# Functional Analysis

Is it possible to form epoxide through this mechanism?

Seeing this mechanism from a waste paper in classroom. It seems like the final product is epoxide. Is this possible? I'm not sure what "E" means, but it is likely to be electric field.

Wu Yi Hsuan · National Central University

Mahdi, Amir and Zakaria
Thank for your help! Really appreciate.

Is it appropriate to give the partial answer to an open problem in the literature?

Is solving an open problem in mathematics with some extra conditions, that do not make it trivial, considered appropriate?

Ahmed Charifi · Université Ibn Tofail

To react, let me say two words. When the question of the problem is clear and that the strategy put in heuristic terms is relevant and fails, Intermediate steps are an orientation mark to the solution the problem. They increase the ability to find the solution posed or the inatendue solution of another problem remained open long enough. In this sense we can say that they are important.

Does any have experience in modeling strategy of relaxation?

Given a PDE model with some constraints restricting the range of values of model quantities, how does one apply the idea of relaxation to such a model? Any idea or references on relaxation idea of modelling, will be appreciated.

Joao A. N. Filipe · University of Cambridge

Following on your reply re ODEs of 19 days ago. I focus on a very simple form of relaxation by assuming that the logistic equation, du/dt = a*u*(k-u), is related to your problem because its solutions do not exceed k if it is initially below k. Here, a, k > 0 are parameters, k is often known as ‘carrying capacity’. If we add an extra term with time-varying per capita rate v(t), we can be recast the equation as a logistic model with time-varying carrying and a thus new long-term bound on u. Specifically: du/dt = a*u*(k-u)+v(t)*u, with v(t)>0. This can be rewritten as: du/dt = a*u*((k+v(t)/a)-u) = a*u(q(t)-u), where q(t)=k+v(t)/a is a time varying carrying capacity. Unlike the basic logistic, this ODE is not likely to have an exact solution (unless v(t) is special, such as a constant), but numerical solution should be straightforward. There are many options for the function v(t), depending on the specific application. Here is an article exploring the case where q(t) also obeys a logistic equation; in this case, there is a form of relaxation if q increases with time.

P.S Meyer and JH Ausubel, Carrying Capacity: A Model with Logistically Varying Limits, Technological Forecasting and Social Change 61(3):209-214, 1999.

This specific choice of v(t) incorporates a time scale parameter characterising the pace of the relaxation, which may be relevant. This is just a concrete example to illustrate a way of thinking about modelling relaxation; many other basic models and modifications thereof to incorporate relaxation would be possible.

To what extent can we say that diversity is her source of performance?

Based on measurement tools, can we measure performance? If it contains some references to the HR function, the analysis would be done on qualitative rather than quantitative

Jaharkanti Dattagupta · Novel Group of Institutes

As I understand, the question is regarding the extent to which " diversity is the source of performance". Diversity may be multidisciplinary experience of the performer, which may certainly help in situation analysis and problem solving. Now with reference to HR functions, the performance measurement is both qualitative and quantitative. While rationality of decisions, human responses to situations,behavioral aspects etc., may not be exactly quantifiable, actual job performances are measurable against set targets in organizations. Multidisciplinary and cross-functional skills are certainly added advantage for managerial performance.

What is the real part of Li_2(i/(2i+x))?
Li_2 is the dilogarithm function of complex argument.
I tried everything I could by using Sec. 5 of Lewin's book, but failed to find an answer.
Mykola Shpot · National Academy of Sciences of Ukraine

Thank you, Ailier,
although I always use Mahematica in my work, I found Maple preferable in dealing with functions of complex variables. Unfortunately, I did not find an answer in both of them.
Best,
Mykola

What is the best introductory book on Functional Analysis?
I'm looking for a book that fits an engineering graduate student.
David E. Stewart · University of Iowa

Peter Lax's book "Functional Analysis", but it is more for mathematics students.

Is anyone familiar with compact-open topology?

How can I verify that the space of all continuous mapping of the interval I into the Tychonoff cube I  with the compact-open topology is not normal.Thanks.

Thanks Klaas.

Does this approximation preserve convexity?
That is, if $f$ is convex, is its approximation convex?

If yes, it can be used to solve in positive a question from the monograph of Deville, Godefroy and Zizler "Smoothness and Renormings in Banach Spaces", if an arbitrary norm on Hilbert space can be approximated by C^2 norm with arbitrary precision. (I can specify the page, but not right now, since I do not have the book near by.)
Rabha Ibrahim · University of Malaya

No, because it is partial sum is not convex in general.

