algorithm analysis – Understanding a summation notation. Sum(j=2 to n) j – 1

I have been reading analysis of insertion sort in the “Introduction to algorithms” and faced a problem with understanding a specific summation notation when the worst case occurs.

I know how one can get formula for arithmetic series when we deal with while loop header, I mean 2+3+…+n equals to (n*(n+1) / 2) – 1. But what I do not understand is how one can get formula for while loop body:

summation notation on page 27

It is obvious, that we execute while loop body one time less, than while loop header, because of the final conditional test. Hence, we should subtract one. But how do we get n * (n – 1) / 2?

real analysis – Compact Sobolev embedding with boundary conditions

Let $X$ be some metric measure space on which Sobolev spaces can be defined in a reasonable way. In many cases, $H^1(X)$ is compactly embedded in $L^2(X)$ (e.g., if $X=Omega$ is a bounded open set of $mathbb R^d$), and in that case, $H^1_0(X)$ is of course compactly embedded in $L^2(X)$, too. In many cases, on the other hand, $H^1_0(X)$ is not compactly embedded in $L^2(X)$ (e.g., $X=Omega=mathbb R^d_+$), let alone $H^1(X)$.

My question is now, whether structures $X$ are known such that the embedding of $H^1_0(X)$ in $L^2(X)$ is compact but that of $H^1(X)$ is not.

fa.functional analysis – When is it true that $Z(A)^{**} = Z(A^{**})$ for a C*-algebra $A$?

For a (unital) C$^*$-algebra $A$ with centre $Z(A)$ the bidual $A^{**}$ is a von Neumann algebra with centre $Z(A^{**})$ for which I believe that $Z(A)^{**} subset Z(A^{**})$ holds by extending the product on $A$ to $A^{**}$ and using that it is separately w$^*$-continuous. My question is if there is anything known about when we have an equality $Z(A)^{**} = Z(A^{**})$? This is true for commutative and finite-dimensional C$^*$-algebras, but I can’t seem to find anything related to this question in the literature.

For example, is the bidual of $B(H)$ (the bounded operators on a Hilbert space) a factor? If not, this would be an example for which the equality does not hold true.

Please note that I am not an expert in the theory of C$^*$-algebras.

real analysis – Fourier and pointwise convergence

Let $f(x)=xsin(x)$, $xin(-pi,pi)$

  • Now, find the Fourier series of $g(x)=f'(x)=sin(x)+xcos(x)$, $xin(-pi,pi)$ and show that it converges pointwise to $g$.

I found the Fourier series of the function $f(x)=xsin(x)$, $xin(-pi,pi)$ (in the just previous problem) to $1-frac{1}{4}cos(x)+sum_{n=2}^{infty} left (frac{2(-1)^{n+1}}{n^2-1} right) cos(nx)$ and shown that it converges uniformly to $f$. Now, is there a trick to finding the Fourier of the derivative of my function given that I have the Fourier for my function? How do I show pointwise convergence to $g$?

*I know about convolutions as I suspect I need that in this problem.

google sheets – How can I build a Predictive Model (analysis)?

In the above spreadsheet…I have imported table containing daily state lotteries results from a website.. I want to confirm or debunk the theory that certain states lottery drawing have some sought of connection. I want to explore this from a mathematical perspective.

In sheet 1 of the linked there are 7 days of lottery results. All possible combinations of the Pick 3 and 4 Ball results are generated. Subsequently, all multiples of 12,13,37,101 and 111 are filtered out and broken down to it’s simplest form e.g. 240/12=020.

On sheet 2…the simplest forms are filtered and each are multiplied via 12,13,37,101,111..

Could I used the historical data in the spreadsheet to establish the following:

  1. Past mathematical correlations or connections between the state lotteries

2.the ability to forecast future draws based on the gathered data.

Thanks in advance.

functional analysis – (left) Shift Semigroup and operator norm

I am reading some lecture notes on strongly continuous semigroups. I am having difficulty understanding an example:
$$X = BUC(mathbb{R}) := {f:mathbb R rightarrow mathbb R : f text{ is uniformly continuous and bounded}} $$
with supremum norm $$|f|_{infty} = sup _{s in mathbb R}|f(s)|$$
$T(t)$ is the shift semigroup; $T(t)f(s) = f(t+s)$, which is indeed a strongly continuous semigroup.
Then it says $T(t)$ is not continuous for the operator norm.
How can it be strongly continuous semigroup on X but not continuous for the operator norm?
I mean it is obviously bounded; what am I missing here?

fourier analysis – How is the Cauchy-Schwarz inequality used to bound this derivative?

