real analysis – Function without fundamental period

Proposition: If a periodic function hasn’t fundamental period then every open interval $)a, b($ contains a period.

My attemption: Let $G$ be the set of periods of $f$. Because of absence basic period $forall x in G$ there is $yin G$ such that $y<frac{x}{2}$. By fixing $x$ and repeating this procedure we can go further and construct other $periods$ less than $frac{x}{2^n} (nin mathbb{N}$) for arbitary higher $n$.

In my opinion choosing arbitary small period isn’t problem. We can do it also by “Dirichlet’s approximation” and may be there are some other ways. But what interests me that is how put $m$ multiple $(min mathbb{N})$ of this period in given interval $)a, b($. It seems a litle bit construction problem.

I would be grateful for help.

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algorithm analysis – Why do researchers only count the number of multiplications when analyse the time complexity of Matrix Multiplication?

It looks like the article was written by someone who does not understand matrix multiplication.

the number of additions is equal to the number of entries in the matrix, so four for the two-by-two matrices and 16 for the four-by-four matrices.

With the classic matrix multiplication algorithm (which is the one explained in the example) between two $4times 4$ matrices, each coefficient of the product requires $3$ additions, so a total of $3times 16 = 48$ additions, not $16$.

Usually, when it is said that only multiplications matter and not additions, it means of matrices, not coefficients.

For example, a “naive” divide-and-conquer strategy compute $Atimes B$ where $A,B$ are matrices of size $ntimes n$ by doing $8$ products of matrices of size $frac{n}{2}timesfrac{n}{2}$ (and some additions of matrices, but those are done in complexity $O(n^2)$, which is negligible in front of the complexity of matrix multiplication). That way, the complexity verifies $C(n) = 8Cleft(frac{n}{2}right) + O(n^2)$ and it is easily proven that $C(n) = O(n^3)$, so this strategy is not an improvement.

The Strassen algorithm use a divide-and-conquer strategy to improve complexity: it does $7$ products of matrices of size $frac{n}{2}timesfrac{n}{2}$ and some additions, so the complexity verifies $C(n) = 7Cleft(frac{n}{2}right) + O(n^2)$ and we get $C(n)simeq O(n^{2.8})$.

In the two examples above, the number of matrices multiplications matter much more than the number of matrices additions. But in both, the number of multiplications/additions on coefficients is not compared.

It is confirmed in the article:

Volker Strassen reportedly set out to prove that there was no way to multiply two-by-two matrices using fewer than eight multiplications. Apparently he couldn’t find the proof, and after a while he realized why: There’s actually a way to do it with seven!

complex analysis – Hadamard factorization of $f(z) = cos( sqrt{z})$

I am trying to self study Complex Analysis by Stein and Shakarchi and I am not quite understanding Hadamard factorization. I came across this problem on the internet that I have no clue about how to solve. Could someone please guide me through a step by step process of finding Hadamard factorization of the function $f(z) = cos( sqrt{z})$?

Also, I read that order of this function is $1/2$ but don’t know how. I’d highly appreciate if you could explain this too.

Thank you very much!

analysis – Entropy for partition with respect to sum of Dirac measures

Let $(X,mathcal{A},m)$ be some probability space where $m=frac{1}{p}sum_{j=0}^{p-1}delta_{f^jx}$ for some fixed $xin X$ that is $p$-periodic with respect to the measure-preserving transformation $fcolon Xto X$, i.e., $f^p(x)=x$. Here the $delta_{T^jx}$ is the Dirac-measure.

Now, let $A={A_1,ldots,A_k}$ be some finite measurable partition of $X$ and define
bigvee_{i=0}^{n}f^{-i}A:=left{bigcap_{i=0}^nf^{-i}A_{j_i}: A_{j_i}in Aright}.

I would like to verify that for the entropy $h(f,A)$ of $f$ with respect to $A$ one has $h(f,A)=0$.

To this end, one needs to argue that

where $Hleft(bigvee_{i=0}^{n}f^{-i}Aright)$ is the entropy of the partition $alpha_n:=bigvee_{i=0}^{n}f^{-i}A$ which is defined as
H(alpha_n)=-sum_{alphainalpha_n}m(alpha)log m(alpha).

Here’s what I’ve tried so far:

Without loss of generality, we can assume that $n>p$. Due to the periodicity of $x$, we have
xin A_{j_0},quad fxin A_{j_1},quadldots,quad f^{p-1}xin A_{j_{p-1}},quad f^pxin A_{j_0},quad,f^{p+1}xin A_{j_1},quadldots

for some unique $A_{j_i}in A, i=0,1,2,ldots$.

