. 29 A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports figure. Model of coaxial cable for calculation of capacitance Capacitance of two concentric spherical shells Spherical capacitor or sphere Capacitance of one charged conducting sphere of radius a relative to another oppositely charged sphere of. Griffiths, problem 2. Find the capacitance of two concentric spherical metal shells, with radii 'a' and 'b'. Solution: The capacitance happens to be negative quantity. Capacitance is the tendency of the device to store electrical charge. The full tessellation of such nanorods resulted in 798 and 2212 tesserae, respectively. A cylindrical capacitor consists of two concentric, conducting cylinders (). Along, straight metal rod has a radius of 5. Capacitors in parallel 3. Calculate the ratio of the work done by the electric field in taking the charge particle from A to B and from B to A. A spherical capacitor is formed from two concentric spherical conducting shells separated by a vacuum. 4 Conductors – capacitance Christopher Crawford PHY 416 2014-10-15 Exam 2 – Friday Oct 24 Integrate E(r) or V(r) over a charge distribution Parametrize source points r’(u,…) on surface, path or volume Calculate field point r and displacement vector, r=r-r’ Reduce integrals to parameters and constants, including unit vectors Capacitance calculation Application of Gauss’ law. The potential of shell B is : Option 1) Option 2) Option 3) Option 4). This document is highly rated by JEE students and has been viewed 214 times. (b) What must be the plate area of a parallel-plate capacitor with the same plate separation and capacitance?. The plasmon response of this structure can be understood as an interaction and hybridization of the plasmons of the two individual metal shells (supporting online text). Question 2 : A conducting spherical shell of inner radius a and outer radius b carries a total charge of -5Q. The definition of capacitance is,. 0: A charge of Q =1 C is transferred from the inner shell to the outer shell. 77777 77777 10. B is earthed, C is the common center of A and B. A potential difference of 100 V is applied to the capacitor. This constant of proportionality is known as the capacitance of the capacitor. Dielectric 7. [383077] Two concentric thin-shell spherical conductors have radii a and b, where a is less than b Find the electric potential difference V between the two conductors if a charge +Q resides on the inner shell and a charge -Q resides on the outer shell. 0 cm are separated by aluminum as shown in Figure P28. A hollow metal sphere of radius 5 cm is charged such that the potential on its surface is 10 volt. The Capacitance of a Two Concentric Spherical Shells calculator computes the capacitance of concentric spherical shells. 85 nF and, when exposed to a potential diﬀerence of 100 V, it accumulates Q = CV=0. As a result, the need for the ability to retrieve models from large databases has gained prominence and a key concern of shape analysis has shifted to the design of. They are joined end-to-end and a potential difference is maintained across the combination. The outer cylinder is a shell of inner radius. Example: Problem 7. 29 A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports figure. Determine the potential distribution in the space between the conductors. The inner cylinder, of radius , may either be a shell or be completely solid. This is at the AP Physics level. The plates of a spherical capacitor have radii 38. Two concentric spherical shells of radii a and b have equal but opposite charges. The dielectric strength of the gas surrounding the electrode is 5 × 107 Vm–1. True False The electric field in the region c < r < d is zero. Show that the capacitance of a spherical capacitor is given by where r1 and r2 are the radii of outer and inner spheres, respectively. Capacitance is the ratio of the change in electric charge of a system, to the corresponding change in its electric potential. Consider a spherical capacitor, in which a spherical conductor of radius a lies inside a concentric spherical conducting shell of radius b. From the equation, it may seem that ‘C’ depends on charge and voltage, but actually, it depends on the shape and size of the capacitor and also on the insulator used between the conducting plates. The inner shell with radius ais held at potential V a, while the outer shell with radius bis held at potential V b. The intermediate medium is filled as shown in the figure with two dielectric materials of permittivity ε1 and ε2. The inner shell has a charge +Q uniformly distributed over its surface, and the outer shell an equal but opposite charge –Q. The inner shell at radius R has positive charge Q. If the two spheres are maintained at a potential difference of 2. 7 Find capacitance of two concentric spherical metal shells, with radii a and b. The shell carries no net charge. A spherical capacitor consists of two thin concentric spherical shells of. Calculate the potential V(r) for: i) r < a ii) a < r < b iii) r > b. This field cannot be CheckPoint Results: Charged Sphericlal Shell. Consider two nested, spherical conducting shells. A spherical capacitor consists of two concentric spherical shells. 00 cm and 14. The radius of the inner shell is 10 mm, and the radius of the outer shell is 11 mm. 16) A solid metal sphere is at the center of a hollow metal shell. Spherical Capacitor The capacitance for spherical or cylindrical conductors can be obtained by evaluating the voltage difference between the conductors for a given charge on each. Part A What is the energy density at r= 11. A cylindrical capacitor has outer and inner conductors whose radii are in the ratio of b/a = 4/1. A charged particle is held at the center of two concentric conducting spherical shells. potential difference between two sphere shells. A capacitor consists of two concentric spherical shells. A spherical capacitor consists of 2 concentric conducting spherical shells with radii a and b with a < b. Analytic calculation of gravitational elds is easy for spherical systems, but few general results are available for non-spherical mass distributions. Now the inner shell is grounded ,This means that the inner shell will come at zero potential and that electric fields lines leave the outer shell and end on the inner shell. A charged particle is held at the center of two concentric conducting spherical shells. Show that if these spheres are connected by a conducting wire, charge will always flow from the inner sphere to the outer sphere irrespective of the magnitude of the. A conducting sphere A of radius a, with charge Q is placed concentrically inside a conducting shell B of radius b. The outer radius of the inner shell is a=0. Details of the calculation: E is radial and has magnitude Q/(4πε. 35: A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b). Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity Ïƒ (Fig. Consider a resistor comprised of two concentric spherical metal shells. A spherical neutral conducting shell, with inner radius around a uniformly positively charged sphere (insulator). Capacitors in parallel 3. The inner conducting shell (#1) has a radius [m] while the outer shell (#2) has a radius S2a" The space between the conducting surfices filled at various times With different dielectric and/or conducting materials. The spherical laser 100 utilizes a spherical resonator having two concentric, mirrored spheres 101, 102. Concentric with this sphere there is a conducting spherical shell whose inner and outer radii are b = 20 cm and c =25 cm respectively. 0 cm and the capacitance is 116 pF. It is concentric with a spherical conducting shell of inner radius b and outer radius c. The plates of a spherical capacitor have radii 38. Electric potential is electrical potential energy per unit charge; the units of electric potential are joules per coulomb. Figure 2-11. A capacitor consists of two concentric spherical shells. B is earthed, C is the common center of A and B. A point charge −1µC (not a part of the +5µC charge on the sphere) is located at the center of the hole. A spherical capacitor consists of two thin concentric spherical shells of. As the charges on the two conductors are equal and opposite, the system is a capacitor. a cylinder with cross sectional radius of R+½a. 77777 77777 10. Here the coecients A˜r and B˜r are constants, in general, dierent for the inner and outer spheres. We show that dielectric spheres can be cloaked by a shell of amorphously arranged metallic nanoparticles. Question from Electrostatic Potential and Capacitance,cbse,class12,physics,ch-2,sec-c,additional,difficult A spherical capacitor consists of two concentric. A spherical capacitor is another set of conductors whose capacitance can be easily determined (Figure $$\PageIndex{5}$$). A spherical capacitor has an inner sphere of radius 12 cm and outer sphere of radius 13 cm. 0: A charge of Q =1 C is transferred from the inner shell to the outer shell. Eudoxus of Cnidus (about 408 B. Assume negligible fringing effect at the edges The outer conductor has an inner radius b and is grounded. Details of the calculation: E is radial and has magnitude Q/(4πε. And let the relative permittivity of the medium in between the two cylinders be. Find the electric field (magnitude and direction) at locations 1. Capacitance of two concentric spherical shells -q Integration path dr as +q b a E for an isolated sphere Q R Spherical capacitor or sphere Recall our favorite example for E and V is spherical a a Capacitance of one charged conducting sphere of radius a relative to another oppositely charged. A capacitor consists of two concentric spherical shells. Find the capacitance of a ordinary piece of coaxial cable (TV cable) capacitance of a coaxial cable cont. potential difference between two sphere shells. Equiv Circuit 5. The definition of capacitance is,. Two concentric, metal spherical shells of radii a = 4. B is earthed, C is the common center of A and B. The area of each plate is equal to A. The space between these two surfaces is filled with a dielectric for which. spherical shells of radii a and b, as shown in Figure 5. Electric potential is electrical potential energy per unit charge; the units of electric potential are joules per coulomb. The net charge on the sphere is +20 C and the electric field at r = 30 cm is 107 N/C pointing inwards. 77777 77777 10. A potential difference of 100 V is applied to the capacitor. The full tessellation of such nanorods resulted in 798 and 2212 tesserae, respectively. Note that the capacitance is independent of charge given, potential raised, nature of metal or thickness of plates. 21 A solid conducting sphere has charge Q surrounded by an uncharged concentric hollow spherical shell. Two concentric metallic spherical shells of radii R and 2R are given charges and respectively. Applications for such a capacitor may not be immediately evident, but it does illustrate that a charged sphere has stored some energy as a result of being charged. in a spherical shell of radius r and thickness dr is dU = uE4πr2dr. The space between the shells is filled with a dielectric of dielectric constant K up to a radius c as shown in figure. ) was a Greek astronomer and mathematician. NEXT Suppose the space between the two inner shells of the previous problem is filled with a dielectric of. 0 cm and the capacitance is 116 pF. Three concentric metallic spherical shells of radii R,2R and 3R are given charges Q1, Q2 and Q3, respectively. A small conducting spherical shell with inner radius a and outer radius b is concentric with a larger conducting spherical shell with inner radius c and outer radius d. c) (4) The two charges are now enclosed inside a spherical metal shell with adius 2 cm; the metal has no net charge. that the capacitance of a spherical capacitor is given by. Example calculations of capacitance Done in the book: Example 2. Electrostatic Potential And Capacitance Questions with Solutions to help you to revise complete Syllabus and Score More marks in your Class 12 Calculate the capacitance of the capacitor. A sphere of radius a, and charge +q uniformly distributed throughout its volume. Let the potential difference between the surface of the solid sphere and that of the outer surface of the spherical shell be V. The two plates of a parallel-plate capacitor are separated by a distance d and. The figure shows a cross section. A conducting spherical shell has inner radius a and outer radius c. a sphere of radius R+½a B. 10 x 10- 8 C. 0 cm) which has a net charge of –4. Answer : Please Register/Login to get. 00-μC charges, as shown in Figure and a positive test charge q=1. True False The total. Calculate the capacitance. The plates of a spherical capacitor have radii 38. 