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. S

    Stats: Need help understanding where the sum of x^2 comes from

    Homework Statement I have no idea where the sum of x^2 comes from, from the information I posted. I know it must be something pretty simple but its completely going over my head. In the picture that I've attached, I am wondering where the 2431.72, 4901.66, and 3252.44 come from. Thank you...
  2. S

    MHB Truth Table, Canonical Sum and Minimizing the Canonical Sum

    Also would someone mind checking my work on these problems too? My answers are in BOLD 2a)Draw the truth table corresponding to $f$(X,Y,Z) = \piM(2,4,6) ANSWER: x y z | f 0 0 0| 1 0 0 1| 1 0 1 0| 0 0 1 1| 1 1 0 0| 0 1 0 1| 1 1 1 0| 0 1 1 1| 1 2b) Write out the canonical product of sums...
  3. C

    Approximate the angle of weighted sum of complex numbers

    Homework Statement This is not a homework question, but I'm facing this from my research. I have N complex numbers defined as x_{n}=|\alpha_n| \cdot e^{j \theta_n} for n = 1,\ldots,N and my observation is the sum of those numbers r = \sum_{n=1}^{N} x_n . From the observation r, I...
  4. C

    Sum of signal and its probability density (special case )

    Dear, I assume that a signal S is expressed as S = a*S1 + b*S2, where a, b are weight constant, and S1, S2 are the different signals. In addition, S1, S2 have similar distribution such as Gaussian or Laplacian distribution, and theirs pdf is p_S1 and p_S2. In the above...
  5. C

    Sum of signal and its probability density

    Dear, I assume that a signal S is expressed as S = a*S1 + b*S2, where a, b are weight constant, and S1, S2 are the different signals. In addition, S1, S2 have similar distribution such as Gaussian or Laplacian distribution, and theirs pdf is p_S1 and p_S2. In the above assumption...
  6. C

    Finding the amplitude of the sum vector

    Homework Statement A signal E(t) is made up of three terms, each having the same frequency but differing in phase: E(t) = E0cos(ωt) + E0cos(ωt + δ) + E0cos(ωt + 2δ) It is possible to find the amplitude of the sum vector by summing each vector described as a magnitude multiplied by a...
  7. D

    Does formula exist for this sum?

    \sum_{i=1}^{n}[i/2^i] Have looked and looked and cannot find it anywhere. EDITED: To correct mistake.
  8. B

    Direct sum of nullspace and range

    Is this true? I am studying direct sums and was wondering if the following statement holds? It seems to be true if one considers the proof of the dimension theorem, but I need to be sure, so I can steer my proof toward a particular direction. ## N(T) \bigoplus R(T) = V ## where ##V## is the...
  9. J

    Algorithm to find the submatrix with the greatest sum of its elements?

    This is a challenge problem I thought of: Given a real-valued matrix A, develop an algorithm that finds the submatrix with the greatest sum of its element. (If there's a tie, just return an arbitrary submatrix that's tied for the win.) Is there a way other than brute force?
  10. B

    Direct sum complement is unique

    I'm curious about whether a statement I conjecture about direct sums is true. Suppose that ##V## is a finite-dimensional vector space and ##W##,##W_{1}##,##W_{2}## are subspaces of ##V##. Let ## V = W_{1} \bigoplus W ## and ## V = W_{2} \bigoplus W ##. Then is it the case that ## W_{1} = W_{2}...
  11. S

    Riemann sum where n = 3 for both left and right endpoint estimates

    Homework Statement Compute the integral that is highlighted in MyWork.jpg using Riemann sums using left and right endpoints. Homework Equations ##x_i* = a + i Δx## ##*x_i = a + i Δx - Δx## ##Σ_{i=1}^{n} i = n(n+1)/2## ##Σ_{i=1}^{n} i^2 = n(n+1)(2n+1)/6## The Attempt at a Solution My...
  12. I

    How to understand Matsubara frequency sum?

