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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How to show $k$-algebras are isomorphic in practice

I am working through some problems which require me to show when some $k$-algebra ($k$ a field) maps are isomorphisms. Unfortunately, I've got myself a bit confused with definitions and the like, and ...
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291 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
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3answers
188 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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4answers
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Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
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2answers
84 views

How to intuit 'only if'?

I already know, and so ask NOT about, the proof of:   $A$ only if $B$   =   $A \Longrightarrow B$. Because I ask only for intuition, please do NOT prove this or use truth tables. My problem: I try ...
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1answer
50 views

What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
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What is the 'implicit function theorem'?

Please give me an intuitive explanation of 'implicit function theorem'. I read some bits and pieces of information from some textbook, but they look too confusing, especially I do not understand why ...
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325 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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107 views

Why General Leibniz rule and Newton's Binomial are so similar?

The binomial expansion: $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$ The General Leibniz rule (used as a generalization of the product rule for derivatives): $$(fg)^{(n)} = ...
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1answer
58 views

pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
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10answers
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What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
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29 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
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2answers
570 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
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1answer
52 views

Why do the interesting antihomomorphisms tend to be involutions?

Given a semigroup $S$, define that an antihomomorphism on $S$ is a function $$* :S \rightarrow S$$ satisfying $(xy)^* = y^*x^*.$ Examples abound. Consider: Transposition, where $S$ equals the set ...
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3answers
64 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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3answers
468 views

What's the intuition of the transpose of a matrix? [duplicate]

I know the transpose is to swap the columns and rows of a matrix. And $A^T$$A$ is a symmetric matrix which elements are the inner product of each column of $A$. But I didn't understand the intuition ...
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1answer
1k views

Knuth's up-arrow notation - Is there practical use for the numbers involved?

From Wikipedia, Knuth's up-arrow notation begins at exponentiation and continues through the hyperoperations: $a \uparrow b = a^b$ $a \uparrow\uparrow b = {\ ^{b}a} = ...
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1answer
429 views

“Poissonization” and intuition

In a french book, "Calcul des probabilités" from Foata and Fuchs, I found this theorem, which they call "Poissonization". "Let $(I_k)_{k \in \mathbb{N}}$ be a sequence of independent variables with ...
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874 views

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
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1answer
88 views

Notions of consistency / heterogeneity in sets of vector values

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
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1answer
308 views

If sup A < sup B, there exists an element b ∈ B that's an upper bound for A. (S.A. pp 18 q1.3.8)

My Figure: By definition of $\sup B$, $\sup B$ is an upper bound for $B$. Set $e = \sup B − \sup A > 0$. By Lemma 1.3.7, there exists an element $b ∈ B$ satisfying $\begin{align} & \sup B − ...
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Intuitive Explanation of Bessel's Correction

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive ...
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1answer
176 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://math.uga.edu/~pete/2400calc2.pdf. This theorem can be illustrated ...
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48 views

Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
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1answer
102 views

What is an example of a proof that explicitly relies on the law of excluded middle?

I was talking with a friend about logic and I realized she might be an intuitionist. I was looking online for a proof that explicitly uses the law of excluded middle to see if she would have an issue ...
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1answer
29 views

intuition about cubic splines vs quadratic splines (degree 3 vs degree 2).

my intuition about quadratic(degree 2) splines is that by the help of its three variables (in each sub-interval) you can make a piecewise differentiable function on the whole interval. in the process ...
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5answers
3k views

What is a Markov Chain?

What is a intuitive explanation of a Markov Chain, and how they work? Please provide at least one practical example.
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2answers
75 views

How to motivate vectors as derivations?

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space ...
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3answers
61 views

Where did the idea of hermite interpolation came from?

I am given the Hermite interpolation formula directly in my text book without ANY explanations about how it was first made (obviously it was somehow constructed for the first time with some sort of ...
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40 views

How to prove that $2^\sqrt2$ is irrational [duplicate]

I know the result is intuitive, but is there a way to mathematically prove it. I tried taking logarithm or tried proving by contradiction, but I don't know how to proceed. Thanks.
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Intuition about $v\otimes w$

In Physics and Differential Geometry usually tensors of type $(k,l)$ on a vector space $V$ over $\mathbb{F}$ are defined as multilinear functions $$f : \underbrace{V\times\cdots\times V}_{k \ ...
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28 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
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4answers
82 views

How to visualize permutations?

