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. M

    MHB Program to calculate the sum of polynomials

    Hey! 😊 A polynomial can be represented by a dictionary by setting the powers as keys and the coefficients as values. For example $x^12+4x^5-7x^2-1$ can be represented by the dictionary as $\{0 : -1, 2 : -7, 5 : 4, 12 : 1\}$. Write a function in Python that has as arguments two polynomials in...
  2. docnet

    Trying to find an easier way to compute the double sum

    I computed the double sum $$\sum_{i=0}^n\sum_{j=i+1}^n j = \sum_{i=0}^n\big(\frac{n(n+1)}{2}-\frac{i(i+1)}{2}\big)=\frac{n(n+1)(2n+1)}{6}$$ and realized the double sum is equal to $$\sum_{i=1}^ni^2$$ which leads to $$\sum_{i=0}^n\sum_{j=i+1}^n j = \sum_{i=1}^ni^2$$ Is there a proof of this...
  3. Afo

    Total KE = Sum of Translational & Rotational KE: Proving the Equation

    Why is the total energy energy equal to the sum of translational kinetic energy and rotational kinetic energy? I understand the derivation KE = 1/2 I w^2 for a rigid object rotating around an axis: sum 0.5 * m_n * (v_T)^2 = sum 0.5 * m_n * (wr_n)^2 = 0.5 * w^2 * sum m_n r_n^2 = 0.5 * I * w^2...
  4. S

    I Finding a polynomial that has solution (root) as the sum of roots

    AIUI, an algebraic is defined as a number that can be the solution (root) of some integer polynomial, and is any number that can be constructed via any binary arithmetic operation or unary root operation with arguments that are themselves algebraic numbers. I have been able to prove this for...
  5. M

    I Using a probability argument I got the sum. Can it be done directly?

    ##\sum\limits_{k=0}^{n-m} \frac{\binom{n-m}{k}}{\binom{n}{k}}\frac{m}{n-k}=1##. Can be derived from question. For ##n\ge m##, pick ##m## marbled out of a set size ##n## labeled from ##1## to ##n##, what is probability distribution of minimum of the number labels on the marbles? The terms is...
  6. A

    Proving this equation -- Limit of a sum of inverse square root terms

    Hi I was working on a physics problem and it was almost solved. Only the part that is mostly mathematical remains, and no matter how hard I tried, I could not solve it. I hope you can help me. This is the equation I came up with and I wanted to prove it: $$\lim_{n \rightarrow+ \infty} {...
  7. LCSphysicist

    Proving that the incident intensity is not the same as the sum of others

    I was supposed to find the relation among the coefficients $T$ and $R$ which represent the amplitude of the reflected electric field and the transmitted electric field respectively, that is, $$E_{R} = E_{i} R, E_{T} = E_{i} T$$ as well as the coefficients $t$ and $r$, that represent the...
  8. M

    Game theory: dice game to aim for sum no greater than 9

    Hi, I was attempting the following question and would appreciate any insight on how others would approach this game theory/probability-type question. Question: You have been chosen to play a game involving a 6-sided die. You get to roll the die once, see the result, and then may choose to...
  9. R

    Show that the sum of the finite limits of these two series is also finite

    In the homework I am asked to proof this, the hint says that I can use the triangle inequality. I was thinking that if both series go to a real number, a real number is just any number on the real number line, but how do I go from there,
  10. F

    A Cl's : sum into a chi^2 when we have a sum of chi^2

    1) If I take as definition of ##a_{lm}## following a normal distribution with mean equal to zero and ##C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})##, and if I have a sum of ##\chi^2##, can I write the 2 lines below (We use ##\stackrel{d}{=}## to denote equality in distribution)...
  11. K

    Why it doesn't sum to one in this simple naive Bayes classification?

    Summary:: When we have only three classes (Orange, Banana and Other) and three features (Long, Sweet and Yellow), why P(Other|Long, Sweet, Yellow) + P(Banana|Long, Sweet, Yellow) is not equal to 1 when P(Orange|Long, Sweet, Yellow) = 0 ? In this example...
  12. Eclair_de_XII

    B Any square matrix can be expressed as the sum of anti/symmetric matrices

    Let ##A## be a matrix of size ##(n,n)##. Denote the entry in the i-th row and the j-th column of ##A## by ##a_{ij}##, for some ##i,j\in\mathbb{N}##. For brevity, we call ##a_{ij}## entry ##(i,j)## of ##A##. Define the matrix ##X## to be of size ##(n,n)##, and denote entry ##(i,j)## of ##X## as...
  13. U

