# Functional Analysis

2
What are some Implicit Function type Theorems which have weaker assumptions?

I can think of many situations in which the assumptions of the Implicit Function Theorem do not hold but most of the results of the theorem can be recovered. The example of f(x,a) = a + x^3 comes to mind. At (x,a) = (0,0) the derivative is zero and therefore the implicit function theorem cannot be used. On the other hand, there is a unique continuous solution given by x = (-a)^1/3 for all real a.

I am looking for a review of some literature in which the authors have either developed a theory or have worked with a specific cases in which they side-step the Implicit Function Theorem to recover similar results.

My particular situation is as follows:

I have a nonlinear operator between a Banach spaces whose Frechet derivative is injective but is not bounded below and therefore it cannot have a bounded inverse. I am looking to show that a zero of the function can be continued for a small parameter value but because the derivative is not invertible at this specific point I cannot use the Implicit Function Theorem.

I am familiar with Nash-Moser Theory although through quite a bit of work I am convinced it cannot be applied in this situation.

If you are familiar with set-valued maps, this book may help you to reformulate your problem and see some possible theorems.

Implicit Functions and Solution Mappings: A View from Variational Analysis
By  A. L. Dontchev and R. Tyrrell Rockafellar

I hope you find it useful.

15
I am working on "Extensions of Special functions and their applications ". Is anyone interested in collaboration then they are most well come?

Since last 15 years, I am working on Special Functions and their applications. Recently, number of applications of special functions found in many fields. Now, I am interested in extensions of special functions?  Those are interested please contact to me at <goyal.praveen2011@gmail.com>

Dear Prof. John Gill,

Can we do some joint work?

Regards,

PA

17
What are the engineering applications of fixed point theorems?

Dear RG friends:

In two weeks time, I am all set to conduct a technical session on Analysis.

I plan to deliver a long lecture on "Fixed Point Theorems". Of course, Banach fixed point theorem is useful to establish the local existence and uniqueness of solutions of ODEs, and contraction mapping ideas are also useful to develop some simple numerical methods for solving nonlinear equations. Are there any other interesting science / engineering applications?

Kindly let me know! Thank you for the kind help.

With best wishes,
Sundar

Dear friends:

I thank you for the useful comments! I included some interesting applications of Banach fixed point theorem in my talk on Dec. 2015. I wrote a report on the same. Kindly check the enclosed link for the same. Your comments are welcome.

Thank you!

With regards

Sundar

8
Could anyone show me some references on topological degree for mappings between spaces of different dimensions?

Some problems of differential equations could be reduced to fixed points of mappings between Banach spaces of different dimensions, in which case the classical Leray-Schauder degree does not apply. So, I am wondering whether there exists topological degree of this type? Could someone tell me some references on this topic?
Best wishes,
Liangping Qi

The question could be more clear. There are two possible meaning of the problem: 1) degree of a map between two finite dimensional spaces - it is the homotopical type of this map; 2) degree for the pair (LN) where L is a linear Fredholm mapping of any positive index. This second theory generalizes the coincidence degree for L-compact maps where L has index 0 and it has origin in a paper by Nirenberg (1972 as far as I remember).

1
Is any stimulation/treatment ineeded to make transfected HEK293 cells to release soluble form of a transmembrane type II protien?

Dear friends

Being interested in doing research about the apoptosis induction by TRAIL (TNFSF10), I cloned the full length TRAIL cDNA in to PCR3 vector. After transfecting HEK293 cells with the mentioned construct, I have done fluorescent microscopy and the result was positive for the cell line (I could observe the cytoplasmic diffusion of TRAIL protein as well as accumulation close to the membrane). As for functional analysis, I performed Annexin-V experiment after treating my target cells (K562) with the supernatant, collected from the HEK293 culture flask. However, the result of this experiment was negative. Namely, there was no significant difference between the negative control and my treated cells, while the positive control (K562 cells, treated for 24 hours with H2O2 worked very well). Now, I want to find out about the reason. Simply stated, I want to know, if is this possible that the protein which was intracellularly expressed in HEK293, could not cleaved on the cell membrane and released as the soluble form in the supernatant? As you know, TRAIL, as a ligand, is a transmembrane type II protein which contains the signal peptide sequence and should be able to come to the membrane, cleaved by Cathepsin E and secreted. Should I add Cathepsin E to my HEK293 cells?