    We know a standard matsubara frequency sum that -\sum_{n}\frac{\xi}{2\pi ni-\xi}=n_B(\xi), but this looks contradictory to the well-known formula \sum_{n}\frac{\xi}{n^2+\xi^2}=\pi \coth(\pi \xi) if we take the imaginary part of the former sum. I know this matsubara frequency sum depends on...
  13. T

    Finding the Value of S in the Sum Equals Product Problem

    Homework Statement If the product of the numbers R and 11/S is the same as their sum, find the value of S. Homework Equations N/A The Attempt at a Solution I am suspecting that the only set of 2 numbers that have the same sum and product is 2 and 2. So I guess R is 2, 11/S is...
  14. anemone

    MHB Sum of Series: Let $x$ with x^{2011}=1 & x$\ne$1

    Let $x$ be a complex number such that x^{2011}=1 and $x\ne1$. Compute the sum \frac{x^2}{x-1}+\frac{x^4}{x^2-1}+\frac{x^6}{x^3-1}+\cdots+\frac{x^{4020}}{x^{2010}-1}.
  15. C

    Finding Two Vectors from Vector Sum / Difference

    Finding Two Vectors from Given Linear Combination Homework Statement If v + w = (5,1) and v - w = (1,5), compute and draw v and w. Homework Equations v + w = (5,1) v - w = (1,5) The Attempt at a Solution I understand how to find the sum of two vectors, but I'm confused on how to find...
  16. Jameson

    MHB What is the proof for the residual sum of squares problem?

    Problem: Through transformation with orthogonal matrix $O$, the problem \hat{b}=\underset{b}{\operatorname{arg min}}||y-Xb||^2 is equivalent to \hat{b}=\underset{b}{\operatorname{arg min}}||y^{*}-X^{*}b||^2, where $y$ and $y^{*}$ are in $\mathbb{R}^m$, $X$ and $X^{*}$ are in $\mathbb{R}^{m...
  17. anemone

    MHB Sum of Factorial Series: Find the Answer!

    Find the exact value of the series \frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+\cdots\cdots
  18. D

    Passing an integral through an infinite sum

    Homework Statement I want to show that $$ \tan^{-1}(x)=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}x^{2n+1}. $$ Homework Equations I start with $$ \int\frac{1}{1+x^{2}}dx. $$ The Attempt at a Solution I want to be able to do the following: $$...
  19. caffeinemachine

    MHB Sum of squares of average change is less than

    Hello MHB, I have no good ideas on how to go about solving the following: Let $f:[a,b]\to\mathbb R$ and $g:[a,b]\to\mathbb R$ be real values functions both of which are differentiable in $(a,b)$. Show that there is an $x\in(a,b)$ such that...
  20. M

    Sum of a serie involving Fourier coefficients

    Homework Statement Let \hat{u}_k the Fourier coefficients of 2-periodic function u(t)=t with t\in [0,2). Evaluate the sum of the serie: \sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2 Ok, I think there is a trick that I don't know... \sum_{k=-\infty}^{\infty}\hat{u}_k...
  21. M

    Finding the sum of multiples of 3 and 5 below 1000 (Matlab)

    Homework Statement Project Euler Problem 1. If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23. Find the sum of all the multiples of 3 or 5 below 1000. Hello, there I do understand what a multiple is, and...
  22. jacobi1

    MHB Sum of Cosines: Find the Infinite Series

    Find \sum_{n=0}^\infty \frac{\cos(nx)}{2^n}.
  23. anemone

    MHB Prove: One of $a, b$ or $c$ is Sum of Other Two

    If $a, b$ and $c$ are three real numbers such that $|a-b|\ge|c|$, $|b-c|\ge|a|$ and $|c-a|\ge|b|$, then prove that one of $a, b$ or $c$ is the sum of the other two.
  24. 8

    Mathematica Mathematica, Plot Implicit function contain Sum (sigma)