    I Prove series identity (Alternating reciprocal factorial sum)

    This alternating series indentity with ascending and descending reciprocal factorials has me stumped. \frac{1}{k! \, n!} + \frac{-1}{(k+1)! \, (n-1)!} + \frac{1}{(k+2)! \, (n-2)!} \cdots \frac{(-1)^n}{(k+n)! \, (0)!} = \frac{1}{(k-1)! \, n! \, (k+n)} Or more compactly, \sum_{r=0}^{n} (...
  14. A

    I Index and bound shift in converting a sum into integral

    Considering the below equality (or equivalency), could someone please explain how the bounds and indices are shifted? $$\sum_{i=2}^{k}(h_i/f_{i-1})=\int_{1}^{k}(h(i)/f(i))di$$
  15. chwala

    Proving an integer problem about the sum of a square number and a prime number

    i do not seem to understand part ##ii## of this problem...mathematical induction proofs is one area in maths that has always boggled me :oldlaugh: let ##n=3, p=7, ⇒m=4## therefore, ##7=(4-3)(4+3)## ##7=1⋅7## ##1, 7## are integers...##p## is prime. i am attempting part ##iii## in a moment...
  16. JD_PM

    I Understanding the concept of direct sum

    Given two subspaces ##U_1, U_2##, I understand the concept of direct sum $$ W= U_1 \oplus U_2 \iff W= U_1 + U_2, \quad U_1 \cap U_2 = \{ 0 \}$$ Where ##W## is a subspace of ##V##. I am trying to generalize it for more than ##2## subspaces, say ##3##. I thought of the following. $$ W= U_1...
  17. V

    Dimension of orthogonal subspaces sum

    ##| V_1 \rangle \in \mathbb{V}^{n_1}_1## and there is an orthonormal basis in ##\mathbb{V}^{n_1}_1##: ##|u_1\rangle, |u_2\rangle ... |u_{n_1}\rangle## ##| V_2 \rangle \in \mathbb{V}^{n_2}_2## and there is an orthonormal basis in ##\mathbb{V}^{n_2}_2##: ##|w_1\rangle, |w_2\rangle ...
  18. tworitdash

    I Can a Gaussian distribution be represented as a sum of Dirac Deltas?

    We know that Dirac Delta is not a function. However, I just talk about the numerical version of it that we use every day. We can simply represent the Dirac delta function as a limiting case of Gaussian distribution when the width of the distribution ##\sigma->0##. $$ \delta(x - \mu) =...
  19. Rabindranath

    I Meaning of terms in a direct sum decomposition of an algebra

    Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...
  20. K

    B Magnitudes of the sum of two vectors

    This is a question that I saw in a textbook: "If the magnitude of a+b equals the magnitude of a+c then this implies that the magnitudes of b and c are equal. Is this true or false?" The textbook says that this statement is true, but I'm inclined to believe it is false. I made a quick sketch to...
  21. anemone

    MHB Can the Inequality of the Sum be Proven Using the Cube Root of -1?

    Assume that $x_1,\,x_2,\,\cdots,\,x_n\ge -1$ and $\displaystyle \sum_{i=1}^n x_i^3=0$. Prove that $\displaystyle \sum_{i=1}^n x_i\le \dfrac{n}{3}$.
  22. S

    I Confused about identity for product of cosines into a sum of cosines

    What I mean is the way that a product of cosines in which the angles increment the same amount is equal, with some extra terms, of the sum of the cosines. It is discussed here...
  23. A

    MHB Probability of Rolling Sum > 3 with Two Dice

    Two fair dice are rolled. What is the probability of rolling a sum that exceeds 3?
  24. S

    Riemann Sum to find the time to fill a container

    (a) I imagine there are several rectangles to represent the area under graph of p vs t then I try to make equation for the total area. Since the question asks about time when the container holds 22 fewer liters than it does at time t = 9, I think the total area of rectangles starting from t = b...
  25. Greg Bernhardt

    I Math Myth: The sum of all angles in a triangle is 180°

    From @fresh_42's Insight https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/ Please discuss! We all live on a globe, a giant ball. The angles of a triangle on this ball add up to a number greater than ##180°##. And the amount by which the sum extends...
  26. G

    Python Is it possible to evaluate a sum in SymPy without breaking the expression?

