# Functional Analysis

4
What are the current application of Riemann Zeta Function in analysis of elementary particles?

I want to know the applications of Riemann zeta function in analysis of elementary particles and other high energy physics phenomenon.

PLEASE CAN YOU PROVIDE ME A BASIC INTRODUCTION TO THE E-INFINITY THEORY AND HOW COMPARATIVELY IT IS BETTER THAN THE RIEMANN ZETA FUNCTION  FOR ANALYZING CASIMIR ENERGY

39
Is there a book in English where one can find characterizations of zero-derivative (stationary) points ?

In non-English literature two  such characterizations for C2 functions of the single variable can be found in the text Neralic, Sego: Matematika (second edition), Element, Zagreb, 2013 (ISBN 978-953-197-644-2) but they do not seem to be widely known. They appear to be important in analysis, calculus, optimization and other areas.  Where can one find such results in functional analysis ?

Hello Miodrag,

It is now clear to me, after our discussions, that the "quadratic envelope property" and "Morse property" are two essentially different notions. The former characterizes an  interior  point x* of a  compact convex subset in the domain of a C1 function f,  with Lipschitz derivative (!), where f'(x*) = 0. The latter does not do that, even for C^2 functions. Agree ?

Note that the assumption on Lipschitz derivative in the quadratic envelope property  cannot be omitted (see my JOGO paper "Characterizing ..." for a counter example).

Cheers, Sanjo

8
For a subset X of RxR with the property that every continuous function f:A-->R attains its maximum in R. Is X compact?

For a subset X of RxR with the property that every continuous function f:A-->R attains its maximum in R. Is X compact? What if f is bounded but does not attain maximum in X.

Topological spaces on which every continuous function is bounded are called pseudo-compact.

By Theorem 30 in

1. E. Hewitt, Rings of real-valued continuous functions. I.
Trans. Am. Math. Soc. 64, 45-99 (1948). (Theorem 30)

a normal pseudo-compact space is compact. Since metric spaces are normal it follows  that a pseudo-compact metric space is compact.

The paper  [1] contains also other caharcaterizations of pseudo-compact spaces.

2. R. M. Stephenson, Pseudocompact spaces, Trans. Am. Math. Soc. 134, 437-448 (1968).

and the book

3. L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, New York, 1960.

8
A converse of the implicit function theorem?

I have a function in terms of a vector x, and a parameter a, say F(x,a). I also know that there exists a smooth unique parametrized curve x(a) such that F(x(a),a) = 0 for all a. I want to know if the Jacobian matrix of F differentiated with respect to x is nonsingular when evaluated on the curve x(a).

In other words, if we have a dynamical system can we say that local solvability of an equilibrium guarantees that stability cannot change assuming a Hopf bifurcation does not take place?

Consider simple form  of the implicit function theorem:

Suppose  that  (i):  F maps a  nbg  V in   R^2 into R , F(0,0)=0 and that   F’_x(x,a)  different from 0  in  V .

Then   (ii)  there is b>0    such that  the set  of points  F(x,a)=  0      in  Q= [-b,b]^2,   is graph   of a  function x=x(a),   -b <=    a <=b.

For example   if  F(x,a)= x-  a^3, then the set of points F(x,a)= 0    is graph  G1 of function x(a)=a^3.

The same graph is also  solution   of equation  F1(x,a)= (x- a^3)5=0.

In this example, the Jacobian matrix of F with respect to x is non singular on G1,    but  the Jacobian matrix of F1  with respect to x is   singular  on G1.

Thus the implicit function theorem  states : (i) implies (ii) .

What is   a converse of the implicit function theorem?

As I understand    (ii) implies (i) ?

Suppose (ii):

In other words  we suppose  that  we have smooth graph  G  given by  a function  x=x(a)   which maps [-b,b]  into itself.

Can we find a function F(x,a)  such that  a form of (i) holds:

the Jacobian matrix of F with respect to x is non singular on   Q= [-b,b]^2  and

F( x(a),a )= 0  for  in [-b,b].

Take  F(x,a)=  x-x(a).  Then   F’_x (x,a)=1.

Is this related to the question.

To Jason Bramburger  and Giovanni Dore:

Do you have something different in mind?

1
How to prove that <x, A*Ax> > 0 ?

Let H1, H2 be Hilbert spaces and

A : H1 → H2 be a compact operator with the singular

system (λn, vn, un).

How to prove that <x, A*Ax> > 0  where <...,...> denotes the inner product ?

