Science topic

# Real and Complex Analysis - Science topic

For discussion about the analytic properties of real and complex sequences and functions.
Questions related to Real and Complex Analysis
Question
Hello
Can someone help me to solve this?
Because I really don't know about these problems and still can't solve it until now
But I am still curious about the solutions
Hopefully you can make all the solutions
Sincerely
Wesley
Hi In what area was the issue raised? Euclidean space, Hilbert space, Banach space?
Question
The question details are contained in the attached pdf file.
W.H. Young, “A note on monotone functions,” The Quarterly Journal of Pure and Applied Mathematics (Oxford Ser.) 41 (1910), 79–87.
Question
Is the series in the picture convergent? If it is convergent, what is the sum of the series?
Dear Joachim,
you've right. Just ignore the first part of my answer. May be the double series calculation will lead to some result.
Rgrds,
Tibor
Question
I has made proposition and prove that (f o f ')(x)= (f 'o f )(x) if f is additive real function and (f o f) (x)=f (x) but I still difficult get example non-trivial. Could you help me? example trivial is an identity function.
I apologize for the inconvenience this
By "additive function" do you mean function on (-\infty, + \infty) that satisfies equation f(x+y) = f(x) + f(y)? Then all such continuous functions are of the form f(x) = cx.
Since in your question you are speaking about the derivative, then you must deal only with continuous functions. So, f(x) = cx and you cannot get other examples for the identity (f o f ')(x)= (f 'o f )(x) with additive functions.
There are only two functions of the form f(x) = cx that satisfy the second identity
(f o f) (x)=f (x), namely f(x) = x and f(x) = -x.
Question
I was looking for examples of first order sentences written in the language of fields, true in Q (field of rational numbers) and C (field of complex numbers) but false in R (field of real numbers). I found the following recipe to construct such sentences. Let a be a statement true in C but false in R and let b be a statement true in Q but false in R. Then the statement z = a \/ b is of course true in Q and C, but false in R.
Using this method, I found the following z:=
(Ex x^2 = 2) ---> (Au Ev v^2 = u)
which formulated in english sounds as "If 2 has a square-root in the field, then all elements of the field have square roots in the field." Of course, in Q the premise is false, so the implication is true. In C both premise and conclusion are true, so the implication is true. In R, the premise is true and the conclusion false, so the implication is false. Bingo.
However, this example is just constructed and does not really contain too much mathematical enlightment. Do you know more interesting and more substantial (natural) examples? (from both logic and algebraic point of view)
Something algebraic, implicitly talking about ordering:
"for every nonzero number x, x or -x is a square but not both."
This holds in R (it is essentially an axiom of real closed fields) but not in Q or C (x=2 is a counterexample for both). Now you can take the logical negation.
Question
If we accept the classical differentiability of spacetime, for all cosmological time, a critical value of that expansion may prevent the sea of virtual particles from ever recombining.  This would convert "virtual" mass to "measurable" mass, keeping the critical density constant and thus solve the horizon problem of the Standard Model.
This possibility may also admit an explanation of why the CMBR is so uniform.  Galaxies would not "wink out" one by one, and a "time averaged" Cosmological Constant and Gravitational Constant may both be necessary consequences.
I am not sure I have seen prof. Nash's work. There were several such attempts in the past from Neo-classical (hydrodynamic) ones and those based on Nelson's original stochastic mechanics. Check for instance Ord's work http://www.math.ryerson.ca/~gord/research.html
Italian "emergent QM" school  http://www.emqm15.org (related with original T' Hooft's attempts for a semi-discrete interpretation http://arxiv.org/abs/1405.1548)
The strangest of all -and probably deeper- is I think the Geneva-Brussels school originating in Diederik Aert's work https://en.wikipedia.org/wiki/Hidden-measurements_interpretation https://en.wikipedia.org/wiki/Diederik_Aerts
True problem behind all this: where do "hidden" variables reside? To make matters worse, recent findings in optics by Christensen (Illinois) showed that not even the good old Bell inequalities are sufficient to fully characterize non-locality.
Possible hint: the so called "entanglement' and non-locality appear now to be two totally disproportionate if disparate qualities. For the abstract logician this could beg the question of how the heck does nature manages to fill a continuum with a set of countable combinatoric structures without an additional stabilizing axiom/"external" constraint? It seems more and more like asking from a field theory to behave as an "automaton" running a "program". It is simply impossible without a very delicate imposition of a rather peculiar initial/boundary condition at the least. But that would again bring about the awkward question on "externality". Who/ what/ how would ever be able for such a delicate fine tuning exactly at the BB event? Unless there is no such "exact' event that is!
Question
Is there a formula to enumerate the lattice paths from (1,1) to (m,n), confined to the region: y=x/2 and y=2x contruct to a triangle,  which step set is {(1,0),(0,1)}?
Denote p(m,n) the number of paths from (1,1) to (m,n).
If the region was x>0, y>0, p(m,n) would copy the coefficients in the Pascal's triangle.
I don't know if an explicit formula already exists for the cone x>0, y>0 and 2x≥y≥x/2 (maybe you have one?).
Anyway, a recursive formula is possible:
1) p(0,0)=0, p(1,2)=p(2,1)=1
2) p(m,n)=p(m-1,n)+p(m,n-1) if m≤2n-2 and n≤2m-2
3) p(2n,n)=p(2n-1,n)=p(2n-2,n) and p(m,2m)=p(m,2m-1)=p(m,2m-2)
(see the attached figure)
Thanks to the symmetry of the region, 1),2),3) can be simplified.
A better simplification is obtained if we put p(m,n)=0 for (m,n) outside the cone:
3) may be omitted, 2) holds for any (m,n) in the cone.
Question
A NSWE-path is  a path consisting of North, South, East and West steps of length 1 in the plane. Define a weight w for the paths by  w(N)=w(E)=1 and w(S)=w(W)=t. Define the height of a path as the y-coordinate of the endpoint. For example the path NEENWSSSEENN has length 12, height 1 and its weight is t^4.
