# Functional Analysis

Is every neighborhood of identity in an abelian topological group absorbing?

Let G be an abelian topological group and U be a neighborhood at identity. Do we have the identity that the union of nU in which n is a positive integer equals to whole of G? We have the same situation in a topological vector space. I am interested to know if it is true for topological groups, too.

Omid Zabeti · University of Sistan and Baluchestan

Dear Dona,

Is it possible to show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is related with each $x\in\partial B_{r}(0)$?

We know that an open ball $B_{r}(0)\subseteq R^{n}$ is a smooth domain. It follows that this is a Lipschitz domain. Is it possible to show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is related with each $x\in\partial B_{r}(0)$?

Christopher Jason Larsen · Worcester Polytechnic Institute

You could use identical functions, oriented correctly, so all Lipschitz constants are the same. You could also make the constant arbitrarily small by taking small arcs.

Is there any criterion for k-positivity of an operator acting on a C*-algebra?

Suppose that H is an n-dimensional Hilbert space. B(H) is the set of bounded operators on H forming a C*-algebra. Given a linear mapping L : B(H) -> B(H).

Choi's theorem helps us determine whether L is a completely positive mapping. Is there any similar criterion on the k-positivity of L?

Hong-Bin Chen · National Cheng Kung University

Thanks a lot!!

Can we have Heine-Borel property in topological groups?

We know that Heine-Borel property and related theorems for topological vector spaces. Do we have similar notions for abelian topological groups? For example when a topological group has this property or not? Can you introduce me a reference about that?

To James Peters

Is available  this Thesis:  X. Shi, Graev Metrics and Isometry Groups of Polish Ultrametric Spaces, Ph.D. thesis, University of North Texas, 2013:?.

Can someone help me with DNA polymerase activity analysis?
I've expressed Taq polymerase in E.coli. I have found a few reports for its purification also using sephadex columns. Next, I need to check its efficacy preferably by methods other than PCR. Can anybody suggest such functional tests?
Wang he · Beijing Genomics Institute

Hi!   Have you solved this problem? I have met the same problem , and I tried a  method according to a paper which used Picogreen to maesure the dsDNA. Unfortunately, the results could not be stable, so I want to know much about the method measuring radioactivity, such as the amount of DNA ,the reaction system. How many is the least amount  of radioactivity that can be detected. Waiting for your reply . Thank you !

Anyone familiar with the Arzela-Ascoli Theorem?

n the proof of the Arzela-Ascoli Theorem, it seems that only pointwise boundedness rather than uniform boundedness is needed. Is it right?

James F Peters · University of Manitoba

This is a good question.

A general form the Arzela-Ascoli theorem is given in

J. Conradie, J. Swart, A general duality result for precompact sets, Indag. Mathem, N.S., 1 (1990), no. 4, 409-416:

http://ac.els-cdn.com/001935779090009C/1-s2.0-001935779090009C-main.pdf?_tid=b7efb76a-7d41-11e4-8f86-00000aacb360&acdnat=1417868374_9514a8e75ac181a3976aaeb2ff31ee65

See Theorem 3.2 (with proof), page 413.

A basic introduction to the Arzela-Ascoli theorem is given in

F. Botelho, Topics of Functional Analysis, Calculus of Variations and Dualilty, Acad. Pubs., 1991:

See Section 1.7, starting on page 20 (Theorem 1.7.2).   The proof of Theorem 1.7.2 is starts on pare 21, extending to page 22.

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.
Vittorino Pata · Politecnico di Milano

Let C be a compact set of the same cardinality of R (say, C=[0,1]), with the metric d. Let f:R->C a bijection. Then define on R the metric D(x,y):=d(f(x,f(y)). This D should be compact.

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.

Which is the best way to characterize an oncogene?

Hi, now I´m working with a protein wich behaives as a oncogene ( after expression microarrays, functional analysis showed that promotes proliferation, growth and avoid apoptosis). I have some ideas about in vitro studies but, as we know,  two heads are better than one. Thanks

Wojciech Kalas · Polish Academy of Sciences

Stable overexpression and trying to grow tumour in mice is definitely the best method, but is costly and laborious.

Before, you can get some answer by introducing it to non-tumorigenic cell line and performing colony formation assay or trying to grow them to confluence, to find out if the cells losed the contact inhibition - common feature of cancer cells. You can try also compare expressing and non-expressing cell lines.It will be only circumstantial, but it can give you a hint if it behaves as oncogene and will ease decision to invest more time and money.

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?

Parasuraman Basker · California Department of Public Health

Dear Sir

How could I express my joy as you have given extra energies to explore further in mathematics. I have been initiated to learn mathematics from its alphabet onwards as I am the pure Biologist. But one thing I firmly confident that mathematics alone give ultimate solution like E= mc squared. In this regard please kindly suggest to read basic mathematics books.

sincerely Yours

Hi all, how can I convert scale axis to frequency in wavelet transform?

