A specific example of the GNS construction
general

A specific example of the GNS construction

In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a repres...

Published: Sep 04, 2026

The transitive closure of a relation $R$ on set $A$ whose relation matrix
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The transitive closure of a relation $R$ on set $A$ whose relation matrix

The transitive closure of a relation $R$ on set $A$ whose relation matrix My attempt: If I assumed node $(1, 2, 3)$, then given is: $R=\{(1, 2), (2, 3), (...

Published: Sep 04, 2026

Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction
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Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction

Question: Let $P(n)$ be the statement that $1+\dfrac{1}{4}+\dfrac{1}{9}+\cdots +\dfrac{1}{n^2} <2- \dfrac{1}{n}$. Prove by mathematical induction. Use ...

Published: Sep 04, 2026

How to solve $\int_C Im(z) dz$ where $C$ is the unit circle
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How to solve $\int_C Im(z) dz$ where $C$ is the unit circle

I want to solve $\int_C Im(z) dz$ where $C$ is the unit circle. The first thing is to parameterize the curve, $C$: $$ z(t)=e^{it} $$ where $0\leq t \leq 2...

Published: Sep 04, 2026

Isometries and Orthogonal Matrices
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Isometries and Orthogonal Matrices

I know how to show that multiplying by an orthogonal matrix preserves the angle and distance between two vectors. I have seen everywhere that Orthogonal m...

Published: Sep 04, 2026

If Type Theories are all Logics.
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If Type Theories are all Logics.

So it sounds like Higher Order Logic (HOL) and Type Theory are equivalent. Then there is Intuitionistic Logic and Intuitionistic Type Theory, but I'm...

Published: Sep 04, 2026

Universal factoring method or list of methods (trinomials)
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Universal factoring method or list of methods (trinomials)

I am a student in calculus II. I'm now failing tests solely because I cannot factor; I understand everything else. This is compounded by the fact it ...

Published: Sep 04, 2026

For which angles we know the $\sin$ value algebraically (exact)?
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For which angles we know the $\sin$ value algebraically (exact)?

For example: $\sin(15^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ $\sin(18^\circ) = \frac{\sqrt{5}}{4} - \frac{1}{4}$ $\sin(30^\circ) = \frac{1}{2}$...

Published: Sep 04, 2026

Finding Linear Dependence Relation
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Finding Linear Dependence Relation

Find a linear dependence relation between the following vectors: x1 = (1, -1, 2) x2 = (-3, 2, 1) x3= (1, 2, -3) x4= (2, 3, 1) I've already created a ...

Published: Sep 04, 2026

How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$
general

How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$

the question How to simplify $\sin(x-y)\cos(y)+\cos(x-y)\sin(y)$ my steps I tried to use trig identities on the $\sin(x-y)$ and $\cos(x-y)$ and tried to d...

Published: Sep 04, 2026

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