linear algebra – Proof of an Identity using Bilinear Maps

I am attempting to prove the following statement: if $phi$ is a bilinear map which takes $V_1 times V_2$ to $W$ where $V_1$ and $V_2$ are vector spaces of dimension $l_1$ and $l_2$ respectively and $phi(v_1,v_2) in W$ is non-zero for every $v_1 in V_1$ and $v_2 in V_2$, then the image of $phi$ spans a subspace of $W$ with dimension greater than $l_1 – l_2 -1$.

My idea to prove this is to consider that tensors of rank $1$ ${ v_1 otimes v_2 }$ form a subvariety of dimension $l_1 + l_2 – 1$ in $V_1 otimes V_2$, and then the kernel of $phi: V_1 otimes V_2 rightarrow W$ only intersects this subvariety at $0$.

real analysis – Proof verification: Fourier Inversion theorem

I want to prove Fourier Inversion theorem:
$$int_{mathbb{R}^n}widehat{f}(xi)e^{2pi ixcdotxi}dxi=f(x)$$
almost everywhere, where $f,widehat{f}in L^1(mathbb{R}^n)$.

To prove it, we can get a equation
$$int_{mathbb{R}^n}widehat{f}(xi)e^{2pi ixcdotxi}e^{-pi|varepsilon x|^2}dxi=int_{mathbb{R}^n}f(xi)varepsilon^{-n}e^{-pivarepsilon^{-2}|xi-t|^2}dxi$$

for any $varepsilon>0$. For the left side of the equation, we apply the Lebesgue dominated convergence theorem.

(Lebesgue dominated convergence theorem)$~~$Let ${h_k}$ be a sequence of measurable functions on a measurable set $E$. Suppose that the sequence converges pointwise to a function $h$ and is dominated by some integrable function $g$ in the sense that $$|h_k(x)|le g(x)$$for all numbers $kinmathbb{N}_+$ and all points $xin E$. Then $h$ is integrable and $$int_E h(x)~dm=lim_{ktoinfty}int_E h_k(x)~dm.$$

In our case, let $$h(xi):=widehat{f}(xi)e^{2pi ixcdotxi}~~~mbox{ and }~~~g(xi):= |widehat{f}(xi)e^{2pi ixcdotxi}|= |widehat{f}(xi) |$$
and we construct a sequence of measurable functions ${h_k}$ by $h_k(xi):= widehat{f}(xi)e^{2pi ixcdotxi}e^{-pi|k^{-1}x|^2}$. Then clearly $$|h_k(xi)|le g(xi)$$for all numbers $kinmathbb{N}_+$ and all points $xiin {mathbb{R}^n}$. Since $g$ is also integrable, we have that
$$lim_{varepsilonto 0^+} int_{mathbb{R}^n}widehat{f}(xi)e^{2pi ixcdotxi}e^{-pi|varepsilon x|^2}dxi= lim_{kto infty} int_{mathbb{R}^n} widehat{f}(xi)e^{2pi ixcdotxi}e^{-pi|k^{-1}x|^2}dxi= left(lim_{kto infty} e^{-pi|k^{-1}x|^2}right)cdotint_{mathbb{R}^n} widehat{f}(xi)e^{2pi ixcdotxi}dxi= int_{mathbb{R}^n} widehat{f}(xi)e^{2pi ixcdotxi}dxi.$$

My question is: Is my reasoning right? I’m not sure about it. For example, the construction of $h_n(xi)$ seems a little wired to me, but I think I must do it if I want to apply Lebesgue dominated convergence theorem. There was no sequence ${h_n}$ in our case originally, which is required in the dominated convergence theorem.

geometry – “Heron’s proof of AL,BK.CF in Euclid’s Figure meet in a point” in God created the Integers.

