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I can find the Feynman-Kac formula for boundary value problem of elliptic equation and for Cauchy problem of parabolic equation. Is there any reference about the elliptic equation in whole space? In particular, I am looking for Feynman-Kac formula of
$$ a(x)u''(x)+b(x)u'(x)+c(x)u(x)=f(x)$$
in $\mathbb{R}$ for regula... |
The following is loosely related to this [question](https://math.stackexchange.com/questions/4860279/what-is-the-correct-sign-for-the-four-vector-potential-gauge-transform-a-mu-t/4860666?noredirect=1#comment10362260_4860666):
>[...], the most general renormalisable Lagrangian that is invariant under both Lorentz trans... |
I want to prove Proposition 10.23 from Tomas Björk's ''Arbitrage theory in continuous time'' in the snippet below.
[![][1]][1]
[1]: https://i.stack.imgur.com/zqZQP.png
My attempt: For simplicity, assume everything is one-dimensional, with one risky asset $S$ and a risk-less asset $S^0$. Assume also that $r... |
Is this the correct proof of Proposition 10.23 in Björk's ''Arbitrage theory in continuous time''? |
The following comes from Page 57 of Davenport's Multiplicative Number Theory. My question is: why is (6) being convergent necessary in deducing the result from (5)?
[![enter image description here][1]][1]
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/nHgeJ.png
[2]: https://i.stac... |
The following is loosely related to this [question](https://math.stackexchange.com/questions/4860279/what-is-the-correct-sign-for-the-four-vector-potential-gauge-transform-a-mu-t/4860666?noredirect=1#comment10362260_4860666):
>[...], the most general renormalisable Lagrangian that is invariant under both Lorentz trans... |
> And btw is there some technique to derive a closed-form (integral is in desire) representation for almost any transcendental equations? I read some articles but they gave too many restrictions on the transcendental equation. (Riemann's method)
$\DeclareMathOperator WW$Here is another representation that did not se... |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
|
How to check whether $f(X)=X^4-6X^2+1$ is irreducible or not in $\mathbb{Q}[X]$?
I have tried to apply Eisenstein's Criterion to a linear substitution, to reduce the $f(X)$ modulo a prime and to write it as $(X^2+aX+c)(X^2+bX+d)$, but all without success.
I know that $f(X)$ has four zeros, namely $\pm\sqrt{-3},\p... |
How to check whether $f(X)=X^4-6X^2+1$ is irreducible or not in $\mathbb{Q}[X]$? |
> And btw is there some technique to derive a closed-form (integral is in desire) representation for almost any transcendental equations? I read some articles but they gave too many restrictions on the transcendental equation. (Riemann's method)
$\DeclareMathOperator WW$Here is another representation that did not se... |
If I have a sphere with radius 1, and a section of the sphere defined by having a polar angle bounded by some constant θ and an azimuthal angle bounded by some constant φ, how do I generate random points on this surface such that the points are uniformly distributed? |
I want to define a sequence for the Proth numbers (these are numbers of the form $h\cdot 2^k+1$ for a natural $k$ and an odd $h<2^k$).
I want it ordered at first by the exponent, then by the coefficient.
So I get the sequence $(a_n)_n = (1\cdot 2^1 + 1, 1\cdot 2^2+1, 3\cdot 2^2+1, 1\cdot 2^3+1\dots).$
But I am re... |
Have you studied the proof of Dirichlet's theorem enough to appreciate how a sum over the primes subject to a congruence condition $a \bmod q$, where $\gcd(a,q) = 1$, can be broken up into a linear combination over all Dirichlet characters mod $q$ of sums over all primes, where the term associated to a prime $p$ is wei... |
Let ° be a closed binary operation on $\mathbb{R}^2$ such that ° is commutative, associative and has a neutral element. Assume that for all pairs (a,0) ,(b,0) $(a,0)°(b,0)=(a,0)+(b,0)$ where $+$ is the usual componentwise sum for pairs. Is it true that ° and + are equal? If not what's a counterexample? |
Is a commutative and associative with neutral element operation on R^2 necessarily componentwise sum? |
I was shocked to watch Anton Petrov's latest video, ["Wow, Incredible Evidence That Universe Is Not Symmetric After All"](https://youtu.be/d15b7udf_ac), where he says that the Tetrahedron is the simplest object that is not Z(2)-symmetric.
