Why are these 2 algebraic expressions equivalent February 14    2

I just solved a long problem for my physics w/calculus homework that required a simplification using a quadratic formula. The "textbook" (flipItPhysics) came up with a different simplification than mine but it turns out they are equivalent. I ca

If $$\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+\ldots+\frac{a_{n-1}}{2}+a_{n}=0,$$ then the maximum possible number of roots of the equation $${a_0}{x^n}+{a_1}{x^{n-1}}+{a_2}{x^{n-2}}+\ldots+{a_{n-1}}{x}+{a_n}=0$$ in $(0,1)$ will be...?This seems

If $f(0)=0$ and $f(x)=\frac x{1+e^{1/x}}$ for $x\neq 0$, find the left and right hand derivatives for f = 0 February 14    1

This problem is for my Mathematical Analysis class, and I'm looking for any help I can get. The equation I am using to solve this problem is: (for left hand only)$$f'(x)=\lim_{x_o\to 0^-}\frac{f(x)-f(x_o)}{x-x_o}$$ I believe that I am applying the fo

How to express a variable as a quadratic polynomial in another variable February 14

I came across following problem in my textbookA definition of efficiency ($E$) is the ratio $Vc/Va$. Obtain a model which expresses $E$ as a quadratic polynomial in $Va$ (i.e., a model in $Va$ and $Va^2$;).Kindly tell me how to do it?

find the derivative of sq(A) February 14

Consider the function sq : Rn×n → Rn×n , A

Finding orthonormal basis February 14    1

I need to find an orthogonal basis for certain symmetric bilinear form $g$ such that matrix of g is diagonal in these basis.Given basis ${b_1,b_2,b_3}$I am applying the following process (called Gramm Schmidt I guess, but I am confused because doesnt

Finding the orthogonal basis, picture included February 14

I decided to share a picture of what I have so far. I am not sure if I did it correctly and sorry if it is not readable. Ask me if anything is unclear. In the exercise I am basically just asked to find the ortogonal basis. I did it by finding the red

Metric Spaces Whose Diameter is Achieved at Every Point. February 14

Suppose $(X,d)$ is a metric space with diameter $\sup \{ d(x,y) \colon x,y \in X\}=1$. Call the point $x \in X$ an edge point to mean that $d(x,y)=1$ for some $y \in X$. Call the metric spac ...

Berry's curvature equation February 14

The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation [1] $$V_{m} = {- 1 \over B^2 } * i * \sum { (<m,B|S|n,B> ∧ <n,B|S|m,B>) \over A^2}$$

exponentiating a matrix and sum of elements February 14

$$M= \begin{bmatrix} 1&1&0\\0&1&1\\0&0&1\\ \end{bmatrix}$$ Then the sum of all entries of $e^{M}$ i just don't know how to calculate this sum as this would be an infinite series.please help

Computation of general continued fractions by $2 \times 2$ matrix multiplicationis it the best way February 14

There are two main ways to compute a continued fraction (or its $n$th convergent). Let's say we have a general fraction:$$x= a_0 + \dfrac{b_1}{a_1+\dfrac{b_2}{a_2+\dfrac{b_3}{a_3+\dfrac{b_4}{a_4+...}}}}$$To compute the $n$th convergent we can start

(i) Show that $|(R/I)| = 1$ if and only if $R = I$.(ii) Show that if $R$ has an identity 1 then (if $I \neq R$) so does $R/I$, and if $R$ is commutative, then so is $R/I$.I know that the quotient ring of $R$ by $I$ is defined to be $R/I = \{a+I : a\ Cancellation law for invertible matricies February 14 2 Show that the cancellation law holds for invertible matrices. i.e. if$A \in GL_n(R), B, C \in M_{n×m}(\mathbb{R})$and$AB = AC$, then$B = C$.What I tried:I know that I can prove this by actually looking at each element of the matrix and multiplyin Orientation on a manifold as a sheaf February 14 1 I am thinking about orientation of a connected manifold$M$of dim$n$as a sheaf.There are two definitions I could use, the first is the sheaf associated to the presheaf$$U\mapsto H_n(M,M- ... Orientation of manifold in topological sense February 14 What do we mean by orientability of a topological manifold? How do we orient two dimensional Euclidean space and why is Moebius band non-orientable? And it would be a great favour to me if you can tell me a good book that explains this idea. I read " Orientable manifolds and R-orientability February 14 Why is it that when a topological manifold is orientable it is also R-orientable, where$R$is an arbitrary ring. Note that we are using orient ability in the sense of homology here. (Non)Existence of limits February 14 When We say that limit of a function does not exist in R(or some metric space) does it make sense to say that it might exist somewhere else? [I am trying to think along lines of existence of imaginary roots]If yes. Then give examples especially regar Let U and V be vector spaces of dimensions n and m over K. Find the dimension and describe a basis of Homk(U,V) February 14 I am given vectors spaces U and V of dimensions n and m over K.How can I find the dimension and basis of Homk(U,V) ? You Might Also Like • Assume given a probability measure$P$on$(\mathbb{R}^p,\mathcal{B}_p)$, where$\mathcal{B}_p$denotes the ... • Given two conics in general form$A_ix^2 + B_ixy + C_iy^2 + D_ix + E_iy + F_i = 0$for$i = 1, 2$, I want to ... • Let$A$be the reflection of the plane$\mathbb R^2$in the line$y=-x$. Find the matrix of$A$in the stand ... • So the question was basically " Suppose that there are n teams in a rugby league competition. Every tea ... • It is known that open sets in real line can be written as a countable union of disjoint open intervals. (lin ... • Before all, good morning; I have just seen an exercise of number systems and equations... I hope you could h ... • I am studying the famous integral of$\dfrac{\sin x}{x}$in complex analysis. My lecturer integrated$e^{iz} ...
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