What is Sum: Definition and 1000 Discussions

Sum, sumu, sumon, and somon (Plural: sumd) are the lowest level of administrative division used in China, Mongolia, and Russia. The word sumu is a direct translation of a Manchu word niru, meaning ‘arrow’ Countries such as China and Mongolia, have employed the sumu administrative processes in order to fulfil their nations economic, social and political goals. This system was acted in the 1980s after the Chinese Communist Party gained power in conjunction with their growing internal and external problems. The decentralisation of government included restructuring of organisational methods, reduction of roles in rural government and creation of sumu’s.

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  1. Math Amateur

    I Characterization of External Direct Sum - Cooperstein

    I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... In Section 10.2 Cooperstein writes the following, essentially about external direct sums ... ... Cooperstein asserts that properties (a) and (b) above "characterize the space ##V## as the direct sum of...
  2. Math Amateur

    MHB Characterization of External Direct Sum - Cooperstein, pages 359 - 360

    I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ... In Section 10.2 Cooperstein writes the following, essentially about external direct sums ... ... Cooperstein asserts that properties (a) and (b) above "characterize the space V as the direct sum of the...
  3. P

    Finding sum of roots of trigonometric equation

    Homework Statement Question: Sum of all the solutions of the equation: ##tan^2 (33x) = cos(2x)-1## which lie in the interval ## [0, 314] ## is: (a) 5050 π (b) 4950 π (c) 5151 π (d) none of these The correct answer is: (b) 4950 π Homework Equations ## cos(2x) = 2cos^2(x) -1 ## The Attempt...
  4. terryds

    Find the Sum of Roots^16 | x3 - x + 1 = 0 Equation | Homework Solution

    Homework Statement Roots of the equation x3 - x + 1 = 0 are a, b, and c. Determine the value of a16+b16+c16 ! Homework Equations For ax3+bx2+cx+d = 0 x1+x2+x3 = -b/a x1 * x2 * x3= -d/a The Attempt at a Solution [/B] I know how to determine the value of a + b+ c but not a^16+b^16+c^16... I...
  5. T

    MHB Find a formula for the Riemann sum and take the limit of the sum as n->infinite

    For the function given below find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then take a limit of thissum as n-> infinite to calculate the area under the curve over [1,5]. Below you...
  6. D

    Sum Digits & Exit with 'Q': A Program Solution

    Homework Statement Write a program that reads in any characters from the keyboard, sums up only characters corresponding to a digit and prints the result on the screen. The program will exit when the character 'Q' is entered (upper or lowercase). Example: Input = 9 8 q Output = The total is...
  7. Blockade

    Are the sum of all forces written correctly for this system?

    Can someone check if my Sum of All Forces is setup correctly? Problem: Diagram: http://puu.sh/o6pLY/285d7b12c1.jpg Sum of all Forces:
  8. anemone

    MHB Evaluating Sum Compute: n+1 Roots

    Compute \sqrt[n+1]{\frac{n+1}{n}}+\sqrt[n]{\frac{n}{n-1}}+\cdots+\sqrt[4]{\frac{4}{3}}+\sqrt[3]{\frac{3}{2}}+\sqrt{2}.
  9. kostoglotov

    An infinite sum of the Heaviside function

    I'm not sure where to put this question. It is by itself pretty basic, but it's a preamble to a Laplace Transform exercise, and I'll probably want to ask some follow up questions once the current query is resolved. 1. Homework Statement Unit stair-case function: f(t) = n, \ if \ \ n-1 \leq t...
  10. karush

    MHB Find the sum of the first 17 terms of the arithmetic series

    Find the sum of the first 17 terms of the arithmetic series $$8+\sqrt{7}, \ 6,\ 4-\sqrt{7}$$ $$u=8+\sqrt{7}$$ $$S_{17} =\frac{u\left(1-\frac{{6}^{17}} {u} \right)}{u}$$ My first shot at this
  11. Kernul

    Exercise with intersection and sum

    My professor did this exercise that I didn't quite get how she went through all of it. We have a ##U = {(x, y, z, t) : x+y+z+t = 0}## and ##B_{Im(f)} = \left[ \begin{pmatrix} 7 \\ -3 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 3 \\ -3 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 5 \\ 0 \\ 1 \\ -5...
  12. J

    A Stats: would the sum of the variances be 1 in this case?