By the definition of the adjoint operator, <x, A*Ax> = <Ax, Ax> = ||Ax||2. Thus, <x, A*Ax> > 0 provided x does not belong to the kernel of A and it is zero otherwise.

2
Is it correct to say that $div(A^t \nabla y)\in L^2(\Omega)$ provided $div(A \nabla y)\in L^2(\Omega)$,
Here
$A$ is bounded nonsymmetric matrix and $y$ belongs to the Sobolev space $H^1_0(\Omega)$?

Thanks a lot, Gianni. Your arguments sound perfect!

• Yuriy Borisovich Zelins’kyi asked a question:
Open
Anyone familiar with m-convex compacts in R n?

Definition. A compact K ⊂ R n is called to be m-convex (m < n) if for each x ∈ Rn \ K there exists an m-plane T(x) such that x ∈ T(x) and T(x) ∩ K = ∅.

Problem. Let K ⊂ R n be a compact. Suppose that for each hyperplane T ⊂ R n the intersection T ∩K is (m−1)-convex. Find a condition on K which together with the above one would be sufficient for K to be m-convex.

http://www.usc.es/dmle/pdf/EXTRACTAMATHEMATICAE_2005_20_01_05.pdf

3
How to use the Moser iteration technique to improve the regularity of very weak solution?

The very weak solution like

-\Delta u=f in \Omega, u=0 on \partial\Omega,

when f \in L^p with 1\le p <2

\int_\Omega u(-\Delta ) \xi dx=\int_\Omega f \xi dx,   \xi\in C^{1.1}_0(\Omega),

so in this sense, how can choose the test functions to improve the regularity?

Give a look at this paper

http://www.ams.org/mathscinet/search/publdoc.html?arg3=1992&co4=AND&co5=AND&co6=AND&co7=AND&dr=pubyear&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=di%20fazio&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq

and let me know if you get trouble.

Best,

Giuseppe

15
Does the collection of all self adjoint operators have any property?
There are unitary, self adjoint and normal operators in operator theory. Do each collection posses nice property?

For a self-adjoint operator, the notion of a positive self-adjoint operator has sense: <Ah,h> >=0 for all h in the Hilbert space H. The following result is very useful in applications: for any positive self-adjoint operator A, there exists a unique positive square root of A. Examples of positive self-adjoint operators are: TT* and T*T, where T is an arbitrary linear bounded operator acting on a Hilbert space. In particular, there exist the positive square roots of these two operators.

4
Is there such a norm on any totally disconnected local field?

how can i prove the following statement? In other words, how can i prove existence of the following norm?

Let K be any totally disconnected local field. Then there is an integer q=pr, where p is a fixed prime element of K and r is a positive integer, and a norm ∣⋅∣ on K such that for all x∈K we have ∣x∣≥0 and for each x∈K other than 0, we get ∣x∣=qk for some integer k.

I second the reference suggested by Thong Nguyen Quang Do, it is better than the one I initially suggested.

10
Can a sequence of local diffeomorphisms with attracting periodic points have a limit with only expanding periodic points?

We are in a compact C1 manifold. Consider a sequence (gn) of local C1 diffeomorphisms with a periodic point qn which is attracting, each gcan have only attracting or expanding periodic points and gn converges in Ctopology to f, where f is also a local C1 diffeomorphism but all its periodic points are expanding. Can the existence of that sequence lead a contradiction?

Remark:

A periodic point p of h is attracting (expanding) if all eigenvalues of Dhtau (p) (p) have moduli less (greater) than 1.

In my question there exists a neighbourhood  C1 U of f such that all g in U can't be approximated by local diffeomorphisms  with a saddle type periodic point.

13
How can one characterize the boundary of a convex set?

I am working on a part of a paper related to topological properties of boundary points. It is important for me to realize the topological and algebraic behavior the boundary points of the convex sets. I would be grateful if someone could help me around this issues by giving some ideas or references related to it.

The general question provided in following;

Let B be a closed set in n−dimensional Euclidean space. What other properties B should have in order to be guaranteed that there exist the closed convex set A such that ∂A=B. How about infinite dimensional spaces?

This is a good question.

