Articles of reference request

Logarithm of differential operator

Using the Taylor series expansion we have (for a sufficiently regular function $f$): $$ f(x+a)=\sum_{k=0}^n \frac{f^{(k)}(x)a^k}{k!} $$ So, defining the differential operator $D=\frac{d}{dx}$ and using the series expansion definition of the exponential function, we can write: $$ S_a f(x)=\exp(aD) f(x) $$ where $S_a f(x)=f(x+a)$ is the shift operator. This gives an ”intuitive” meaning to the […]

Resources for learning fixed point logic

As the title says, I am looking for resources to learn some fixed point logic, especially partial fixed point logic. I have basic knowledge of propositional calculus and predicate logic, but sadly not much beyond that, and have need of PFP for my bachelor’s thesis. Thanks in advance.

How does integration over $\delta^{(n)}(x)$ work?

For a math paper I need to be able to evaluate $\int_{-a}^{a}\delta^{(n)}(x)\ f(x)\ dx$ for differentiable $f$. I know that it is ‘supposed’ to equal $(-1)^nf^{(n)}(0)$: $$\int_{-a}^a\delta^{(n)}(x)\ f(x)\, dx=\int_{-a}^a \frac{d^n}{dx^n}\delta(x) f(x)\, dx =\int_{-a}^a(-1)^n\delta(x) \frac{d^nf}{dx^n}\, dx=(-1)^nf^{(n)}(0).$$ This argument seems like a hack and I have no idea what is actually going on. I want to write […]

Book Recommendation – Hard problems for Multivariable Calculus w/ Solutions

I’m looking for a text that covers roughly what’s sometimes called “Calculus III” or multivariable calculus.* But this text must satisfy certain additional criteria: (1) It must be more in-depth (and consequently have harder exercises) than usual; (2) It must contain many solutions to the exercises; (3) It must not engage in the sort of […]

Looking for a very gentle first book on number theory

My gf is the classic math-phobe, totally traumatized by math, etc., which surprises me, since she’s whip-smart. The only explanation I can think of is that she got off to a bad start in elementary school. My own love for mathematics was kindled by an old book on number theory that I found in my […]

A Kleinian group has the same limit set as its normal subgroups'

It should be well known that a Kleinian group and all its normal (non-elementary) subgroups have the same limit set. Do you know any book/article where I could find the proof? Thank you.

$ \cos(\hat{A})BC+ A\cos(\hat{B})C+ AB\cos(\hat{C})=\frac {A^2 + B^2 + C^2}{2} $

What more can be said about the identity derived from law of cosines (motivation below)$$ \cos(\widehat{A})BC+ A\cos(\widehat{B})C+ AB\cos(\widehat{C})+=\frac {A^2 + B^2 + C^2}{2} \tag{IV}$$ RHS seems as if operator $\cos(\widehat{\phantom{X}})$ is being applied consecutively to terms of ABC, I tried to represent it in an analogous way to the Laplacian operator convention, but maybe there […]

Which book is appropriate for a Chemistry student that needs to learn basics about integrals?

A friend of me who is not studying mathematics now needs to deal with integrals, double integrals and triple integrals within his study of chemistry. He asked me to give him a suggestion for a basic book that explains basic facts, rules etc., about integrals. I think this means Riemann integral and that it should […]

Wavelet through the lens of Harmonic analysis

I need some references for starting to study about wavelet. I have enough information about abstract Harmonic analysis. Thanks!

Show that Polynomials Are Complete on the Real Line

Consider the Hilbert Space of weighted-square-integrable functions f(x): $$ \begin{equation} \int_{-\infty}^{\infty}\frac{f(x)^2}{e^{x}+e^{-x}}dx<\infty. \end{equation} $$ Note this integral is taken over the real line in both directions, and the weighting is roughly exponential. Hilbert Polynomials are complete using a different weighting, while Laguerre polynomials use a similar exponential weighting, but only on the positive real line. What […]