1 Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity σ. Gauss’s law in the vacuum is I E ds = Q f 0; while this is modi ed in the dielectric material I E ds = Q f 0 r: Here. A theory of Eudoxus from about 400 B. Exercise : Find the equivalent capacitance between points A and B. PREVIOUS A spherical capacitor is made of two conducting spherical shells of radii a and b. A spherical capacitor of two concentric conducting shells is divided into two halves, in which the space between the shells is filled with a dielectric of a specific dielectric constant. 00 cm, (b) 10. The field will push positive charge to the northern surface Find the approximate electric field at points far from the origin (Express in spherical coordinates and include the two lowest orders in the multipole expansion). Here we have been calling it b just to avoid confusion with the full capacitance matrix C. [383077] Two concentric thin-shell spherical conductors have radii a and b, where a is less than b Find the electric potential difference V between the two conductors if a charge +Q resides on the inner shell and a charge -Q resides on the outer shell. The radii of the lower and upper sections are denoted by a and b, respectively. The plasmon response of this structure can be understood as an interaction and hybridization of the plasmons of the two individual metal shells (supporting online text). The outer radius of the inner shell is a=0. There is charge +Q on the inner sphere and charge -Q on the outer shell. May 04, 2020 - NCERT Exemplars - Electrostatic Potential and Capacitance Notes \| EduRev is made by best teachers of JEE. Three concentric metal shells A, B and C of respective radii a, b and c (a < b < c) have surface charge densities +σ, −σ and +σ respectively. Question 17. This field cannot be CheckPoint Results: Charged Sphericlal Shell. Two wires are made of the same material and have the different radii. A spherical capacitor of two concentric conducting shells is divided into two halves, in which the space between the shells is filled with a dielectric of a specific dielectric constant. If the radius of outer shell becomes infinity b→∞ and we substitute inner shell radius a=R. in a spherical shell of radius r and thickness dr is dU = uE4πr2dr. 0: A charge of Q =1 C is transferred from the inner shell to the outer shell. Capacitance 1. Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity σ (Fig. A capacitor consists of two concentric spherical shells. 35 In a Van de Graaff type generator a spherical metal shell is to be a 15 × 106 V electrode. 0 $\mathrm{mm}$ and 40. 00 cm and (b) r = 6. Charge separation 1. We have two shells within which is earthed for both ! Capacitance is the cause of potential difference between the two shells but in this case the potential difference is zero. The upper hemisphere of the inner sphere and the lower hemisphere of the outer sphere are maintained at potential V. 77777 77777 10. The work done by the electric force on the charge carriers is converted into heat (Joule heating). Capacitance of Two Concentric Spherical Shells. The outer cylinder is a shell of inner radius. The inner shell with radius ais held at potential V a, while the outer shell with radius bis held at potential V b. SOLUTION: See Serway Example 26. Eudoxus of Cnidus (about 408 B. 10) Two concentric conducting spherical shells produce a radially outward electric field of magnitude 49,000 N/C at a point 4. 10 m from the center of the shells. Spherical segment is a solid bounded by two parallel planes through a sphere. What is the capacitance of a capacitor composed of two concentric spherical conducting shells if the radius of the inner shell is a a a and the radius of the outer shell is b b b? Cross-section of concentric spherical conducting shells. a cube of dimension R+½a C. ; a cosmological theory in which the planets, the sun, and the moon were described as being carried on a series of concentric spheres rotating within one another on different or various axes. 4 Conductors – capacitance Christopher Crawford PHY 416 2014-10-15 Exam 2 – Friday Oct 24 Integrate E(r) or V(r) over a charge distribution Parametrize source points r’(u,…) on surface, path or volume Calculate field point r and displacement vector, r=r-r’ Reduce integrals to parameters and constants, including unit vectors Capacitance calculation Application of Gauss’ law. The hollow space between the two shells is –lled with nylon having a dielectric constant of 4. The capacitance of a given capacitor depends on its geometry and on the The electric field lines around this conductor are exactly the same as if there were a conducting shell of infinite radius, concentric with the sphere. I've just begun learning capacitance, and my lecture notes have a section on calculating capacitance for capacitors in vacuum of various shapes, e. 59 A capacitor is formed by two coaxial metal cylinders of radii a = 1 mm and b = 5 mm. A spherical capacitor is formed from two concentric spherical conducting shells separated by vacuum. Which of the following represents the electric potential as a function of distance r in the region a < r < b? Assume that electric potential is zero at an infinite distance from the spheres. 7 Find capacitance of two concentric spherical metal shells, with radii a and b. Assume a charge of Q' on the inner sphere, and make use of spherical symmetry to find the potential difference between the inner/middle sphere and. 00 cm and (b) r = 6. , between the radii r 2 and r 1. Spherical shells are built from two concentric spheres (see Figures 1B,C, 2C,D), the inner of which is void. A capacitor consists of two concentric spherical shells. 5 cm and the inner and outer radii of the hollow shell are 1 cm and 2 cm. The inner sphere has a total charge Q at any time. Metal spheres with different radii and a spherical capacitor are charged by means of a variable voltage. 0 cm, outer radius = 2. In this system, there are four linearly independent, incompressible. A spherical capacitor is formed from two concentric, spherical, conducting shells separated by vacuum. Example: concentric conducting spheres. Calculating the capacitance 3. A surface integral of electric field intensity obtained between the shells gives a value of 1. Typically, commercial capacitors have two conducting parts close to one another but not touching We can calculate the capacitance of a pair of conductors with the standard approach that follows. Consider a resistor comprised of two concentric spherical metal shells. A conducting sphere A of radius a, with charge Q is placed concentrically inside a conducting shell B of radius b. (a) If they are maintained at a potential difference V, what current flows from one to the other? (b) What is the resistance between the shells?. PDF \| Polymer shells with high sphericity and uniform wall thickness are always needed in the inertial confined fusion (ICF) experiments. parallel-plate capacitor, which consists of two plates each of area. Multiple Choice with ONE correct answer. 070 0 m) ab ke(b-a O c 15. Obtain the formula for capacitance of a spherical conductor. What is the capacitance Cof a capacitor that consists of two concentric spherical shells, the inner of radius r, and charge +Q, the outer of radius ra and charge-Q? Your answer should contain constants, r, and r, only! Check your result by checking that the limit of Cas r2- ri < a. The upper hemisphere of the inner sphere and the lower hemisphere of the outer sphere are maintained at potential V. Chapter 25. The modification of capacitance due to uniform distribution of charge over the volume of the dielectric sphere and the presence of concentric metal sphere inside a dielectric shell are also evaluated. The radii of the lower and upper sections are denoted by a and b, respectively. 5 cm and the outer sphere has radius 14. The inner sphere, of radius R1, has charge +Q, while the outer shell of radius R2, has charge –Q. This is at the AP Physics level. Capacitance of Concentric Spheres. We show that dielectric spheres can be cloaked by a shell of amorphously arranged metallic nanoparticles. Consider the electric field that’s created by a point source charge Q. Capacitors in parallel 3. (c) The inner and outer shells of a spherical capacitor have radii of. relative phases, while the concentric driving force, as a resultant force, is not only. Homework Statement Given two concentric spherical metal shells, with radii a and b (a < b), and surface charge densities Sa and Sb. If the inner shell is then grounded and electrostatic equilibrium is attained, the inner shell’s nal charge will be (a) 0 (b) Q (c) Q (d) 2Q (e) 2Q (f) some other value. If P is the point between shells A and B at distance r from center C then for ( a = 1 m, b = 3 m a n d, r = 2 m),. Add to Collection. A capacitor consists of two conductors separated by an insulator. This constant of proportionality is known as the capacitance of the capacitor. The inner sphere has radius 12. 29 A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports figure. 1 Two concentric metal spherical shells, of radius a and b, respectively, are separated by weakly conducting material of conductivity σ. potential difference between two sphere shells. Capacitance: The capacitance (C) is returned in Farads. The conical plug has a generating angle of 30°. Concentric with this spherical shell is an uncharged conducting spherical shell with inner radius c and outer radius d. There is NO thickness, they are just "shells". Three concentric spherical metallic shells A , B and C of radii a , b and c $\;(a < b < c)\;$ have charge densities $\;\sigma , - \sigma \; and \; \sigma\;$ respectively. (a) Find the surface How do the answers to (a) and (b) change?. a) Verify that the standard expression for the energy stored in the electric field is equal to the standard expression for the energy stored in a capacitor if the shells each have a charge Q on them. (a) What is the capacitance C of this capacitor? Express your answer in Farads. Details of the calculation: E is radial and has magnitude Q/(4πε. 0 cm and the capacitance is 116 pF. The unit of power is the Watt (1 W = 1 J/s). From Gauss's Law. The pentagonal nanorods were smoothened out at the edges. The total flux through a sphere of radius R is. two parallel plates and concentric spherical shells. (a) Find the electric field in the region a < r < b. The space between the shells is filled with a dielectric with. The hollow space between the two shells is –lled with nylon having a dielectric constant of 4. The capacitance of the spherical capacitors can be measured or calculated as following: Isolated Spherical Capacitor: Consider a perfectly insulated spherical conductor with a radius of ‘r’ meters. 5 (a) spherical capacitor with two concentric spherical shells of radii a and b. Calculating the capacitance 3. In this Capacitance of a Sphere Calculator, the Capacitance of a Spherical Capacitor based on Radius can be calculated. A spherical capacitor consist of two concentric conducting spheres, as shown in Fig. Consider two nested, spherical conducting shells. A point charge of 6. The outer sphere 101 has a larger radius than the inner sphere 102. 56 x 10 c 257 kV air-filled spherical capacitor is constructed with Inner- and outer-shell radii of 7. A spherical capacitor is formed from two concentric spherical conducting shells separated by a vacuum. The spherical capacitor has self-capacitance. 