I was going through the section “Heron’s proof of AL,BK.CF in Euclid’s Figure meet in a point”, in God Created the Integer’s book . The statement of problem and its proof are both in very vague form. While searching the internet, all I found was link to the “The Thirteen Books Of Euclid”, which also has exact same vague statement and proof. It seems Hawkins lifted it straight from that book. Any how, I want to know the exact problem statement and proof.

real analysis – A question regarding the proof of theorem 1.21 in Baby Rudin

On page 10 of Rudin’s Principles of Mathematical Analysis, Rudin makes a claim that the identity

$b^{n} – a^{n} = (b – a)(b^{n-1} + b^{n-2}a + cdots + a^{n-1})$

yields the inequality $b^{n} – a^{n} < (b – a)nb^{n-1}$ when $0 < a < b$, for real $a$ and $b$, and natural $n$.

Is this result obvious or does it actually require proving? Rudin gives no explanation for why it’s true and I failed at finding any obvious reasons for it being true.

set theory – Equivalence classes properties proof, when [a] ∩ [b] ̸= ∅ then [a] = [b]?

Fix an inetger n > 1. Let R ⊆ Z × Z be the equivalence relation on Z defined by
R = {(a, b) : a = b + kn for some k ∈ Z}. Recall, for a ∈ Z we have (a) = {b ∈ Z : (a, b) ∈ R}.
Fix a, b ∈ Z show that if (a) ∩ (b) ̸= ∅ then (a) = (b) as follows:
(i) What does (a) ∩ (b) ̸= ∅ mean?
(ii) If x ∈ (a) then there is y ∈ (b) such that (x,y) ∈ R; deduce that (a)⊆(b)
(iii) Repeat argument of (ii) to show that (b) ⊆ (a).

linear algebra – Jordan matrix form and polynomial proof.

let $ fin F(x) $ be polynomial. and prove that the matrix $ fleft(J_{n}left(lambdaright)right) $ satisfies

$ (fleft(J_{n}left(lambdaright)right))_{ij}=begin{cases}
frac{1}{left(j-iright)!}f^{(j-i)}left(lambdaright) & 1leq ileq jleq n\
0 & else
end{cases} $

when $ f^(j-i) $ is the (j-i) deriviative of $ f $.

Here’s what i tried:

step 1:
I proved that

$ (left(J_{n}left(0right)right)^{k})=begin{cases}
1 & j=i+k\
0 & else
end{cases} $

step 2:
Using the binom formula, I proved that

$ left(J_{n}left(lambdaright)right)^{k}=sum_{i=0}^{k}binom{k}{i}lambda^{k-i}left(J_{n}left(0right)^{i}right) $

Now assume $ fleft(xright)=sum_{j=0}^{k}a_{j}x^{j} $ then,

$ fleft(J_{n}left(lambdaright)right)=sum_{j=0}^{k}a_{j}left(J_{n}left(lambdaright)^{j}right)=sum_{j=0}^{k}a_{j}sum_{i=0}^{j}binom{j}{i}lambda^{j-i}left(J_{n}left(0right)right)^{i}=sum_{j=0}^{k}sum_{i=0}^{j}a_{j}lambda^{j-i}left(J_{n}left(0right)right)^{i} $

Im not sure how to recognize the (j-i) deriviative out of the expression. And I’m not sure how to continue. Any thoughts will help.
Thanks in advance.

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proof techniques – Lambda calculus without free variables is as strong as lambda calculus?

First question: How would one prove that by removing free (unbound) variables from lambda calculus, and allowing only bound variables, its power is not reduced (it is still Turing-complete) ?

Second question: Is the proposition given above really true? Is lambda calculus sans free variables really Turing-complete?

$text{DSPACE}(O(1))=text{REG}$ Proof?

I want to know why $text{DSPACE}(O(1))=text{REG}$, especially in the direction of why all languages in $text{DSPACE}(O(1))$ can be recognized by a finite automaton. I’ve thought for some time and know that the idea is to encode the finite possibilities of memory as finite states, and also have read https://cs.stackexchange.com/a/57938/112750 . However, I still have a question that I’m unable to answer. A finite automaton needs to read the characters one by one and never look back, but a TM, though restricted to use only $O(1)$ memory, may choose to go back and forth on its input tape. How can I eliminate this kind of looking back problem and construct an automaton that is equivalent to this TM?