I apologize for the long message and thanking you in advance for your help.

Best

Zoya

Until someone else comes up with a more positive reply, this group published a similar finding as yours: treatment of a TRAIL gene sensitive cell with any of the supernatants of 3 cell lines transduced with the full-length TRAIL gene had no effect on cell death of the treated cells:

Antitumor Activity and Bystander Effects of the Tumor Necrosis Factor-related Apoptosis-inducing Ligand (TRAIL) Gene. Shunsuke Kagawa, et al. Cancer Res 2001.

Unfortunately, they didn't have any positive control in the supernatant transfer experiment like you did. They did mention that their 293 cells (listed as not Fast or T antigen expressing like yours) were sensitive to death upon TRAIL plasmid transfection.

5
Bounded linear operator with dense range, is it surjective?

Hi, I have a linear bounded operator A between tow Hilbert spaces with a dense domain and dense range, is it surjective? If the answer is no, are there more conditions on A to be surjective?

Any compact operator in  l_2 with dense range gives a counterexample.

15
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?

Dear followers,

Suppose that
(i) $B$ is homeomorphic to $n-1$ dimensional sphere
and
(ii) for every point $x$ in $B$ there is a closed
half-spaces $L(x)$ (sets of point in space that lie on and to one
side of a hyperplane) which contains $B$.

It seems at a first glance that properties (i) and (ii)
guarantee hat there exist the closed convex set A such that
$\partial A=B$.

Please tell me if I miss something here.

2
How to obtain slope from inverse square law function?

I have inverse third order that have equation y=y0+a/x+b/x2+c/x3. usually the slope, for example cubic function equation y=y0+ax+bx2 + cx3 is taken from the constant with a with x degree of 1. so, for the inverse equation, is the slope can be calculated from constant a also?

Just differentiate it and obtain for x≠0 that

y'=dy/dx=-a/x^2-(2*b)/x^3-(3*c)/x^4

2
How can I plot a time frequency plot of DWT?

I need to plot frequency, time, and amplitude of DWT. Can anyone help me with this?

Thank you sir
9
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.

The above arguments show that such a subset X in R^n is bounded. In order to show that it is also closed, let y be in the closure of X. Then consider the continuous function f(x) = - d(x,y), x in X. This is a continuous function on X, and because y is in the closure of X, its least upper bound on X equals zero.  Since it is attained,  there exists x_0 =(x_0)(y) in X such that - d(x_0,y)=0, that is y = x_0 is in X. Thus X is also closed.

11
What is Monotonus function?

When I study the properties of autonomous differential equation, they said that the solutions to the autonomous equations are monotonus functions.

I can't undestand what is monotonus function?

what is the difference between monotonic function and monotonus function?

Thank you for all your suggestions.....

6
Is it correct to say that using wavelet transform , a sampled signal is expressed as sum of approximation and detail coefficients ?

I am working on a project based on ECG signal denoising using wavelet transform. Just as we do in empirical mode decomposition , a signal is expressed as sum of IMFs and residue, can a signal be expressed as a sum of approximation and detail coefficients in wavelet transform ?

Dear Rastogi,

from theoretical point of view,  the main merits of wavelet transform is the ability to capture the signal in time-frequency components. By other words,  if you have a signal that is very flat or doesn't have much variation, then you may care more about the coefficients of the wavelets that have large support..

In addition, choice of wavelet basis function affects “compactness” of wavelet decomposition. so the approximation with  a filtered version of your original signal that will have frequency components is mainly based on the filtering order.