    Often in empirical studies you see statements that factor X explains some fraction of the variance in some other variable V, and thinking about what this means intuitively made me curious about the following question. Suppose you have a model where the values of some set of factors X1, X2, ...
  13. amind

    Sum involving reciprocal of binomial coeffients

    Homework Statement If $$a_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}}$$ Find $$\sum_{r=0}^{n} \frac{r}{\binom{n}{r}}$$ in terms of an and n 2. The attempt at a solution Let $$f(x) =\sum_{r=0}^{n} \frac{x^r}{\binom{n}{r}}$$ Then, an = f(1). Observe that f'(1) is the required sum. I was thinking...
  14. K

    How Does the Direct Sum Relate to Unique Decomposition in Vector Spaces?

    During lecture, the professor gave us a theorem he wants us to prove on our own before he goes over the theorem in lecture. Theorem: Let ##V_1, V_2, ... V_n## be subspaces of a vector space ##V##. Then the following statements are equivalent. ##W=\sum V_i## is a direct sum. Decomposition of...
  15. G

    Sum of eigenvectors of linear transformation

    Homework Statement Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2. Homework Equations -Eigenvalues and...
  16. Tone L

    MATLAB Solving MATLAB Sum Loop Issue with 2,187 Data Points

    So i want to calculate an r value 5 different times then find the mean of the 5 calculated values. I have 2,187 data points. the first line of code generates 100 random points 1-2187. The code has a bug but my major issue is it calculates r then loops again calculates r, loops again calculates...
  17. B3NR4Y

    Prove an infinite sum exists and its sum

    Homework Statement Let {b k } be a sequence of positive numbers. Assume that there exists a sequence {a k}, such that a k is greater than or equal to 0 for all k, a_k is decreasing, the limit of a_k is 0 and b_k = a_k - a _(k+1). Show that the sum from k=1 to infinity of b k exists and equals...
  18. evinda

    MHB How to Express a Force as a Sum of Parallel and Perpendicular Components?

    Hello! (Wave) We suppose that a force that is given by the vector $2i+j$ is applied at an object that moves at the direction $i+j$. How can we express this force as a sum of a force that has the direction of the movement and a force that is perpendicular to the direction of the movement?
  19. L

    Is the Subset Sum Problem Truly Solved by Generating Large Random Sets?

    Hello, My hobby is to design algorithms especially data compression algorithms, but when I can't find a solution to my problems I usually go find myself a different problem to solve because it helps me think differently or maybe it lights a bulb about the original problem …today I stumbled on...
  20. anemone

    MHB What is the Sum of Roots for $P(x)=x^3-2x^2-x+1$ with $x_1>x_2>x_3$?

    Let $x_1,\,x_2,\,x_3$ be the three real roots of $P(x)=x^3-2x^2-x+1$ such that $x_1>x_2>x_3$. Evaluate $x_1^2x_2+x_2^2x_3+x_3^2x_1$.
  21. I

    MHB Sum of an Infinite Arithmetic Series

    Somewhere I saw that the sum of the infinite arithmetic series \sum_{n=1}^{\infty}n = \frac{-1}{12} Why exactly is this? I thought infinite arithmetic series had no solution? Also... WHY is it negative? Seems counter-intuitive that the sum of all the NATURAL numbers is a decimal, a negative...
  22. C

    Why is differential equation equal to sum of partials?

    My current understanding of differential equations is extremely shaky, and my vocabulary is probably very incorrect, but I'm curious about something I've recently seen in some Khan Academy videos (specifically this one) and in other situations with differential equations. It seems that the...
  23. Greg

    MHB Trigonometric sum with a product as the argument

    Prove $$\sum_{n=0}^N\cos(nx)=\csc\left(\dfrac x2\right)\sin\left(\dfrac{(N+1)x}{2}\right)\cos\left(\dfrac{Nx}{2}\right)$$ I've tried working from the RHS with various identities but haven't managed to come up with anything that works. I suspect this problem involves some trigonometry that I...
  24. Euler2718

    Showing the sum of this telescoping series

    Homework Statement Determine whether each of the following series is convergent or divergent. If the series is convergent, find its sum \sum_{i=1}^{\infty} \frac{6}{9i^{2}+6i-8} Homework Equations Partial fraction decomposition \frac{1}{3i-2} - \frac{1}{3i+4} The Attempt at a Solution...
  25. Steve Turchin

    Limit of Sum: Understanding the Equation and Correcting Common Mistakes

    Homework Statement ## lim_{n \rightarrow \infty}{\frac{1}{n^2} \sum_{k=1}^{n} ke^{\frac{k}{n}}} ## Homework EquationsThe Attempt at a Solution ## lim_{n \rightarrow \infty}{\frac{1}{n^2} \sum_{k=1}^{n} ke^{\frac{k}{n}}} \\ = lim_{n \rightarrow \infty}{\frac{1}{n^2}...
  26. 1

    Understanding dx in Integration

    Hello, I am currently in my first year of college, and I already took calculus in high school. I was able to solve all the problems, but I feel like I didn't understand everything conceptually. When integrating dy/dx=x you get, ∫x dx=1/2x2. But what exactly happened to the dx, why did it...
  27. X

    How Do You Solve a Discrete Convolution Sum with Step Functions?