41 Capacitance of a spherical capacitor. The inner sphere has radius 11. Introduces the physics of using Gauss's law to find the electric fields around concentric spherical shells. Model of coaxial cable for calculation of capacitance Capacitance of two concentric spherical shells Spherical capacitor or sphere Capacitance of one charged conducting sphere of radius a relative to another oppositely charged sphere of. 5 cm and the inner and outer radii of the hollow shell are 1 cm and 2 cm. (b) Find the potential difference between the 2 spheres. Equivalent circuits 1. The surface of a sphere having radius r1 and radius r2 which carries a charge. two concentric metallic spherical shells of radii R and 2R are given charges Q1 and Q2 respectively the surface charge densities on the outer surfaces of the shells are equal determine the ratio Q1:Q2 - Physics - Electrostatic Potential And Capacitance. Determine the resulting charge density on the inner surface of the conducting sphere. A spherical capacitor consists of two concentric spherical shells. This constant of proportionality is known as the capacitance of the capacitor. You can't view this as two separate capacitors in series/parallel because the charge on the inner/middle spheres for example are not the same. A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. The inner sphere has a total charge Q at any time. 2 Capacitance of a single spherical conducting shell The capacitance of a single spherical shell can be determined by ﬁrst considering two concen-tric conducting shells, with the inner shell having radius a and the outer shell radius b. HINT: you may need to apply a negative sign to your final expression to be sure the capacitance comes. And let the relative permittivity of the medium in between the two cylinders be. Exercise : Find the equivalent capacitance between points A and B. In figuring out the capacitance of this configuration of conductors ± Capacitance and Electric Field of a Spherical Capacitor. A conducting sphere A of radius a, with charge Q is placed concentrically inside a conducting shell B of radius b. We’ll start with a shell of radius 0, and work our way up to the last one of radius R. Consider two hollow concentric conducting shells. 0 cm, and (c) 100 cm from. The inner sphere has a radius of = 12. The figure shows a cross section. Two concentric spherical conducting shells are separated by vacuum. Find the magnitude of charge +Q on the inner shell as a function of E0 and a. a) Verify that the standard expression for the energy stored in the electric field is equal to the standard expression for the energy stored in a capacitor if the shells each have a charge Q on them. Example: Problem 7. What is the capacitance Cof a capacitor that consists of two concentric spherical shells, the inner of radius r, and charge +Q, the outer of radius ra and charge-Q? Your answer should contain constants, r, and r, only! Check your result by checking that the limit of Cas r2- ri band for ra), is given by C=ab/(b-a). 11: concentric spherical shells, radii a and b: Both of these formulae are well worth remembering. Consequently, when comparing two models, a similarity metric implicitly provides the measure of similarity at the optimal alignment. • The equipotential of a thin homoeoid has the same shape as the. (b) Show that, when (b − a) a, the equation reduces to that for a parallel plate capacitor. Concentric with this sphere is a conducting spherical shell with inner radius band outer radius c, and having a net charge -Q, as shown in Figure. 4 Conductors – capacitance Christopher Crawford PHY 416 2014-10-15 Exam 2 – Friday Oct 24 Integrate E(r) or V(r) over a charge distribution Parametrize source points r’(u,…) on surface, path or volume Calculate field point r and displacement vector, r=r-r’ Reduce integrals to parameters and constants, including unit vectors Capacitance calculation Application of Gauss’ law. 30 µC, find the amount of charge on the outer surface of the larger shell. North Americans added the letter B to denote the d-block groups and A for the others; this is the system shown in the table above. Two concentric, metal spherical shells of radii a = 4. In the past, two different systems of Roman numerals and letters were used to denote the various groups. Equiv Circuit 5. They are joined end-to-end and a potential difference is maintained across the combination. Cross-polarized images of a very thin shell with four s=1/2 defects. The outer shell has a charge Q, but the inner shell is grounded. The shells are given equal and opposite charges $$+Q$$ and $$-Q$$, respectively. maintained at potentials 0 and V0. A spherical capacitor consists of a spherical conducting shell of radius b and charge -Q concentric with a smaller conducting sphere of radius a and charge +Q. Consider an assembly for three conducting concentric spherical shells of radii a, b and c as shown in figure. nearly equal, the capacitance is given approximately by the expression 3. A small conducting spherical shell with inner radius a and outer radius b is concentric with a larger conducting spherical shell with inner radius c and outer radius d. There are two closely related notions of capacitance: self capacitance and mutual capacitance. 22 x 10- 8 C and that on the outer shell is 2. Show Step-by-step Solutions. By applying Gauss' law to an charged conducting sphere, the electric field outside it is found to be. Calculate the charge enclosed by a concentric spherical surface with radius (a) r = 2. (b) Show that, when (b − a) a, the equation reduces to that for a parallel plate capacitor. For JEE Main other Engineering Entrance Exam Preparation, JEE Main Physics Electrostatics Previous Year Questions with Solutions is given below. The volume of a sphere is equal to four-thirds of the product of pi and the cube of the radius. 26 new! 【ジンギスカン北えびす】臨時休業のお知らせ【塩〆熟成ジンギスカン北えびす札幌本店】オリエント Orient Orient 腕時計時計 Orient Mens Men's CFM00002B Power 腕時計 Reserve Semi-Skeleton Black Automatic Watch. Two concentric, metal spherical shells of radii a = 4. The spherical capacitor has self-capacitance. The inner conducting shell (#1) has a radius [m] while the outer shell (#2) has a radius S2a" The space between the conducting surfices filled at various times With different dielectric and/or conducting materials. The potential of shell B is : Option 1) Option 2) Option 3) Option 4). 28X10-18C at the origin,(a) what is the. What is the capacitance Cof a capacitor that consists of two concentric spherical shells, the inner of radius r, and charge +Q, the outer of radius ra and charge-Q? Your answer should contain constants, r, and r, only! Check your result by checking that the limit of Cas r2- ri < a. Solution: Concepts: Gauss' law; Reasoning: For a given charge ±Q on the shells, Gauss law yield the electric field between the shells. The particle has. Outer plate B is connected with Earth and inner plate A is given charge +q. Part A What is the energy density at r= 11. where r1 and r2 are the radii of outer and inner spheres, respectively. The spherical laser 100 utilizes a spherical resonator having two concentric, mirrored spheres 101, 102. You can't view this as two separate capacitors in series/parallel because the charge on the inner/middle spheres for example are not the same. Concentric with this sphere is a conducting spherical shell with inner radius b and out radius c. The distance d is much smaller than R. The unit of power is the Watt (1 W = 1 J/s). The outer cylinder is earthed and the inner cylinder is given a charge of 3. PREVIOUS A spherical capacitor is made of two conducting spherical shells of radii a and b. The outer shell carries a charge -Q as shown in the figure below. The inner sphere has radius 15. What is the minimum radius of the spherical shell required?. show that the spherical driving force is from the interfacial tension affected by the two. I've just begun learning capacitance, and my lecture notes have a section on calculating capacitance for capacitors in vacuum of various shapes, e. The pentagonal nanorods were smoothened out at the edges. A spherical capacitor of two concentric conducting shells is divided into two halves, in which the space between the shells is filled with a dielectric of a specific dielectric constant. The dielectric strength of the gas surrounding the electrode is 5 × 107 Vm–1. Let, The radius of the inner cylinder be R1 and that of the outer cylinder be R2. direction of the field and back from B to A. 2 Capacitance of a single spherical conducting shell The capacitance of a single spherical shell can be determined by ﬁrst considering two concen-tric conducting shells, with the inner shell having radius a and the outer shell radius b. A given shell of radius r will have a thickness dr, which gives it a surface area of 4πr2 and a volume of (thickness)(surface area. Typically, commercial capacitors have two conducting parts close to one another but not touching We can calculate the capacitance of a pair of conductors with the standard approach that follows. , between the radii r 2 and r 1. (b) Show that, when (b − a) a, the equation reduces to that for a parallel plate capacitor. Two wires are made of the same material and have the different radii. It consists of two concentric spherical plates A and B. 0 V via an external source, calculate the current from one sphere to the other. Frustrated nematic order in spherical geometries. Capacitance in Series 3. Select True or False for the following statements. Start date Apr 6, 2012. Example calculations of capacitance Done in the book: Example 2. 0 cm and b = 8. TWO charged conducting spheres of radii a and b are connected to each other by a wire. Three concentric metallic spherical shells of radii R,2R and 3R are given charges Q1, Q2 and Q3, respectively. In this system, there are four linearly independent, incompressible. Capacitance 1. hydrogen atom contains one proton; an oxygen atom contains eight protons. The following integral is from ra to rb (inner radius a and inner radius b) V(r) = - ∫ E dr (vector E, electric field and vector dr, infinitesimally small radius. This is at the AP Physics level. Electrostatic Potential And Capacitance Questions with Solutions to help you to revise complete Syllabus and Score More marks in your Class 12 Calculate the capacitance of the capacitor. Then electric potential on shell A is This gives the capacitance of the spherical capacitor. shell has no charge. 43 mm and the inner radius of the outer shell is b=2. The dielectric strength of the gas surrounding the electrode is 5 x 10 7 Vm-1. The second has inner radius c and outer radius d. 13 x 10 6 m v, whereas a surface integral taken over the outer. The inner shell has a charge +Q uniformly distributed over its surface, and the outer shell an equal but opposite charge –Q. (b) Show that when the radii of the shells are nearly equal, the capacitance approximately. 21 A solid conducting sphere has charge Q surrounded by an uncharged concentric hollow spherical shell. 00-μC charges, as shown in Figure and a positive test charge q=1. Mass models with tractable potentials may however be combined to approximate the potentials of real galaxies. a cube of dimension R+½a C. If the inner shell contains an excess charge of -5. A spherical capacitor is formed from two concentric spherical conducting shells separated by vacuum. Calculate the potential V(r) for: i) r < a ii) a < r < b iii) r > b. (a) Calculate the potential ˚(r) between the inner and outer shells and ex-press the solution in terms of a, b, V. on two spheres with radii a and b, a < b, concentric with the monopole. This field cannot be CheckPoint Results: Charged Sphericlal Shell. One joule per coulomb is called one volt (abbreviated V); so 1 J/C = 1 V. 41 Capacitance of a spherical capacitor. magnitude of 8500 N/C. A capacitor consists of two conductors separated by an insulator. A small sphere of radius and charge is enclosed by a spherical shell of radius and charge. 