7
Can we say that a linear space X is not reflexive?
Suppose X be a linear space. My question is: How can we say that X is not reflexive in any norm. Reflexive means X is linearly isometric to its second dual X''

Here is an answer to a related problem. Namely, i will try to show that if a linear space X has an algebraic countable Hamel basis, then there exists a locally convex topology on X which makes X reflexive. Take as the desired topology the finest locally convex topology on X (in this topology every linear functional is continuous, so that the topological linear dual X '  equals the algebraic dual  X*). This locally convex space is a direct sum of a countable collection of "lines", so it is the strict inductive limit of a sequence of finite dimensional spaces. All these spaces are reflexive, so that their inductive limit X is reflexive too (see "Topological Vector Spaces" by H. H. Schaefer, third edition, Section 5.8, page 146). It is known that this locally convex topology cannot be generated by a norm. Moreover, this topology is not metrizable, but it is complete. The conclusion is valid for any strict inductive limit of a sequence of (complete) reflexive locally convex spaces. Thus we avoid the assumption on countable algebraic dimension on X.

4
Why small shape parameter is recommended in radial basis function (RBF) interpolation and application?

I would like to initiate a discussion on the role of shape parameter in RBF interpolation and applications. By shape parameter, I mean the one in new convention $\epsilon$ as suggested by Prof. G E Fasshauer. We all know that keeping the shape parameter small creates the problem of ill-conditioning and several works have been done to resolve this. My question is then, why do people recommend small shape parameters ? I have seen good accuracy in 2D RBF interpolation when I use large shape parameter with Gaussian RBF. Please provide your opinion on related aspects of the role of the shape parameter in this context.  I have read the literature, but I want to know the views of people working with RBFs.

I can add that there are theoretical results where the errors depend  on the shape parameter.

6
What is the easiest or most useful characterization of dual Banach spaces?
There is a famous characterization of reflexive Banach spaces by R.C. James, saying B is reflexive if every bounded linear functional attains its norm on the boundary of the unit ball. Is there a characterization of dual Banach spaces in the spirit of this theorem?

Some characterizations of conjugate spaces, sufficient conditions and examples can be found in  "Geometric Functional Analysis and its Applications", by R. B. Holmes, section 23, pages 211-221. The main theorem is that stated at page 213 (Dixmier-Goldberg-Ruston theorem). Passing over these characterizations, I would like to recall an interesting class of conjugate spaces, mentioned in the same book. Let X, Y be Banach spaces and consider the space B=B(X, Y*) of all bounded (linear) operators applying X into Y*. Then any such operator - space is a conjugate space. Concerning reflexive Banach spaces, it is obvious that such a space is conjugate. On the other hand, on every non-reflexive Banach space X there is an equivalent norm such that  X so normed is not congruent to a conjugate space.

13
Can someone share examples of topological vector space?
Examples with explanation will be much better

Dear Sheba,

You can find such examples in every chapter and section of the book "Topological Vector Spaces" by H. H. Schaefer (Springer, third edition). In Chapter V of this monograph, the connection with topological ordered spaces is investigated. Inductive and projective limits of families of locally spaces which appear in applications are also studied.

Sincerely,

Octav

3
Can anyone help me with Banach lattices?

We know that in a Banach lattice, if |x|\le|y| then, ||x||\le||y||. Can we have the converse. I  mean if we have ||z||\leq 1, can we find natural number n such that |z|<\frac{1}{n} a, for some positive element a in the Banach lattice?