I'm getting a warning that this is a subjective question, and it very well probably is. But nevertheless, it is still a valid question that helps in the studying of mathematics from my point of view. ...
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$h$ invariant of forms and its relation to Birch rank

Let $k$ be a field. Given a form $F_i \in k[x_1,..., x_n]$ of degree $d$, the $h$-invariant $h_k(F_i)$ is defined to be the least positive integer $h$ such that $F_i$ can be written as $$ A_1B_1 + ...
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Explanation of Gauss sums' preference for the North-East?

Define the Gauss sums algebraically as follows: $$\mathcal{G}_q = \sum_{a=0}^{q-1} e^{2 \pi i a^2 / q}$$ Then the result ends up being $\sqrt{q}, 0, i \sqrt{q}, (1+i)\sqrt{q}$ depending on $q \equiv ...
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Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
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391 views

Novel approaches to elementary number theory and abstract algebra

As a part of a university course, I'll have to study Herstein's Topics in algebra and Hardy&Wright's Introduction to the theory of numbers. Can you suggest some books (to be used as companions) ...
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Intuition behind spectral radius of a graph

Suppose that I have a graph G, along with its respective adjacency matrix A. The definition of how one can compute the spectral radius of this graph is not hard to grasp, but I was wondering about the ...
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1answer
27 views

Is there an interpretation for writing a polynomial in $x$ as a polynomial in $(x-b)$?

Let $Q(x)$ be a polynomial in $x$ of order $n$. The Taylor polynomial of $Q(x)$ of order $n$ developed around $x=b$ (denoted by $P_{n,b}(x)$ ) corresponds to $Q(x)$ written in $(x-b)$. This can be ...
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106 views

Intuitive Aproach to Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
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3answers
147 views

$\int_{0}^{\infty}e^{-st}h(t)dt=0 \Rightarrow h(t)=0.$

Suppose $h(t)$ is continuous function and $\int_{0}^{\infty}e^{-st}h(t)dt=0 ~\forall~ s>s_{0}$, then prove that $h(t)=0$. I know "if a function is continuous, non-negative or non-positive, and its ...
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51 views

Intuition behind non-continuous derivatives

Although the intuition behind continuity and derivatives seem pretty obvious to me, I cannot figure out what it means to have a differentiable function with a non-continuous derivative. I'm trying to ...
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44 views

Find the area bounded by the hypercycloid

Parametrization : $x = acos^3(t), y = asin^3(t)$ $a>0$ If you can solve it for me that would be awesome.:D If not, can you give me some hints? Tell me how to set it up and stuff. It's solveable ...
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'If…then…' and '…if…' and '…only if…' and 'If… only then…' statements?

Suppose you have two statements A and B and "If A then B". I am trying to think of what this implies and alternative ways of writing this. I think "If A then B" = A$\rightarrow$B = "A is ...
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859 views

How to explain to a 14-year-old that $\sqrt{(-3)^2}$ isn't $-3$?

$\require{cancel}$ I had this problem yesterday. I tried to explain to the kid this: $$\sqrt{(-3)^2} = 3,$$ and he immediately said: "My teacher told us that we can cancel the square with the square ...
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1answer
235 views

Intuition on Wald's equation without using the optional stopping theorem.

The Wald's equation even at its simplest form as stated below simplifies many problems of calculating expectation. Wald's Equation: Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of real-valued, ...
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What does continuity *in general* mean?

I am looking from : http://en.wikipedia.org/wiki/Lipschitz_continuity Continuously differentiable $\subseteq$ Lipschitz continuous $\subseteq$ α-Hölder continuous $\subseteq$ uniformly continuous ...
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I can't understand logical implication

I just started studying logic (high school) anyway...for the truth table of logical implication If sentence $A$ is true and $B$ is true then $A\implies B$ is true. does that mean if $A$ and $B$ are ...
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Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
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1answer
43 views

What is an intuitive explanation for Birkhoff's ergodic theorem?

If I'm not familiar with measure theory, what is a good way to understand the idea behind the definitions involved, the interpretation of the theorem, and the proofs thereof? Particularly, it's not ...