In “Hardy’s Uncertainty Principle, Convexity and Schrödinger Evolutions” (link) on page 5, the authors state that they are using the Cauchy-Schwarz inequality to bound the derivative of the $L^2(mathbb{R}^n)$ norm of a solution to a certain differential equations, but I am not sure how exactly they applied it.

Some context: Let $v$, $phi$, $V$, $F$ be nice enough functions of $x$ and $t$ so that the following integrals are well-defined, $A>0, Binmathbb{R}$ be constants, and $u=e^{-phi} v$ solve
$$partial_t u = (A+iB)left( Delta u + Vu+Fright).$$
Denote the $L^2$ inner product on $mathbb{R}^n$ between some $f$ and $g$ as $(f, g) = int f g^{dagger} dx$, where $g^dagger$ is the complex conjugate of $g$, and define $f^+=mathrm{max}{f, 0}$.

We know the equality
$$partial_t vert !vert v vert!vert^2_{L^2} = 2,mathrm{Re}left(Sv,vright) + 2,mathrm{Re}left((A+iB)e^phi F, vright),$$
$$mathrm{Re}left(Sv,vright) = -Aint |nabla v|^2 + left(A|nabla phi|^2+partial_t phi right) |v|^2 + 2B ,mathrm{Im}, v^{dagger} nablaphicdotnabla v + left( A,mathrm{Re},V – B,mathrm{Im}, Vright)|v|^2dx,$$
holds true. The authors go on to conclude that the Cauchy-Schwarz inequality implies that
$$partial_t vert !vert v(t) vert!vert^2_{L^2} le 2vert !vert A , left(mathrm{Re},V(t)right)^+ – B,mathrm{Im}, V(t) vert !vert_{infty}vert !vert v(t) vert !vert^2_{L^2} + 2 sqrt{A^2+B^2} vert !vert F e^phi vert !vert_{L^2} vert !vert v(t) vert !vert_{L^2}$$
$$left(A+frac{B^2}{A}right)|nablaphi|^2 + partial_t phile 0, ,,,,mathrm{in}, mathbb{R}_+^{n+1}.$$
However, I am not sure how the authors used the C.S. inequality to arrive at this conclusion, and am especially confused as to where the factor of $B^2/A$ came from, and why we only need the constraint to hold over $mathbb{R}_+^{n+1}$ when we are integrating over all of $mathbb{R}^{n}$, though I understand why we only care about positive time.

Does anyone have any insight here?

analysis – Construct an increasing function f on R that is continuous at every irrational number and is discontinuous at every rational number.

Construct an increasing function f on R that is continuous at every irrational number and is discontinuous at every rational number.

Solution: Let ($r_n$) be a sequence with distinct terms whose range is
Q. Let f: $R to R$ be given by $f(x)= Sigma_{r_n<x} frac{1}{2n}$

If $x_1 < x_2$, then the series yielding $f(x_2)$ has additional positive terms than the series whose sum is $f(x_1)$. Thus f is increasing.

I don’t understand this function at all. Can anyone tell me about the construction of function?

What are the modern systems analysis and design tools?

In college and university (longer ago than I care to admit!) we had SAD (systems analysis and design) modules which primarily gave a set of methods and tools to design a system, often using UML modelling. Flow charts, entity relationship diagrams, etc, were used to take an idea or an existing/paper based system and come up with the design documents for developers to create the new system.

How much are these kind of tools used these days and is there a shift towards different methods of systems analysis and design? If I were to study now, or go into a large developer, what would they likely be using?

I am particularly interested from a web development point of view.

A question about Functional Analysis : The linear operator is surjective

The Question says:

Let be $varphi: X rightarrow mathbb{C}$ linear. If $varphi$ isn’t null then $varphi$ is surjectvive.

I have no idea how to do this.
The only thing that I know is that ‘ how $varphi$ isn’t null, then exists a $xi in X$ such that $varphi (xi ) = a+ i b neq 0. $
But I can’t see how this implies that for all $a+ib in mathbb{C}$ exists $ x in X$ satisfying $varphi (x ) = a+ i b. $