Thus, for $0leq ileq p-1$,
delta_{f^ix}(alpha)=begin{cases}1, & alpha=A_{j_i}cap f^{-1}A_{j_{i+1}}capldotscap f^{-(p-1)}A_{j_{i+p-1}}cap f^{-p}A_{j_i}cap f^{-(p+1)}A_{j_{i+1}}capldotscap f^{-n}A_{j_{i+n}}\0, & textrm{otherwise}end{cases}

ca.classical analysis and odes – Almost-differential functional equations

The ODE $y'(x)+P(x)y(x)=Q(x)$ has solution $$I(x)y(x)=int I(x)Q(x),dx$$ where $I(x)=expint P(x),dx$. Equivalently, $$Y(x)+P(x)int_0^xY(t),dt=Q(x)tag1$$ has solution $$Y(x)=frac d{dx}frac{int I(x)Q^*(x),dx}{I(x)}=Q^*(x)-frac{I'(x)int I(x)Q^*(x),dx}{I(x)^2}$$ where $Y=y’$ and $Q^*(x)=Q(x)+P(x)y(0)$. Equation $(1)$ gives the limiting case, where $$int_0^xY(t),dt=lim_{ntoinfty}frac xnsum_{k=0}^nYleft(frac{kx}nright).$$ Given $P(x),Q(x)$, what could be said about the solutions of the functional equation $$Y(nx)+frac{xP(x)}nsum_{k=0}^nY(kx)=Q(x)tag2,$$ where $n$ is no longer under the limit? That is, what is the behaviour of the families of solutions to $(2)$ as $n$ increases?

(Cross-posted on MathSE but received no input.)

calculus and analysis – It would be possible to visualize the value of the integral together with the graph (in the same Manipulate command)

f(x_, y_) := x – 3 y Manipulate(RegionPlot( 0 <= x <= a && 0 <= y <= b (1 – (x/a)), {x, 0, a + 2}, {y, 0, b + 2}), {a, 1, 20, 1}, {b, 1, 20, 1})

Hi, I would like to show with manipulate, apart from the domine (which is what I have done as it can be seen), also the integral of the function with manipulate, changing the values a and b, if it is posible. I think it is easy but I am a beginner. Thank you

Rouches Theorem in Complex Analysis on the relation of the number of zeros and poles of meromorphic functions in a region

This question is from my son referenced in my earlier question, Need advice or assistance for son who is in prison. His interest is scattering theory . He asked me to post this question:

Hello and thanks to everyone for help finding papers thus far. I am currently looking for some further information and applications of Rouches Theorem in Complex Analysis on the relation of the number of zeros and poles of meromorphic functions in a region. I have the basic statement, but am looking for some more advanced or peripheral results, reformolulations, extensions, etc. Any other theorems with conditions for the relation of the poles and zeros of two functions in a region would also be helpful.
To be very specific, if f=g+h, with all functions meromorphic in the plane, I’m looking for conditions on f, g, h, so that f and g have the same number of poles and zeros in a region. The form of the particular functions I’m dealing with are generally highly oscillatory, nonlinear Fourier transforms of smooth, compactly supported functions where the nonlinearity can cause poles, but sometimes their real and imaginary parts can be controlled well, so conditions relating their arguments, or real/im parts might be useful. Thanks.

real analysis – Lipschitz hypersurface

I asked this already on Math SE. Maybe this definition is not quite common, but I’m asking myself what a Lipschitz hypersurface is. Intuitively this is a hypersurface which can locally be parametrized by Lipschitz functions. I hope this intuition is correct.
Nevertheless I didn’t find any precise definition. Does anybody know an exact definition of what a Lipschitz hyersurface is ?

real analysis – Attempt at portion of problem presented earlier

I was looking at this problem posted earlier Let $I$ be a generalized rectangle and let $f: I to mathbb{R}$. Show that $$lim_{ptoinfty}left(int_I|f|^pright)^{1/p} = max|f|$$

and struggling to prove LHS $leq$ RHS

Since the integral of a continuous function on a generalized rectangle is integrable, the integral is well defined. Take an archimedean sequence of partitions $textbf{P}_k$ of $f$. Since $|f|^p$ is integrable $int_{textbf{I}}|f|^p=limlimits_{k to infty}U(|f|^p,textbf{P}_k)$. $|f|$ is bounded above by $M=textbf{max}|f|$. so for each rectangle $textbf{J} in textbf{P}_k$ the largest value of $f$ say $mid M_textbf{J} mid leq M$. Can I say largest of $|f|^p=|M_{textbf{J}}|^p$?

Can someone help me with this. I trying the following
limlimits_{p to infty} bigg{(}int_{textbf{I}}|f|^pbigg{)}^{frac{1}{p}}& =limlimits_{p to infty}(limlimits_{k to infty}U(|f|^p,textbf{P}_k))^{frac{1}{p}} \
& =limlimits_{p to infty}(limlimits_{k to infty}sumlimits_{textbf{J} text{in} textbf{P}_k}|M_textbf{J}|^p text{Vol} textbf{J})^frac{1}{p} \
&leq limlimits_{p to infty}(limlimits_{k to infty}sumlimits_{textbf{J} text{in} textbf{P}_k}M^p text{Vol} textbf{J})^frac{1}{p}\
& = M limlimits_{p to infty}(limlimits_{k to infty}(text{vol} textbf{I})^{frac{1}{p}}) \
& = M\
&= text{max}|f| \