Let B(n,k) be the weight of all non-negative NSEW-paths of length n (i.e. those which never cross the x-axis) with endpoint on height k.
With generating functions it can be shown that for each n the identity
(*)            B(n,0)+(1+t)B(n,1)+…+(1+t+…+t^n)B(n,n)=(2+2t)^n
holds. The right-hand side is the weight of all paths of length n.
Is there a combinatorial proof of this identity?
For example B(2,0)=1+3t+t^2 because the non-negative paths of length 2 with height 0 are EE with weight 1, EW+WE+NS with weight 3t, and WW with weight t^2.
B(2,1)=2+2t because the non-negative paths are NE+EN with weight 2 and NW+WN with weight 2t. And  B(2,2)=1 because w(NN)=1.
In this case we get the identity
B(2,0)+(1+t)B(2,1)+(1+t+t^2)B(2,2)=(2+2t)^2.
In the meantime I have found a combinatorial proof in the literature: Naiomi T. Cameron and Asamoah Nkwanta, On some (pseudo) involutions in the Riordan group,  J. Integer Sequences   8 (2005), Article 05.3.7, proof of identity 1. https://www.researchgate.net/profile/Asamoah_Nkwanta/publications?sorting=newest
Instead of NSWE-paths they use bicolored Motzkin paths, but their proof can easily be translated to the situation of NSWE-paths. So my question has been answered.
Question
Let f1:ℕ⭢ℕ be the identity, that is, ∀n∈ℕ: f1(n)=n. For each n in ℕ, let fn:ℕ⭢ℕ be the map defined recursively as follows.
fn(m) = fn-1(m+1) if fn-1(m) = 2
fn(m+1) = fn-1(m) = 2 if fn-1(m) = 2
fn(m) = fn-1(m) otherwise.
Since every map fn is defined recursively with a transposition in the image
of fn-1, the map fn is again a bijection that satisfies the following properties.
1) ∀n ≤ m: fm(n)≠ 2.
2) ∀n ≤ m the map fn:ℕ⭢ℕ is a bijection.
If the recursion process never ends, both properties are compatible.
However, assuming the actual infinity existence, every infinite process can be completed. Under this assumption the recursive proces give rise to the following properties for m =∞.
1) ∀n∈ℕ: f(n)≠ 2.
2)  The map f:ℕ⭢ℕ is a bijection.
which is a contradiction, unless we assume that every inifinity is potential, and
n keeps always finite.
Taneli Huuskonen
I think that your answer is true in the scope of potential infinity, but my question requires the actual infinity axiom. Recall that, according to  actual infinity concept,  every process can be ended. Thus, actual infinity is not synonimous of endless. To avoid any wrong interpretation of my words, I write both axioms below.
Axiom 1. Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already.
Henri Pointcaré (1854-1912).
Axiom 2. Every mathematical construction can form an actual and completed totality.
G. Cantor (1845-1918)
Question
Any idea or comment is appreciated!
Dear all,
That is true dear Octav.
The sum can be computed for any number s > 0 where the question is put when s is 1/2, from a transform definition. The story of it is very important as the discrete Laplace like transform is used to generate beautiful sequences of difference equations including the Fibonacci sequence.
Question
What are completely monotonic functions on an interval $I$? See the picture 1.png
What is the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? See the picture 1.png
My question is: is there an anology on the finite interval $I=(a,b)$ of the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? In other words, if $f(x)$ is a completely monotonic function on the finite interval $I=(a,b)$, is there an integral representation like (1.2) in the picture 1.png for the completely monotonic function $f(x)$ on the finite interval $(a,b)$?
The answer to this question is very important for me. Anyway, thank everybody who would provide answers and who would pay attention on this question.
The formula (1.2) for completely monotonic function on an interval is not valid. For example, f(t) = e-t - e-1 is completely monotonic on [0, 1] with f(1) = 0, and the latter property is impossible for a non-zero function satisfying the integral representation (1.2). In order to have (1.2) you need a function that has completely monotonic extension on the semi-axes (0, \infty). There is really a lot of information about absolutely monotonic / completely monotonic functions in the original Bernstein's paper
S. N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679
Maybe some of this information can be of use for you. S. N. Bernstein is one of the most famous mathematicians from my University, and it is always a pleasure for me to mention his works: they contain much more material than the textbooks citing his results.
Question
See the attached file for mathematical description of a PDE system steaming from Flash Photolysis. I'm interested in analytical (symbolic) solution but if that is not possible then numerical method would suffice. Thanks.
Dear Mikael,
Your system admits kink-solutions (travelling waves which connect lower energy values to higher). See the attachment, which was produced in Mathematica.
Cheers,
Sotiris
Question
Greeting and salutation!
I have a initial value differential equation with a unknown parameter, how can i solve it With Matlab ODE solver or other software????
With Best Regards
Hamed
Dear Prof Abram
attached figure is attached for clarification; don't hesitate if more explain is necessary
Hamed
Question
We have at least two different definitions of “infinite” (“actual infinite” and “potential infinite”) since antiquity, and these two different “infinites” with different natures unavoidably become the foundation of present classical infinite related science theory system dominating all the infinite related contents as well as all of our infinite related cognizing activities since then.
When one faces an infinite related content in mathematical analysis, is it in  “actual infinite mathematical analysis” or “potential infinite mathematical analysis”?
The infinite small (few) related errors and paradox families (such as the newly discovered Harmonic Series Paradox) and the infinite big (many) related errors and paradox families (such as the newly discovered Cantor’s ideas and operating process of mistaken diagonal proof of “the elements in real number set are more infinite than that in natural number set”) are typical examples of “master pieces of confusing potential infinite and actual infinite”.
Dear Geng Ouyang,
one possible answer is the following:
What is meant by 1 + 1/2 + 1/3 + 1/4 + ...
It is the limes of the sequence
1
1+ 1/2
1 + 1/2 + 1/3
...
