Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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Intuition on $S_4/K_4\cong S_3$ [duplicate]

This is a question I was pondering on my way to class, and may not have an answer. Take the normal copy of $K_4$ in $S_4$ (The group $K_4 \cong\{e,(12)(34),(13)(24),(14)(23)\}$, not the non-normal ...
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1answer
40 views

Why this process is nonergodic?

I am studying a tutorial on stochastic processes and there's an example in it which I don't understand anything of it. First of all there is this criterion for a mean-ergodic random process: For ...
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1answer
31 views

Is tossing a die in 10 consequent days an ergodic process?

IT maybe an elementary question but I'm totally new to the concept. In Wikipedia, ergodicity is defined as follows: In statistics, the term describes a random process for which the time ...
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1answer
20 views

what's the difference between variable and process from a statistical point of view?

I'm reading a tutorial stochastic process: ergodicity and temporal averages and I'm totally confused. It is said that: Suppose an IID random process whose marginal PDF is Gaussian with mean ...
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4answers
107 views

Definition of Equivalent Norms

Two norms $F,G$ are equivalent when there are constants $a,b$ such that $aF \le G \le bF$. I'm reading about this idea, and so far I've seen that equivalence of norms implies that the underlying ...
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30 views

Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...
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1answer
49 views

Can someone help me understand this: integrating over a discrete set of points yields 0 under Lebesgue integral?

Suppose I had some linear function $f(x)$ and then I sampled the function over the integers to form $f(n)$, what would be the evaluation of the Lebesgue integral of $\int_\mathbb{Z_+} f(n) d\mu$? For ...
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62 views

Show that $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2}$ + Constant

I tried to do this integration by parts and got $\int x\mathrm{e}^{-\alpha x^2}\mathrm dx =\dfrac{-1}{2\alpha} \mathrm e^{-\alpha x^2} +\alpha\int x^3\mathrm{e}^{-\alpha x^2}\mathrm dx$ + constant. ...
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26 views

Disk in analytic topology vs. the spectrum of a Henselian DVR in etale topology

In this informative and concise set of notes on vanishing cycles by Donu Arapura, it is stated that the theory of vanishing cycles ports nicely to the etale world if the role of the disk is replaced ...
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1answer
70 views

A matrix as a point in $\mathbb{R}^{nm}$

I just had a really quick question to ask. I was reading a book on linear algebra and have just been trying to wrap my head around what exactly a matrix represents. At one point, the book said "In a ...
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1answer
26 views

Geometric interpretation of the derivative of a Bezier curve

For a given set of control points $b_0, b_1, \ldots, b_n$, the Bezier curve is defined as $$b^n(t) := \sum_{j=0}^n b_j B_j^n(t),$$ where $B_j^n(t):=\binom{n}{j}t^i(1-t)^{n-i}$ are Bernstein ...
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1answer
53 views

Intuition on the Representable Functor

Given a locally small category C, and an object $C$, the functor: \begin{equation} \mbox{Hom}_\textbf{C}(C,-):\textbf{C} \longrightarrow \textbf{Sets} \end{equation} that sends objects to hom-sets ...
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2answers
74 views

Elements of order $3$ in $\text{Aut}\left(\mathbb{Z}/91\mathbb{Z}\right)$

It looks like someone has already been here, but the question I have goes farther. To summarize my work, as well as the work in the above post, we know that ...
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1answer
58 views

What are the Results of the First and Second Axioms of Countability?

What are the consequences of a space being first or second countable? What was the motivation for these axioms in the first place?
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1answer
377 views

What is the intuition behind generating functions? What makes them valuable?

I'm sorry if this question makes no sense. I have been reading generatingfunctionology and I have been able to solve the problems in the first chapters and I understand the mechanism I have to follow ...
2
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1answer
145 views

Can I have a logical explanation for why this number is so ridiculously close to a whole number? [duplicate]

$$e^ {\pi\sqrt{163}}=262537412640768743.9999999999992\cdots$$ Why does this number run so incredibly close to a whole number? Can I have a logical explanation for why this finding? I know how to ...
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1answer
53 views

Is there a way to visualize, like a picture in mind, the $n$-th derivative?

