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:

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.

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
$$partial_t vert !vert v vert!vert^2_{L^2} = 2,mathrm{Re}left(Sv,vright) + 2,mathrm{Re}left((A+iB)e^phi F, vright),$$
where
$$mathrm{Re}left(Sv,vright) = -Aint |nabla v|^2 + left(A|nabla phi|^2+partial_t phi right) |v|^2 + 2B ,mathrm{Im}, v^{dagger} nablaphicdotnabla v + left( A,mathrm{Re},V – B,mathrm{Im}, Vright)|v|^2dx,$$
holds true. The authors go on to conclude that the Cauchy-Schwarz inequality implies that
$$partial_t vert !vert v(t) vert!vert^2_{L^2} le 2vert !vert A , left(mathrm{Re},V(t)right)^+ – B,mathrm{Im}, V(t) vert !vert_{infty}vert !vert v(t) vert !vert^2_{L^2} + 2 sqrt{A^2+B^2} vert !vert F e^phi vert !vert_{L^2} vert !vert v(t) vert !vert_{L^2}$$
when
$$left(A+frac{B^2}{A}right)|nablaphi|^2 + partial_t phile 0, ,,,,mathrm{in}, mathbb{R}_+^{n+1}.$$
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

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 $$varphi (x ) = a+ i b.$$