Each of the elements of this sequence is a finite sum. And therefore in every pair of brackets you can place there can only be finitely many summands. Perhaps this is not an answer to the question you intended, but I was not yet able to read the papers I found on your profile.
Best regards
Johann Hartl
Question
I have a 4x4 density matrix (Trace 1) whose elements are nonzero. Its form is
a  b  c  d
b* e  f  g
c* f* h  j
d* g* j* k
where a+e+h+k=1.
Is there a simple way to find eigenvalues and eigenvectors of this matrix?  I calculated in Mathematicai Maple, MATLAB (There are too much terms). More simple way?
real eigenvalues exist
Question
Dear all
What and where is the formula for higher derivatives of a product of many functions? In other words, how to compute higher derivatives of a product of $n$ functions? Concretely speaking, what is the answer (a general formula) to
$$\frac{d^m}{d x^m}\left[\prod_{k=1}^n f_k(x)\right]=?$$
where $m,n\in\mathbb{N}$. Could you please show me a reference containing the answer? Thank a lot!
Best regards
Feng Qi (F. Qi)
Dear Feng Qi, you can find it on:
Best regards, Viera
Question
In two-dimensional case, we can follow Kohn's definition of type by using holomorphic tangent vector field and the Levi function to define infinite type(cite H. Kang, Holomorphic automorphisms of certain class of domains of infinite type, Tohoku Math. J. (2) 46 (1994), 435–442. MR 95f:32041).  But when we consider in higher dimensions, can we still use this method to define infinite type?
In higher codimension, we cannot define the regular infinite one type with  tangent vector field and the levy form directly like in C^2. But it is in some sense possible : see my paper with J.F Barraud  (at least for define finite type):
Barraud, Jean-François; Mazzilli, Emmanuel
Regular type of real hyper-surfaces in (almost) complex manifolds. (English) Zbl 1082.32017
Math. Z. 248, No. 4, 757-772 (2004). This is done for regular one type. Now for the D'angelo type : singular type, it is possible see the paper :
Barraud, Jean-François; Mazzilli, Emmanuel
Lie brackets and singular type of real hypersurfaces. (English) Zbl 1154.32009
Math. Z. 261, No. 1, 143-147 (2009). I do not know if it is exactly a think like you want but perhaps it is helpfull for you. regards E/.Mazzilli.
Question
How to compute the limit of a complex function below? Thanks.
For $b>a>0$, $x\in(-\infty,-a)$, $r>0$, $s\in\mathbb{R}$, and $i=\sqrt{-1}\,$, let
\begin{equation*}
f_{a,b;s}(x+ir)=
\begin{cases}
\ln\dfrac{(x+ir+b)^s-(x+ir+a)^s}s, & s\ne0;\\
\ln\ln\dfrac{x+ir+b}{x+ir+a}, & s=0.
\end{cases}
\end{equation*}
Compute the limit
\begin{equation*}
\lim_{r\to0^+}f_{a,b;s}(x+ir).
\end{equation*}
Before computing the limit one has to fix the precise meaning of the expression. The functions $g(z) = z^s$ and $h(z) = \ln(z)$ are multi-valued, so the question will be correctly posed only after selecting appropriate branches of all expressions that  include these functions. Outside of this I see no difficulties.
Question
How to define and compute the power-exponential function of a complex variable? For example, how to define and compute the complex function $(\ln z)^{\sin z}$? where $z=x+iy$ is a complex variable. Thank you for your help.
Hi colleague Feng Qi
I just had the intention to help you, but the Professor Yao Liang Chung was faster than me.
Best regards and wishes,
Mirjana Vukovic
Question
Remark : they are several "logics" to make sense additively with simple recognition shapes for instance like dashes. If you want to add shapes in a compositionnal way you get the following equation : One thing that is an horizontal dash  ( ___ )  " + " (drawing) one other thing that is a vertical dash ( / ) can build by composition a cross like the additive symbole (+) that is either one thing as this symbole or 5 things ( 4 segments + a central point.) So you could have 1"+"1=1 or 1"+"1=5. Immediately you see these "counting logics" make fun of the most elementary arithmetical habits !?
JYTA
Hello Jean-Yves Tallet
I think that it is a problem of unit here. You can devide any dash into any number of smaller dashes and points relating any two concecutive of them. On the other hand if we consider dashes and point as unit, the lenght of a dash makes the difference.
Question
As I understand that for differentiable and monotone functions we can partition the period and find the total variation, but what about the case when it's not differentiable ?
For example in this article http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1083433 the authors have essentially mentioned the total variation of signum type function is 4 . But how is it done ? In general it looks to be 2.
Question
I have a doubt regarding completeness of C[0, infinity) with respect to sup norm. Is the space Banach Space? Is it a Banach lattice?
Actually the sup norm is not defined on C[0,\infty) since this space contains unbounded functions. If you consider the space C^b[0,\infty) of bounded continuous functions (or the space suggested yb Luiz) , then the sup norm is well defined, and this space is complete (basically since the uniform limit of continuous functions is continuous).
Question
For example, Circle is not a function. Can any one explain with this inverse and implicit function theorem?
1. The  unit  circle   x^2 + y^2=1   is not a function,  but locally  is  a graph of a function .Set  f(x,y)=x^2 + y^2-1.  Then the partial derivatives  are   f'_x=2x    and   f'-y=2y.Hence  if    (x_0,y_0) is a point on the unit circle    and for example  y_0     is not 0,  by the implicit function theorem  locally  there is  a function y=g(x)   which is solution of the equation    f(x,y)=0.Denote   by   g_0(x) the   square root of 1-x^2, for x between -1 and 1. Actually if y_0  >0 , y=g0(x)  is solution    and    if y_0 <0 , y=-g0(x) is solution.  In a similar way we can discuss the case if   x0  is not 0.