Is there a way to visualize (like a picture in mind) the $n$-th derivative ? For $n=1$ is the tangent line and we can visualize it quite well. More abstractly is it possible to see the geometric ...
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3answers
51 views

Why does the division algorithm work for converting between number bases?

I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this ...
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186 views

What exactly is a differential? [duplicate]

I've seen the formula for differentials alot, namely $$dy=f'(x)dx$$ but what I think when I see this is that someone is manipulating the "formula" $$f'(x)=\frac{dy}{dx}$$ When I think of ...
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1answer
72 views

Problem with Justifying the Formula for First Order Seperable Differential Equations

I am reading this text http://www.math-cs.gordon.edu/courses/ma225/handouts/sepvar.pdf to justify the method to solve first order seperable differentiable equations, where we are told first told that: ...
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1answer
88 views

$A^tA-AA^t$ in Mathematical Physics

In very different contexts of mathematical physics (rigid body mechanics, fluidodynamics, general relativity, quantum field theory,...) I have come across the following expression: $$ A^tA-AA^t, $$ ...
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93 views

Explaining the meaning of equality

I've been tasked with explaining to a group of people what the notion of equality means in mathematics, I've come up with a working explanation, but would appreciate some input, suggestions etc. ...
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58 views

How do I visualize this quotient space?

If $V = [0,1] \times [0,1] \subset \mathbb{R}^2$. We define the equivalence relation $\sim$ on $V$ as follows: every element $(x,y) \in V$ is equivalent with itself and besides that the three ...
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43 views

Verify $\frac {\partial B} {\partial T} =$ $\frac{c}{(e^\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}e^\frac{hf}{kT}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}$$ The correct answer (I believe) ...
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62 views

Is there a physical interpretation of the alternating property?

A map from lists to list-elements is called "alternating" if any list with repeated elements is mapped to zero. This has statistical significance: regressions on collinear data are bad, dependent ...
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2answers
22 views

Intuition - For every % point that rates rise, a bond’s value will decline by its duration in years.

[Source:] Generally speaking, for every percentage point that rates rise, a bond’s value will decline by its duration (stated in years). So if rates climb by one percentage point, the value of a ...
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1answer
90 views

A math puzzle about slow clock

You have the misfortune to own an unreliable clock. This one loses exactly 20 minutes every hour. It is now showing 4:00am and you know that is was correct at midnight, when you set it. The clock ...
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2answers
45 views

Show that $\frac {\partial B} {\partial T} =$ $\frac{c}{(\exp\frac{hf}{kT}-1)^2}\frac{hf}{kT^2}$

Find an expression for $\frac {\partial B} {\partial T}$ applied to the Black-Body radiation law by Planck: $$B(f,T)=\frac{2hf^3}{c^2\left(\exp\frac{hf}{kT}-1\right)}$$ The correct answer is $\frac ...
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0answers
23 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
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1answer
26 views

Solving for a Binary Matrix: A somewhat unusual method needs justification, and mabye interpretation.

Introduction: Define a "Bit Map" to be a matrix whose entries can only be $0$ or $1$. Then numbers above and beside each column and row indicates how many entries are "filled" with a one. For ...
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40 views

Diagonally dominant matrix — geometric interpretation

I like to have a visual interpretation of mathematical concepts. This is simple for many important kinds of matrices: orthogonal matrices are rotations, diagonal matrices scale along the natural ...
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13 views

Examples of Relation Algebras

Would anyone please direct me to a host of examples of relation algebras. Is there an intuition for what these algebras are to model? That is, groups, for example, model a notion of symmetry; ...
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1answer
48 views

Tangent space as derivations exercise

Thinking of the tangent space to a manifold as derivations is a concept which just kind of eludes me. I am comfortable thinking about tangent vectors as equivalence classes of curves and with the ...
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1answer
50 views

(Visual) Intuition: Division and complex fractions

When treating division as "groups of the numerator" (sorry, I don't know the technical term -- see image), why does a complex fraction in the denominator get added together to produce a 1 (number of ...
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1answer
50 views

Uses of stalks of sheaves and germs

I am trying to understand the motivation behind defining stalks of sheaves, but I suppose my complex geometry is a little weak. I know they are meant to represent germs of holomorphic functions at a ...
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1answer
41 views

Losing a dimension when finding intersection between subspaces

Let $F=\mathbb Z_3, V=F^4$. Let $U=sp\{(1,0,0,0),(1,0,1,0),(0,1,1,1) \} \\W=sp\{(0,0,1,0),(-1,1,0,1),(1,1,1,1) \}$ Find $dim (U\cap W)$ we have $v\in U \text{ and } v\in W$ so $v=v$ ...
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55 views

Intuition behind generic point of a scheme?