Do you have questions on the discussion announced on 04/27/14 related to Lyapunov's Functions and Concept of Stability?
The questions can be divided into three groups. The first group consists of the questions, to which the author has the answers directly resulting from the paper. The second group is composed of the questions, for which he has only conjectures or guesstimates. The third group represents the questions, the answers to which the author has no ideas about. The questions that interest me particularly as the author are as follows:
1. What is the mathematical nature (algebraic, geometrical, topological, etc.) of Lyapunov functions? What physical interpretations can be given to them?
2. Are there any direct or indirect relations between Lyapunov functions, first integrals and the right-hand sides of systems of differential equations? If yes, then what kinds they are?
3. How to approach a nonlinear non-autonomous system of the most general form by means of the second Lyapunov method? What and why do we need in the very beginning to know and how to get on with the system from this initial point further using the general procedure of utilization of Lyapunov functions? Is the procedure workable enough to crack the concrete practical problems of stability despite the presence of general nonlinearity, non-autonomousness, structural and coefficient uncertainties?
4. What are advantages and disadvantages of the utilization of Lyapunov functions at the investigation of the stability of nonlinear non-autonomous systems in the light of the results of the paper?
5. What role if any does the Lyapunov concept of stability play for quantum-mechanical and biochemical physical processes? Can it be considered one of the fundamental principles of the creation, formation and existence of living and nonliving matter?
David E. Stewart · University of Iowa

Of course there is a way of constructing Lyapunov functions! Here is one way to do it:

Take x(t;x0) as the solution of dx/dt = f(x), x(0) = x0. Pick a small neighbourhood U of x* (the globally stable equilibrium) and set V(x0) = inf { t>0 | x(t;x0) in U }.

Did you want to construct a Lyapunov function in order to prove x* is globally stable, or find an analytical formula for V? Well then you're out of luck.

Can anyone help me with a problem with a polynomial and a matrix?
Consider the Polynomial x^2+x+1=0 coefficients from the set of real numbers. Let S be the set of all 2 by 2 matrices with integer entries. How many elements of S will satisfy the equation?
Wiwat Wanicharpichat · Naresuan University

The companion matrix C = [-1  -1;1  0]  and all of its similar matrices  P^{-1}CP satisfy the equation.

Can you suggest any material, book or paper on connection of Crossed products of C*-algebras and semigroup C*-algebras?
Can you suggest any material, book or paper on connection of Crossed products of C*-algebras and semigroup C*-algebras?
Vardan H. Tepoyan · National Academy of Sciences of Armenia

Thank you Praveen!

If H is infinite dimensional Hilbert space and A is dense subset in H., does there exist a proper subspace M of H in which A is dense in M?
If yes how we can prove it? If not, give an example in which A is dense in H but for any subspace M, A can not be dense in M.
Milad Karimi · Isfahan University of Technology

No, it's not true. Because  if A be dense in M, then CL(A)=M. By using assumption of question we have CL(A)=H.Since M is a proper subspace of H thus CL(M)=M subset H.Hence M=H and it's contraction.

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.
Carsten Trunk · Technische Universität Ilmenau

I have to support Geoff. Quite often (especially in Russian books or in papers from authors with an Russian background) a subspace is always understood to be CLOSED. And this is just by definition. Unfortunately, often this is not stated explicitely...

On the opposite, in the western literature (like in the US) a subspace generally is just a subset of a vector space which itself is again a vector space but which need not to be closed.

And yes, you are right, this is annoying.

Finding solutions to: det F(z) = 0 in Qp. Any thoughts?

Let Qbe the field of p-adic numbers where p≥2 is a prime number, which we equip with the p-adic valuation and let Zp be the ring of p-adic integers.

Consider the function F: Q\ Zp → Qp, z → F(z), where F(z) is an m×m square matrix whose entries are given by:

δij - aij(z) for i,j = 1, 2, ..., m,

where δij is the usual Kronecker symbols and the functions aij: Qp \ Zp → Qp, z → aij(z) are continuous, for i,j = 1, 2, ..., m.

QUESTION: Find all the solutions to the equation

det F(z) = 0

where det F(z) is the determinant of the square matrix F(z).

Toka Diagana · Howard University

Hi Rogier,

Thank you for your interest in this question. And thank you for these interesting questions.

Let me make the following clarifications:

1) This question originated from a spectral problem, which I recently studied. In fact, with a few collaborators, we computed the spectrum of linear operators of the form A = D + F where D is a diagonal operator and F is a finite rank linear operator in the non-archimedean free Banach space Ew = c0(Qp, w). Precisely, we have shown that the spectrum of A consists of the union of the so-called essential spectrum of D (difficult to compute but we did) and the eigenvalues of A. Further, we have shown that the eigenvalues of A are exactly the zeros of the equation det F(z) = 0 on Qp \ Zp

2) The function aij: Qp \ ZpQp has a specific expression. But I intentionally omitted here as it requires the introduction of lots of other concepts etc. (The specific expression of aij can be found in Example 7.3 of my recent paper: https://www.researchgate.net/publication/263163019_Spectral_Analysis_for_Finite_Rank_Perturbations_of_Diagonal_Operators_in_Non-Archimedean_Hilbert_Space.)