Question
Hello;
I have ended up with a quadratic function of X in the form of:
f(X)= XTAX-XTBX+CTX with A and B as positive definite matrices. Note that the second term has a negative sign. Clearly the function is not necessarily convex, but based on some experimentation and prior calculations, it should convex. I wonder to know if there is any analytical or famous experimental method to prove the convexity of this quadratic function.
Thank you
i agree with Octav;
f is convex iff A-B is positive semidefinite,
f is strictly convex iff A-B is positive definite,
f is  concave iff A-B is negative semidefinite,
f is strictly concave iff A-B is negative definite.
Question
The real number definition requires de limit concept corresponding to the standard topology. If we forget the topological structure of R, then its definition vanishes. I think that, handling non-defined objects it is not an accurate method.
Nevertheless, it is not difficult to find mathematical proofs using the topological structure of R while it is assuming that the result is concerned to the underlying set of R exclusively, that is to say, the structure-free real number set.
Juan Esteban,
My pdf-file describes a construction of the full system of real numbers. Of course you have recognized some traditional features in it. At the same time, it may urge you into an additional sharpening of your criteria. Constructing |R on the fly as in
|R := { x : x in X and P(x) }    (with some available set X and some property P)
is not to be expected. Instead, it may involve intermediate constructions and logical formulae. Where lies the watershed between acceptable formulae/constructions and the ones you find in my file?
Nevertheless, your question hits a point. Both Dedekind cuts and Cauchy sequences connect with topology (order topology and metric topology) and both do assist  in constructing the reals. The real field is also characterized as the only complete and totally ordered field. A structure with such references is expected not to be constructed in a simple way.
Question
Say a definition to be self-referential provided that contains either an occurrence of the defined object or a set containing it. For instance,
Example 1) n := (n∈ℕ)⋀(n = n⁴)⋀(n > 0)
This is a definition for the positive integer 1, and it is self-referential because contains occurrences of the defined object denoted by n.
Example 2) Def := "The member of ℕ which is the smaller odd prime."
Def is a self-referential definition, because contains an occurrence of the set ℕ containing the defined object.
Now, let us consider the following definition.
Def := "The set K of all non-self-referential definitions."
If Def is not a self-referential definition, then belongs to K, hence it is self-referential. By contrast, if Def is self-referential does not belong to K, therefore it is non-self-referential. Can you solve this paradox?
Take into account that non-self-referential definitions are widely used in math.
Juan-Esteban,
That's a nice way to avoid the so-called Russell paradox at any finite stage of the process. It feels as if you avoid the paradox by rejecting infinity. If you read my post carefully, you will see that the problem is solved in a more fundamental way with or without  considerations of infinite sets. Formal logic tells you that there cannot be x such that
(all y) ( not P(y,y) <--> P(y,x) ),
whatever you mean by P(y,x) and whatever your universe of discourse may be. If you take the "barber definition" (anyone  shaving all those people not shaving themselves), even in an imaginary infinite society of humans, it turns out that there is no such barber. In fact, it is not a definition because it deals with nothing. Let me illustrate this with a more obvious contradiction:
Remarkable_Weather := weather with rain and yet without rain.
Such a "definition" is verbosity with (literally) no subject, hence with no meaning.
I prefer Webster's definition of "paradox":
A tenet or proposition contrary to received opinion; an
assertion or sentiment seemingly contradictory, or opposed to
common sense; that which in appearance or terms is absurd,
but yet may be true in fact.
The only difference between having a Remarkable_Weather and the Russell paradox is, that the latter involves a slightly more hidden logical contradiction.
Question
I need these measurement for the diversity analysis of population of vectors
The (vector) norm not the suitable measure in this problem.
Question
Consider the cubic equation:
$t^3 = 3pt + 2 q \qquad (1)$.
We introduce two variables $u$ and $v$ linked by the condition
$u+v=t\,$
and substitute this in the depressed cubic (1), giving
$u^3+v^3+(3uv-3p)(u+v)-2q=0 \qquad (2)\,.$
At this point Cardano imposed a second condition for the variables $u$ and $v$:
$3uv-3p=0\,$, ie $uv=p$.
As the first parenthesis vanishes in (2), we get $u^3+v^3=2q$ and $u^3v^3=p^3$. Thus $u^3$ and $v^3$ are the two roots of the equation
$z^2 -2 qz + p^3 = 0\,.$
$(z-q)^2 =q^2 - p^3=D$. Hence $z= q + \sqrt{q^2 - p^3}$. Denote by $a_1$ and $a_2$ two roots of $D$ and set $z_k=q +a_k$.
Solutions of equations $w^3=1$ are $1,e^{2\pi i/3},e^{-2\pi i/3}$.
Denote by $\underline{A}=\{u_1,u_2,u_3 \}$ and $\underline{B}=\{v_1,v_2,v_3\}$ solutions of equations $u^3=z_1$ and $v^3=z_2.$
Check that $u_k v_j \in \{p , p e^{2\pi i/3},p e^{-2\pi i/3} \}$. Set $p_1=p , p_2= p e^{2\pi i/3}$, and $p_3=p e^{-2\pi i/3}$.
In discussion with my students (Svetlik,Knezevic,Stankovic,Avalic), we prove the following:
Proposition 1.
The set $\underline{X}= \{u_k + v_j:1\leq k,j \leq 3 \}$ has $9$ elements if $q + \sqrt{q^2 - p^3} \neq 0$ and six elements if $D=0$ and $q\neq 0$.
If $u_0\in A$, $v_0\in B$, then $\underline{X}= \{\omega^k u_0 + \omega^l v_0:k,l=0,1,2\}$ and
the points of $\underline{X}$ forms three (respectively two) equilateral triangles if $D\neq 0$(respectively $D=0$ and $q\neq 0$).
Presently, we did not find origin of this result in the literature.
What are possible generalizations of this results?