    Dear friends, Over the past week, I tried to plot implicit function by mathematica but failed. I am very disappointed. Hopefully someone help me at this time. My equation is given by (see below figure): Where z0 := 6 d := 12 k := 11800 w0 := 0.025 w[z_] := sqrt[w0^2*(1 + (z/z0)^2)]...
  25. dexterdev

    To find mean wedivide the sum of no.s with 'n'for variance why 'n-1'

    Hi all, For finding average we take the sum of sequence numbers and divide by the number of elements. Why for variance this changes to number of elements minus 1. -Devanand T
  26. M

    Solving the Infinite Series Sum of a Function

    I don't understand any of the steps that were made in the book. So I will try and solve it on my own so please let me know where I am going wrong. The problem is in the paint document. The general Term for the infinite series given is n(n+1) where n is greater than one and integral. LHS we...
  27. MarkFL

    MHB Maya's question at Yahoo Answers regarding a Riemann sum and definite integral

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  28. B

    Direct sum of the eigenspaces equals V?

    I am following Friedberg's text and having some trouble understanding some of the theorems regarding diagonalizability. The proofs seem to skip some steps, so I guess I need to work through them a bit more slowly. Given a linear operator ## T:V → V ##, with eigenspaces ## \{ E_{...
  29. P

    Hyperbola Fermat, Geometric Infinite Sum

    Hello everybody, I'm trying to understand some steps in the evolution of calculus, and in a .pdf found in the internet I read the document: http://www.ugr.es/~mmartins/old_web/Docencia/Old/Docencia-Matematicas/Historia_de_la_matematica/clase_3-web.pdf , in pags. 14-15. I want to solve the to...
  30. C

    Trouble understanding vector subspace sum

    I'm self-studying Linear Algebra and the book I'm using is Linear Algebra done right by Sheldon Axler but I came across something that I don't understand .- Suppose \mathrm U is the set of all elements of \mathbb F ^3 whose second and third coordinates equal 0, and \mathrm W is the set of all...
  31. A

    Does correlation affect the expectation value of a sum?

    We have for two random variables X and Y (one sum runs over j and one over k): E(X+Y) = ƩƩ(xj+yk)P(X=xk,Y=yk) = ƩƩxjP(X=xk,Y=yk) + ƩƩykP(X=xk,Y=yk) Now this can be simplified to obtain E(X+Y)=E(X)+E(Y) if we use that: P(X=xk,Y=yk) = P(X=xk)P(Y=yk), because then (and same goes the other...
  32. C

    Finding the formula for the partial sum Sn

    Homework Statement Consider the series Ʃ 1/[k(k+2)]; n=1 to infinity Find the formula for the partial sum Sn 2. The attempt at a solution I have calculated the first 5 terms of the sequence as follows, but I can't see any pattern. Am I doing this right? S1=1/3 S2=1/3+1/8=11/24...
  33. Albert1

    MHB What is the Sum of Digits in the Power of Three Raised to 10,000?

    $ X=3^{10000}$ Given $log_{10} 3=0.4771$ All the digits sum of X =A All the digits sum of A =B All the digits sum of B =C please find C
  34. J

    So why does the integral represent an infinite sum?

    In an earlier post i was shown how to represent an integral as an infinite sum. So why is the anti derivative a summation by definition? For example, the derivative dy/dx is found by f(x+h)-f(x)/h.
  35. alyafey22

    MHB Can the Intriguing Harmonic Sum be Solved with Elementary Methods?

    This thread will be dedicated for a trial to prove the following \sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360} \mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2 In this paper the authors give solutions to the sum and others ...
  36. alyafey22

    MHB Can the Harmonic Sum be Proven Using a Newer Method?