3) At the end of your comment; you've probably meant "since Qp is NOT algebraically closed".

• Harish Kumar Kotapally asked a question:
How to find the dimension of the range of an operator and its adjoint?

Proof for dim(R(T))=dim(R(T∗)) for a linear operator in a Hilbert space. T is the operator and T∗ is its adjoint.

I would like to know about the authenticity of the following line of proof of the above fact and get directed to a reference which uses provides this line of proof (if at all it is correct).

Here goes:

Proof: From rank nullity theorem dim(T)+dim(ker(T))=dim(V), where T:V→W. From the fact that ker(T) and dim(T∗) are orthogonal complements, and the fact that ker(T) is a subspace of V, dim(T∗)+dim(ker(T))=dim(V). Thus, dim(T∗)+dim(ker(T))=dim(T)+dim(ker(T)). dim(ker(T)) cancels out and we are left with, dim(T∗)=dim(T)...Q.E.D.

Where can I find the .pdf format of the following book?

Fixed Point Theory: An Introduction by Vasile I. Istratescu

Maria A. Dobritoiu · University of Petrosani

This book exists at our library of University of Petrosani, but not in electronically format and, therefore, you cannot download it. Moreover, this book can not borrow, because it is unique. Search in the Romanian State Universities libraries. Might to find in scanned format or in electronically format.

Can anyone supply me with an efficient protocol for invasion and migration assay?

Hello all,

I want to express miRNA candidates in PC cell line and I would like to know what is the best protocol for cell numbers and the miRNA conc.

Thanks for information.

Sharif U Ahmed · University of Toronto

check ibidi.com

Cpp: question about function pointers and member functions. Can anyone help?

I have a function (not member of a class) which looks as follows:
void functionA(..., double (*pt2Cdf)(const double&),....)

which works wonderful if pt2Cdf just points to normal functions. But I would like that functionA also accepts member functions (of potentially different classes and which use the object's data but all with const double& argument) as input so that I can do calls as follows:

functionA(...,ObjectofClassA.Function,....);
functionA(...,ObjectofClassB.Function,....);
functionA(...,NormalNonMemberFunction,....);

Any hints? Thanks a lot!

Daniel Burren · ProMaSta

Inspired by your suggestions, I came up with a solution, namely to work with a function template as follows:

template<class ClassWithMemberFunction>

void functionA(...,ClassWithMemberFunction& object,...)

{...do something by calling object.memberfunction()...}

This means I renounced on the function pointer which in my case is not really a loss.

How we can describe physical applications of Special Functions?

With the help of physical applications, we will develop more and more properties of these useful functions.

Jesus S. Dehesa · University of Granada

- A.I. Aptekarev, A. Martínez-Finkelshtein and J.S. Dehesa. Asymptotics of orthogonal polynomials entropy.  J. Comp. Appl. Math. 233 (2010) 1355-1365.

- A.I. Aptekarev, J.S. Dehesa, A. Martínez-Finkelshtein and R.J. Yáñez. Discrete entropies of orthogonal polynomials. Constructive Approximation 30 (2009) 93-119

- J.S. Dehesa, R.J. Yáñez, A.I. Aptekarev and V. Buyarov . Strong asymptotics of Laguerre polynomials and information entropies of 2D harmonic oscillator and 1D Coulomb potentials.  J. Mathematical Physics 39 (1998) 3050-3060.

- A.I. Aptekarev, V. Buyarov and J.S. Dehesa. Asymptotic behavior of Lp-norms and entropy for orthogonal polynomials . Russian Acad. of Sci. Sbornik Math. 185(8) (1994) 3-30; English trans- lation 82(2) (1995) 373-395.

- A. I. Aptekarev, J. S. Dehesa, P. Sánchez-Moreno and D. Tulyakov.  Asymptotics of Lp-norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states. Contemporary Mathematics 578 (2012) 19-29

• Let A & B be two square matrices such that A^2 is not equal to B^2, A is not equal to B but A^3=B^3 & A^2B=AB^2. What is determinant of A^2-B^2?
If A and B Are non-singular then we have determinant of (A^2-B^2)=-determinant of AB , Am I correct? If so, then the problem is If A and B are singular, how can we prove?
Prasanth G. Narasimha-Shenoi · Government College Chittur

@ Samuli  Yeah the facts are correct.  Sorry that I am not able to answer

Can anyone help regarding mutation identification and functional validation of mutation in disease?

With NGS technologies, we are able to find the mutations in thousands in many genes but after identification, where do we stand in functionally validating those mutations and associating them with the disease (if I consider a sporadic case with no family history of disease)? How reliable are the bioinformatics tools for functional predictions?