Dear Miodrag:
Please allow me to add just a very simple remark (which does not contribute to your question). There is a more intuitive way to Cardano's formula: Why should someone put $u+v=t$ and assume $uv=p$? Here is the idea: We want to solve the cubic equation
$$t^3 = 3pt + 2q \eqno(1)$$
which is difficult. But other cubic equations are apparently easy:
$$(t-u)^3 = v^3 \eqno(2)$$
is solved by $(*)$ $t - u = v$ or $t = u+v$. Now the idea is to transform the easy equation (2) into the difficult'' form (1). In fact,
\begin{eqnarray}
(t-u)^3 &=& t^3 - 3ut^2 + 3u^2t - u^3 \cr
&=& t^3 - 3u(t-u)t -u^3 \cr
&=& t^3 - 3uvt - u^3 \nonumber
\end{eqnarray}
using $(*)$. Thus (2) is of the form (1) with
$$p = uv,\ \ \ 2q = v^3 + u^3.$$
Solving these equations for $u^3$, $v^3$ gives Cardano's formula.
Best regards
Jost
Question
I study the existence of germs of J-holomorphic curves inside a compact real analytic hypersurface embedding in a J-almost complex manifold (with J analytic). In general for fourth dimensional manifold, we have only two possibilities : the hypersurface is foliated or does not contains any germ of J-holomorphic curves. In dimension bigger than four, is it true that for a Stein J-almost complex manifold, a compact real analytic hypersurface does not contain any germ of J-holomorphic curve ?
U r great dear profesor El Naschie.. thanks for your kind explanations
Question
Suppose we have an analytic function f(z) of a complex variable z and that its Taylor expansion possesses only even order terms. Is the function g(z) = f(sqrt(z)) analytic?
g(w)=tan(w)/w
I guess you first note that g has a removable singularity at 0, so becomes analytic on a neighbourhood of zero. Also noting that g is an even function, the coefficients of odd powers of w are all zero.
Since g(-w)=g(w) away from poles of g, g(\sqrt z) is well-defined except at squares of poles of g, and the discussions above show that g(\sqrt z) is analytic away from these points.
Question
How to construct the system of equations by susbtituting the assumed series solution in the nonlinear partial differential equation? Kindly find the attachment.
Dear Saravanan
First at all, your problem seems to be so easy, if you want to solve it Numerically. i.e. if all constants are given and the function fi(zeta) is explicit. then you can catch the values of the unknown values for a0, a1, d and b.
Furthermore, If you want an explicit form for a1, you can let it to be a polynomial(as example) of suitable degree with unknown constants to be determined.
Is my understanding for your problem is suitable?
Question
I am wondering whether the relative interior of a linear subspace which is not closed is empty or not ? I work in a general Banach space.
Hellow everybody,
What is meant by relative interior ? If you mean the interior with respect to its own topology, it is of course non empty since any topological space is open in itself. If you mean the interior of a subset  A of a subspace F of a Banach space E with respect to the relative topology on F, then Thomas Korimort has given the answer.
Remember that : in any topological vector space, the only subspace which has a non empty interior is E itself, because the 0-neighbourhoods are absorbant..
Also a subset of a topological space can fail to be neither open nor closed.
Question
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?
Pure math is always ahead of the game because we pave the roads and identify the dead ends for applied sciences. Sometimes we pave the roads so far ahead of technology that our  results get lost or forgotten. I'm sure there are answers to many applied questions hidden away in decades old pure math publications but the applied scientists don't have the vocabulary to even do a useful search for this information. There needs to be more communication from both sides.
Question
If we have a matrix and remove for example its last row and column, is
there a condition under which the new matrix has different minimum
eigenvalue than the original matrix?
By Cauchy theorem they might be equal.
Actually the matrix is indefinite and not triangular
Question
Looman–Menchoff theorem states that a continuous complex-valued function defined in an open set of the complex plane is analytic if and only if it satisfies the Cauchy–Riemann equations.
My Question is: Can we drop the assumption of continuity in Looman-Menchoff theorem, if not please provide an example.
The Cauchy-Riemann equations themselves do not imply analyticity! Indeed,the function given by
f(z)=exp(-z-4) when z not equal to 0 and f(z)=0 when z=0
is readily seen to satisfy Cauchy-Riemann equations everywhere but falls to be analytic at origin.
A stronger result than Looman–Menchoff theorem which asserts the Analyticity of complex valued function states that
If
Question
I'm looking for optimizing multivalued  vector valued function.
Your question is not completely clear. Do you mean you are looking for a set-valued map that is not the subdifferential of a function, the word "subdifferential" being taken in one of the existing meanings given by a list of properties?
The application you have in view seem to orient to another interpretation.
Question
As we may know, in the standard approach to the $q$-calculus there are two types of $q$-exponential functions $e_{q}=\sum_{n=0}^{\infty}\frac{z^{n}}{\left[ n\right]_{q}!}$ and $E_{q}(z)=\sum_{n=0}^{\infty}\frac{q^{\frac{1}{2}n\left(n-1\right) }z^{n}}{\left[ n\right]_{q}!}$. Based on these $q$-exponential functions, also, some new functions are defined whose most of their properties are similar to the exponential function in calculus. Despite having interesting properties similar to the exponential function in calculus, for none of them the above property holds. So, can we define a new $q$-exponential function with similar characteristics to the exponential function in calculus which satisfies the aforesaid property?Any help is appreciated.
Unfortunately, if you are interested in continuous solutions of your functional equation F(x+y)=F(x)F(y), then the exponential functions exp(a x) are the only answer, see the links below for details (note that if F satisfies the above equation, f=ln(F) satisfies the Cauchy functional equation f(x+y)=f(x)+f(y)):
Question
I created a model for a coil located over a conductor, coil is a multiturn coil with driving current of 1 amper. I modeled the half of geometry and the quarter of geometry. By decreasing the geometry from half to quarter the coil inductance value is getting half but this does not happen for the coil resistance. In fact I get a wrong value for the resistance of the coil. Could anybody take a look at these models and help me to find the mistake?