Does this mean that you can't connect four equilateral triangles to form a sym... |
Have you studied the proof of Dirichlet's theorem enough to appreciate how a sum over the primes subject to a congruence condition $a \bmod q$, where $\gcd(a,q) = 1$, can be broken up into a linear combination over all Dirichlet characters mod $q$ of sums over all primes, where the term associated to a prime $p$ is wei... |
Techniques for proving this series identity? |
What about if $(a,c)°(b,d)= (a+b,c*d)$?
Then, certainly , it satisfies the fact that ° is commutative(easy to see) and associative (addition and multiplication are associative), and there is an identity element, namely (0,1). Therefore, ° and + are not necessarily equal. |
Why is $\displaystyle{\lim_{h \to 0} \frac{0}{h} = 0}$?
We can re-write the LHS as
$$\left(\lim_{h \to 0} 0\right) \cdot \left( \lim_{h \to 0} \frac{1}{h} \right)$$
$$0 \cdot \left( \lim_{h \to 0} \frac{1}{h} \right)$$
the term in the brackets is undefined, since $1 / h$ does not have a limit at $0$.
I am... |
Why is $\lim_{h \to 0} \frac{0}{h} = 0$? |
I recently found out about the [Steinmetz Solids][1], obtained as the intersection of two or three cylinders of equal radius at right angles. If we set the radius $ = 1 $ the volumes are respectively
$ V_2 = \frac{4}{3}\cdot4 $
$ V_3 = \frac{4}{3}\cdot6(2-\sqrt{2}) $
A natural question is to find the volume of... |
I'm trying to implement the exponentiation of numbers in my [Scheme interpreter](https://lips.js.org/). I'm wondering how one Scheme implementation (Kawa) implemented this equation:
$(10+10i)^{10}$
Kawa give this output:
$+320000000000i$
My implementation (same as few others) return:
$9.79717439317883e-5... |
I am asking if existence, taken as an assumption for a solution $L$ to the linear operator equation $$\mathcal{E}L = F$$ with conditions on $F$, implies further conditions on $F$ and uniqueness of a generic form for the terms in $L$ not annihilated by $\mathcal{E}$.
The variational problem giving rise to the operato... |
How to check whether $f(X)=X^4-6X^2+1$ is irreducible or not in $\mathbb{Q}[X]$?
I have tried to apply Eisenstein's Criterion to a linear substitution, to reduce the $f(X)$ modulo a prime and to write it as $(X^2+aX+c)(X^2+bX+d)$, but all without success.
I know that $f(X)$ has four zeros, namely $\pm\sqrt{\frac... |
I'm trying to implement the exponentiation of numbers in my [Scheme interpreter](https://lips.js.org/). I'm wondering how one Scheme implementation (Kawa) implemented this equation:
$(10+10i)^{10}$
Kawa give this output:
$+320000000000i$
My implementation (same as few others) return:
$9.79717439317883e-5... |
I'm new to matrix multiplication and was trying to generate a sequence of consecutive numbers from the product of two $2 \times 2$ matrices and noticed this pattern. Let
$$
A = \begin{pmatrix} n & n+1 \\ n+2 & n-1 \end{pmatrix}, \quad B = \begin{pmatrix} m & m-1 \\ m-1 & m \end{pmatrix}, \quad C = AB
$$
where
... |
Let $m,n\geq 2$ and consider the rectangular lattice of $mn$ vertices in $\mathbb{R}^2,\ (i,j);\ i\in \{1,2,\ldots,m\},\ j\in \{1,2,\ldots,n\}.\ $ Call these vertices $X_1, X_2, \ldots, X_{mn}.$ Is there a Hamiltonian cycle of these $mn$ points i.e. vertices such that no two lines (i.e. edges) are parallel? [Edges are ... |
I have a straight line segment PQ between two points $P(x_0,y_0),Q(x_1,y_1).$
An arc of a circle is drawn whose tangents at P,Q make angles $\alpha, \beta$ to PQ.