Peter Krawitz · Charité Universitätsmedizin Berlin

For many monogenic forms of e.g. nonsyndromic intellectual disability animal models may not be indicative. Thus most of the papers that report about the identification of new disease genes in this context are purely based on statistics. As a rule of thumb you currently need three or more unrelated cases with a ultra rare disease with rare, potentially pathogenic mutations in the same gene. If you have a patient one of the biggest challenges is finding another one. With that taks also www.gene-talk.de might help. It's an expert network for geneticists that enables them to discuss variants of unkown clinical significance and to identify further patients of the same kind. Check it out!

Is there any metric in which the set of real numbers R is compact?
I want to check those metrics in which the set $R$ of real numbers is compact! Note that if that metric is induced from a norm, then R with that metric is not compact.
Paul Bankston · Marquette University

Regarding the problem of identifying the continuous bijective images of the real line, there is the paper: Anatole Beck, Jonathan Lewin, Mirit Lewin, On compact one-one continuous images of the real line, Colloq. Math. 23 (1971), 251--256.  I don't have access to this paper, but I did read a review on the AMS Math Reviews site.  The main result is quite technical, and there does not seem to be a neat characterization. (One necessary condition for a continuum to be such an image is that it is a nested union of arcs--because R is, and a continuous bijection, when restricted to a compact subset, is a homeomorphism.  This eliminates the circle, but not the figure '8' curve.)

Can anyone help with decomposition of an integrable function?
Suppose f:(0,1) --> (0,1) is non-differentiable and Lebesgue integrable, then under what conditions, if any, is it possible to decompose f as,
f = f_0 + f_1
where:
i) the Lebesgue integral of f_0 = 0
ii) f_1 is differentiable

If possible and the decomposition is non-unique, does there exist an f_1 which is of minimal total variation?
Geoff Diestel · Texas A&M University System

You could also view f as 1-periodic and consider a smooth approximate identity, i.e.

f=lim_{t->inf} f*g_t

Then for any t>0,

f=f*g_t + (f-f*g_t) := f_1 + f_0

This way f_1 is as smooth as the function g generating the family {g_t}_t

Is the Laplace transform a projection onto the set of canonical solutions of linear homogeneous ODE?

The Laplace transform of a function is the inner product between this function and the canonical solution of the linear homogeneous ODE in the domain [0,\infty)

A.-J. Muñoz-Vazquez · Center for Research and Advanced Studies of the National Polytechnic Institute

Yes, one can. Decomposition of a function in Laplace transformable functions, it is just an idea, I need to see deeper.

• Which cloning kit is effective for cloning of large size DNA insert (>20 kb) extracted from hot spring water?
We are dealing with (gene expression/functional analysis) of metagenome isolated from hot spring water of gujarat, India. Can anyone suggest me cloning kit for cloning of metagenome (size >20 kb). We found various kits from literature survey like pGMET, Cosmid cloning kit (contains pWEB-TNC vector) etc. We suggest kit name which shows reproducible results with low cost, less time consuming experiment and efficient results.
Amitsinh Vijaysinh Mangrola · Shri Alpesh N Patel P.G Institute
Thanks Dinesh kumar and Zhaowei Wu for your valuable suggestion
Does a Hölder continuous function have a bounded fractional derivative?
Some nowhere differentiable functions are fractional differentiable and comply with the Hölder condition.
A.-J. Muñoz-Vazquez · Center for Research and Advanced Studies of the National Polytechnic Institute
Also, the Riemann-Liouville derivative can imply the Caputo differentiability since they are related in function of the initial conditions of the function and its derivatives.
Given the information in the attached below, is it possible to show that G^{y_{n}} is quasi-nonexpansive?
See the definitions of total quasi asymptotically nonexpansive and quasi-nonexpansive attached below.
Lawan Bulama Mohammed · Putra University, Malaysia
See the attached below, can I conclude that G^{y_{n}} is quasi nonexpansive, if yes, how?
What is the definition of supporting functional of the convex set C at point x0?
Compare its definition in wiki and in the book (Foundations of Mathematical Optimization Convex Analysis without Linearity) the note follows Proposition 6.1.17. - Is there a contradiction between both definitions?
Mohammad W. Alomari · Irbid Private University, Jordan
As Milen said it is not possible, actually it depends on convexity (f(x) > f(x0)) or concavity (f(x)< f(x0)) of f. Such line exists and uniqe iff f is differentiable.
Can anyone answer my question about norm of operators in a Hilbert space?
Let H be an infinite dimensional Hilbert space and M be a closed subspace of H. If \left\| Tx \right\| > \left\| x \right\| for some x \in H. Does there exist y \in M such that
\left\| Ty \right\| > \left\ |y \right\|?
Gandalf Lechner · University of Leipzig
Assuming T is a linear operator on H, the answer is no, in general: Take T as the direct sum of 0 (on M) and 2 (on the orthogonal complement of M), for example.