A good place to start in answering this question is

S. Alexander, M. Ghomi, The convex hull property of non compact hyper surfaces with positive curvature:

http://www.math.uiuc.edu/~sba/nchp.pdf

Theorem 1.1,  page 1, is helpful.   Let M be a metrically complete, positively curved immersed hyper surface with compact boundary $\partial M$ and let

\begin{allgn*}

f: M &\longrightarrow \mathbb{R}^{n+1}\\

C  &= conv( f (\partial M ) ) \mbox{convex hull of image of boundary of M}\\

f(int M) \cap C &= \emptyset

\end{align*}

which says that the image of the interior of M lies completely outside the convex hull of the image of the boundary.    A neat result.     The proof of Theorem is given in Section 3, starting on page 3.   An example of an immersed surface is given in Fig. 1 (see the attached image).

More to the point, consider

U. Eckhardt, W. Scherl, Z. Yu, Representations of plane curves by means of descriptors in Hough space:

http://www.math.uni-hamburg.de/home/eckhardt/HOUGH.pdf

This thesis has a crisp way to denote the boundary of any set M in the plane ( bd M ).  See Chapter 3, starting on page 6, on the classification of descriptors in terms of a continuous, piecewise smooth curve K in the plane.   Then see Chapter 8, starting on page 17, on convex boundary segments.    In ch. 8, starting on page 20, a curve K is reconstructed up to boundary tangents.   This is an important chapter, since it considers the boundery of a convex set and boundary of the complement of a convex set.

2
Which empirical measures, associated to infinite dimensional stochastic processes, satisfy the moderate deviation principle?

In 1998 we proved that the Antoniadis-Carmona processes satisfy the above requirements, in connection with the tunneling effect. Am interested in the current situation.

Alternatively, the infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion coefficient also satisfy the requirements, see: http://arxiv.org/pdf/1503.05322.pdf

3
Are there characterization of harmonic gradient mapping from the unit ball onto itself in 3-space?

In communication between V. Zorich and the author, the
question was asked to find examples  of harmonic gradient mapping from the unit ball onto itself in 3-space. For example, if $u=x^2 +y^2 - 2 z^2$, then $f= \nabla u=(2x,2y,-4z)$ is injective harmonic gradient mapping from
$\mathbb{B}^3$ onto the ellipsoid.

If $u$ is real-valued function such that $f= \nabla u=(x,y,z)$, then $u= x^2/2 +y^2/2 +z^2/2 +c$.

In particular, $Id$ is not harmonic gradient mapping.

In complex plane, if $u$ is real-valued harmonic function, then $u_z=\frac{1}{2}(u'_x -u'_y)$ is analytic function and therefore
$\nabla u=\overline{F}$, where $F= 2 u_z$ is analytic function.

Znam da je pretpostavka jaka.

• Qingping Zeng asked a question:
Open
Any advice on isolated points of the approximate point spectrum of a bounded operator?

Let $X$ be a complex Banach space and $T$ be a bounded operator acting on $X$. Let $\sigma(T)$ and $\sigma_{ap}(T)$ denote the spectrum and approximate point spectrum of $T$, respectively. Let $\lambda \in \sigma(T)$. It is classical that, with the aid of the spectral projection, $\lambda$ is isolated in $\sigma(T)$ if and only if $T - \lambda$ can be decomposed as a direct sum of a quasinilpotent operator and a invertible one. Now my nature question is as follows: Let $\lambda \in \sigma_{ap}(T)$. Is it true that $\lambda$ is isolated in $\sigma_{ap}(T)$ if and only if $T - \lambda$ can be decomposed as a direct sum of a quasinilpotent operator and a bounded belowness one? Here we say that an operator is bounded below if it is injective and its range is closed. It is also nature to find the answer for other spectra (eg. left spectrum, surjective spectrum, right spectrum, essential spectrum,...). Thank you!

6
Can anyone help with a membership function question?

Can I use different membership functions in Fuzzification layer for ANFIS type model? e.g. U1 = Bell-shape and U2 = Triangular, so that the output of this layer will be wi = U1 x U2. In most of the papers they used the same type of membership function.

Yes you can do that, but there is not a good reason to do that. On the other side, you can do anything using C or C++.

1
Is there any appropriate version of the Rad\'{o}-Kneser-Choquet theorem (RKC-Theorem) in space?

Let $\gamma$ be a closed Jordan curve and $f_0 : S^1 \overset{\text{onto}}{\longrightarrow} \gamma$. The basic question that they address in this paper is
under which conditions on $f_0$ we have that $F=P[f_0]$ is a
homeomorphism of $\mathbb{B}$ onto $D$, where $D$ denotes the
bounded open, simply connected set for which $\partial D = \gamma$. The fundamental benchmark for this issue is a classical
theorem, first conjectured by T. Rad\'{o} in 1926 , which was
proved immediately after by H. Kneser [12], and subsequently
rediscovered, with a different proof, by G. Choquet. Let us recall the result.