    I want to evaluate the sum using Python and SymPy, I'm relatively new using SymPy and the first thing I tried was from sympy import exp, oo, summation, symbols n, x = symbols('n,x') summation(exp(-n*x), (n, 1, oo)) But it doesn't work, it just returns the unevaluated sum. I can make it...
  27. anemone

    MHB What is the Sum of x, y, and z in a Non-Negative Real Number System?

    $x,\,y$ and $z$ are non-negative real numbers that satisfy the following system: $x^2+y^2+xy=3\\y^2+z^2+yz=4\\z^2+x^2+xz=1$ Evaluate $x+y+z$.
  28. M

    I Sum independent normal variables

    (I know how to prove it). Prove that a finite sum of of independent normal random variables is normal. I suspect that independence may not be necessary.
  29. A

    I The sum of S and L quantum numbers

    I am reading the textbook Magnetism and Magnetic Materials by Coey and I am confused about how they grouped the terms and how they ended up getting the sums of L and S. My confusion lies in the two red boxes. Also, how is D even considered here when we have up to $2p_1$? And why would the spin...
  30. U

    Tensor help -- Write out this tensor in a simplified sum

    I managed to write $$F_{\alpha\beta}F^{\alpha\gamma}=F_{0\beta}F^{0\gamma}+F_{i\beta}F^{i\gamma}$$ where $$i=1,2,3$$ and $$\gamma=0,1,2,3=\beta$$. How do I proceed?
  31. S

    MHB Is the Sum of n^2 Terms in an Arithmetic Sequence Limited to 1?

    How many different arithmetic sequences have the sum of the first n terms n^2? solution an= 2n-1.Does that mean there is only one arithmetic sequence?
  32. anemone

    MHB Find the sum of all values of positive integer a

    For a pair of positive integers $(a,\,b)$, $Q(a,\,b)$ is defined by $Q(a,\,b)=\dfrac{a^2b+2ab^2-5}{ab+1}$. Let $(a_1,\,b_1),\,(a_2,\,b_2),\,\cdots, (a_n,\,b_n)$ be all pairs of positive integers such that $Q(a,\,b)$ is an integer. Calculate $\displaystyle \sum_{i=1}^n a_i$,
  33. anemone

    MHB Proving Sum of Tangents $\ge$ Sum of Cotangents in Acute Triangle

    In an acute triangle $ABC$, prove that $\sqrt{\tan A}+\sqrt{\tan B}+\sqrt{\tan C} \ge \sqrt{\cot \left(\dfrac{A}{2}\right)}+\sqrt{\cot \left(\dfrac{B}{2}\right)}+\sqrt{\cot \left(\dfrac{C}{2}\right)}$.
  34. P

    Prove the reflection and transmission coefficients always sum to 1

    Consider polarized light crossing a sharp boundary between two volumes, each of a different but uniform refraction index ##n_1## or ##n_2##. Prove that the sum of the transmission and reflection coefficients of this light ##R+T=1##, where $$R \equiv {I_R \over I_I} = \left( {E_{0_R} \over...
  35. A

    Problem with the sum of a Fourier series

    Good day I really don't understand how they got this result? for me the sum of the Fourier serie of of f is equal to f(2)=log(3) any help would be highly appreciated! thanks in advance!
  36. Leo Liu

    Midpoint Riemann sum approximation

    Can someone please explain why the formula for midpoint approximation looks like the equation above instead of something like $$M_n=(f(\frac{x_0+x_1}2)+f(\frac{x_1+x_2}2)+\cdots+f(\frac{x_{n-1}+x_n}2))\frac{b-a}n$$? Thanks in advance!
  37. A

    Studying the convergence of a series with an arctangent of a partial sum

    Greeting I'm trying to study the convergence of this serie I started studying the absolute convergence because an≈n^(2/3) we know that Sn will be divergente S=∝ so arcatn (Sn)≤π/2 and the denominator would be a positive number less than π/2, and because an≈n^(2/3) and we know 1/n^(2/3) >...
  38. S

    B Is There a Cosmological Model That Considers All Possible Laws of Physics?