I could add an example on this subject. It can be formulated in the ordered vector space setting, endowed with a natural norm, as well as in the Banach lattice framework. Let H be a finite or infinite dimensional Hilbert space, and A(H) the real vector space of all self-adjoint  operators acting on H. The order relation is defined by the convex cone of all positive  elements in A(H), i.e. by the elements U with the property <U(h),h> >= 0 for all h in H. Then for any U in A(H), we have U<= Norm(U)*I, where I is the identity operator, which stands for the strong order unit of the space A(H). In particular, for all U in the unit ball of B(H), we have  - I<=U<= I, so that we have also an uniform order boundedness, in terms of I.Notice that A(H) is not a vector lattice. Now I recall the definition of a related Banach lattice. Let fix U in A(H). Define S=S(U)={V in A(H); V*U=U*V}, Y=Y(U)={W in S; W*V=V*W, for all V in S}. Obviously, Y=Y(U) is a commutative algebra of self-adjoint operators. The interesting and not trivial fact is that Y is an order complete vector lattice, with respect to the order relation mentioned above. Obviously, the identity operator I is a strong order unit in this vector (Banach) lattice, by the same argument written above. For details see R. Cristescu, "Ordered Vector Spaces and Linear Operators", Academiei, Bucharest and Abacus Press, Tunbridge Wells, Kent, 1976, pages 303-305.

3
Can we represent elements of R^n using the base(frame ) of R^n-1?

I am investigating for a algorithm to represent an element for example in R^n using the combination of R^(n-1),R^(n-2),...,R^1 bases(frames).In particular, is it possible to represent an element of space R^n using the base (frame) of space R^m that m<n? For example we can represent any element in R^2 with R^3 frame like Mercedes-Benz frame. However, my question is that vice versa of that case can be true?

In general, one cannot represent any element of R^n by means of a base involving only elements from R^m, m < n. However, we can choose a base in R^m and complete this system of linearly independent elements from R^n up to a base in R^n. Then we can represent any element of R^n in terms of the latter base in R^n, which contains the elements of the first base of R^m.

• Edison Salazar asked a question:
Open
Is it possible to represent the Pauli potential as a functional only of \nabla ^2\rho(\ vec r)?

I would like to know if there is any paper where the Pauli potencial is represented only in terms of \nabla ^2\rho(\ vec r).

8
Does any one have any book/material/papers on fixed point theorem in differential equation, integral equation?

1. Functional Analysis by Walter Rudin

2. Non-linear Functional Analysis byProfessor Wataru Takahashi,

6
Is there any theorem / lemma/ theory regarding closed form expressions which says that we can find out some nth derivative of a function?

Consider a function

x_dot= f (x),  its 1st derivative can be written as   x(1)=f(x),

And its 2nd derivative can be x(2)=f '(x). x_dot,

And recursively, we can find out x(n) nth derivative of the x_dot= f(x) in the case if f(x) is linear, which is a reason for the formation of matrix exponential (eAT) If A is a linear matrix in f (x).

Or

One can also say that if f (x) results in a closed form expression for its Taylor expansion. Then nth derivative can be written. My question is that expression can be written for nonlinear systems if they come to have a closed form expression in their Taylor expansion.

Of course if your function is analytic, there is also Cauchy's integral formula, though I don't know if it qualifies for a "closed-form" expression (it is often pretty useful).

11
Does there exist a norm which belong to C^1(R)?

Real Analysis, Functional Analysis, Numerical Analysis, Mathemathics

Dear All,

let me return to the original question (Is there a norm on the space C^1(R)?). Of course the answer is yes, since one can construct'' a norm with the help of a Hamel basis. But as Luis has already pointed out, one might wish to look for a norm having some natural'' properties.

Let's do the first step before the second and investigate the space C(R) of all continuous functions on the real line first. There is no translation-invariant norm \|.\| on this space: Take f(x)=e^x and f_a(x)= e^{x+a}. Then all \|f_a\| = \|f\|, but f_a=e^a f.

Dirk

3
How can I accelerate B3LYP hybrid functional calculations?

These are the tags I have used in my VASP INCAR to use B3LYP functional to calculate DOS from pre-converged WAVECAR.

LHFCALC = .TRUE.
GGA = B3
AEXX = 0.2
AGGAX = 0.72
AGGAC = 0.81
ALDAC = 0.19

and replacing the LEXCH = PE tag in POTCAR by LEXCH = B3. I have also replaced GGA = PE by GGA = B3 in the INCAR.