First: How to find the center $(x0,y0)$ and radius of is circular arc $R$?
Second: How to find intersection(s) between two such arcs PQ and RS? ... |
I'm new to matrix multiplication and was trying to generate a sequence of consecutive numbers from the product of two $2 \times 2$ matrices and noticed this pattern. Let
$$
A = \begin{pmatrix} n & n+1 \\ n+2 & n-1 \end{pmatrix}, \quad B = \begin{pmatrix} m & m-1 \\ m-1 & m \end{pmatrix}, \quad C = AB
$$
where
... |
I'm trying to implement the exponentiation of numbers in my [Scheme interpreter](https://lips.js.org/). I'm wondering how one Scheme implementation (Kawa) implemented this equation:
$(10+10i)^{10}$
Kawa give this output:
$+320000000000i$
My implementation (same as few others) return:
$9.79717439317883e-5... |
I was thinking about this concept the other day, but I couldn't reach a solid conclusion.
Visualise your right hand in front of you, and put your thumb, second and third finger in a configuration such that they are all perpendicular to one another(like the right hand rule). Draw 3d axes, and allow the hand to rotate... |
Special Isomorphism? |
I'm new to matrix multiplication and was trying to generate a sequence of consecutive numbers from the product of two $2 \times 2$ matrices and noticed this pattern. Let
$$
A = \begin{pmatrix} n & n+1 \\ n+2 & n-1 \end{pmatrix}, \quad B = \begin{pmatrix} m & m-1 \\ m-1 & m \end{pmatrix}, \quad C = AB
$$
where
... |
Can this Non-Linear Residual Function, $R(x)=((x+1)^3-1)^{1/3}-(x+1)$, be expressed exactly using a infinite series (using integer powers in $x$) or a finite polynomial function in $x$?
For example I have found one approximate series / polynomial function that is a reasonably accurate approximation above $x=2$
$$... |
Can this Non-Linear Residual Function, $R(x)=((x+1)^3-1)^{1/3}-(x+1)$, be expressed exactly using an Infinite Series or Polynomial Function in $x$? |
Consider a projective bundle $\pi:\mathbb P(E)\to X$ that comes from a vector bundle of $E$ on a variety $X$. My question is about what is the line bundle "$\mathcal{O}(1)$" on $\mathbb P(E)$. To my understanding, this line bundle defines $\mathcal{O}(1)$ on each fiber and satisfies some universal properties, and shoul... |
Does $\mathcal{O}(1)$ on a projective bundle depends on the presentation of vector bundle? |
I recently found out about the [Steinmetz Solids][1], obtained as the intersection of two or three cylinders of equal radius at right angles. If we set the radius $ = 1 $ the volumes are respectively
$ V_2 = \frac{4}{3}\cdot4 $
$ V_3 = \frac{4}{3}\cdot6(2-\sqrt{2}) $
A natural question is to find the volume of... |
I am trying to find the discrepancy between an integral and a discrete approximation.
The "ideal" inner product between two signals $f(x)$ and $g(x)$ is an integral,
$$
\langle f, g \rangle_\text{continuous} \equiv \int\limits_{-\infty}^{\infty} f(x) g(x) dx.$$
However, in my application, $f$ and $g$ are someho... |
Relate a continuous inner product to a discret one by expanding a kernel around a Dirac delta function? |
What about this one?
>$$\int_{0}^{1} \left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right) - 1\right) \, dx = e^{\pi}.$$
[1]: https://en.wikipedia.org/wiki/Gelfond%27s_constant |
What's $\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\text{erf}\left(\frac{\sqrt 2 R\cos\theta}{\sigma}\right)\text d\theta$? |
Is there a violation example that rejects the continuity of the derivative of a function at the relative extremum points, the meaning is to assume that the function in the open interval l is the derivative of a point c is the relative extremum in this interval, in this case the derivative of this point will be zero, bu... |
I'm new to matrix multiplication and was trying to generate a sequence of consecutive numbers from the product of two $2 \times 2$ matrices and noticed this pattern. Let
$$
A = \begin{pmatrix} n & n+1 \\ n+2 & n-1 \end{pmatrix}, \quad B = \begin{pmatrix} m & m-1 \\ m-1 & m \end{pmatrix}, \quad C = AB
$$
Comput... |
There is a statement about locally convex spaces in Reed & Simon, Methods of Modern Mathematical Physics (Vol I, Section V.1) that is given without a proof.