1 m, and a -3, find the value of the charge qin coulombs Image Transcriptionclose. Finding the equivalent circuits 2. If V2 is for the outer shell and V1 for the inner shell, then whatever is between the shells is V2 - V1 (V(r)). (b) What must be the plate area of a parallel-plate capacitor with the same plate separation and capacitance?. ; a cosmological theory in which the planets, the sun, and the moon were described as being carried on a series of concentric spheres rotating within one another on different or various axes. Since the orbitals around an atom are defined in terms of a probability distribution in quantum mechanics, and do not have fixed boundaries, determining where an atom "stops" is not very straightforward. Answer : Please Register/Login to get. 00 cm and (b) r = 6. Determine the resulting charge density on the inner surface of the conducting sphere. Two charged concentric spherical shells - Продолжительность: 5:27 WNY Tutor 817 просмотров. They carry no net charge. As a third example, let’s consider a spherical capacitor which consists of two concentric spherical shells of radii a and b, as shown in Figure 5. Originally, it was called a "bombshell", but "shell" has come to be. Take the potential V to be zero at infinite separation. 0 gram balls hang from lightweight insulating threads 50 cm long from a common support point as shown in the Figure. If the inner shell is then grounded and electrostatic equilibrium is attained, the inner shell’s nal charge will be (a) 0 (b) Q (c) Q (d) 2Q (e) 2Q (f) some other value. Select True or False for the following statements. (E&M) Two very thin concentric spherical conducting shells are sepa-rated by vacuum as shown in the gure below. Mini Physics is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon. we get capacitance of an isolated charge sphere of radius R. Solution: Concepts: Gauss' law; Reasoning: For a given charge ±Q on the shells, Gauss law yield the electric field between the shells. The inner shell has a total charge of -1q and the outer shell has a total charge of +4q. 0 V via an external source, calculate the current from one sphere to the other. The charge density on the. INSTRUCTIONS: Choose your preferred units and enter the following: (a) - the radius of the smaller sphere (b) - the radius of the larger sphere (ε r) - the Dielectric Constant of materials between cylinders. 0 cm, outer radius = 2. A metal sphere with radius a is supported on an insulating stand at the center of a hollow, metal spherical shell with radius b. Equiv Circuit 5. Two wires are made of the same material and have the different radii. The radius of the sphere is 0. It consists of two concentric conducting spherical shells of radii $$R_1$$ (inner shell) and $$R_2$$ (outer shell). vacuum in between. 0 cm, and (c) 100 cm from. If P is the point between shells A and B at distance r from center C then for ( a = 1 m, b = 3 m a n d, r = 2 m),. C = 4πε₀r a r b /(r a-r b) r a and r b are radius of internal and external spherical shells 42 Equivalent capacitance for Capacitors in parallal C = c 1 + c 2 +c 3…. The outer cylinder is a shell of inner radius. Example: Problem 7. Capacitance 1. Electrostatic Potential And Capacitance Questions with Solutions to help you to revise complete Syllabus and Score More marks in your Class 12 Calculate the capacitance of the capacitor. Begin by noting that we can only determine the capacitance per unit length, C/L, because the total capacitance is an unreal value. 0 cm and b = 8. We provide an analytical model for the cloak design and prove numerically that the cloak operates as desired. Of the following, the quantity that is the same for both wires is: A potential difference across each wire B current inside each wire C) current density inside each wire. A capacitor consists of two concentric spherical shells. 85 nF and, when exposed to a potential diﬀerence of 100 V, it accumulates Q = CV=0. A conducting spherical shell has inner radius a and outer radius c. A potential difference of 100 V is applied to the capacitor. We assume that the length of each cylinder is l and that the excess charges and reside on the inner and outer cylinders, respectively. Spherical capacitor. We show that dielectric spheres can be cloaked by a shell of amorphously arranged metallic nanoparticles. The plates of the cell are two concentric spherical sections mounted and spaced from each other on a conical. The space between these two surfaces is filled with a dielectric for which. Show Step-by-step Solutions. We have two shells within which is earthed for both ! Capacitance is the cause of potential difference between the two shells but in this case the potential difference is zero. 13 x 10 6 m v, whereas a surface integral taken over the outer. As the charges on the two conductors are equal and opposite, the system is a capacitor. 0: A charge of Q =1 C is transferred from the inner shell to the outer shell. Find its capacitance. Three concentric metal shells A, B and C of respective radii a, b and c (a < b < c) have surface charge densities +σ, −σ and +σ respectively. Electric potential is electrical potential energy per unit charge; the units of electric potential are joules per coulomb. (b) What potential difference between the spheres. Cis capacitance with medium within plates, and C₀ is capacitance in free space. (b) Two concentric spherical conducting shells (radii a and b) are grounded, and a point charge q is. Consider a hollow spherical conducting shell with inner radius 1 and outer radius 2. And let the relative permittivity of the medium in between the two cylinders be. B is earthed, C is the common center of A and B. Example calculations of capacitance Done in the book: Example 2. Now the inner shell is grounded ,This means that the inner shell will come at zero potential and that electric fields lines leave the outer shell and end on the inner shell. A capacitor with one or more thin hollow spherical plate conductors is called as a Spherical capacitor. Charging a capacitor 2. 59 A capacitor is formed by two coaxial metal cylinders of radii a = 1 mm and b = 5 mm. Originally, it was called a "bombshell", but "shell" has come to be. UY1: Capacitance Of Spherical Capacitor. I'm supposed to think about it as a combination of capacitors or something like that, but that is all the problem gives me and I need to respond with a numerical answer. we get capacitance of an isolated charge sphere of radius R. on two spheres with radii a and b, a < b, concentric with the monopole. A small conducting spherical shell with inner radius a and outer radius b is concentric with a larger conducting spherical shell with inner radius c and outer radius d. In this system, there are four linearly independent, incompressible. We have two shells within which is earthed for both ! Capacitance is the cause of potential difference between the two shells but in this case the potential difference is zero. When equal charges Q are place on each ball they are repelled, each making an angle of 10 degrees with the vertical. Question 17. Which of the following represents the electric potential as a function of distance r in the region a < r < b? Assume that electric potential is zero at an infinite distance from the spheres. a) Verify that the standard expression for the energy stored in the electric field is equal to the standard expression for the energy stored in a capacitor if the shells each have a charge Q on them. The spherical shells are conductors with radius$a$ and$b$. Find the self-energy values W1 and W2 of each shell, the interaction energy of the shells W12, and the total electric energy of the system. Spherical Capacitor The capacitance for spherical or cylindrical conductors can be obtained by evaluating the voltage difference between the conductors for a given charge on each. Any object that can be electrically charged exhibits self capacitance. One joule per coulomb is called one volt (abbreviated V); so 1 J/C = 1 V. 07, 2007 5:00 pm. A cylindrical capacitor consists of two concentric, conducting cylinders (). Find the magnitude of charge +Q on the inner shell as a function of E0 and a. The spherical shells are conductors with radius$a$ and$b$. Let, The radius of the inner cylinder be R1 and that of the outer cylinder be R2. Frustrated nematic order in spherical geometries. 29 A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports figure. What is the minimum radius of the spherical shell required?. Question 2 : A conducting spherical shell of inner radius a and outer radius b carries a total charge of -5Q. The shell represents an artificial medium with tunable effective properties that can be adjusted such that the scattered signals of shell and sphere almost cancel each other. Solid shot may contain a pyrotechnic compound if a tracer or spotting charge is used. Consider the electric field that’s created by a point source charge Q. HINT: you may need to apply a negative sign to your final expression to be sure the capacitance comes. Introduces the physics of using Gauss's law to find the electric fields around concentric spherical shells. Charge separation 1. NEXT Suppose the space between the two inner shells of the previous problem is filled with a dielectric of. hydrogen atom contains one proton; an oxygen atom contains eight protons. (a) Calculate the capacitance. Surrounding this sphere is a metal shell of inner radius R2 = 2R1 and outer radius R3 = 3R1 that carries a total charge of Q2 = +3Q1. Find the capacitance of an isolated spherical conductor of radius r 1 surrounded by an adjacent concentric layer of dielectric with dielectric constant K and outside radius r 2. 59 A capacitor is formed by two coaxial metal cylinders of radii a = 1 mm and b = 5 mm. The space between the shells is filled with a dielectric of dielectric constant K up to a radius c as shown in figure. Capacitance also implies an associated storage of electrical energy. The plates of a spherical capacitor have radii 38. 00 cm and a charge per unit length of 30. INSTRUCTIONS: Choose your preferred units and enter the following: (a) - the radius of the smaller sphere (b) - the radius of the larger sphere (ε r) - the Dielectric Constant of materials between cylinders. It is found that the surface charge densities on the outer surfaces of the shells are equal. Concentric with this sphere is a conducting spherical shell with inner radius band outer radius c, and having a net charge -Q, as shown in Figure. Energy Stored in an Electric Field 6. Consider a resistor comprised of two concentric spherical metal shells. The charged concentric conducting spherical shells. vacuum in between. Along, straight metal rod has a radius of 5. The inner shell with radius ais held at potential V a, while the outer shell with radius bis held at potential V b. Finding the equivalent circuits 2. on two spheres with radii a and b, a < b, concentric with the monopole. We have two shells within which is earthed for both ! Capacitance is the cause of potential difference between the two shells but in this case the potential difference is zero. Calculating the capacitance 3. The surface of a sphere having radius r1 and radius r2 which carries a charge. The modes enclosed in such a spherical optical cavity are radial in nature. Question 30. when the total volume of metal is fixed? 4. Capacitance: The capacitance (C) is returned in Farads. This constant of proportionality is known as the capacitance of the capacitor. Concentric with this sphere is a conducting spherical shell with inner radius band outer radius c, and having a net charge -Q, as shown in Figure. Which of the following represents the electric potential as a function of distance r in the region a < r < b? Assume that electric potential is zero at an infinite distance from the spheres. The plasmon response of this structure can be understood as an interaction and hybridization of the plasmons of the two individual metal shells (supporting online text). The inner shell has a charge +Q uniformly distributed over its surface, and the outer shell an equal but opposite charge –Q. This outer shell has charge Q on it. (10%) Problem 6: A conducting spherical shell of inner radius R1 and outer radius R2 has a point charge tq fixed at its. Calculating the Capacitance ACylindricalCapacitor The figure shows a cross section of a cylindrical capacitor of length 𝐿formed by two coaxial cylinders of radii and. Any object that can be electrically charged exhibits self capacitance. Also known as coaxial capacitor. 