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Mar
30
comment Why is the delta function the continuous generalization of the kronecker delta and not the identity function?
@Qmechanic: I mean multiplication.
Mar
30
comment Why is the delta function the continuous generalization of the kronecker delta and not the identity function?
@Qmechanic: What I mean is the map/operator $x -> x$ can be represented as scalar number "1" in the sense that $x*1 = x$.
Mar
20
comment What are curves (generalized ellipses) with more than two focal points called and how do they look like?
This link is also interesting: en.wikipedia.org/wiki/Generalized_conic#Multifocal_oval_curves
Mar
15
comment Computing the limits of integration when doing a nontrivial coordinate transform
Maybe related: math.stackexchange.com/questions/7807/…
Mar
10
comment Do functional integrals have countable or uncountable infinite dimensions?
Thanks. I followed that link and found that in in chapter 2.1.2 of users.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic02.pdf they say one takes a "continuum limit" from an expression with a product symbol to get a path integral. I guess that means that functional integrals (at least for path integrals) have uncountable infinite dimensions.
Jan
15
comment Is every square matrix a tensor of 2nd order?
"The total number of indices required to uniquely select each component is equal to the dimension of the array, and is called the order, degree or rank of the tensor." (see en.wikipedia.org/wiki/Tensor#As_multidimensional_arrays). But searching through the article the term "order" seems to be the most used one.
Jan
12
comment Why is there only a complex conjugate, but no real conjugate?
Aren't the roots of $X^2 -1 = 0$ also symmetric, so $X = 1$ and $X=-1$ is an arbitrary choice?
Jan
12
comment Why is there only a complex conjugate, but no real conjugate?
@null Yes. I corrected the text.
Dec
23
comment If $(-1) \cdot (-1) = +1$ shouldn't $(+1) \cdot (+1) = -1$?
Could one not assume Positive × Negative == Zero and Negative × Positive == Zero ?
Dec
23
comment If $(-1) \cdot (-1) = +1$ shouldn't $(+1) \cdot (+1) = -1$?
@Pierre-GuyPlamondon The most symmetric (commutative) solution would be $(+1) \cdot (-1) = 0$.
Dec
9
comment What is the advantage of the Fourier Transform over the Hartley Transform?
I don't think so. A cosine transform is simply the real part of a Fourier transform. But the real part of the Fourier transform can be computed from a Hartley transform like $Re[F(\omega)] = (H(\omega) + H(-\omega))/2$. So only the even part of the Hartley transform is equivalent to a cosine transform. This is because the Hartley transform kernel is a shifted cosine function, which is not symmetric around the origin.
Feb
13
comment What does it mean to represent a number in term of a $2\times2$ matrix?
This can be even be done for en.wikipedia.org/wiki/Split-complex_number, see math.stackexchange.com/questions/3510/…
Feb
11
comment Why is the imaginary part of the logarithm of the gamma function a square wave?
You are right. The definition of the logarithm for negative values in Mathematica is simply $Im(\log(x))=\pi$ for $x<0$ and $0$ for $x>0$$, see also wolframalpha.com/input/?i=plot+log%5Bx%5D+x%2C+-10%2C+10 . I simply didn't expect such a step in the definition of the logarithm.
Feb
9
comment Why is the imaginary part of the logarithm of the gamma function a square wave?
Ok, the gamma function has poles. This explains maybe the period of the square wave. But why is the imaginary part of the logarithm of the Gamma function a constant (0 or $\pi$) between the poles so that one sees a square wave?
Feb
5
comment What's the connection between the Laplace transform and the Fourier transform?
It seems the Laplace transform given by the LCT is the bilateral Laplace transform or two-sided Laplace transform (en.wikipedia.org/wiki/Two-sided_Laplace_transform).
Jan
26
comment Relationship between Levi-Civita symbol and Grassmann numbers?
Wouldn't it then be much less confusing to call these $\theta_i$s "Grassmann vectors" instead of "Grassmann numbers"?
Jan
24
comment Relationship between Levi-Civita symbol and Grassmann numbers?
Ok. But then isn't the product $\theta_i\theta_j$ a tensor (more precisely a pseudotensor, since it anticommutes)?
Jan
24
comment Relationship between Levi-Civita symbol and Grassmann numbers?
Do I get this right: The notation of a Grassmann number like $\theta_i$ does not imply that $\theta$ is a vector with $i$ components. Instead $\theta_i$ is a scalar number like e.g. the imaginary number $i$, correct? So the product of two Grassmann numbers $\theta_i\theta_j$ is not an outer product of two vectors, but just a product of two scalars.
Apr
10
comment complex numbers and 2x2 matrices
Should we not have $a^2+b^2 \ge 0$, because the absolute value of a complex number cannot be smaller than zero?
Apr
9
comment What does it mean to multiply a real matrix by a complex scalar?
I'm confused about the sentence "Multiplication with a fixed complex number is a linear transformation on this two-dimensional vector space". I thought that a complex number itself (not only the multiplication operation) can be represented as a 2x2 matrix, see en.wikipedia.org/wiki/…. Doesn't that mean that the complex numbers are not a vector space, but a matrix space over the real numbers?