I've been reading a little about algebraic geometry and how there seems to have existed this notion of "generic point" on a variety which wasn't carefully defined at first. But often times, ...
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1answer
46 views

How to geometrically interpret intertia of primes in field extensions?

I am trying to understand the intuition of thinking about number theoretic ideas in terms of geometric ones. For example, ramification is something that happens when a "covering" space of a Riemann ...
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0answers
32 views

Motivation for the name “vertical subspace” in the context of fiber bundles.

Let $p:E\to B$ be a smooth fiber bundle with fiber $F$. Consider the vector spaces $V_u=\{x\in T_uE: p_*(x)=0\}$. We call $V_u$ the vertical subspace of the tangent space $T_uE$. How can we see that ...
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1answer
33 views

Problem in deducing gradient in spherical coordinates.

I know the differential displacement in spherical coordinate as $$dr \cdot \widehat{r}+ r d\theta\cdot\widehat{\theta} + r\sin\theta d\phi\cdot \widehat{\phi}$$. But I can't figure out how the ...
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0answers
32 views

What is the intuitive meaning if multiplying by fractional 1?

first post ever on stack exchange in years of using it. Can anyone provide a historical or logical deduction of the reasoning behind multiplication by 1 via a fraction? For instance, in finance ...
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0answers
19 views

Custom equation guidance

i was having a tough time deciding on which SE site to post this rather unique question being that it would help me with a program I'm writing. I decided to come to you guys, not hoping that you would ...
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1answer
34 views

Need some help understanding the condition of the implicit function theorem

The condition for the implicit function theorem is that the (smooth) map $f: \mathbb R^n \to \mathbb R^m$ is locally a (smooth) map of $n-k$ variables if there are locally smooth maps $g_i , i \in ...
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91 views

What is the correct analogue of $\mathbb N$ in a ring of integers?

Question: Let $K$ be a number field. The proper intuitive motivation for the ring of integers $\mathcal O_K$ is that $\mathbb Z$ is to $\mathbb Q$ as $\mathcal O_K$ is to $K$. But what plays the role ...
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1answer
64 views

Why should we expect duality to give useful concepts in category theory?

Why should we expect the abstract notion of flipping arrows in a category to generate useful concepts from other useful ones? What exactly does flipping the direction of arrows mean and why is it ...
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48 views

The shortest path in a metric space with a given metric

My questions seem to be very basic and intuitively correct but I can't formally prove them. Before learning metric spaces, for $R^2$, we always define the distance between 2 points as $d_2 = ...
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149 views

How to intuitively understand prolongations

This question is concerned with the algebraic side of the theory of prolongations as explained in this paper by V. Guillemin and S. Sternberg. Let me first introduce my notation. We're working with a ...
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5answers
642 views

What is the difference between necessary condition & sufficient condition?

My book says : For having extreme point $a$ of function $f$, the necessary condition is that $f'(a) = 0$. However, it isn't a sufficient condition. Now, what is the difference between necessary ...
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1answer
66 views

Is there any intuitive way to think about the gamma function?

Is there a way to realize the gamma function intuitively? My first (and probably correct) guess is no, because, for example, $\Gamma(\frac 12)=\sqrt{\pi}$ doesn't make any intuitive sense at all. ...
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3answers
29 views

Why $\omega$ in $x = \cos(\omega t + \alpha)$ , $\omega$ isn't considered an arbitrary constant?

We know the SHM differential equation is of second-order $$\dfrac{d^2 x}{dt^2} = -{\omega}^2 x$$ . So, the solution of this equation must contain two arbitrary constants. And also we know that $x = ...