0
votes
1answer
21 views

How to eliminate coefficients from a sum

For given random values $$X_i \sim\mathcal{N}(0,1)$$ and $$\frac{X_i-\mu}{\sigma}=\tilde{X_i}\sim\mathcal{N}(\mu,\sigma),\,\mu\in\mathbb{R},\,\sigma>0$$ prove ...
0
votes
0answers
39 views

problems with applying a $f(x)=x^2$ curve

I am applying a curve over time (two seconds), to transition from one value to another. The formula I am using is: $$x = \left(\frac {\text{time}}{2.0}\right)^2$$ ...
0
votes
1answer
65 views

Prove with combinatorial arguments this equation [duplicate]

Prove with combinatorial arguments, that, $\forall n \in \mathbb{N}$. $$\sum_{k=0}^n (-1)^k {n \choose k} =0$$
0
votes
0answers
27 views

How to construct a function to map coefficients?

Surely this question is known by many people but I lack of enough maths knowledge so I prefer ask here. I have a triangular matrix that represent coefficients, all of them are rational numbers ...
2
votes
1answer
45 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
1
vote
1answer
28 views

Summation of series- substitution

If we have $\sum_{n=0}^{\infty}nf(n)=C, C\ne0\tag 1$, C is a constant, can we find a closed form for f(n)?. NB : Given condition is that $\sum_{n=0}^{\infty}f(n)$ converges to a constant value $K$ ...
0
votes
0answers
29 views

Finding alternating series for Power series

Given data and conditions I have a power series, $PS(x) = \sum_{n=0}^\infty R_nx^n$. I have a infinite GP,something like G(x) = $\sum_{k=0}^\infty ax^k = \frac{a}{1-x} $ . Never take G(x),such ...
0
votes
1answer
25 views

Rank of a simple matrix series

Problem Specifications and Given conditions I have a matrix $L$ with rank 3 and dimension $ 3 \times 3$. $L = K_0+\sum_{n=1}^{\infty}K_i $ . Rank of $K_0$ is 3 and rank of L is also 3. Rank of ...
1
vote
2answers
56 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
0
votes
0answers
59 views

Showing that two sums are equivalent

given \begin{gather} U_d(x,y,q\mid i_1,\ldots,i_k)=\sum\limits_{n,m\geq0}x^ny^m\sum\limits_{\sigma = i_1\ldots i_k\sigma_{k+1}\ldots\sigma_m\in C_{[d]}(n,m)}q^{v(\sigma)}. \end{gather} show ...
1
vote
1answer
38 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
3
votes
2answers
33 views

Find value of $n$ with given conditions

The 4-digit positive number $n$'s digit sum is $20$. The sum of the first two digits is $11$, the sum of the first and the last digit as well. The first digit is the last digit $+3$. What is the ...
-1
votes
2answers
70 views

How to approximate this large sum of exponential terms

Is there any way to approximate the following sum: $$ \sum\limits_{i_1=1}^N\sum\limits_{i_1=2}^N \cdots \sum\limits_{i_k=1}^N \cdots\sum\limits_{i_N=1}^N \exp(-r_{i_1}-r_{i_{k+1}}-r_{i_{2k+1}}- ...
1
vote
1answer
48 views

Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
1
vote
1answer
74 views

${n \brack k} = \sum_{m=0}^{k}2^{(k-m)^2}{n-k \brack k-m}{k \brack m}$

$${n \brack k} = \sum_{m=0}^{k}2^{(k-m)^2}{n-k \brack k-m}{k \brack m}$$ I need hint to prove this. ${n \brack k}$ is the number of $k$ dimension subspaces of $n$ dimension space over field $F_2$. I ...
1
vote
1answer
20 views

help with simplifying this sum

Problem I need help with simplifying following sum: $$ 1 + \sum_{i=1}^{\infty}{\frac{1}{i!} * (-1)^i * a * (a + b)^{i-1}} $$ and can get the $a$ out to get $$ 1 + ...
0
votes
0answers
42 views

How can this equation be solved with respect to mu?

How can this equation be solved with respect to mu? I'm not even sure whether this equation can be solved or not. If it's not possible, could you explain why?
1
vote
1answer
39 views

What can I do to this expression to lose the summations?

I'm at the end of a past paper question and need to derive this answer: I am very close and have got to this by doing d/dx to the * equation: What can I do to get rid of these summation signs ...
1
vote
3answers
53 views

Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$ [duplicate]

The question: Using $S_n = \sum_{k=1}^{n}H_k$ where $H_k$ are the harmonic numbers, show $S_n = (n+1)H_n - n$. So far I have $S_n = \sum_{k=1}^{n} H_k = \sum_{k=1}^{n} ...
1
vote
4answers
297 views

Algebra-sum of entries in each column of a sqaure matrix = constant

This is a question from an algebra homework and I am just looking for some tips. The question is: We have: $M$: an $n\times n$ matrix with real entries $c$: a real constant the ...
0
votes
2answers
197 views

Raising $e$ to the power of a matrix

Does there exist a definition for matrix exponentiation? If we have, say, an integer, one can define $A^B$ as follows: $$\prod_{n = 1}^B A$$ We can define exponentials of fractions as a power of a ...
0
votes
2answers
148 views

Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
3
votes
1answer
110 views

Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...
6
votes
2answers
142 views

What can be computed by axiomatic summation?