The statement is:
Given two families of seminorms $\{\rho_\alpha\}_{\alpha \in A}$ and $\{d_\beta\}_{\beta \in B}$ over a locally convex space X, the familie... |
Let $\mathcal{M}$ be a model category with class of weak equivalences $W$ and $\mathcal{J}$ a small category. For a (small) category $\mathcal{J}$ the category of diagrams $\operatorname{Fun}(\mathcal{J}, \mathcal{M})$ inherits a class of weak equivalences $W_{\mathcal{J}}$, those natural transformations which are obje... |
Melissa O'Neill, (in her PCG prng papers), makes reference to prng "party tricks" where she says that you can seed a prng appropriately in order to force it to generate a sequence that contains, (at some point), a desired subsequence of values. She goes on to show a prng that "contains" the contents of a .zip file in i... |
Can you seed a seekable prng in order to force it to generate a given subsequence of values? |
I encountered this sum
$S(N,j)= \sum _{n=1}^{\frac{N}{2}}
-\frac{2 \sqrt{2} h \sin ^2\left(\frac{\pi j n}{N+1}\right)}{(N+1) \sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}}$
where $h\in \mathbb{R}^+$, $\{n,j\}\in \mathbb{Z}^+$ and $N\to \infty$
which has a curious property that for a fixed $h$ (say 5/... |
> Let $p$ be prime, $n \in \mathbb{Z}$. Show that if $\Phi_{5}(n) = n^4 + n^3 + n^2 + n + 1 \equiv 0 \pmod{p}$, then $p = 5$ or $p \equiv 1 \pmod{5}$.
If $p = 5$, then we are done. If $p \neq 5$, then I believe I may follow an argument [here][1] by Thomas Andrews, whose answer I do not fully understand. We notice t... |
I can't speak to your professors intent but, generally it is best practice when introduced to a subject to either write completely algebraic sentences or complete prose. This is because it is often not entirely obvious when to use which one, and so switching between them can create awkward sentences that are repetitive... |
$B_0$ is 1, $B_1$ is $\pm 1/2$, and so on (I'm not going to list every single Bernoulli number). The Bernoulli numbers $B_n$ are defined for every integer $n \ge 0$. But what about $B_{1/2}$? And $B_i$? How can the Bernoulli numbers be extended beyond integer terms? So far the only analytic continuation of $B_n$ I coul... |
Analytic continuation of Bernoulli numbers: $B_1/2$? |
I see that a typical definition of the cartesian product $A \times B$ is $\{ x \in \wp\wp A \cup B: \exists a \exists b (a \in A \wedge b \in B \wedge x=(a,b)) \}$. I have come up with an alternative that does not use the quantifiers and I don't think is entirely equivalent, logically speaking, to the traditional one a... |
I encountered this sum
$S(N,j)= \sum _{n=1}^{\frac{N}{2}}
(-)^j\frac{2 \sqrt{2} h \sin ^2\left(\frac{\pi j n}{N+1}\right)}{(N+1) \sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}}$
where $h\in \mathbb{R}^+$, $\{n,j\}\in \mathbb{Z}^+$ and $N\to \infty$
which has a curious property that for a fixed $h$ (sa... |
This is based on my rough understanding, so let me know which part if any is wrong. Suppose $M$ is a $(G,X)$-manifold, where $X$ is a homogeneous $G$-space. Pullback a $(G,X)$ structure from $X$ to its universal cover $\widetilde{X}$. The $(G,X)$-automorphisms $\mathrm{Aut}(\widetilde{X})$ form a Lie group acting trans... |
For $(G,X)$ - manifolds can we assume $X$ is simply connected? |
My book says
>Prove that if $z\in\mathbb C$ and $\lvert\sin z\rvert\le 1$, then $z\in\mathbb R$.