Hi Ehsan,
Thanks for the link to the comsol blog on the use of symmetry which I read with interest. This actually confirms what I thought might be happening. From what is described in the blog, the multi-turn coil feature in comsol appears to use what is usually called the 'current sheet' approximation. This replaces the current flowing through the multi-turn coil with a uniform current sheet that flows around the coil perimeter. This is an approximation which, once in place, permits the use of symmetry as a means to reduce the model size.
However, this approximation is only good for tightly wound coils and only for estimating the coil's inductance. The reason is that the inductance is primarily determined by the magnetic flux external to the coil (giving rise to the external inductance) which depends on the total current flowing in the wire. Therefore, using a uniform current sheet doesn't alter the result for the inductance too much from that obtained using the actual coil geometry.
However, the internal inductance (that due to the magnetic flux inside the wire) and - most importantly for what you are looking at - the coil's resistance will not be accurately determined by the current sheet approximation.
To get an accurate simulation of the resistance, the spatial distribution of current in the actual wire geometry must be detemined. In an isolated single turn of circular wire, this distribution of current will be cylindrically symmetric and follow the same kind of distibution as the electric field, that is, it will be concentrated near the surface of the wire and decay exponentially into the wire - the familiar 'skin effect'.
When additional turns of wire are added and brought close together (as in a tightly wound coil) the current distribution in the wire will be altered due to the 'proximity effect' which results from the Lorentz Force on the electrons in the wire due to the magnetic field in neighbouring parts of the coil. The net result is that the current gets concentrated in the parts of the wire cross section that are nearest to the other turns. The current is then no longer cylindrically symmetric within each turn of wire. This proximity effect therefore increases the resistance over and above that due to the skin effect since the current now flows in a reduced cross sectional area.
The proximity effect isn't modelled by the current sheet approximation. Therefore, one cannot utilise symmetry in the comsol multi-turn coil model for calculating resistance (or internal inductance).  A real multi-turn coil doesn't have symmetry with regard to the actual current distribution in the wire.
Regards,
Ray
Question
We focus on the “deep structural relationship” between “nonstandard one” and “standard one”. Let’s exam following facts:
1, as “monad of infinitesimals” has much to do with analysis; nonstandard analysis is much more a way of thinking about analysis, as a different analysis------simpler than standard one.
2, CONSERVATIVE is the nature and a must for Nonstandard Analysis or Nonstandard Mathematics, it is called a conservative extension of the standard one.
3, because of the “deep structural CONSERVATIVE”, the “provable” equivalence are guaranteed.
If there are “no defects” in the “standard one”, the “CONSERVATIVE guaranteed nonstandard” work would be really meaningful.
Now the problem is “nonstandard one” inherits all the fundamental defects disclosed by “infinite related paradoxes” from “standard one” since Zeno’s time 2500years ago------guaranteed by the “deep structural CONSERVATIVE” .
Theoretically and operationally, “nonstandard one” is exactly the same as those of “standard one” with suspended infinite related defects in nature.Simpler or not weights nothing here.
Hi Akira, I think a firm theoretical bridge between discrete systems and continuous approximations to them may not be possible any time soon.  Again, if a physicist can find an excuse to treat a large discrete system (e.g., a container of gas) as a continuous entity, she will.  "Anything that gets answers."   As for supporting/refuting physical theories, only degrees of confidence are required to underpin belief.  I don't believe empirical data near as much as I believe a mathematical argument;  and the more complicated the argument, the more difficult it is for me to believe.
Question
after importing geometry file (I tried with parasolid, iges, iam, stp), the structure is imported in 5 bodies separately with contacts. but it can be see that exists some separations that DM shows. then when I try to mesh, there is no conformal association between nodes and meshing.
Hello,
Have you tried performing Boolean operations? In DesignModeler it is: create > booleans >  unite.
Hope it helps.
Question
Does anyone know an example of a Cauchy continuous function on a bounded subset of R?
Right! By the same arguement scetched above you hae the following theorem: let f: X -> Y be Cauchy continuous and asume that Y is a complete metric space as well as X is a precompact metric space (as for example a bounded subset of R).. Then if f is Cauchy continuous it is uniformly continuous. For the completion of X is compact and since f ix Cauchy continuous it possesses a unique extension to a continous function from the completion of X to Y. But the completion of X is compact, so the extensioin and a fortiory f itself is uniformly continuous.
Manfred
Question
By space of measurable functions, I mean L_0(m) where m is a non-atomic sigma-finite measure space.
In my opinion, you can start with the book by Martin Vaeth "Ideal Spaces" LNM 1664. But, of course, you should think about classes of measurable functions...
Question
I have one equation as shown in the following. I also discussed the integration region for 6.21. What I am interested to know about is the integration region of 6.22.
Integration region is the one kinds of metric distance.so it may be the error(comparison) between two things(images, cars,samples,...)
Question
X is an Inner product space and U is a subset(space) of X. U^\perp is the space of all vertices orthogonal to all the elements of U. Give an example of a space U^\perp\perp is not a subset of U. Can it be from l^2?
If U is a subspace, then U^{\perp\perp} is its closure (by the Hahn-Banach theorem), so you may take any subspace that is not closed as an example. For instance, if X=\ell^2, then you may take U to be the space of all finitely supported sequences.
Question
If we have an n by n matrix called A. How do we know if there is an inverse matrix A^-1 such that the product A * A^-1 is the n by n identity matrix?
Yes Jose Vegas is right. Infact, what you should have if det(A) is non-zero A.Inv(A) = Inv(A).A = I the identity matrix. However, for large value of n it is difficult to find det(A). If you apply, Gauss elimination method, then during elimintion process t some point your diagonal element becomes zero can not be made non-zero by elementary row exchange then the matrix is singular and the inverse does not exist.
Question
Union of sets
There is a condition called 'local finiteness': suppose that every point has a neighborhood that intersects only finitely many of those closed sets. In that case, the union of the closed sets will be closed.
Question
Assume the underlying measure is a probability measure. I think I've heard this is true but I could be wrong.
