Science topics: AnalysisReal and Complex Analysis

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# 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

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

The question details are contained in the attached pdf file.

Is the series in the picture convergent? If it is convergent, what is the sum of the series?

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

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)

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.

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)}?

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.

Let f

_{1}:ℕ⭢ℕ be the identity, that is, ∀n∈ℕ: f_{1}(n)=n. For each n in ℕ, let f_{n:}ℕ⭢ℕ be the map defined recursively as follows.f

_{n}(m) = f_{n-1}(m+1) if f_{n-1}(m) = 2f

_{n}(m+1) = f_{n-1}(m) = 2 if f_{n-1}(m) = 2f

_{n}(m) = f_{n-1}(m) otherwise.Since every map f

_{n}is defined recursively with a transposition in the imageof f

_{n-1}, the map f_{n}is again a bijection that satisfies the following properties.1) ∀n ≤ m: f

_{m}(n)≠ 2.2) ∀n ≤ m the map f

_{n}: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.

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.

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.

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

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”.

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?**

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)

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?

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*}

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.

Hello dear reader,

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 !?

Waiting for your feedback,

JYTA

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.

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?

For example, Circle is not a function. Can any one explain with this inverse and implicit function theorem?

Hello;

I have ended up with a quadratic function of X in the form of:

f(X)= X

^{T}AX-X^{T}BX+C^{T}X 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

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.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.

I need these measurement for the diversity analysis of population of vectors

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?

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 ?

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?

How to construct the system of equations by susbtituting the assumed series solution in the nonlinear partial differential equation? Kindly find the attachment.

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.

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?

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.

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.

I'm looking for optimizing multivalued vector valued function.

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.

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?

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.

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.

Does anyone know an example of a Cauchy continuous function on a bounded subset of R?

By space of measurable functions, I mean L_0(m) where m is a non-atomic sigma-finite measure space.

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.

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 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?

Assume the underlying measure is a probability measure. I think I've heard this is true but I could be wrong.

This looks "reasonable" from a geometrical point of view. If one of the conditions does not hold, there are obvious counterexamples.

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's related to wavelet analysis and I want to know what problem arises in those usual basis.

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.

Can we define a set of complete Jacobi SN orthonormal functions?

By using comparison tests, how we can explain the convergence and divergence of these two integrals?

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

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.

In the attached file, there are two inequalities. When will these two different inequalities be true for real value of w?

I need information about Pridmore-Brown equation. How one can solve such equations?

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.

Let $\alpha\in[0,\infty)$ be any nonnegative irrational number. Do for any $\varepsilon>0$ there exist $m,n\in\mathbb{N}$ such, that $|n-\alpha m|<\varepsilon$?

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.

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.

I want to know theorems/results related to Entire functions (like following question) and suggest me on-line resource for the same.

Answer the following.

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.

Do you have any idea about 2-quasinormal operators?

Hello

The series is 1/2 - 1/4 + 1/8 + 1/16 - 1/32 - 1/64 + 1/128 + 1/256 + 1/ 512 - 1/1024 - and continue this + + - - + + + - - - + + + + - - - - + + + + - - - - ... does this converge and if so what is the limit?