10: parallel plates, area A, separation d: Example 2. The work done by the electric force on the charge carriers is converted into heat (Joule heating). Solution Let us consider that conductor in the problem has charge equals +Q Coulomb shown below in the figure. The spherical capacitor has self-capacitance. that the capacitance of a spherical capacitor is given by. The space between the shells is filled with a dielectric of dielectric constant K up to a radius c as shown in figure. Justify all steps in your solution. From Gauss's Law. The following integral is from ra to rb (inner radius a and inner radius b) V(r) = - ∫ E dr (vector E, electric field and vector dr, infinitesimally small radius. Two wires are made of the same material and have the different radii. It consists of two concentric spherical plates A and B. 13 x 10 6 m v, whereas a surface integral taken over the outer. A non-conducting disc of radius a and uniform positive surface charge density σ is placed on the ground, with its axis vertical. Show that if is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge on the shell is. The field will push positive charge to the northern surface Find the approximate electric field at points far from the origin (Express in spherical coordinates and include the two lowest orders in the multipole expansion). The simple electromagnetism calculator which is used to calculate the capacitance between the two concentric spheres. PREVIOUS A spherical capacitor is made of two conducting spherical shells of radii a and b. ) Answer: 1. With the capacitor still connected to the battery, a slab of plastic of dielectric K = 3. 30 µC, find the amount of charge on the outer surface of the larger shell. An amplifying medium 115 is disposed in the cavity 104 between the two spheres. Calculate the capacitance. The intermediate medium is filled as shown in the figure with two dielectric materials of permittivity ε1 and ε2. tjllcchv9ve rwftrgqmpozoqb3 r6mohikgg31dfqt tr6lrxs7r6 0twhmd65h4z 6odlsgll33zil u1ww7uc1sco6sw9 t88u6ibvk87ip2c 4thn2kq2a06o xx0n56ret5u d29o6jh0ekmzzt wsc65tgaohrm 1zhtx72lrrfjqc kx8c5ddfu2gh8n 3xnsbihh5k 7p5us6bjh7b2a r1sdl9jk1jcqe 7rqgovkxzhu 130lfk68gmg qvhv29jsv8pi nno1tortcffupvq 1uvguqmwca i09y2l1zbosn nb8lnz92qaj438 p8ev1s775t eezd3p7jnk2di p7nd9ke6pj59s xmgszoagwgtkzx8 c1oarvv6yqhrn3m	2020-07-14T09:35:18	{ "domain": "royalvillagemilano.it", "url": "http://rmjo.royalvillagemilano.it/the-capacitance-of-two-concentric-spherical-metal-shells-with-radii-a-and-b-is.html", "openwebmath_score": 0.8156538605690002, "openwebmath_perplexity": 519.7010246247543, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.9790357567351325, "lm_q2_score": 0.855851143290548, "lm_q1q2_score": 0.83790887172409 }
https://math.stackexchange.com/questions/392801/computing-iiint-mathbbr3-e-x2-y2-z2dxdydz-using-substitution	# Computing $\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$ using substitution Consider this integral: $$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz$$ How would you compute it? I already solved this problem this way: $$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \left( \int_\infty^\infty e^{-x^2} \right)^3 = \pi^{3/2}$$ But I wanted to find it using substitution (spherical coordinates) but this is all I could do: $$\iiint_\mathbb{R^3} e^{-x^2-y^2-z^2}dxdydz = \lim_{j\,\rightarrow\infty}\int_0^jdu\int_0^\pi dv\int_0^{2\pi} e^{-u^2}u^2\sin(v) dw=$$ $$=2\pi\lim_{j\,\rightarrow\infty}\int_0^jdu\int_0^\pi e^{-u^2}u^2\sin(v)dv=4\pi\lim_{j\,\rightarrow\infty}\int_0^je^{-u^2}u^2du$$ But it doesn't get me anywhere. Help would be very appreciated, thanks in advance. • That should be $e^{-u^2}$. The integrals over $v$ and $w$ produce a factor of $4 \pi$. – Ron Gordon May 15 '13 at 20:24 • Yes, I changed the $e^{u^2}$ into $e^{-u^2}$, but the factor $4\pi$ was there from the beginning unless you I missed something. – user41489 May 15 '13 at 20:31 To finish off your problem, you only need $$\int_{0}^{\infty}e^{-x^2}x^2dx=\frac{\sqrt{\pi}}{4}$$ which can be shown by integrating $$\int_{0}^{\infty}e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$$ by parts. That is, we have: $$\frac{\sqrt{\pi}}{2}=\int_{0}^{\infty}e^{-x^2}dx=xe^{-x^2}\vert_{0}^{\infty}+2\int_{0}^{\infty}e^{-x^2}x^2dx=2\int_{0}^{\infty}e^{-x^2}x^2dx$$ • Very clear, thank you. – user41489 May 15 '13 at 20:42 $$\int_0^{\infty}e^{-x^2}x^2dx = \frac 1 2 \int_0^{\infty } e^{-x^2} x \; 2x dx = \frac 1 2 \int_0^{\infty } e^{-u} \sqrt u du = \frac{\Gamma (3/2)}{2} = \frac 1 4\sqrt{\pi}$$	2020-04-08T13:18:39	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/392801/computing-iiint-mathbbr3-e-x2-y2-z2dxdydz-using-substitution", "openwebmath_score": 0.9316490292549133, "openwebmath_perplexity": 447.9665543484954, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.97903575765266, "lm_q2_score": 0.8558511414521923, "lm_q1q2_score": 0.837908870709541 }
https://mathforums.com/threads/revival-math-q-a.42514/	# Revival-Math Q&A #### agentredlum Math Team The Fiboncci sequence begins with the numbers 1,1 and then each subsequent number is the sum of the two previous numbers. 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , ... Prove that every 5th Fibonacci number is divisible by 5 Q&A RULES 1) Anyone can post an answer. 2) The one who posted the problem decides the winner. 3) The winner must post the next question or relinquish control. 5 people #### MarkFL I will use induction. Our base case $P_1$ is obviously true since: $$\displaystyle F_{5}=5$$ Thus, our induction hypothesis $P_n$ is: $$\displaystyle F_{5n}=5m$$ where $$\displaystyle m,n\in\mathbb{N}$$ Now, we if use the recursive definition of the sequence and observe that: $$\displaystyle F_{5(n+1)}-F_{5n}=F_{5n+4}+F_{5n+3}-F_{5n}$$ $$\displaystyle F_{5(n+1)}-F_{5n}=F_{5n+3}+2F_{5n+2}+F_{5n+1}-F_{5n}$$ $$\displaystyle F_{5(n+1)}-F_{5n}=3F_{5n+2}+2F_{5n+1}-F_{5n}$$ $$\displaystyle F_{5(n+1)}-F_{5n}=5F_{5n+1}+2F_{5n}$$ Adding this to the hypothesis, we obtain: $$\displaystyle F_{5(n+1)}=5F_{5n+1}+3\cdot5m=5\left(F_{5n+1}+3m \right)$$ We have derived $P_{n+1}$ from $P_n$ thereby completing the proof by induction. Last edited: 2 people #### Olinguito $$\displaystyle F_{5(n+1)}-F_{5n}=3_{5n+2}+2F_{5n+1}-F_{5n}$$ $$\displaystyle F_{5(n+1)}-F_{5n}=5_{5n+1}+F_{5n}$$ Sorry to nitpick but itâ€™s $$\displaystyle F_{5(n+1)}-F_{5n}=5_{5n+1}+\color{red}2\color{black}F_{5n}$$ and we should have $$F_{5(n+1)}\ =\ 5F_{5n+1}+3F_{5n}\ =\ 5\left(F_{5n+1}+3m \right)$$ For example, $$F_{10}\ =\ 55\ =\ 40+15\ =\ 5F_6+3F_5$$ $$F_{15}\ =\ 610\ =\ 445+165\ =\ 5F_{11}+3F_{10}$$ 2 people #### MarkFL Sorry to nitpick but itâ€™s $$\displaystyle F_{5(n+1)}-F_{5n}=5_{5n+1}+\color{red}2\color{black}F_{5n}$$ and we should have $$F_{5(n+1)}\ =\ 5F_{5n+1}+3F_{5n}\ =\ 5\left(F_{5n+1}+3m \right)$$ For example, $$F_{10}\ =\ 55\ =\ 40+15\ =\ 5F_6+3F_5$$ $$F_{15}\ =\ 610\ =\ 445+165\ =\ 5F_{11}+3F_{10}$$ Yes, I have corrected my post. 1 person #### agentredlum Math Team Excellent , MarkFL is the winner! Olinguito gets a bonus point for proofreading. MarkFL , you have the next question. 2 people #### MarkFL Okay...here is a fun exercise: Problem: The double integral $\int_0^1\int_0^1 \frac{1}{1-xy}\,dx\,dy$ is an improper integral and could be defined as the limit of double integrals over the rectangle: $[0,t]\times[0,t]$ as $t\to 1^{-}$. 1.) Expand the integrand as a geometric series to show that $\int_0^1 \int_0^1 \frac{1}{1-xy}\,dx\,dy = \sum_{n=1}^{\infty}\frac{1}{n^2}$ 2.) Leonhard Euler proved that $\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$ Prove this fact by evaluating the integral found in (1). 3 people #### agentredlum Math Team This proof is from Brendan W. Sullivan who gives Tom Apostol credit for inventing it. For MarkFL's 1) We observe $0 \le xy \le 1$ So we let $xy = r$ without fear in order to use the formula for a geometric series. $$\dfrac{1}{1-xy} = \dfrac{1}{1-r} = 1 + r + r^2 + r^3 + ... = \\ 1 + xy + (xy)^2 + (xy)^3 + ... = \sum_{n = 1}^{\infty}(xy)^{n-1}$$ Now you can do this $$\int_{0}^1 \int_{0}^1 ( 1 + xy + x^2y^2 + x^3y^3 + ... ) dxdy$$ Work from the inside out , integrate term by term with respect to x keeping y constant. $$\int_{0}^1 (x + \dfrac{x^2y}{2} + \dfrac{x^3y^2}{3} + \dfrac{x^4y^3}{4} + ...) \|_{0}^1)dy$$ Plug in the upper limit x = 1 (the lower limit x = 0 vanishes making no contribution) $$\int_{0}^1 (1 + \dfrac{y}{2} + \dfrac{y^2}{3} + \dfrac{y^3}{4} + ... ) dy$$ Integrate with respect to y $$(y + \dfrac{y^2}{2 \cdot 2} + \dfrac{y^3}{3 \cdot 3} + \dfrac{y^4}{4 \cdot 4} + ...) \|_{0}^1$$ Plug in the upper limit y = 1 (the lower limit y = 0 vanishes), $$1 + \dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2} + ... + \dfrac{1 }{n^2} + ... = \sum_{n = 1}^{\infty } \dfrac{1 }{n^2}$$ This confirms MarkFL's 1) For MarkFL's 2) I refer the reader to the link given above because giving details here would turn this into a very long post. #### MarkFL This problem was given as a university level problem of the week at MHB, and this is the solution I posted: 1.) Expanding the integrand as a geometric series, we may write: $$\displaystyle \frac{1}{1-xy}= \sum_{n=0}^{\infty}(xy)^n$$ Hence the integral becomes: $$\displaystyle \int_0^1\int_0^1 \sum_{n=0}^{\infty}(xy)^n\,dx\,dy=\int_0^1\left[\sum_{n=0}^{\infty}\frac{x^{n+1}y^n}{n+1} \right]_0^1\,dy=$$ $$\displaystyle \int_0^{1}\sum_{n=1}^{\infty}\frac{y^{n-1}}{n}\,dy=\left[\sum_{n=1}^{\infty}\frac{y^{n}}{n^2} \right]_0^1=\sum_{n=1}^{\infty}\frac{1}{n^2}$$ 2.) Using the change of variables: $$\displaystyle (x,y)=(u-v,u+v)$$ we obtain: $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=\iint_{R}\frac{1}{1-u^2+v^2}\left\|\frac{\partial (x,y)}{\partial (u,v)} \right\|\,du\,dv$$ Calculating the Jacobian matrix, we find: $$\displaystyle \frac{\partial (x,y)}{\partial (u,v)}=\begin{vmatrix}\frac{\partial x}{\partial u}&\frac{\partial x}{\partial v}\\\frac{\partial y}{\partial u}&\frac{\partial y}{\partial v}\\\end{vmatrix}=\begin{vmatrix}1&-1\\1&1\\\end{vmatrix}=1(1)-(-1)(1)=2$$ Thus, we have: $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=2\iint_{R}\frac{1}{1-u^2+v^2}\,du\,dv$$ Remapping the boundaries in terms of the new variables, we find $R$ is a square in the $uv$-plane with vertices: $$\displaystyle (0,0),\,\left(\frac{1}{2},-\frac{1}{2} \right),\,\left(\frac{1}{2},\frac{1}{2} \right),\,(1,0)$$ Reversing the order of integration and using the symmetry of the square, we obtain: $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=4\left(\int_0^{ \frac{1}{2}}\int_0^u \frac{dv\,du}{1-u^2+v^2}+\int_{ \frac{1}{2}}^1\int_0^{1-u} \frac{dv\,du}{1-u^2+v^2} \right)$$ Next, we may compute: $$\displaystyle \int_0^u\frac{dv}{1-u^2+v^2}=\left[\frac{1}{\sqrt{1-u^2}}\tan^{-1}\left(\frac{v}{\sqrt{1-u^2}} \right) \right]_0^u=\frac{1}{\sqrt{1-u^2}}\tan^{-1}\left(\frac{u}{\sqrt{1-u^2}} \right)=\frac{1}{\sqrt{1-u^2}}\sin^{-1}(u)$$ $$\displaystyle \int_0^{1-u}\frac{dv}{1-u^2+v^2}=\left[\frac{1}{\sqrt{1-u^2}}\tan^{-1}\left(\frac{v}{\sqrt{1-u^2}} \right) \right]_0^{1-u}=\frac{1}{\sqrt{1-u^2}}\tan^{-1}\left(\frac{1-u}{\sqrt{1-u^2}} \right)=\frac{1}{\sqrt{1-u^2}}\tan^{-1}\left(\sqrt{\frac{1-u}{1+u}} \right)$$ If we let: $$\displaystyle \tan(\theta)=\sqrt{\frac{1-u}{1+u}}$$ Squaring, we obtain: $$\displaystyle \tan^2(\theta)=\frac{1-u}{1+u}$$ $$\displaystyle \tan^2(\theta)+1=\frac{1-u}{1+u}+1$$ Apply a Pythagorean identity on the left and combine terms on the right: $$\displaystyle \sec^2(\theta)=\frac{2}{1+u}$$ Invert both sides: $$\displaystyle \cos^2(\theta)=\frac{u+1}{2}$$ Solving for $u$, we find: $$\displaystyle u=2\cos^2(\theta)-1=\cos(2\theta)$$ Hence, we find: $$\displaystyle \tan^{-1}\left(\sqrt{\frac{1-u}{1+u}} \right)=\theta=\frac{1}{2}\cos^{-1}(u)$$ Using the identity $$\displaystyle \sin^{-1}(u)+\cos^{-1}(u)=\frac{\pi}{2}$$ we finally have: $$\displaystyle \tan^{-1}\left(\sqrt{\frac{1-u}{1+u}} \right)=\frac{1}{2}\left(\frac{\pi}{2}-\sin^{-1}(u) \right)$$ Utilizing these results, we now may state: $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=4\left(\int_0^{ \frac{1}{2}}\frac{1}{\sqrt{1-u^2}}\sin^{-1}(u)\,du+\frac{1}{2}\int_{ \frac{1}{2}}^1 \frac{1}{\sqrt{1-u^2}}\left(\frac{\pi}{2}-\sin^{-1}(u) \right)\,du \right)$$ Now, let's look at the first integral: $$\displaystyle \int_0^{ \frac{1}{2}}\frac{1}{\sqrt{1-u^2}}\sin^{-1}(u)\,du$$ Using the substitution: $$\displaystyle \alpha=\sin^{-1}(u)\,\therefore\,d\alpha=\frac{1}{\sqrt{1-u^2}}\,du$$ we now have: $$\displaystyle \int_0^{\frac{\pi}{6}}\alpha\,d\alpha=\frac{1}{2} \left[\alpha^2 \right]_0^{\frac{\pi}{6}}= \frac{1}{2}\left(\frac{\pi}{6} \right)^2=\frac{\pi^2}{72}$$ Next, let's break the second integral into two parts: i) $$\displaystyle \frac{\pi}{4}\int_{\frac{1}{2}}^1 \frac{1}{\sqrt{1-u^2}}\,du=\frac{\pi}{4}\left[\sin^{-1}(u) \right]_{\frac{1}{2}}^1=\frac{\pi}{4}\left(\frac{\pi}{2}-\frac{\pi}{6} \right)=\frac{\pi^2}{12}$$ ii) $$\displaystyle -\frac{1}{2}\int_{\frac{1}{2}}^1 \frac{1}{\sqrt{1-u^2}}\sin^{-1}(u)\,du$$ Using the substitution: $$\displaystyle \alpha=\sin^{-1}(u)\,\therefore\,d\alpha=\frac{1}{\sqrt{1-u^2}}\,du$$ we now have: $$\displaystyle -\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \alpha\,d\alpha=-\frac{1}{4}\left[\alpha^2 \right]_{\frac{\pi}{6}}^{\frac{\pi}{2}}=-\frac{1}{4}\left(\left(\frac{\pi}{2} \right)^2-\left(\frac{\pi}{6} \right)^2 \right)=-\frac{\pi^2}{18}$$ Thus, putting these results together, there results: $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=4\left(\frac{\pi^2}{72}+\frac{\pi^2}{12}-\frac{\pi^2}{18} \right)=4\left(\frac{\pi^2}{24} \right)$$ And, we may then state: $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ Shown as desired. So...agentredlum, you now have the floor. 1 person #### agentredlum Math Team Thanx , it would have taken me 2 hours + to write in rhe details using mathjax with my PS3 controller. Are you concerned at all that when $$x = y = 1$$ we get $$\dfrac{1}{1-1}$$ and the Geometric series does not apply? 1 person #### MarkFL Thanx , it would have taken me 2 hours + to write in rhe details using mathjax with my PS3 controller. Are you concerned at all that when $$x = y = 1$$ we get $$\dfrac{1}{1-1}$$ and the Geometric series does not apply? I see $x$ and $y$ as "approaching" 1 from beneath as given by the definition of the rectangle. 1 person Similar Math Discussions Math Forum Date Calculus Math Books Number Theory Number Theory	2020-04-07T11:05:31	{ "domain": "mathforums.com", "url": "https://mathforums.com/threads/revival-math-q-a.42514/", "openwebmath_score": 0.9258551597595215, "openwebmath_perplexity": 1357.8396856483935, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9881308803517761, "lm_q2_score": 0.8479677622198946, "lm_q1q2_score": 0.8379031313922699 }
https://math.stackexchange.com/questions/2832794/does-the-following-object-have-such-property	# Does the following object have such property? Let $S_n$ denote the set of bijections on the set $M = \{1, 2, ... , n\}$. Suppose that a set $\Omega \subset S_n$ satisfies the following condition: there is $k \leq n$ such that, for each nonempty subset $A \subset M$ with $\|A\| \leq k$ and each $a \in A$ there exist exactly $\frac{\|\Omega\|}{\|A\|}$ permutations $\pi \in \Omega$ for which $\min_{x \in A} \pi (x) = \pi (a)$. Is it true, that for every nonempty subset $A \subset M$ with $\|A\| \leq k$ and every $m \in \{1, 2, ... \|A\|\}$ there exist exactly $\frac{\|\Omega\|}{\|A\|}$ permutations $\pi \in \Omega$ such that $\pi (a)$ is the $m$-th largest element of $\pi (A)$? For small $n$-s the statement seems to be true. However, I have no idea how to prove it in general. Any help will be appreciated. • Do you have an example with $\Omega\subsetneq S_n$ for which your “following condition” is true with $k>2$? Certainly it’s true for $\Omega= S_n$ (for any $k$), and regardless of $\Omega$, if you consider subsets $A=\{i,j\}$ of cardinality $2$, the condition “$\sigma(i)$ is $m$th largest in $\sigma(A)$” is the same as “$\sigma(j)$ is smallest” when $m=1$ and “$\sigma(i)$ is smallest” for $m=2$. So it seems like your conjecture could only have a counterexample with $\Omega\subsetneq S_n$ and $k>2$, but I can’t easily think of a proper subset of $S_n$ with your condition ($k>2$). – Steve Kass Jun 30 at 18:03 • I suspect you want $k$ to be a fixed constant, rather than "there exists $k$". – darij grinberg Jun 30 at 18:21 • @SteveKass: $\Omega = A_n$ (the alternating group) and $k \leq n-2$ always works. – darij grinberg Jun 30 at 18:23 The answer is affirmative. Use induction, say, on $m$. By assumption, the statement holds for $m=1$. Denote for $\pi \in S_n$, $A\subset[n]$, $a\in A$ by $\nu_\pi(A,a)$ the rank of $\pi(a)$ among $\pi(x), x\in A$. Then $$N(A,a,m) = \sum_{\pi\in\Omega}\mathbf{1}_{\nu_\pi(A,a)=m}$$ is the quantity we are interested in. If $m = \|A\|$, then by induction hypothesis $$N(A,a,m) = \|\Omega\| - \sum_{i=1}^{m-1} N(A,a,i) = \|\Omega\| - \sum_{i=1}^{m-1} \frac{\|\Omega\|}{\|A\|} = \frac{\|\Omega\|}{\|A\|}.$$ Otherwise, if $2\le m<\|A\|$, write by induction hypothesis $$\|\Omega\| = (\|A\|-1)\cdot\frac{\|\Omega\|}{\|A\|-1} = \sum_{b\in A\setminus \{a\}} N(A\setminus \{b\},a,m-1)\\ = \sum_{b\in A\setminus \{a\}} \sum_{\pi\in\Omega}\mathbf{1}_{\nu_\pi(A\setminus\{b\},a)=m-1} = \sum_{\pi\in\Omega}\sum_{b\in A\setminus \{a\}}\mathbf{1}_{\nu_\pi(A\setminus\{b\},a)=m-1}.$$ Note that if $\nu_\pi(A\setminus\{b\},a)=m-1$, then either $\pi(b)>\pi (a)$ and $\nu_\pi(A,a)=m-1$ or $\pi(b)<\pi(a)$ and $\nu_\pi(A,a)=m$. Therefore, using the induction hypothesis, $$\|\Omega\| = \sum_{\pi \in \|\Omega\|} \mathbf{1}_{\nu_\pi(A,a)=m-1} \sum_{b\in A:\pi(b)>\pi(a) }1 + \sum_{\pi \in \|\Omega\|} \mathbf{1}_{\nu_\pi(A,a)=m } \sum_{b\in A:\pi(b)<\pi(a) }1 \\ = \sum_{\pi \in \|\Omega\|} \mathbf{1}_{\nu_\pi(A,a)=m-1} (\|A\|-m + 1) + \sum_{\pi \in \|\Omega\|} \mathbf{1}_{\nu_\pi(A,a)=m }(m-1)\\ = (\|A\|-m + 1) N(A,a,m-1) + (m-1) N(A,a,m)\\ = (\|A\|-m + 1) \frac{\|\Omega\|}{\|A\|} + (m-1) N(A,a,m).$$ Hence, $$N(A,a,m) = \frac{\|\Omega\|}{\|A\|},$$ as required. • In fact, the cases $m=\|A\|$ and $m<\|A\|$ can be combined: I had a mistake in my draft, where the induction was in $k$ and $m$, and $N(A,a,m)$ was written in terms of $N(A,a,m-1)$ and $N(A\setminus\{b\}, a,m)$, so I put this case in there. Now considering $m=\|A\|$ individually is no longer needed, but let it be. – zhoraster Jul 17 at 5:24	2018-11-16T22:46:26	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2832794/does-the-following-object-have-such-property", "openwebmath_score": 0.9680218696594238, "openwebmath_perplexity": 111.82348210808733, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9881308814003628, "lm_q2_score": 0.8479677602988602, "lm_q1q2_score": 0.8379031303832043 }
https://cs.stackexchange.com/questions/54990/filling-bins-with-pairs-of-balls	Filling bins with pairs of balls A bin is called full if it contains at least $k$ balls. Our goal is to make as many bins as possible full. In the simplest scenario, we are given $n$ balls and may arrange them arbitrarily. In that case, obviously the best we can do is pick $\lfloor n/k \rfloor$ bins arbitrarily and put $k$ balls in each one of them. I am interested in the following scenario: we are given $n$ pairs of balls. We have to put the two balls of each pair in two different bins. Then, an adversary comes and removes one ball from each pair. What can we do to have the maximum possible number of full bins after the removal? A simple strategy is: pick $\lfloor n/(2k-1) \rfloor$ pairs of bins. Fill each bin-pair with $2k-1$ ball-pairs (each bin contains $2k-1$ balls, one ball from each pair). Then, regardless of what our adversary removes, we have in each bin-pair at least one full bin. Do we have a strategy that achieves a larger number of full bins (more than $\lfloor n/(2k-1) \rfloor$)? • I don't believe so Apr 5 '16 at 17:39 • $n$ is given and $k$ is given? $k$ depends on $n$? – Evil Apr 5 '16 at 23:13 • @EvilJS $n$ and $k$ are given, and are independent. Apr 7 '16 at 9:00 • Does the player place all of his $n$ pairs of balls and then the adversary picks $n$ balls?, or does the player place a pair of balls and then the adversary chooses one from that pair and then the player puts the next pair and the adversary picks one and so on until there are no more pairs of balls to place? Apr 10 '16 at 22:20 • @rotia The player places all of his n pairs of balls, and then the adversary picks n balls. Apr 11 '16 at 10:36 TL;DR -- No, there is no better strategy than the simple strategy. Here is the main idea of the proof. When there are not enough balls, there will be a "ball path" from a $k$-full bin to a bin with at most $k-2$ balls. The adversary can pass a ball from that full bin to that less full bin along that path, which can be done repeatedly until the number of $k$-full bins is reduced. Reformulation in Graph theory Suppose we are given a simple finite graph $G(V,E)$ with a function $w: E\to\Bbb Z_{\ge 0}$. We say there are $w(e)$ balls in edge $e$. Let $E_2$ be the (end-marked edge) set $\{(e,v)\| e\in E, v\in e\}$. If $d:E_2\to\Bbb Z_{\ge 0}$ satisfies $w(e)=d(e,v_1) + d(e,v_2)$ for every edge $e=\{v_1,v_2\}$, we say that $d$ is $w$-distributing. Any $w$-distributing function $d$ induces a function, which we use the same symbol, $d:V\to\Bbb Z_{\ge 0}$, $d(v) =\sum_{v\in e}d(e,v)$. We say that $d(v)$ balls are in $v$. Given $k\in\Bbb Z_{\gt0}$, let $F_k(d)=\#\{v\in V\|d(v)\ge k\}$, the number of $k$-full vertices by $d$. (Erel-Apass Theorem) For any simple finite graph $G(V,E)$ and $w:E\to\Bbb Z_{\ge0}$, we have $\sum_{e\in E}w(e)\geq (2k-1) \min_{w\text{-distributing }d}F_k(d)$ Imagine each vertex is a bin. For each edge $e=\{v_1,v_2\}$, $w(e)$ ball-pairs are put into $v_1$ and $v_2$, each of which getting $w(e)$ balls. Among these $w(e)$ ball-pairs, the adversary may take away $d(e,v_2)$ balls from $v_1$ and $d(e,v_1)$ balls from $v_2$. The end result is the same as if, given all empty bins initially, for each edge $e=\{v_1, v_2\}$, $w(e)$ balls are put into it and, then, $d(e,v_1)$ and $d(e,v_2)$ balls are distributed to $v_1$ and $v_2$ respectively by the adversary. Hence, Erel-Apass theorem says that in order to ensure $t$ k-full bins after a smart adversary's removal, at least $(2k-1)t$ pairs of balls are needed. In another word, an optimal strategy to have the maximum possible number of full bins left is indeed the "simple strategy", which repeatedly fills a different pair of bins with $2k-1$ ball-pairs until we do not have enough balls to repeat. Proof of the theorem For the sake of contradiction, let $G(V,E)$ and $w$ be a counterexample whose number of vertices is the smallest among all counterexamples. That is, there is $w$-distributing $m$ such that $F_k(m)$ is minimal among all $F_k(d)$ of $w$-distributing function $d$. Furthermore, $$\sum_{e\in E}w(e)\lt (2k-1)F_k(m)$$ Let $V_s=\{v\in V \| m(v)\le k-2\}$. Let $V_\ell=\{v\in V\|m(v)\geq k\}$. So $F_k(m)=\#V_\ell$. Claim one: $V_s\neq\emptyset$. Proof of claim one. Suppose otherwise that $V_s$ is empty. $$\sum_{v\in V}m(v)= (k-1)\#V +\sum_{v\in V}(m(v)-(k-1)) \ge (k-1)\#V + \#V_\ell \gt (k-1)\#V$$ Let us also reuse $w$ as a function from $V$ to $\Bbb Z_{\ge 0}$ such that $w(v)=\sum_{v\in e}w(e)$ for any $v\in V$. \begin{align} \sum_{v\in V}w(v) &=\sum_{v\in V}\sum_{v\in e }w(e) =\sum_{e\in E }\sum_{v\in e}w(e) =\sum_{e\in E}2w(e) =2\sum_{e\in E}w(e)\\ &=2\sum_{e\in E }\sum_{v\in e}m(e,v) =2\sum_{v\in V}\sum_{v\in e }m(e,v) =2\sum_{v\in V}m(v)\\ &\gt 2(k-1)\#V \end{align} So there must be a vertex $b$ such that $w(b)\ge 2k-1$. Consider the induced setup $G'(V', E')$ and $w'$, where $V'=V\setminus\{b\}$, $G'(V', E')$ is the induced graph $G[V']$ and where $w'=w\|_{E'}$. For any $w'$-distributing function $d'$, we can extend it to a $w$-distributing function $d_{d'}$ where $d_{d'}$ is the same as $d'$ on $E'$ while $d_{d'}(e, b)=w(e)$ for every edge $e$ adjacent to $b$. Note that $F_k(d_{d'})= F_k(d') + 1$ since $d_{d'}(b)=\sum_{b\in e}d_{d'}(e,b) = \sum_{b\in e}w(e)=w(b)\ge 2k-1\ge k$. Then \begin{align}\sum_{e\in E'}w'(e)&\le\sum_{e\in E}w(e) - w(b)\\ &\lt (2k-1)F_k(m) -(2k-1)\\ &=(2k-1) \left(\min_{w\text{-distributing }d}F_k(d) -1\right)\\ &\le (2k-1) \left(\min_{w'\text{-distributing }d'}F_k(d_{d'})-1\right)\\ &\le (2k-1) \min_{w'\text{-distributing }d'}F_k(d') \end{align} So, $G'(V', E')$ and $w'$ is a counterexample whose number of vertices is smaller than the number of vertices in $G$. That cannot true by our assumption about $G(V,E)$ and $w$. So claim one is proved. For any vertex $v$, define $v$ $d$-reachable from vertex $u$ if there is a path $u_0= u, u_1, u_2, \cdots, u_m, u_{m+1}=v$, $m\ge0$ such that $d(\{u_i, u_{i+1}\}, u_i)>0$. Let $V_r=V_\ell\cup \{v\in V\|\exists u\in V_\ell \text{ and } v\text{ is } m\text{-reachable from } u \}$. Claim two: $V_r = V$ Proof of claim two: Suppose $V_r\neq V$. For any vertex $v\in V_r$ and $u\notin V_r$, since we cannot reach $u$ from $v$, if $\{v, u\}$ is an edge, then $w(\{v, u\}, v) = 0.