Dear Geoff Diestel
I am not used to play with weird measured spaces, but on the part of the set where the measure has no atoms, one may use a theorem of Liapounov (not the famous one who worked on ODE) for which Zvi Artstein has given a quite simple proof in the mid 70s (Look for the Extreme Points), and construct a sequence of characteristic functions converging weakly * in L^{\infty} to 1/2 (hence a sequence like the Rademacher one mentioned by Anton Schep).
Luc TARTAR, mathematician
Question
.
Perhaps the simplest illustration of the differences between the integrals of Riemann and Lebesgue is the following. Imagine that you have a lot of coins of different denominations and you need to count how much money you have. Riemann integral answers this question as follows. He consistently adds dignity of another coin to the amount already recorded. Lebesgue integral first splits the set of all coins on the sets of coins of the same denomination. Then calculates the cost of each of the resulting subsets. That is quite simply. And then finds the sum of the resulting values.
Question
This looks "reasonable" from a geometrical point of view. If one of the conditions does not hold, there are obvious counterexamples.
If $\Delta=\sum_i D_{ii}u$ I think you mean non-negative Laplacian, since any concave function has a non-positive Laplacian. However even with non-negative the answer is not. Take u radial in the unit ball in 2d such that $ru_r =\int_0^r f$ with positive f. Then $\Delta u=u_{rr}+u_r/r=f/r \ge 0$. However $u_{rr}=f/r-(1/r^2) \int_0^r f$ can be negative somewhere (take f with compact support in [a,b], then u_{rr} <0 for r >b) and u is not convex.
Question
In the paper by Hwa Kil Kim published in June 14, 2012. What is the meaning of (R^D) and the meaning of (J:(R^D)--->(R^D) is a matrix satisfying (
(J_v )_|_v) for all v in (R^D) ). The name of the paper is:Moreau-Yosida approximation and convergence of Hamiltonian systems on Wasserstein space, and it is on RG.
It looks like D is a natural number and the condition on J means that v and Jv are orthogonal for each vector v, i.e. their scalar product is 0.
Question
It's related to wavelet analysis and I want to know what problem arises in those usual basis.
Hamel bases, well, they exist. And are not really that useful. Except to demonstrate that continuity is, even in very weak forms, a useful condition.
Question
We have the following two equations, where A_1, A_2, B_1 and B_2 are the coefficients that we are interested in to find the values. We need to calculate them so that, for example, p_1(R) and A_1 p_1(q) + B_1 p_2(q) represent the same function. We could do this by requiring that, at some point R_0, both sides of this equation have the same value and the same derivative.
Perhaps, you should define r_c^{+} and r_c^{-}. :)
If you have a differential equation with a set of fundamental solutions, then, of course, the Wronskian is a useful tool.
The constants arising in expressing solutions as linear combinations of fundamental solutions can quite often be obtained from normalization and/or boundary conditions.
Thus, it could well be that your sought relations for A_i,B_i can be obtained from some pointwise information.
However, you should be careful here: It seems that you are using your ODE in terms of two different coordinates q and R, i.e. the ODE has different coefficient functions upon transforming between q and R, which means that you cannot use the above argument so easily, having essentially two different ODEs and two different fundamental solution sets, one as a function of q and one as a function of R. I only can guess that your four functions are meant to be fundamental solutions.
Therefore, let me add some further remarks (hopefully, you will not misunderstand me :): no harm is intended, it is only a recommendation)
* While asking for help with/discussion of mathematical problems, it does not hurt to provide at least pointers to such properties of the functions under consideration ("is solution of the ODE given by ...") Please note also: The "pridmore-brown equation" is not uniquely defined, since there are also two-dimensional versions of it, and there is always the possibility of using a variety of coordinates and/or scaling factors etc.. Also, this ODE is not known to numerical analysts, say.
* It would also be very helpful, if you would use a different notation for the function p_1(q) and p_1(R), e.g. P_1(R) instead of p_1(R). The functions P_1: x -> P_1(x) and p_1: x -> p_1(x) differ by a factor x^3 as is plain from the series expansion. Mutatis mutandis this applies also to the functions P_2 and p_2. Often, a mathematically sound notation helps a lot to clarify the problems.
Question
Can we define a set of complete Jacobi SN orthonormal functions?
Shouldn't k be equal or less than 1?
Question
By using comparison tests, how we can explain the convergence and divergence of these two integrals?
First one obviously diverges as far as the function under the integral is equivalent to \frac{constant}/{1 - w} as w \to 1-0.
The second one converges iff 0 \neq [\alpha_1; \alpha_2]
Question
The problem is that the scientific community places more value on absolute answers, but wicked problems are unanswerable because solutions involve tradeoffs and compromised objectives. There is no point of finality in the solution of wicked problems (Kunz & Rittel, 1972).
Refs:
Kunz, Werner, & Rittel, Horst. (1972). Information science: on the structure of its problems. Information Storage Research, 8, 95-98. doi: 10.1016/0020-0271(72)90011-3
Soft systems / problem structuring methods / soft OR deal with messy problems. For a little primer, have a look at
Fran Ackermann, Problem structuring methods ‘in the Dock’: Arguing the case for Soft OR, European Journal of Operational Research, Volume 219, Issue 3, 16 June 2012, Pages 652-658
Question
We have a waveguide as shown in the attached figure.
At point A it has the singularity that gives the divergence of the integral related to this wave-guide. Now in order to make this integral convergence we have the following three ways.
1-Circumvent the singularity by a contour inside the wave-guide.
2-Circumvent the singularity by a contour outside the wave-guide.
3-Pass the singularity but use the Cauchy principal value.
These three methods can be used to get the convergent result of the integral related to this wave-guide.
Now I am confused about these three methods, I know Cauchy Principal value, but I have less information about the other method. One more thing that I wanted to know is that method one is physical realistic, that is why it is used by them, but I want some argument as to why the other two are not used.