    Homework Statement Find ##x[n] \ast h[n]## when ##x[n] = 3 u[2-n]## and ##h[n] = 4\left( \frac{1}{2} \right)^{n+2}u[n+4]## where ##u[n-k]## is the unit step function. Homework Equations None really The Attempt at a Solution So I know this is probably simple but I am confused. So the...
  28. C

    Sum of range of numbers divisble by 6 but not by 9

    Homework Statement Find the sum of the numbers between 200 and 800 inclusive, which are multiples of 6, but not multiples of 9. Homework EquationsThe Attempt at a Solution Numbers that are multiples of 6 should be: a = 6n, n ∈ ℤ and a is any multiple of six. 200 = 6n → n1 = ##\frac{200}{6}## =...
  29. Math Amateur

    MHB Sum of Two Subspaces and lub - Roman, Chapter 1, page 39

    I am reading Steven Roman's book, Advanced Linear Algebra and am currently focussed on Chapter 1: Vector Spaces ... ... In discussing the sum of a set of subspaces Roman writes (page 39) ...In the above text, Roman writes: " ... ... It is not hard to show that the sum of any collection of...
  30. anemone

    MHB Prove the sum of a² and d² is equal the sum of b² and c².

    Let $a,\,b,\,c,\,d\in \mathbb{R}$ with condition that $a+b\sqrt{2}+c\sqrt{3}+2d\ge \sqrt{10(a^2+b^2+c^2+d^2)}$. Prove that $a^2+d^2=b^2+c^2$.
  31. anemone

    MHB Prove the sum is less than 2016

    Prove the inequality \sqrt{\frac{1\cdot 2}{3^2}}+\sqrt{\frac{2\cdot 3}{5^2}}+\sqrt{\frac{3\cdot 4}{7^2}}+\cdots+\sqrt{\frac{4032\cdot 4033}{8065^2}}<2016
  32. H

    MHB Solving 3r(r+1)=r(r+1)(r+2)-r(r-1)(r+1) and Finding r(r+1) Sum

    Hey, Please may someone help me. How can I show that 3r(r+1) is equal to r(r+1)(r+2) - r(r-1)(r+1) and then I would find the total sum of r(r+1). Thanks in advance for any help.
  33. RealKiller69

    Help with Sum ∑n!/(3*4*5...*n)

    Homework Statement ∑n!/(3*4*5...*n) s1=1/3 sn=1/3+2/(4*3)+3!/(5*4*3)+...+n!/(3*4*5*...n) so i multiplied the sum with 1/2sn=1/6+1/(4*3)+1/(5*4)+1/(6*5)...+1/((n+2)(n-1)) got blocked here,i don't know how to continue, help please
  34. evinda

    MHB Find Min Sum of Finite Seq. of Ints Using Dynamic Prog.

    Hello! (Wave) The finite sequence of integers $Y_1, \dots, Y_M$ takes both positive and negative values, where $M$ is a fixed positive integer. Could you help me to find a formulation using dynamic programming that solves the problem of finding integers $i_1, i_2$ with $1 \leq i_1 \leq i_2...
  35. A

    Integral equivalent to fitting a curve to a sum of functions

    Hello, I am searching for some kind of transform if it is possible, similar to a Fourier transform, but for an arbitrary function. Sort of an inverse convolution but with a kernel that varies in each point. Or, like I say in the title of this topic a sort of continuous equivalent of fitting a...
  36. S

    Sum of null and time-like vectors

    Homework Statement Show that the sum of two future-pointing null vectors is a future-pointing time-like vector, except when the two null vectors have the same direction. Conversely, show that any time-like vector can be expressed as a sum of two null vectors. For a given time-like vector the...
  37. L

    Sum of area bounded by the curve

    Why we sometimes take the area bounded by the curve is sum of positive area and absolute of negative area(e.g. ∫\int_0^2π sin(x)\, dx is equal to 4 or area of ellipse )?But sometimes we just sum positive and negative areas which is equal to 0(e.g. area of cycloid →when we integrate we get...
  38. A

    How many ways a number can be written as components sum?