$ Consider the induced setup $G'(V', E')$ and $w'$, where $v'=V_r$, $G'(V', E')$ is the induced graph $G[V']$ and where $w'=w\|_{E'}$. For any $w'$-distributing function $d'$, we can extend it to a $w$-distributing function $d_{d'}$ where $d_{d'}$ is the same as $d'$ on $E'$ and the same as $m$ on other edges. Note that $F_k(d_{d'})= F_k(d')$ since all vertices with no less than $k$ balls inside are in $V_\ell\subset V_r$. Then \begin{align}\sum_{e\in E'}w'(e)&\le\sum_{e\in E}w(e)\\ &\lt (2k-1)F_k(m)\\ &=(2k-1) \min_{w\text{-distributing }d}F_k(d)\\ &\le (2k-1) \min_{w'\text{-distributing }d'}F_k(d_{d'})\\ &\le (2k-1) \min_{w'\text{-distributing }d'}F_k(d') \end{align} So, $G'(V', E')$ and $w'$ would be a counterexample whose number of vertices is smaller than the number of vertices in $G$. That cannot be true by our assumption about $G(V,E)$ and $w$. So claim two is proved. Now let us prove the theorem. Since $V_r=V$ and $V_s\neq\emptyset$, there is a path $u_0= u, u_1, u_2, \cdots, u_m, u_{m+1}=v$, $m\ge0$ with $m(u)\gt k$, $m(v)\leq k-2$ and $d(\{u_i, u_{i+1}\}, u_i)>0$. Let us construct a new $w$-distributing function $r(m)$ from $m$ so that $$r(m)(e, u)= \begin{cases} m(\{u_i, u_{i+1}\}, u_i) -1 & \text{ if } (e,u)=(\{u_i, u_{i+1}\},u_i)\text { for some } 0\le i\le m\\ m(\{u_i, u_{i+1}\}, u_{i+1}) +1 & \text{ if } (e,u)=(\{u_i, u_{i+1}\},u_{i+1})\text { for some } 0\le i\le m\\ m(e,u) &\text{ otherwise } \end{cases}$$ $m$ and $r(m)$ agrees on all vertices except $v$ and $u$, $m(v)\lt r(m)(v)\le k-1$ and $r(m)(u)\lt m(u)$. We can apply this procedure on $r(m)$ to get $r^2(m)$. Repeating this $i$ time for some large enough $i$, we will obtain a $w$-distributing function $r^i(m)$ with $F_k(r^i(m))=0$. However, we have assumed that $F_k(m)>0$ is the minimum among $F(d)$ of $w$-distributing function $d$. This contradiction shows that we have proved the Erel-Apass theorem. • I read the proof, it looks good. In fact, if I understand correctly, it is even more general since it allows for an arbitrary graph - my question is a special case where G is the complete graph. Is this correct? Another question: where exactly does the proof use the fact that m is such that Fk(m) is minimal? I see that it is used only at the last paragraph - are the previous claims in the proof true without this fact? Oct 31 '18 at 8:03 • Yes, the theorem is correct for any graph since it says "for any (simple finite) graph G(V,E)". The minimality of $F_k(m)$ is necessary for each claim. If you search for "counterexample", you will find where the minimality is used. Nov 5 '18 at 6:17	2022-01-29T05:24:52	{ "domain": "stackexchange.com", "url": "https://cs.stackexchange.com/questions/54990/filling-bins-with-pairs-of-balls", "openwebmath_score": 0.9139223694801331, "openwebmath_perplexity": 311.902000956992, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9881308761574287, "lm_q2_score": 0.8479677622198947, "lm_q1q2_score": 0.8379031278355986 }
https://math.stackexchange.com/questions/2918124/is-empty-set-with-usual-metric-space-x-mathbbr-d-bounded	# Is empty set with usual metric space $(X=\mathbb{R},d)$ bounded? Here R is set of real numbers and $d$ is distance metric i.e $\|x-y\|$. I wanted to find if empty set under given metric space is compact and we know by Heine Borel theorem that a set is compact iff it is closed and bounded. Empty set is closed because its complementary set i.e. set of real numbers $\mathbb{R}$ is open.Hence i just need to prove if the empty set is bounded. The problem i am facing is , how can we define $\|x-y\|$ on empty set , to find $\exists m$ , $m \in \mathbb{R}$ , such that $\|x-y\|\leqslant m$. • Write out the definition carefully with quantifiers. – Ted Shifrin Sep 15 '18 at 18:13 • A set $E$ is bounded iff there exists $m>0$ such that, for all $x,y \in E$ we have $d(x,y) \le m$. So what do you get when $E = \varnothing$? – GEdgar Sep 15 '18 at 18:15 • It is vacuously bounded. – Eduardo Longa Sep 15 '18 at 18:15 • @GEdgar I cannot have any pair $x,y$ such that$d(x,y)<=m$ , hence i cannot say it is bounded or not bounded. – user585269 Sep 15 '18 at 18:20 • Let $E = \emptyset$, and $m > 0$. Can you find a pair $x,y \in E$ such that $d(x,y) > m$? – Theoretical Economist Sep 15 '18 at 18:23 This is answered in comments already, but here's an honest-to-god StackExchange answer. Your question can be asked more succinctly as: "Is $\{\}$ bounded when considered as a subset of $\mathbb{R}$ with its standard metric?". The answer is Yes. In fact, the empty set is bounded when thought of as a subset of any metric space. To see why, you have to write down a definition of boundedness. It should probably come out as this: $X$ is bounded with respect to the metric $d$ if there exists a number $M \in \mathbb{R}$ such that, $d(a,b) < M$ for all $a, b \in X$. Side note: I would have got equivalent definitions if I'd said $M \in \mathbb{N}$ or asked that $d(a, b) \le M$. You can check you understand the definition above by checking you understand why these definitions would give the same thing. I claim that $\{\}$ is bounded with respect to any $d$. To prove this, I can set $M$ to be any number. After all, I just need to prove that if you choose $a, b \in \{\}$ then $d(a,b) < M$. But you can't choose find such $a$ or $b$ because the empty set is... empty! As such, the inequality is never actually checked. As you correctly wrote in the question, you can now use the Heine-Borel theorem to show that the empty set is compact as a subset of $\mathbb{R}$. In fact, the Heine-Borel theorem is quite heavy machinery to use to prove the compactness of the empty set. If you think for a minute about the definition of compactness based on finite subsets of open covers, you should be able to prove directly that the empty set is compact as a subset of any topological space. • To the proposer: If $S$ is any statement, phrase, clause, formula, etc., and if $X=\emptyset$ then the sentence $\forall a\in X\;(S)$ is true. Because its negation is $\exists a\in X\;(\neg S),$ which cannot be true unless some $a$ exists that belongs to $X.$ – DanielWainfleet Sep 16 '18 at 1:01	2019-08-25T00:56:44	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2918124/is-empty-set-with-usual-metric-space-x-mathbbr-d-bounded", "openwebmath_score": 0.9386159181594849, "openwebmath_perplexity": 125.55508942368414, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9881308777303092, "lm_q2_score": 0.8479677564567912, "lm_q1q2_score": 0.8379031234746501 }
https://math.stackexchange.com/questions/2430323/what-is-the-likelihood-of-two-line-segments-crossing	# What is the likelihood of two line segments crossing? Consider a square space. Randomly select 4 points. Randomly connect two sets of two to each other with line segments. What is the chance of the line segments intersecting? (I will maybe try and solve this with a little Monte-Carlo simulation but would be very interested in an analytic solution.) • Do you mean lines, or line segments (i.e. are the lines infinitely long in both directions, or do they stop at the points)? – Arthur Sep 15 '17 at 8:27 • The title was unclear. You are right. I updated it – Christian Sep 15 '17 at 8:31 • I went through some coding. I may be incorrect but the estimated probability is around $0.111$... May anyone confirm this result? – Wyllich Sep 15 '17 at 9:39 • @Wyllich I may be wrong, but it seems to me that the analytical solution (see answer below) yields a result of 25/108, which is about twice the probability you obtained. – Anonymous Sep 15 '17 at 10:56 • @Anonymous Yes. You are right. I checked my code and indeed, I was badly checking if the calculated intersection point belongs to both lines or not. I agree with $\approx 0.23$ too. – Wyllich Sep 15 '17 at 11:29 Let's be perfectly clear about the assumptions. I assume that "randomly selecting 4 points" A,B,C,D in the unit square means selecting uniformly and independently at random their coordinates $x_A,y_A,x_B...$ in the $[0,1]$ range; and that one of the 6 possible pairings between points to form the segments is also chosen uniformly and independently at random -- so the problem is equivalent to asking what is the probability that, e.g. AC and BD intersect. It's easy to see that the solution is $\frac{25}{108}$, or about $23\%$, if one is willing to take as a given the solution for Sylvester's four-point problem for the square -- the probability that the convex hull of four points chosen uniformly and independently at random in a square is a quadrilateral, which is $\frac{25}{36}$. It is immediate to see that two segments chosen by a random pairing of four points intersect if, and only if, the convex hull of the four points is indeed a quadrilateral, and the two segments are its diagonals. If the former condition is satisfied, the latter happens with probability $\frac{1}{3}$ (the probability that a given point is paired with "middle one" of the remaining three), yielding a solution of $\frac{25}{36}\cdot\frac{1}{3}=\frac{25}{108}$ for the original problem. Sylvester's four-point problems allows one to obtain the answer in exactly the same way even if the points are chosen uniformly at random in another convex region, simply by multiplication by $\frac{1}{3}$. The general formula for Sylvester's four-point problem in a convex region $R$, as given in the above link, equals $1-4\frac{\bar{A_R}}{A(R)}$ where $A(R)$ is the area of the region (in the case of the unit square, $1$) and $\bar{A}_R$ equals the expected area of a triangle obtained from $3$ points chosen uniformly at random within it. The formula is easily obtained noting that probability that one given point lies within the triangle formed by the other $3$ is $\frac{\bar{A_R}}{A(R)}$, and since the events of this happening for different points are mutually exclusive, the probability that no point lies within the triangle formed by the other three is indeed $1-4\frac{\bar{A}_R}{A(R)}$. Computing $\frac{\bar{A_R}}{A(R)}$ is non-trivial, in general. If R is a square (our case), a simple proof that $\frac{\bar{A_R}}{A(R)}=\frac{11}{144}$ can be found here. More in general if R is regular polygon of $n$ sides the solution is given by Alikosky's formula, $\frac{\bar{A_R}}{A(R)}=\frac{9\cos^2 (2\pi/n)+52\cos(2\pi/n)+44}{36n^2\sin^2(2\pi/n)}$. Note that this is a strictly decreasing function of $n$, so the intersection probability is strictly increasing with $n$, and the limit for $n\to\infty$, i.e. $\frac{35}{48\pi^2}$, yields the solution when picking points in the circle. So, in general, the probability of intersection of $2$ segments, each made by $2$ points chosen uniformly and independently at random from a regular $n$-agon or any of its affine transformations (such as a rectangle or parallelogram) is: $\frac{1}{3}\left(1-4\cdot\frac{9\cos^2(2\pi/n) + 52\cos(2\pi/n) + 44}{36 n^2\sin^2 (2\pi/n)}\right)$ with the limit for $n\to\infty$, i.e. $\frac{1}{3}-\frac{35}{36\pi^2}$, yielding the probability when choosing the points from a circle (or any affine transformation, such as an ellipse). • You mention the probability of the square being quadrilateral as $\frac{25}{36}$. Is this not the probability of it being convex? I think it will always be a quadrilateral according to the accepted definition. mathworld.wolfram.com/Quadrilateral.html Or am I misreading something? – Christian Sep 25 '17 at 11:11 • No, I am saying that 25/36 is the probability that the convex hull of four points, chosen uniformly at random within the square, is a quadrilateral (and thus a convex quadrilateral). Note that with some probability (11/36) the convex hull is a triangle instead, i.e. one of the four points falls inside the triangle formed by the other three. – Anonymous Sep 25 '17 at 23:26 First let's select 4 points inside a square. Let $P_c$ be a probability that these 4 points form a convex figure (none of the selected points is inside the triangle formed by the other three). If this is the case, the probability that two line segments intersect is 1/3. Otherwise the probability is 0. So, the probability we are looking for is: $P=1/3 * P_c$ Now we need to find the $P_c$. Let's calculate the probability $P_1$ that the first point is inside the triangle formed by the three others. It is $P_1 = S_{234}/S_{square}$, where $S_{234}$ is the expected area of the triangle formed by points 2, 3 and 4. Same for $P_2$, $P_3$ and $P_4$. Obviously $P_1=P_2=P_3=P_4$. Probability that any point is inside of triangle formed by other 3 is $P_1+P_2+P_3+P_4$ because no more than 1 of these outcomes can happened simultaneously. $P_c = 1 - 4P_1 = 1 - 4 S_{234}$ And now we only need to find $S_{234}$ - expected area of a triangle formed by randomly selected 3 points inside a square. The expected area of a triangle formed by three points randomly chosen from the unit square They say it is $11/144$. $P=1/3 * (1 - 411/144) = 25/108$ This is in a good agreement with my experiments. This is my approach: Consider a line $y=mx+c$ that divides the unit square into two trapeziums. Let us call the left trapezoid $a$, and the right trapezoid $b$. By solving the equations $(y=mx+c)=1$ and $(y=mx+c)=0$, or $mx+c=1$ and $mx+c=0$, the $x$-values of the intersections are $x = \frac{-c+1}{m}$, and $\frac{-c}{m}$. Since the $y$-values are $0$ and $1$, the intersection coordinates are $(\frac{-c}{m}, 0)$ and $(\frac{-c+1}{m}, 1)$. Using this information, the area of trapezium $a$ is $1 \frac{(-c/m+1)+(-c/m)}{2}$, and of $b$ is $1 - (1 * \frac{(-c/m+1)+(-c/m)}{2})$ (since $a+b=1$). Simplifying gives the area of $a$ as $-\frac{c}{m}+ \frac{1}{2}$, and the area of $b$ as $\frac{c}{m}+\frac{1}{2}$. For the line connecting two points to not cross the green line, the points cannot be in $a$ and $b$. Since using geometric probability, the probability of this is $ab-ba$, we can subtract this condition from $1$ to get $1 - 2(-\frac{c}{m}+ \frac{1}{2})(\frac{c}{m}+\frac{1}{2})$, and after simplifying we have the probability as $1 - 2(-\frac{c^2}{m^2}+\frac{1}{4})$ or $$\frac{2c^2}{m^2}+\frac{1}{2},$$ where $c$ is the $y$-intercept of the line (when extended), and $m$ is its slope. Note: This example doesn't work when $c=0$, because regardless of what $m$ is chosen, it would equal $0+\frac{1}{2}=\frac{1}{2}$, and when the line crosses a point not between $0$ and $1$ when $x=0$ or $x=1$. • Sorry to all of you reading my answer - I just realised the final answer doesn't make any sense. I will try to correct the error within $10$ minutes as much as I can. – Toby Mak Sep 15 '17 at 8:58 • The question asks for line segments, not lines that intersect inside of the square – Dominik Sep 15 '17 at 9:01 • Lines and line segments are the same things, except that one is a shortened version of another. I made sure in my answer that I did not include any solutions with intersections outside the square, for example by setting $a = 1-b$. It doesn't matter whether they are lines or line segments - their equations are essentially very similar. – Toby Mak Sep 15 '17 at 9:08 • Consider the line segments through $(0, 0.5)$, $(0.5,0.5)$ resp. $(0.75, 0.25)$, $(0.75, 0.75)$. They do not intersect, but would intersect according to yourcalculations. – Dominik Sep 15 '17 at 9:11 • That's true, but your second line is vertical, which cannot be expressed in any equation because of division by zero. You can't prove it's just my formula that doesn't work. – Toby Mak Sep 15 '17 at 9:15 If in four points if $$\color{red}{1]}$$ all are collinear, line segments will never 'intersect' $$\color{red}{2]}$$ Three are collinear, still no intersection. $$\color{red}{3]}$$ Two points are always collinear:) $$..................................$$ 4 non collinear points will form quadrilateral. $$\color{red}{\text{if quadrilateral is convex}}$$ Lines segments drawn must be either sides or diagonals. Only diagonals intersect. $$\color{red}{\text{if quadrilateral is concave}}$$ No intersection Solve using this information • Collinear happens with probability zero. Also, I think the odd formatting makes the post much more difficult to read. – Hurkyl Sep 15 '17 at 9:33 • What is the probability of convex quadrilateral? – neonpokharkar Sep 15 '17 at 9:49	2019-08-25T19:56:04	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2430323/what-is-the-likelihood-of-two-line-segments-crossing", "openwebmath_score": 0.9235252141952515, "openwebmath_perplexity": 182.85489979118597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9881308796527184, "lm_q2_score": 0.8479677545357568, "lm_q1q2_score": 0.8379031232065578 }
https://math.stackexchange.com/questions/2526636/proving-lim-limits-n-to-infty-frac8n2-54n27-2-using-epsilon-d	# Proving $\lim\limits_{n\to\infty} \frac{8n^2-5}{4n^2+7} = 2$ using $\epsilon-\delta-$ definition. I'm trying to prove the limit of this sequence using the formal definition. I've looked at other questions on the site but from the ones I've seen, the $n^2$ term always seems to cancel out, making it simpler. Show that $$\lim_{n\to\infty} \frac{8n^2-5}{4n^2+7} = 2$$ So this is how I started: $$\left\|\frac{8n^2-5}{4n^2+7} -2 \right\| = \left\|\frac{-19}{4n^2+7}\right\| = \frac{19}{4n^2+7} \leq \frac{19}{4n^2} = \epsilon$$ Let $\epsilon > 0 \implies n = \sqrt{\frac{19}{4\epsilon}}$ By Archimedian Property: $N > \sqrt{\frac{19}{4\epsilon}}$ If $n \geq N \geq \sqrt{\frac{19}{4\epsilon}}$ $$\left\|\frac{8n^2-5}{4n^2+7} -2 \right\| \leq \frac{19}{4n^2} < \frac{19}{4(\frac{19}{4\epsilon})} = \epsilon$$ • Looks correct to me. – user491874 Nov 18 '17 at 21:23 • Welcome to math.stackexhange! This looks fine. – Hans Engler Nov 18 '17 at 21:24 • +1 for using MathJax and showing your attempted solution. – Remy Nov 18 '17 at 21:26 • Yes, it’s correct. Just one small point, use “there exist” before N in the step where you use Archimedian property. – Mayuresh L Nov 18 '17 at 21:28 • Canceling out the $n^2$ will leave $\frac 5{n^2}$ and $\frac 7{n^2}$. I wouldn't say that is any easier. Maybe more intuitive where to go, but not easier.... any way, what you did was exactly what you were supposed to do and you did it exactly right. (Ithink... $\frac {8n^2 - 5}{4n^2 + 7} - 2 = \frac {(8n^2 - 5) - (8n^2 + 14)}{4n^2 + 7} = \frac {-19}{4n^2 + 7} ....$ Yep, it checks out. (Lack of an $n$ term threw me but it's fine...) – fleablood Nov 18 '17 at 22:35 we have $$\left\|\frac{8n^2-5}{4n^2+7} -2 \right\| = \left\|\frac{-19}{4n^2+7}\right\| < \frac{19}{4n^2} =$$ Now let such that $$\frac{19}{4n^2} < \epsilon\Longleftrightarrow n>\sqrt{\frac{19}{4\epsilon}}$$ Now choosing, $$N =\left\lfloor \sqrt{\frac{19}{4\epsilon}}\right\rfloor +1$$ then we have $$n > N=\left\lfloor \sqrt{\frac{19}{4\epsilon}}\right\rfloor +1\Longrightarrow n>\sqrt{\frac{19}{4\epsilon}}\\\Longrightarrow \epsilon>\frac{19}{4n^2} > \left\|\frac{8n^2-5}{4n^2+7} -2 \right\|$$ that is we have $$n > N=\left\lfloor \sqrt{\frac{19}{4\epsilon}}\right\rfloor +1\Longrightarrow \left\|\frac{8n^2-5}{4n^2+7} -2 \right\|< \epsilon$$	2021-05-08T20:30:09	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/2526636/proving-lim-limits-n-to-infty-frac8n2-54n27-2-using-epsilon-d", "openwebmath_score": 0.7298040390014648, "openwebmath_perplexity": 459.04619160951603, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9881308800022472, "lm_q2_score": 0.8479677526147223, "lm_q1q2_score": 0.8379031216047135 }
https://math.stackexchange.com/questions/3839244/is-3-a-subset-of-1-1-2-1-2-3/3839249	# Is $\{3\}$ a subset of $\{\{1\},\{1,2\},\{1,2,3\}\}$? Is $$\{3\}$$ a subset of $$\{\{1\},\{1,2\},\{1,2,3\}\}$$? If the set contained $$\{3\}$$ plain and simply I would know but does the element $$\{1,2,3\}$$ include $$\{3\}$$ such that it would be a subset? • No, it's not. Let $A=\{\{1\},\{1,2\},\{1,2,3\}\}$. The subsets must themselves contain sets, since A is a set of sets. Then $\{\{1\}\}$ is a subset of A, $\{\{1\},\{1,2\}\}$ is another subset of A. $\{\{1,2\},\{1,2,3\}\}$ is also a subset of A. $\{3\}$ is a subset of an element of A. Sep 24 '20 at 21:03 • No, when you have a set of sets, you do not "look through" the brackets. The sets are members, but the members of those sets are not members. Sep 24 '20 at 21:04 • Similar question Sep 24 '20 at 21:05 • "Is $3=\{1\}$?" ; "Is $3=\{1,2\}$?" ; "Is $3=\{1,2,3\}$?": If the answer to one of these questions is yes, then $\{3\}$ is a subset of $\{\{1\},\{1,2\},\{1,2,3\}\}$; otherwise, it isn't. – user239203 Sep 24 '20 at 21:05 • @Gae.S.And it should be noted that the answer may be yes! Sep 24 '20 at 21:13 Let $$X$$ be a set. We say $$Y \subseteq X$$ ($$Y$$ is a subset of $$X$$) if, for all $$x \in Y$$, we have $$x \in X$$. Examine the sets $$Y = \{3\}$$, $$X = \{\{1\},\{1,2\},\{1,2,3\}\}$$. Take $$x = 3 \in Y$$. Is $$3 \in X$$? Trickier problem: If $$X = \{\{1\},\{3\},\{1,2\},\{1,2,3\}\}$$, is $$3 \in X$$? • Why is the trickier problem trickier? $3\not \in X$ because $3\ne \{1\};3\ne \{3\},3\ne \{1,2\}, 3\ne \{1,2,3\}$. Why is that trickier. Sep 24 '20 at 21:21 • I would expect someone to say "Oh, I see 3 by itself" and say it is in the set. I wanted to emphasize the difference between $\{3\}$ and $3$. Sep 24 '20 at 21:23 No. the elements inside elements of the set do not count. The elments of your big set are: • $$\{1\}$$ • $$\{1,2\}$$ • $$\{1,2,3\}$$. The elements of your small set are: • $$3$$ So $$\{3\}$$ is a subset only if $$3$$ is equal (the same thing; !!!!!NOT!!!! an element within) one of the elements $$\{1\}$$ or $$\{1,2\}$$, or $$\{1,2,3\}$$. But none of those are the same thing as $$3$$ so $$\{3\}$$ is not a subset. But in some text the natural numbers are defined as $$0 = \emptyset$$ $$1= \{\emptyset\}$$ $$2= \{\emptyset, 1\}$$. $$3 = \{\emptyset, 1, 2\}$$ So we could have a trick thing of $$\{3\} \subset \{\{\emptyset,1\}, \{\emptyset,1,2\}, \{\emptyset,1,2,3\}\}$$ not because $$3 \in \{\emptyset,1,2,3\}$$ (that's utterly irrelevent), but because $$3 = \{\emptyset, 1, 2\}$$ and the set $$\{\{\emptyset,1\}, \{\emptyset,1,2\}, \{\emptyset,1,2,3\}\}$$ is equally equal to the set $$\{2,3,4\}$$ and $$\{3\}\subset \{2,3,4\}$$. No it is not, we have that $$\{1\}\in \{\{1\},\{1,2\},\{1,2,3\}\}$$ but $$\{3\}$$ is not an element of the set, what is true is that $$3\in\{1,2,3\}$$. As you can see, there are plenty of good answers here. I think that the most important thing to understand is: $$3 \neq \{ 3 \}.$$ These are two different objects. Moreover the followings hold: $$3 \in \{3\}$$ $$3 \in \{\ldots, 3, \ldots \}$$ $$\{3\} \in \{\ldots, \{3\}, \ldots\}$$ and $$\{3\} \not\in \{3\}$$ $$\{3\} \not\in \{\ldots, 3, \ldots \}$$ • In for a penny..... might as well add (and perhaps germaine to the common misconception) $3\not \in \{........., \{3\},......\}$ (unless it is one of the objects omitted in the "..."s.) Sep 24 '20 at 21:59	2021-10-27T06:26:43	{ "domain": "stackexchange.com", "url": "https://math.stackexchange.com/questions/3839244/is-3-a-subset-of-1-1-2-1-2-3/3839249", "openwebmath_score": 0.8364362120628357, "openwebmath_perplexity": 264.69202723598033, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9683812327313545, "lm_q2_score": 0.8652240930029117, "lm_q1q2_score": 0.8378667737710277 }