But I think this can't be true as
$$\lvert\sin z\rvert^2=\sin^2x+\sinh^2y$$
and so if $\lvert\sin z\rvert\le 1$, then, $$\sinh^2y\le1-\sin^2x=\cos^2x.$$
Clearly we can find some $y\neq 0$, such that $\sinh^2y\l... |
I encountered this sum
$$S(N,j)= 2 \sqrt{2}\cdot\sum _{n=1}^{\frac{N}{2}}
(-1)^j\frac{h \sin ^2\left(\frac{\pi j n}{N+1}\right)}{(N+1) \sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}},$$
where $h\in \mathbb{R}^+, \{n,j\}\in \mathbb{Z}^+$ and $N\to \infty.$
It has the curious property that for a fixed $h$ (... |
Say you have an proportion such that
a is proportional to b^2 * c^4. If one doubles a, but leaves c constant, how much does b change by? This is just a sample question, but is there a rule that I can use to find the relation between these three? For example, if c doubles as well as a, how does b change? This is jus... |
I am given a function $f(x)=x^2$ on $(0,1)$. First, I found the cosine series for the even periodic extension on $[-1,1)$ which is $\frac{3}{2}+\sum_{n=1}^{\infty}\frac{4(-1)^n}{(n\pi)^2}cos(n\pi x)$. I then found the sine series for the odd periodic extension on $[-1,1)$ which is $\sum_{n=1}^{\infty}\frac{4(-1)^n +4}{... |
I encountered this sum
$$S(N,j)= \frac{2 \sqrt{2}h(-1)^j}{N+1}\cdot\sum _{n=1}^{\frac{N}{2}}
\frac{\sin ^2\left(\frac{\pi j n}{N+1}\right)}{\sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}},$$
where $h\in \mathbb{R}^+, \{n,j\}\in \mathbb{Z}^+$ and $N\to \infty.$
It has the curious property that for a fixed ... |
It's said that the primitive concepts of set theory are those of "set" and "membership", then all axioms of set theory must begin with "Let $A$ be a set" or "Let $x\in A$", but they don't. For example, let us consider the subset axiom:
<b>Subset Axiom.</b> Let $\varphi(x)$ be a formula and let $A$ be a set. Then the... |
If set theory only contains the notions of “set” and “is a member of” as primitives, how can an axiom of set theory refer to a “formula”? |
I would like to find an entire function satisfying $f(0)=1$ and $f(n^{1/3}-n^{1/6})=0$ for every $n\geq 2$. My initial thought was to use the Weierstrauss Factorization Theorem which states if {$a_n$} is a sequence of distinct complex numbers such that $a_0=0$ and $a_n \rightarrow \infty$ as $n\rightarrow \infty$ and $... |
Is the matrix $A^2-A+I$ invertible?