Theorem 1.1 (T. Rad\'{o} H. Kneser G. Choquet ) If $D$ is convex, then $F$ is a
homeomorphism of $\mathbb{B}$
onto $D$.

Laugesen (see Duren's book}, p. 54-56) constructed a homeomorphisam
of the the unit sphere in $\mathbb{R}^3$ onto itself, whose
Poisson extension to a vector-valued harmonic function fails to
be univalent in the ball.

None that I am aware of.

3
Minkowski type inequality in Banach algebras
Under which circumstances it is true that ||(A+B)ⁿ||¹/ⁿ ≤ ||Aⁿ||¹/ⁿ + ||Bⁿ||¹/ⁿ for elements A and B in a Banach algebra and a natural number n?

How can you even talk about the relation $\leq$ in a Banach algebra? Unless you mention it as an ordered Banach algebra or Banach lattice algebra.

4
Does a Hölder continuous function have a bounded fractional derivative?
Some nowhere differentiable functions are fractional differentiable and comply with the Hölder condition.

By example, boundedness of the Caputo derivative of f at every interval, implies Hölder continuity of f in the same order

1
Can someone help with the interpolation between the Bloch and the Moebius invariant H^1 space?

The space $H'$ consists of functions $f$ analytic in the unit disc such that $f'\in H^1$. The Bloch $B$ space is defined by the requirement

$$\sup (1-|z|^2)|f'(z)| <\infty,$$

Endowed with the obvious semi-norms, both $H'$ and $B$ are M-invariant in the strong sense: $\|f\circ \m\|=\|f\|,$ for every Moebius transformation $m$ of the disc.

The question is: What is the real or complex interpolation space $(H',B)_...$ ?

7
Can anyone prove how Legendre wavelet forms an orthonormal basis for L^2(R)?

http://planetmath.org/orthogonalityoflegendrepolynomials
http://physicspages.com/2011/03/18/legendre-polynomials-orthogonality/

18
If A is a subspace of a normed space X and \{x_n\} is a sequence in A such that \{x_n\} converges to z, does z \in A?
Please note that A is a subspace.

No, Carl L. Devito is right. For any z in B there exists a sequence in E which converges to z. This is due to the properties of a completion: If B is a completion of E, then, by definition of a completion, E is dense in B which is precisely the property mentioned by Carl, i.e. for any z in B there exists a sequence in E which converges to z.

But this is rather a property connected with completions. It is not so much connected with closed subsets...

2
Any suggestions on the classification of Moebius invariant Besov spaces?

The Besov space $B^{p,q}_s$ of analytic functions on the unit disc D consists of those f for which

$$\int_0^1 M_p^q(r,D^n f)(1-r)^{(n-s)q-1} dr < \infty,\quad 0<p,q\le\infty$$

where $D^nf$ is the $n$-th derivative of $f$, $s$ is an arbitrary real number,and $n>s$. The definition is independent of a particular choice of $n$. It is known that the space $B^{p,p}_{1/p}$ is M-invariant for all $p$.

What is about the general case?

This is a good question.   I also agree with @Romesh Kumar that this question is interesting.

Moebius invariant Besov spaces are considered in terms of boundary behaviour in

K.T. Hahn, E.H. Youssft, Tangential boundary behaviour of M-harmonic Besov functions in the unit ball,  J. of Math. Anal. and Applications 175, 1993,  206-221:

The space of holomorphic Besov functions in the diagonal Besov space are shown to be Moebius invariant subsets of the Bloch space (see remark p. 208).   Also see Corollary 1.5, page 209: the elements of the Besov space  extend continuously to the closed unit ball.    See, also,

F. Beatrous, J. Burbea, Holomorphic  Soblev spaces on the ball, Diss. Math., 276, 1989

and

K.T. Hahn, E.H. Youssfi, Moebius invariant Besov p-spaces and Hankel operators in the Bergman space on the ball in $\mathcal{C}^n$, Complex Variables 17, 1991, 89-104.

Another good place to look for answer is

Journal of Computational Analysis and Applications 10, 2008, no. 1:

http://www.eudoxuspress.com/images/JOCAAA08-VOL10.pdf

15
A power series summation a_n z^n such that a_n tends 0 as n goes infinity. How can we show it does not have pole on unit circle?

A power series summation a_n zn such that a_n tends 0 as n goes infinity. How can we show it does not have pole on unit circle?