I have observed that it didn't calculate a single energy by B3LYP functional even after 37 hours of job submission, and I am using 4 nodes with 48 cores. Moreover, I am using small k-mesh.

Could anyone guide me that how can I accelerate the calculations with B3LYP hybrid functional?

I am using from 30 to 100 atoms in different 2D models and using ENCUT similar to ENMAX of carbon i.e. 400 eV with 5x5x1 K mesh. I am using NPAR = 1 for parallelization, only, Doing static calculations with pre-converged WAVECAR from GGA =PE. Furthermore, PRECFOCK = Fast doesn't work with B3LYP hybrid functional.

17
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?

My previous answers show that, in general, such power series might have poles on the unit circle (the pole was z_0=1). So, the answer is negative. If we are looking for power series which have no poles on the unit circle, we have to consider power series having the convergence radius R>1 (this is trivial), or power series with R = 1, which are uniformly convergent in the closed unit disc. Such sums define analytic functions in the open unit disc, which are continuous in the closed unit disc. Then |f(z)| < = Sum(a_n)=f(1) < infinity, for all z on the unit circle, so that there are no poles in this case.

1
Does any logistic kernel (e.g. the sigmoid) reproduce a Hilbert space?

I would like to use the logistic function as distance operator between two functions from a set of them. These two functions live in a Hilbert space, Im not sure if the result of successive measures is a set living in a Hilbert space also. In the case it is not true, how can be mathematically proved the computational consistence of this measure? How this consistence can be theoretically ensured before empirically test it? Thank you

If the logistic kernel is a radial function, say g(r), it will reproduce a Hilbert space in a space variables of any dimensionality if and only if the univariate function f(t) = g(\sqrt{t}) is a completely monotonous function on (0,\infty) and is bounded at zero. Complete monotonity means that (-1)^m f^{(m)} is nonegative on (0,\nfty) for any m. In other words f should be nonnegative, f' should be nonpositive, f'' should be nonnegative, and so on. Excluding the case when f(t)=t^n for natural number n.

If the dimensionality of space of independent variables is limited (say 2-dimensional or 3-dimensional), it is enough that the function and its first derivatives satisfy the monotonity property.

See detail in Wedland, Holger, "Scattered Data Approximation", Cambridge Monographs on Applied and Computational Math., 2005, Chapters 7 and 8.

15
Can anyone help me with topological fields?

I am looking for examples of topological fields. It seems that they are scarce in the literature. continuous real functions on a compact space are just topological ring. In fact, I am interested in knowing a topological field with sequences. Simple examples are R or C. But sequences of these spaces are not topological field.

Your question remains for me unclear. If you are looking for a field consisting of sequences on some algebra with coordinatewise operations, it cannot exist because any element admitting a nonzero coordinate must be invertible, but this is not true.

Now, if you are looking for a field with a linear topology (even with continuous multiplication), you can find them. Actually, every field with its strongest locally convex topology is a topological field (here multiplication need not be continuous in both variables but only separately continuous). You can also have a fields with a linear topology such that the multiplication is continuous in both variables, but the topology need not be locally convex. You can have a look at  somes (old)  papers of L. Waelbrock, and Williamson.

Anyway, because of Gelfand-Mazur theorem, if you ask a "topological" field to have some stronger properties you just get $\mathbb{C}$ in the complex case. You can have a look at some papers of Mati Abel.

16
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?

Unitary operators U (U*=U^(-1)), have important geometric properties. Namely, U and U* preserve the scalar products (hence the norms and the angles), so both of them are isometries.. The spectrum of a unitary operator is contained in the unit circle. Self-adjoint operators have real spectrum. They admit  an integral representation associated to a spectral measure. Normal operators T have the form T = A + i B, with A, B commutting self - adjoint operators. Unitary operators have a similar representation, where A^(2)+B^(2)=I, (AB=BA). Normal and unitary operators admit spectral measures and associated integral representations too.