$A^2=A-I \cdot A$
$ A^3 = A^2 - A$
$A^2 = A$
Thus, I conclude that $\lambda=0$ is one of the eigenvalues of this matrix and it isn't invertible. My conclusion comes from the fact that the eigenvalues of the matrices $A,A^2,\ldots$ are all connected, and if $A^3=O$ that me... |
If $A^3=O$ is the matrix $A^2-A+I$ invertible? |
Say you have an proportion such that
a is proportional to b^2 * c^4. If one doubles a, but leaves c constant, how much does b change by? This is just a sample question, but is there a rule that I can use to find the relation between these three? For example, if c doubles as well as a, how does b change? This is jus... |
Is there a counter example that rejects the continuity of the derivative of a function at the relative extremum points? The meaning is to assume that the function in the open interval l is the derivative of a point c is the relative extremum in this interval, in this case the derivative of this point will be zero, but ... |
I would like to find an entire function satisfying $f(0)=1$ and $f(n^{1/3}-n^{1/6})=0$ for every $n\geq 2$. My initial thought was to use the Weierstrass Factorization Theorem which states if {$a_n$} is a sequence of distinct complex numbers such that $a_0=0$ and $a_n \rightarrow \infty$ as $n\rightarrow \infty$ and $k... |
I'm new to matrix multiplication and was trying to generate a sequence of consecutive numbers from the product of two $2 \times 2$ matrices and noticed this pattern. Let
$$
A = \begin{pmatrix} n & n+1 \\ n+2 & n-1 \end{pmatrix}, \quad B = \begin{pmatrix} m & m-1 \\ m-1 & m \end{pmatrix}, \quad C = AB
$$
Comput... |
A common schema in intuitionistic, constructive set theories is that of *$\it{\Delta_0}$ separation*:
$$
\forall x\exists y\forall z(z\in y\leftrightarrow z\in x\wedge\phi(x, z))\text{ provided $\phi$ is a $\Delta_0$ formula}
$$
So chosen as $\Delta_0$ sets are clearly predicatively defined, and full separation... |
In the scenario you're asking about, you're dealing with variables in **direct proportion** to one another, which simply means that if you increase one, then the other also increases (as opposed to **inverse proportion**, where an increase in one variable results in a decrease of the other). So we may conclude that inc... |
I'm confused about independence of events in the following question from Blitzstein and Hwang, chapter 3 (Random Variables):
> Let X be the number of Heads in 10 fair coin tosses. Find:
(a) the conditional PMF of X, given that the first two flips landed Heads, and
(b) the conditional PMF of X, given that at least ... |
In "Interpretable Machine Learning: A Guide For Making Black Box Models Explainable", I found the following for Friedman's H-statistic:
$$PD_{jk}(x_j, x_k) = PD_j (x_j) + PD_k (x_k),$$
where $PD_{jk}(x_j, x_k)$ is the two way partial dependence function of both features and $PD_j (x_j)$ and $PD_k (x_k)$ the partial d... |
Is there an opposite of a dirac delta function, a function that is infinitely wide and infinitesimally high?
The dirac delta function is defined as:
$$
\delta(x) =
\begin{cases}
0, & \text{if } x \neq 0\\
\infty, & \text{if } x = 0
\end{cases}
$$
where
$$
\int_{-\infty}^{\infty} \delta(x) \, dx = 1
\\
\i... |
Is there an opposite of a dirac delta function, a function that is infinitely wide and infinitesimally high? |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
***
Notice that double sums over rational functions can ... |
Through the usual Riemann sum developement, it holds that as $N \to \infty,$ $$ \sum_{n = 1}^{N/2} \frac{1}{N+1} \frac{\sin^2 ( \pi j \cdot(n/N+1)) }{ \sqrt{2 h^2 +1 + \cos(2\pi j (n/N+1))} } \to \int_0^{\frac 12} \frac{\sin^2(\pi j x)}{\sqrt{2h^2 +1 + \cos(2\pi jx)}} \mathrm{d}x.$$
Note that $\cos(2\pi j x) = 2\cos... |
I tried to prove this the following way, but not sure if this is correct?
n^k \leq c \cdot 2^n
\log_2(n^k) \leq \log_2(c \cdot 2^n)
\log_2(n^k) \leq \log_2(c) + \log_2(2^n)
\log_2(n^k) \leq \log_2(c) + n
\text{If } n = k, \text{ then } \log_2(k^k) \leq \log_2(c) + k
\text{If } c = k^k, \text{ then } \... |
Can you have a graph/plot with more than 3 sides? |
I tried to prove this the following way, but not sure if this is correct?
$n^k \leq c \cdot 2^n$
$\log_2(n^k) \leq \log_2(c \cdot 2^n)$
$\log_2(n^k) \leq \log_2(c) + \log_2(2^n)$
$\log_2(n^k) \leq \log_2(c) + n$
$\text{If } n = k, \text{ then } \log_2(k^k) \leq \log_2(c) + k$
$\text{If } c = k^k, \tex... |
Through the usual Riemann sum developement, it holds that as $N \to \infty,$ $$ \sum_{n = 1}^{N/2} \frac{1}{N+1} \frac{\sin^2 ( \pi j \cdot(n/N+1)) }{ \sqrt{2 h^2 +1 + \cos(2\pi j (n/N+1))} } \to \int_0^{\frac 12} \frac{\sin^2(\pi j x)}{\sqrt{2h^2 +1 + \cos(2\pi jx)}} \mathrm{d}x.$$
Note that $\cos(2\pi j x) = 2\cos... |
About a year ago, while using GeoGebra, I discovered a beautiful property of a triangle with sides 1,2,2.