A related nice property holds. Assume that the power series is convergent in the whole closed unit disc, and denote by "f" its sum. If the coefficients are non negative, then z_0=1 is a maximum point for  |f| on the closed unit disc.

2
Does anybody know any continuous function in time domain whose frequency response looks like the attached figure?

I want a function in time domain whose frequency response should be of the form shown in the following figure.

Yes, interpolate the polynomial and do invers Fourier transform.

Good luck

18
What are the differences between "Rieman integrable" function and "Lebesgue integrable" function?
.

Also you can compare with the following. The main difference between the Lebesgue and Riemann integrals is that the Lebesgue method takes into account the values of the function, subdividing its range instead of just subdividing the interval on which the function is defined. This fact makes a difference when the function has big oscillations or discontinuities. However, the Lebesgue method needs to compute the measure of sets that are not intervals.

Reference: http://demonstrations.wolfram.com/RiemannVersusLebesgue/

1
Space of rapidly decreasing test function is not invariant with respect to fractional integral. Is there any example to understand this?

Lizorkin space is a subspace of a rapidly decreasing function. The space of a rapidly decreasing test function is not invariant with respect to fractional integral. Where as lizorkin space is invariant w.r.t. fractional integral and differentiation.

I am not getting an example of this failure so please suggest few examples so that I can get this clearly.

Look at the definition of the fractional derivative , and you will see that it will have a singularity and power behaviour for Riemann-Liouville derivative. Lizorkin space implies a special definition of derivative related to multiplication by (1+x^2)^r for Fourier image which is multiplicator in space of r.d.t.f.

3
Convex bodies with equi-measurable radial functions. Does regularity of bodies imply the bodies are rotations of each other?

If K and L are convex bodies whose radial functions have the same distribution function, i.e. equi-measurable, what can be said? In particular, does this mean that the two radial functions(of the sphere) are equal up to composition with some measure-preserving transformation? If so, does the convexity of the two bodies imply any regularity of this measure-preserving transformation of the sphere? If possible, how much regularity must the bodies possess to force this transformation to be a rotation or orthonormal linear transformation of the sphere?

true. as an engineer  who learned this the hard way i say it with some regret.

2
Is functional alpha diversity equal to functional richness?

Pool et al (2014) quantify alpha functional diversity as the volume of the convex hull filled by the fish species of each community in two-dimensional functional space using the values from the first two functional axes.

But I wonder taxonomic alpha diversity is simply the species richness, so the alpha functional diversity can be functional richness...

• Source
##### Article: Species contribute differently to the taxonomic, functional, and phylogenetic alpha and beta diversity of freshwater fish communities
[Hide abstract]
ABSTRACT: AimWe examined the current biogeographical patterns of native fish communities throughout France, using a multifaceted taxonomic, functional and phylogenetic diversity approach. We then identified the contribution of individual species to each facet of watershed's native fish diversity.LocationContinental France.Methods The taxonomic, functional and phylogenetic diversity of the fish communities were quantified at the watershed-scale (i.e. alpha diversity approach), and congruencies between diversity facets were assessed. Variation between watersheds was then quantified (i.e. beta diversity approach) using Jaccard's dissimilarity index for all three facets of diversity, and congruencies were assessed. We subsequently determined the relationship between alpha and beta diversity for each diversity facet. Lastly, the mean relative contribution of each species to watershed's alpha taxonomic, functional and phylogenetic diversity was quantified. The conservation status of each species was considered to determine if threatened and endangered species contributed more significantly to watershed alpha diversity than common species.ResultsAcross all watersheds, taxonomic, functional and phylogenetic diversity facets were found to be highly congruent using both the alpha and beta diversity approaches. In contrast, the relationship between the watersheds' alpha and beta diversity was primarily negative; watersheds with decreased beta diversity tended to have increased alpha diversity for all three facets. Individual species also rarely contributed prominently to more than one diversity facet, with conservation status insignificantly influencing species relative contributions.Main conclusionsWe found that native fish diversity ‘hotspots’ exist in France; exhibited in our results by areas of high, overlapping taxonomic, functional and phylogenetic diversity. Consequently, conservation planning approaches supported by species-based metrics may concurrently target areas of increased ecological and evolutionary importance at the watershed-scale. Interestingly, a diverse mosaic of species accounted for the different facets of diversity, suggesting that future reductions in species richness could have differential effects on each watershed's diversity facets.
Diversity and Distributions 07/2014; 20(11). DOI:10.1111/ddi.12231