This integral is not just numerically, but mathematically divergent because of the singularity as $w \to 1$! Are you sure the integral is correct or do you want to integrate to $1 - \alpha^2$ for example? The $4l$ is a constant and integrates trivially to $4l(1 - \alpha^2)$ so it looks suspect, or I just don't understand the notation and $l$ depends on $w$ in some implicit way.
The good news is that if you integrate $w$ from $\alpha^2$ to $1- \epsilon$, and let $\epsilon \to 0$, then it diverges like $\int_\alpha^2^ 1 dw \sqrt{(1 - \alpha^2)(1 - 1/\alpha^2)}/(1-w) \approx \sqrt{(1 - \alpha^2)(1 - 1/\alpha^2)} \log(1/\epsilon))$.
Question
In the attached file, there are two inequalities. When will these two different inequalities be true for real value of w?
If I am not too wrong, in problem 1, there are three distinct cases to consider:
1) w < 0
2) w > 1/a²
3) 0 < w < a²
The sign of the denominator is < 0 for case (1) and (2) and > 0 for case (3). From there, we can simplify further the problem to N > 1 or N < 1, where N is the numerator.
Moreover, since (w-a²)(w-1/a²) = w² - (a² + 1/a²)w + 1 then equations such as N > 1 or N < 1 become straightforward to solve.
Question
I need information about Pridmore-Brown equation. How one can solve such equations?
The primary source journal article can be found which has the original development of the equation: D. C. Pridmore-Brown, "Sound propagation in a fluid flowing through an attenutating duct", Journal of Fluid Mechanics, Volume 4, 1958, pp 393 - 406. Solutions to Pridmore-Brown can be found on page 3 of this journal article.
Question
The question is contained in the attached image.
The substitution x=tanh(u) is immediate because of the term (x+y)/(1+xy)
in the functional equation, but then we get (1+tanh(u)tanh(v)) on the right
side of the equation. It happens that
$(1+\tanh(u)\tanh(v))=\cosh(u+v)/(\cosh(u) \cosh( v))$
so it is a natural thing to divide both sides by $\cosh(u+v)$ and then you see the pattern.
Finally, the logarithm transforms the multiplication into addition and one gets
the Cauchy functional equation.
Question
I would like to know about the convergence and divergence of the integral described above. Where dF/dw and dw/dz are the conformal transformation from one plane to another plane. In the attached file, one can see the integral.
Don't quite understand what you mean the singularity. Once you derive to the results, they will tell you whether x=0 is an issue or not. Be aware that the integral you write is an even function. So, it can be written as 2* integral(|x|^p dx) from 0 to +1.
Question
Calculus
I always liked the related volume - area question - rotate the curve y = 1/x for x > 0 around the x axis - it is easy to show that the volume is finite and the surface area infinite - that means you can fill it with paint, but you can't paint it *smile*
what these examples show is that our naive notions of how lengths, areas, and volumes relate to the corresponding integral formulas must be taken with a grain of salt (i.e., very carefully)
Question
Let $\alpha\in[0,\infty)$ be any nonnegative irrational number. Do for any $\varepsilon>0$ there exist $m,n\in\mathbb{N}$ such, that $|n-\alpha m|<\varepsilon$?
From Dirichlet's approximation theorem, for every irrational number α, there are infinitely many fractions p/q (p, q are integers ) such that | α-p/q|<1/q^2 By taking q big enough one can deduce |q α –p|<ε
Question
I also need to check for which real of p the integrals converge or diverge
Integral(|x|^p), with lower limit -1 and upper limit 1.
1/{Sqrt(x^4-1)}<2/x^2
and the latter is easily integrable.
As for the second question, google "principal value".
Question
This question is connected with the oscillations of differential equations.
Since every solution of the linear equation with constant coefficients is the linear combination of exponents this combination is oscillatory if and and only if it has infinitely many zeros.
Well, it looks like you can transform the equation into the form
e^z_1 + e^z_2 = 1
(for a different z_1 and z_2, of course),
and then use some of the reasoning that goes into the first statement
the imaginary parts of the two exponentials still have to add to 0,
and the real parts to 1 -
that should give you some useful constraints,
Question
I want to know theorems/results related to Entire functions (like following question) and suggest me on-line resource for the same.
1) Let f be an entire function. If Re f is bounded then
a) Im f is constant
b) f is constant
c) f = 0
d) f' is a non-zero constant.
Note: More than one option may be correct.
2) Let f be entire function such that lim |f(z)| = 00 (infinity)
|z| -> 00 then
a) f(1/z) has an essential singularity
b) f cannot be a polynomial
c) f has finitely many zeros
d) f(1/z) has a pole at zero.
1. Since Re f is bounded, the range of f is contained in a half plane. We know from Picard's Little theorem that the range of a non constant entire function is the whole plane minus possibly one point. Therefore in our case f is a constant function. So options (a) and (b) are correct. If we donnot wish to use Picard's theorem we can argue in a different manner also. We know that f is either a constant function or a polynomial or a transcendental function. A transcentental function has an essential singularity at infinity. A function having an essential singularity has a range which is dense in the plane.This is the gist of Casorati-Weirstrass theorem on essential singularity. So f is not a transcendental function.The range of a polynomial is the whole plane by Fundamental theorem of algebra. So in our case f is also not a polynomial. So f is a constant function.
2. Answer to this question is also in similar lines. Since lim f(z) = infinity as z tends to infinity, f is a non constant function. So it is either a polynomial or a transcendental function. But it cannot be transcendental function because f shall have an essential singularity at infinity. But then lim f(z), as z tends to infinty along different paths, can have different values. This is again the gist of Casorati-Weirstrass theorem on essential singularity. Therefore f is a polynomial. So options (c) and (d) are correct.
Question
Do you have any idea about 2-quasinormal operators?
$S=\root{9/4}\of{2} \vartheta_2(1/2)-4\vartheta_3(1/2)-1$
$2^(9/4) EllipticTheta[2, 0, 1/2] - 4 EllipticTheta[3, 0, 1/2]$