It is that many of the important centers of this triangle are located at equal distances in a row, so much so that I called this triangle the “numbers line triangle.”
You can consider the following two imag... |
Let $\Sigma$ an alphabet. We define the primitive recursive set as follows:
$PR_{0}^{\Sigma}$ contains the succesor, predecessor, projection and constant functions. It also contains the $d_a$ function, defined if $a \in \Sigma$, s.t. $d_a(\alpha) = \alpha a$. Then, recursively,
\begin{align*}
PR_{k}^{\Sigma}... |
**I. Question**
We have a quartic in $v$ to be made a square,
$D^2 = 4(-6 - 2u + u^2)(2 - 2u + u^2) - 8(-2 - 4u + u^2)(2 - 2u + u^2)v - 16u(4 - 3u + u^2)v^2 - 4(4 - 12u + 4u^2 - 2u^3 + u^4)v^3 + (4 - 8u - 4u^3 + u^4)v^4$
If there is rational $(u,v)$, then it is birationally equivalent to an elliptic curve. An... |
Consider the figure
[![complex plane][1]][1]
Which represents the vectors $z$ and $w$ in the complex plane (the drawing is an approximation).
I want to prove that $z+w$ is just the diagonal of the parallelogram formed by $z$ and $w$.
If we define the diagonal of such a parallelogram by
$d=\sqrt{w^2+z^2-... |
About a year ago, while using GeoGebra, I discovered a beautiful property of a triangle with sides 1,2,2.
It is that many of the important centers of this triangle are located at equal distances in a row, so much so that I called this triangle the “numbers line triangle.”
You can consider the following two imag... |
About a year ago, while using GeoGebra, I discovered a beautiful property of a triangle with sides 1,2,2.
It is that many of the important centers of this triangle are located at equal distances in a row, so much so that I called this triangle the “numbers line triangle.”
You can consider the following two imag... |
Analytic continuation of Bernoulli numbers: $B_{1/2}$? |
I am working on a project and need to apply a certain (Lorentzian) conformal transformation, $\psi:\mathbb{R}^2\mapsto \mathbb{R}^2$, to a figure which I have generated numerically in Mathematica. I know that the new metric will be conformally flat. Hence, I know that the pullback metric will be $g_\text{ab}=\psi^*\eta... |
How could ‘two angles cannot be equal, but two angles can be congruent? ‘ My understanding was if a shape is congruent then it would be equal in size and shape. If two angles are congruent, how could they not be equal? |
Angles can be congruent but not equal? |
Let $X$ be a connected topological space, and $Z$ be a subspace of $X$. Suppose $f:X \to Z$ is a retraction, i.e. a mapping of topological spaces such that $f \circ i = id_Z$, where $i:Z \to X$ is the inclusion.
Is it true that all fibers of $f$ are connected? |
Are the fibers of a retraction always connected? |
Can a shape be both similar and congruent? |
In homogeneous second-order ordinary differential equations (ODE) with constant coefficients, of the form:
\begin{equation}
ay''+by'+cy=0,
\end{equation}
is there any conclusion about the effects of each of the parameters $a$, $b$ and $c$ on the graph of an ODE solution? It's a curiosity I've been having, but I've ... |
Suppose $S$ is a convex set in $\mathbb{R}^2$. A point $w$ satisfies that $w=\lambda_1 x+\lambda_2 y+\lambda_3 z$, where $\lambda_1+\lambda_2+\lambda_3=1$, all three coefficients are positive, and $x, y, z \in S$. I want to prove that $w$ is not a vertex of $S$.
I checked the definition of an extreme point, but it ... |
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