    If we have a positive integer, how many ways can this number be written as a sum of its components? By components, I mean all numbers less than that number. For example, 5 has 6 ways to be written; 5x1, 3x1+2, 2x2+1, 2x1+3,1+4 and 2+3. In digits form; [11111, 1112, 221,113, 14, 23] So there are...
  39. J

    Why does (-1)^n appear in the power series for 1/(1+(z-1))?

    Homework Statement So I'm checking my solutions to past question and there's one bit that throws me. 1/(1+(z-1)) = Σ(-1)n(z-1)n (for 0<|z-1|<1) I don't know where the (-1)n factor came from. Is it just something that always happens that I didn't know about / forgot about, or is there some...
  40. P

    Frobenius Method When Initial Value of A Sum is not 1

    Homework Statement Solve \begin{equation*} 36x^2y''+(5-9x^2)y=0 \end{equation*} using the Frobenius method Homework Equations Assume a solution of the form \begin{equation*} y=\sum_{n=0}^{\infty}{a_nx^{n+s}} \end{equation*} then \begin{equation*}...
  41. ajayguhan

    Is sum centrifugal force and centripetal force zero?

    While balancing rotating mass we consider the inertia force (centrifugal force) is equal and opposite to centripetal force which causes the rotation. if both force(applied external force on rotating mass) which causes the motion and force which resist motion (inertia force) are equal and...
  42. A

    What is the closed form for the sum of binomial coefficients over any interval?

    Is there a way to find the following sum in closed form: ∑K(N,n) , where K(N,n) is the binomial coefficient and the sum can extend over any interval from n=0..N. I.e. not necessarily n=0 to N in which case on can just use the binomial theorem.
  43. S

    Finding Fourier Series for (-π, π): Sketch Sum of Periods

    Homework Statement Find the Fourier series defined in the interval (-π,π) and sketch its sum over several periods. i) f(x) = 0 (-π < x < 1/2π) f(x) = 1 (1/2π < x < π) 2. Homework Equations ao/2 + ∑(ancos(nx) + bnsin(nx)) a0= 1/π∫f(x)dx an = 1/π ∫f(x)cos(nx) dx bn = 1/π ∫f(x) sin(nx) The...
  44. alexmahone

    MHB Sum of binomial coefficients multiplied by k^2

    Evaluate \sum\limits_{k=1}^{12} {12\choose{k}}k^2 The answer is 159744.
  45. A

    What is the closed form expression for the sum: Σx2/n! from 0 to N?

    Is there a closed form expression for the sum: Σx2/n! from 0 to N for N=∞ it is an exponential but what about the finite case?
  46. G

    Evaluating Finite Sum: Homework Statement

    Homework Statement Find \sum\limits_{k=0}^{n}k^2{n\choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k} Homework Equations -Binomial theorem The Attempt at a Solution I am using the binomial coefficient identity {n\choose k}=\frac{n}{k}{{n-1}\choose {k-1}}: \sum\limits_{k=0}^{n}k^2{n\choose...
  47. P

    Sum the even numbers between 1000 and 2000 inclusive

    this is just an arithmetic series but with a small difference. i will show that below The attempt at a solution the general arithmetic formula ## S_N=\sum_{n=1}^\infty n## for my problem ## S_N=\sum_{n=1000}^{2000} n ## i have to rewrite it so i will just add the even numbers ##...
  48. Taryn1

    MHB Sum or difference formula (sin, cos, and tan)

    So I'm supposed to find the exact values of the sine, cosine, and tangent of an angle by using a sum or difference formula ( i.e. sin(x+y)=sin(x)cos(y)+cos(x)sin(y) ), but this is the angle I was given: ${-13\pi}/{12}$. How do I use a sum or difference formula to get the sin, cos, and tan of that?
  49. ognik

    MHB Matrix Sum of Squares: Rotate Coord System to Express as Diagonal

    Maybe I just need help understanding the question ... write $ x^2 + 2xy + 2yz + z^2 $ as a sum of squares $ (x')^2 -2(y')^2 + 2(z')^2 $ in a rotated coord system. The 1st expression $ = \left[ x, y, z \right]M \begin{bmatrix}x\\y\\z\end{bmatrix} $ and I get $ M =...
  50. C

    Error Propagation in Mass Flow Rates

    I tried posting this question in this forum a couple of weeks ago, but didn't get an answer to my question. I'm going to try posting it again using the formatting template so that it is hopefully clearer. I am also not sure if this is the right forum to be posting this in. It is a problem I ran...
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