    Prove the following \sum_{k\geq 1} \frac{H^2_k}{k^2}=\frac{17}{4}\zeta(4)=\frac{17\pi^4}{360} \mbox{where }\,\,H^2_k =\left( 1+\frac{1}{2}+\frac{1}{3}+\cdots \frac{1}{k}\right)^2
  37. anemone

    MHB Finding the Sum of Real Numbers Satisfying Cubic Equations

    The real numbers x and y satisfy x^3-3x^2+5x-17=0 and y^3-3y^2+5y+11=0. Determine the value of x+y.
  38. Fernando Revilla

    MHB Solving an Integral with a Right Endpoint Riemann Sum

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  39. S

    MHB How to Graphically Determine the Minimum of a Function Under a Constraint?

    Problem: The sum of two real numbers is 1. What is the minimum value of the sum of the squares of the two numbers? I have already managed to solve the problem algebraically (by substitution and completing-the-square we arrive at a minimum value of 0.5), but what I am interested in is a...
  40. Farmtalk

    MHB Prove Sum and Difference Sine?

    I was wondering if someone could take the time to show a proof for the sum and difference identity for sine. I've seen and learned to understand some other identities, but never this one. I've been trying to understand more of the "why" than the "how" of mathematics, and this one is very...
  41. N

    The Sum of the Grandi's Series

    I've looked over this peculiar problem and the many different forums that have tried to resolve it yet I'm confused as to why the answer simply isn't zero. for those not familiar with this problem, it is this Ʃ of (-1)^n as n goes from 0=∞ which...
  42. C

    Integers as the sum of 3 integers.

    Homework Statement Prove that every integer >17 can be written as the sum of 3 integers >1 that are pairwise relatively prime. The Attempt at a Solution I already proved the case for even integers. Now I am just working on the case for odd integers. I know that it has to be the sum of 3...
  43. C

    How can integers be written as sums of relatively prime integers?

    Homework Statement Prove that every integer bigger than 6 can be written as a sum of 2 integers bigger than 1 which are relatively prime. The Attempt at a Solution Ill first look at the case where our number is odd. Let x be an odd integer. I will just add (x-2)+2=x since x is odd so is x-2...
  44. mathworker

    MHB Sum of Products of N Natural Numbers

    If we are given with set of n natural numbers and asked to find the sum of all possible products of two number.. E.G: {1,2,3} is given then , 1*1+1*2+1*3+2*2+2*3+3*3=s how to find the general formula of SUM for given n-tuple containing consecutive natural numbers not necessarily starts with 1 or...
  45. mathworker

    MHB How Do You Calculate the Sum of Fourth Powers Up to N?

    how to find the sum, 1^4+2^4...n^4 i tried myself but couldn't find an approach , when i googled it ifound the formula but not proof
  46. B

    How many ways to sum vectors in a plane to zero

    Homework Statement (a) Suppose you have two arrows of equal length on a tabletop. If you can move them to point in any direction but they must remain on the tabletop, how many distinct patterns are possible such that the arrows, treated as vectors, sum to...
  47. I like Serena

    MHB AM-GM inequality for sum of 3 square roots

    Let $a,b,c$ be positive real numbers with sum $3$. Prove that $√a+√b+√c≥ab+bc+ca$.
  48. anemone

    MHB Sum of a Complex Fraction Sequence

    Find \sum_{x=0}^{101}\frac{\frac{2x}{101}-1}{\frac{3x^2}{10201}-\frac{3x}{101}+1}.
  49. A

    Calculating Ʃn(E) with Boson States

    My book wants to sum the number of particles in each energy states for a gas of bosons, that it calculate the infinity sum: Ʃn(E) now if the E's are narrowly spaced it says we can approximate this with an integral. ∫n(E)ρ(E)dE Now can anyone tell me using the definition of the...
  50. L

    Calculate sum. Fermionic operators

    Homework Statement How to calculate? ## \sum _{i,j} \langle 0|\prod_n \hat{C}_n \hat{C}^+_i\hat{C}_j \prod_n \hat{C}^+_n|0 \rangle ##Homework Equations ##\hat{C}^+, \hat{C}## are fermionic operators. ##\{\hat{C}_i,\hat{C}^+_j\}=\delta_{i,j}##The Attempt at a Solution I have a question. What is...
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