I wanna to produce time frequency representation after using wavelet  transform.

Reza Mehrnia · Payame Noor University

Hi, you should transform all values into frequency responses by spatial based software (such as Geo-soft , Surfer and Arc-Soft Families) according to Wavelet algorithm facilities. therefore it is automatically caused changing your scale bar into frequency measurements..

Can anyone suggest a method to solve the following constrained functional minimization problem?

When a functional of one function is linear in the derivative of the function, I=int(f(y) + A(y)y'), the Euler-Lagrange equation leads to an algebraic equation as the terms involving y' cancel each other. But if the functional depends on more than one function this doesn´t happen:
I=int(f(y,z) + A(y,z)y' + B(y,z)z'), the E-L equations are:

df/dy + dB/dy*z' - dA/dz*z'=0
df/dz + dA/dz*y' - dB/dy*y'=0

These are first-order differential equations. Can this problem be solved, considering that there are at least two boundary conditions (imposed or natural)?

Actually my problem comes from a functional I=int(f(y,z)), subject to a constraint linear in the derivatives, so the E-L equation of the augmented functional using the linear constraint lead to first order differential equation. BUT, if I re-write the linear constraint:

A(y,z)y' + B(y,z)z' + C=0

as

C/(A(y,z)y' + B(y,z)z') + 1=0

and I construct the augmented functional with this constraint, the augmented functional is no longer linear in the derivatives and the E-L equation is a second order differential equation.
Does it make sense? Can the same problem lead to differential equations of different orders? What happens then with the boundary conditions?

Ezequiel Soule · Universidad Nacional de Mar del Plata

Ha, it was much easier (and more obvious) than I thought... Thank you very much Vladimir!

The relation between the orthogonal condition and the angle 45 degrees in convex cone?
We discover that the orthogonal condition is related with the angle 45 degrees for convex cone. And we also cite three examples about the application of the angle 45 degrees.the three examples are:
1.real symmetric orthogonal matrix
2.bounded norms in vector spaces related with the function xsin(1/x),the function is used to study the angle 45 degrees of convex cones and conversely we can use the property of convex cone to study the positive and negative to the continuous function
3.by studying the positive root of the quadratic function we can verify the relation between the orthogonal condition and the angle 45 degrees,conversely we can use this example to study the quadratic function associated with convex cone.
( page1-13)
James F Peters · University of Manitoba

This is a good question.

Convexity and orthgonality are considered in ch. 8, starting on page 169 in

D. Drusvyatskiy, Slope and Geometry in Variational Mathematics, Ph.D. thesis, Cornell University, 2013:

http://www.math.washington.edu/~ddrusv/Thesis_Drusvyatskiy.pdf

Vectors orthogonal to a cone (called facet nomals) are considered, starting on page 18, in

K. Garaschuk, Linear methods for rational triangle decompositions, Ph.D. thesis, University of Victoria, 2014:

https://dspace.library.uvic.ca:8443/bitstream/handle/1828/5665/Garaschuk_Kseniya_PhD_2014.pdf?sequence=4&isAllowed=y

Convex cones are considered in 17 places, starting on page xv, in

J. Gandini, Simple linear compactifications of spherical homogendeous spaces, Ph.D. thesis, Sapienza University, Rome, 2010:

http://www1.mat.uniroma1.it/ricerca/dottorato/TESI/ARCHIVIO/gandinijacopo.pdf

Can someone share examples of topological vector space?
Examples with explanation will be much better
George Stoica · University of New Brunswick

Dear Sheba,

Yet another example, not mentioned above: vector lattices (or Riesz spaces), with the topology therein generated by an order relation. Very well explained is the following: http://thesis.library.caltech.edu/3955/1/Chow_tky_1969.pdf

Sincerely,

George

What are eigenvectors of Markov operator?
.
George Stoica · University of New Brunswick

Dear Cezary,

It goes back to the Perron-Frobenius theorem, with various developments, please see the references at: http://en.wikipedia.org/wiki/Perron–Frobenius_theorem

Sincerely,

George

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

Learn a lot!  Thank you very much.

Does anyone determine the notion of a bounded set in a topological field?

Bounded sets are defined on general topological vector spaces, topological modules, topological rings and topological groups. But, I could not find a suitable definition of a bounded set in a topological field.

Any nice material regarding the question will be appreciated.

Liaqat Khan · King Abdulaziz University

In fact, one can also define a bounded subset of any topological space. A subset A of a topological space X is called bounded (or functionally bounded) if f(A) is a bounded subset of R for ever f∈C(X).

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

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 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.

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.)

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