Here are three simple properties one might require of a summation method for divergent series: A stable summation scheme is one in which (assuming also each sums are defined iff the other is) ...
0
votes
3answers
106 views

Proof involving a summation

How would I go about proving that $\sum_{i<j} 1 $ = $ n\choose 2$ and $$ \sum_{i<j}(x_i +x_j) = (n-1)\sum x_i $$ I understand the intuition behind the statements. I'm just unsure of how to ...
0
votes
2answers
83 views

Is there any formula for summation?

$$0.01\sum_{x=1}^{30}(0.99)^{x-1} = 1-0.99^{30}$$ I wonder if there is a formula for summation and I want to know. Would anyone mind telling me? It would be better for me to solve problems, like the ...
2
votes
3answers
64 views

Why do I see i and k as the indices of summation?

I'm working on linear algebra and just wanted to clear up an uncertainty regarding whether there is a difference in the use of i and k as the dummy variables for the index of summation? ...
1
vote
2answers
52 views

Help with summation question?

I have a problem involving sums: I just have no idea how to solve this. I know how to solve sums but this does not make sense to me. I need the value of the 'a' but in the first one, 'a' must be ...
2
votes
1answer
40 views

Summing infinite numbers of matrices

Let $\mathbf{I}$ be a identity matrix, and $\mathbf{A}$ be a symmetric matrix in which every entry $a_{ij}$ follows $0 \le a_{ij} < 1$. I want to get $\mathbf{S} = \mathbf{I} + \mathbf{A} + ...
0
votes
3answers
423 views

Derivative of a summation in order to minimize

I am asked to minimize $\sum^n_{i=0}(x_i - C)^2$ with respect only to C so I know I have to take the derivative respect to C, set it equal to 0, and then solve. I have never done summation in my ...
2
votes
1answer
152 views

Finding the summation of the floor of the series.

Can someone help me the summation of the given series. $$\sum_{i=1}^n\left\lfloor\frac{n}i\right\rfloor$$ Negative of the above summation looks similar to the expansion of the $\log(1-x)$ without ...
2
votes
0answers
97 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
1
vote
2answers
687 views

How to calculate weight of positive and negative values.

We have used formula formula to calculate weight as, $$ w_1 = \frac{s_1}{s_1 + s_2 + s_3};$$ $$ w_2 = \frac{s_2}{s_1 + s_2 + s_3};$$ $$ w_3 = \frac{s_3}{s_1 + s_2 + s_3};$$ However, their is ...
2
votes
1answer
50 views

(CHECK) Cardinality of Terms in the Expansion of a Product of Multinomials

QUESTION: How many terms are there in the expansion of $$(x+y)(a+b+c)(e+f+g)(h+i)$$ I'd like some help with this one, but I'd also like to discuss a method of generalization on the problem, ...
2
votes
1answer
106 views

Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
1
vote
0answers
84 views

Simplifying expression

I am looking for a way to simplify this expression: $$ \sum_{i=0}^{n-k-1} \sum_{j=0}^{k-1} \left[ {n-k-1 \choose i} {k-1 \choose j} ((-1)^{k-1-j} - (-1)^{n-k-1-i}) \times {(n+0.5)! \over ...
0
votes
0answers
34 views

Finding solutions to the equation

I want to find possible solution satisfying both the equation: $\sum_{i=1}^{n} f_i^{2} = n$ $\sum_{i=1}^{n} f_i=0$ As the number of equations less than number of variables can we just comment on ...
5
votes
1answer
219 views

eigenvalues of the sum of a matrix with known eigenvalues and a diagonal matrix

Suppose $B = A+D$, where all the eigenvalues of $A$ are already known and $D$ is a diagonal matrix, how to compute the eigenvalues of B without diagonalizing $B$ directly?
2
votes
3answers
928 views

What's the rule for solving nested sums?

I have the following nested double sum : $\sum_{t=1}^{T}\sum_{u=t}^{T} Z(u) \cdot (1+i)^{-u}$ with $0<i<1$ and $Z(u)$ being a non-specified function. By working with an example of $T=3$ I ...
1
vote
1answer
89 views

Simplify the expression

The below expression has three summations (sigmas) and $L$ is a real-matrix and symmetric, $X$ is a real matrix with $n$ rows and $X_{p\mathbb{.}},X_{q\mathbb{.}}$ denote the $p$ and $q$ rows of ...
0
votes
1answer
120 views

find the multiplicative factor for get a specific amount of sum on sin

i am not a math guru so please sorry if this is a silly question. i'm not sure on how to latexize this question so i've done a spreadsheets with openoffice (and i'm interest also in the best way to ...
4
votes
2answers
2k views

Smallest eigenvalues of Sum of Two Positive Matrices

Let $C = A + B$, where $A$, $B$, and $C$ are positive definite matrices. In addition, $C$ is fixed. Let $\lambda (A)$, $\lambda (B)$, and $\lambda (C)$ be smallest eigenvalues of $A$, $B$, and $C$, ...