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1500 questions
73
votes
10 answers

Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$? Edit: note that while…
Isaac
  • 37,057
73
votes
3 answers

Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar. I think finding the distance between two given matrices…
Synex
  • 995
73
votes
2 answers

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have gained quite some intuition for tensor products and…
Martin Brandenburg
  • 181,922
73
votes
5 answers

What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the Millennium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even fully grasp the problem, but seeing the hypothesis and the other problems I wonder: what practical use will a…
Mythio
  • 967
73
votes
5 answers

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of why modular forms are important is correct a)…
Alex Youcis
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73
votes
16 answers

Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?

I don't know if my textbook is written poorly or I'm dumb. But I can't bring myself to understand the following definition. A real number is a cut, which parts the rational numbers into two classes. Let $\mathbb{R}$ be the set of cuts. A cut is a…
God bless
  • 2,095
73
votes
14 answers

What isn't a vector space?

I'm really confused about vector spaces. We're learning about them in Linear Algebra, and my book doesn't give good examples of what a vector space is. I understand sets and vectors, but I don't understand vector spaces. From the definitions they've…
Aleksandr Hovhannisyan
  • 3,163
73
votes
3 answers

First-Order Logic vs. Second-Order Logic

Wikipedia describes the first-order vs. second-order logic as follows: First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables…
Sadeq Dousti
  • 3,371
73
votes
2 answers

When can we interchange the derivative with an expectation?

Let $ (X_t) $ be a stochastic process, and define a new stochastic process by $ Y_t = \int_0^t f(X_s) ds $. Is it true in general that $ \frac{d} {dt} \mathbb{E}(Y_t) = \mathbb{E}(f(X_t)) $? If not, under what conditions would we be allowed to…
Jonas
  • 2,563
73
votes
7 answers

The relation between trace and determinant of a matrix

Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?
TPArrow
  • 976
73
votes
4 answers

How to know if a point is inside a circle?

Having a circle with the centre $(x_c, y_c)$ with the radius $r$ how to know whether a point $(x_p, y_p)$ is inside the circle?
Ivan
  • 939
73
votes
9 answers

What is the most expensive item I could buy with £50?

I was set the following question during the discrete mathematics module of my degree and despite my instructor explaining his working to me I still disagree with the answer he says is correct. Can someone please help me either understand where my…
Sam
  • 788
73
votes
10 answers

Why is the tensor product important when we already have direct and semidirect products?

Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product, Semi-direct products, so after all why do we need Tensor Product? The Definition of Tensor…
anonymous
73
votes
8 answers

Is linear algebra laying the foundation for something important?

I'm majoring in mathematics and currently enrolled in Linear Algebra. It's very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear Algebra being the foundation for Applied…
Mallory
  • 1,187
73
votes
13 answers

What is the definition of a set?

From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition. My question is what the definition of a set is? I have…
John Doe
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