    Hawking and Hartle proposed a well-known model which postulated a sum over all possible histories considering all compact euclidean metrics to explain the origin of the universe (this is called the No Boundary model). I was wondering whether there is any model or theory (related to cosmology)...
  39. Arman777

    Dynamical Programming - Sum of numbers / Advanced Problem

    Let us suppose we have an equation such that $$N = \sum_{i=1}^N ix_i = x_1 + 2x_2 + 3x_3 + ...+Nx_N$$ and we also know that the solutions (i.e ##x_i##) ranges from ##\{0, N\}##. For example, if ##N=4## we would have $$x_1 + 2x_2 + 3x_3 + 4x_4 = 4$$ and ##x_1,x_2,x_3,x_4## will range from...
  40. anemone

    MHB Finding 3-Digit Numbers with Sum of Digits Squared = 2

    For any natural number $n$, let $S(n)$ denote the sum of the digits of $n$. Find the number of all 3-digit numbers $n$ such that $S(S(n))=2$.
  41. Eclair_de_XII

    Cartesian sum of subspace and quotient space isomorphic to whole space

    Let ##n=\dim X## and ##m=\dim Y##. Define a basis for ##X: y_1,...,y_m,z_{m+1},...,z_n##. The first ##m## terms are a basis for ##Y##. The remaining ##n-m## terms are a basis for its complement w.r.t ##X##. Let's call it ##Z##. ##X## is the direct sum of ##Y## and ##Z##; denote it as ##X=Y+Z##...
  42. H

    I Sum over histories implies no definite history?

    What does Feynman's sum over histories mean to the interpretation of our world? Does it mean that we (or a particle) do not have a definite history, but only the most probable one?
  43. M

    MHB Sum of normalized B-splines

    Hey! :giggle: Let $N_j$, $j=-k,\ldots , m-1$ the normalized B-splines of the set of nodes $x_0, \ldots , x_m$ of degree $k$. Show that $$\sum_{j=-k}^{m-1}N_j(x)=1 \ \text{ for all } x\in [x_0, x_m]$$ A formula with divided differences is \begin{align*}&N_j(x)=(x_{j+k+1}-x_j)B_j(x) \\&...
  44. Leo Liu

    Sum of oscillating frequency and angular velocity

    The "egg" initially spun around axis 1 with at ##\omega_s##. After being disturbed, it has started to possesses angular velocities along 2 and 3. The question is to find the rotational speed of ##\vec \omega=\vec\omega_1+\vec\omega_2+\vec\omega_3## to a fixed observer. It is calculated that...
  45. LCSphysicist

    Find this sum involving a polynomial root

    if x_{I}, I = {1,2,...,2019} is a root of P(x) = ##x^{2019} +2019x - 1## Find the value of ##\sum_{1}^{2019}\frac{1}{1-\frac{1}{X_{I}}}## I am really confused: This polynomial jut have one root, and this root is x such that 0 < x < 1, so that each terms in the polynomial is negative. But the...
  46. karush

    MHB Gre.al.13 sum of even and odd numbers

    $\tiny{gre.al.13}$ For which of the following conditions will the sum of integers m and n always be an odd integer.? a. m is an odd integer b. n is an odd integer c. m and n both are odd integers d. m and n both are even integers e. m is an odd integer and n is an even integerI chose e just...
  47. anemone

    MHB Prove that the sum of 6 positive integers is a composite number

    Let $a,\,b,\,c,\,d,\,e,\,f$ be positive integers and $S=a+b+c+d+e+f$. Suppose that the number $S$ divides $abc+def$ and $ab+bc+ca-de-ef-df$, prove that $S$ is composite.
  48. K

    Show that V is an internal direct sum of the eigenspaces

    I was in an earlier problem tasked to do the same but when V = ##M_{2,2}(\mathbb R)##. Then i represented each matrix in V as a vector ##(a_{11}, a_{12}, a_{21}, a_{22})## and the operation ##L(A)## could be represented as ##L(A) = (a_{11}, a_{21}, a_{12}, a_{22})##. This method doesn't really...
  49. S

    I Divergent series sum, versus integral from -1 to 0

    Some popular math videos point out that, for example, the value of -1/12 for the divergent sum 1 + 2 + 3 + 4 ... can be found by integrating n/2(n+1) from -1 to 0. We can easily verify a similar result for the sum of k^2, k^3 and so on. Is there an elementary way to connect this with the more...
  50. S

    Mathematica Create real function from symbolic sum formula

    In this code, I define a function of x as the sum of the first x integers. In[7]:= fnSum[x_] := Sum[k, {k, 1, x}] In[8]:= fnSum[x] Out[8]= 1/2 x (1 + x) In[9]:= fnSum[3.5] Out[9]= 6 I would like now to take the symbolic formula underlying fnSum, and use it with real arguments. How can I...
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