Questions tagged [problem-solving]

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Is it possible to find UNSATisfiable solutions to a SAT problem with a SAT problem?

I'm working with several problems, which can have special unsatisfiable configurations. For example, consider the simple function $f(x,y)=x+y+2$ with $n$-bit unsigned inputs and $(n+2)$-bit unsigned ...
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1 vote
0 answers
191 views

How does one build and refine strong technical skills relevant to research?

I am an early career researcher working in an area of "hard" analysis but this is a fairly broad question. My technical skills are likely below par and my greatest hindrance to my research ...
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4 votes
0 answers
132 views

Finding roots of equation with gamma functions

Encountered this function in one of my research problems $$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\...
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3 votes
0 answers
320 views

Hard problems solving tricks

This question is motivated by this one that I posted on math.stackexchange. When I fail to solve a hard math problem (like the ones I presented in the linked post), I read a solution and I noticed ...
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1 vote
0 answers
59 views

How to proceed in this Boundary value problem where Eigen values are calculated numerically?

While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$) \begin{eqnarray} \lambda_h F''' - 2 \...
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  • 47
18 votes
3 answers
2k views

Simplest diophantine equation with open solvability

What is the simplest diophantine equation for which we (collectively) don't know whether it has any solutions? I'm aware of many simple ones where we don't know (whether we know) all the solutions, ...
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  • 323
1 vote
0 answers
89 views

Coupled (Solid-Fluid) Heat transfer problem in a Heat Sink

I am trying to solve a coupled heat transfer problem between a solid and fluid (I have under braced the governing equations and labelled them). Eqn. (3) is the partio-integral differential equation I ...
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  • 47
12 votes
1 answer
337 views

Covering the disk with a family of infinite total measure - the convex sequel

Let $(U_n)_n$ be an arbitrary sequence of open convex subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). ...
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17 votes
1 answer
524 views

Covering the disk with a family of infinite total measure

Let $(U_n)_n$ be an arbitrary sequence of open subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). Does ...
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4 votes
0 answers
219 views

How many arrangements of $n$ points with $k$ edge lengths exist in $d$ dimensions?

[Asking on behalf of a high school mathematics course, but responses written at any level are welcome!] I was recently reading over a nice puzzle called the four points, two distances problem: ...
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2 votes
0 answers
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Using weak maximum principle to prove continuous dependence of the boundary data?

I am currently looking at the following ingomogenous Dirichlet problem over an open, bounded domain $\Omega \subset \mathbb{R}^2$ with continuous boundary: \begin{align} \begin{cases} -\operatorname{...
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0 votes
1 answer
76 views

Solving an recursive sequence [closed]

I have an recursive sequence and want to convert it to an explicit formula. The recursive sequence is: $f(0) = 4$ $f(1) = 14$ $f(2) = 194$ $f(x+1) = f(x)^2 - 2$
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4 votes
0 answers
220 views

Looking for U.K. problem column (?) from 1980s

While digging through some dusty corners of my file cabinet, I found a photocopied sheet of eight (handwritten) problems from 1985 that I recall receiving from my secondary school mathematics teacher ...
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-2 votes
1 answer
61 views

I need help with snake's position bounds based on center point(rounded) and the length of the snake problem [closed]

First of all, if it's an existing problem just tell me the name, please. To solve the problem a formula/algorythm which receivs a center point of a snake (snake game type (points on a grid connected ...
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1 vote
2 answers
180 views

Constructing an n-node DAG, with exactly k paths between node 1 and node n [closed]

Pretty straight forward, yet I didn't find how to approach such a problem. I tried constructing a solution from the reverse problem (Given a DAG count the number of paths between node 1 and node n), ...
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15 votes
1 answer
523 views

Characterization of a sphere: every "sub-sphere" has two centers

Let me ask this question without too much formalization: Suppose a smooth surface $M$ has the property that for all spheres $S(p,R)$ (i.e. the set of all points which lie a distance $R\geq 0$ from $p ...
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5 votes
1 answer
232 views

Classifying functions up to suitable pre-composition and/or post-composition

What's a name for a general technique I've seen used many times? Given any family $\mathcal{F}$ of functions such that $f:X\to Y$ for all $f\in \mathcal{F}$ when one wishes to study in general for an ...
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16 votes
2 answers
1k views

Is there any published summary of Erdos's published problems in the American Mathematical Monthly journal?

As we know Erdos has proposed a considerable number of problems in the "American Mathematical Monthly" journal. Is there any published summary of Erdos's published problems in the American ...
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1 vote
1 answer
70 views

A problem with elementary inequality involving probabilities and Brier scoring rule

I am trying to prove certain relations between certain values of the so called Brier inaccuracy measure (Brier scoring rule). Given a vector $p = (p_1, \ldots p_n)$, where $p_1 + \ldots p_n = 1$ and $...
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13 votes
0 answers
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Identifying poisoned wines, with a twist

(This is a joint musing with Andrew Gordon and Wyatt Mackey) There is a classic, elementary riddle, discussed before on MO and math.SE: suppose you have 1000 bottles of wine, and one is poisoned. The ...
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1 vote
5 answers
711 views

Inequality with symmetric polynomials [closed]

How to prove the inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$?
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1 vote
0 answers
678 views

The derivative of an integral function with indicator and max function as integrand

I encounter the following type of problem: \begin{equation} F(x) = \int_a^b \mathbf{1}_{\{v+x-h(v)\geq 0\}}\max\{h(v)-y-x,0\}dv \end{equation} where $\mathbf{1}_{\{z\geq 0\}}=1$ if event $z\geq 0$ ...
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1 vote
0 answers
83 views

Diophantine equation $z=(ax+by+c)/(dxy)$, references? [closed]

I am looking for some sources (books or papers) which discuss the Diophantine equation $$ z=\frac{ax+by+c}{dxy} $$ where $a,b,c,d$ are given positive integers. Could anyone give some references? ...
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0 votes
2 answers
162 views

Characterize the Monotonicity of a root of a cubic equation

I have a cubic function: \begin{equation*} h(x)\triangleq \eta+x-\frac{V(\eta-x)^3}{c\eta} \end{equation*} we know that $x\in[0,\eta)$ and all letters are positive and $V>c/\eta$. Hence we know ...
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  • 191
2 votes
1 answer
294 views

the sum of fractional parts times the ordinary powers

Is there any way to compute/express $\sum\limits^m_{i=0}\{\frac{q*i}{m}\}(\frac{i}{m})^n$ ? Here $q,m,n$ are natural numbers, one can assume $gcd(q,m)=1$. Furthermore, $n$ can be treated as a ...
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16 votes
0 answers
380 views

Division of a square and value of a disk

[Full disclosure] I asked this question on math.stackexchange with little success : https://math.stackexchange.com/questions/1866295/division-of-a-square-and-value-of-a-disk I cam across this problem ...
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  • 477
1 vote
0 answers
66 views

Competitive functions: uniqueness of solution [closed]

Let $f_i(x_1, x_2, ..., x_n)$ for $i=1,...,n$, be real-valued differentiable functions with the following properties: 1) $f_i(x_1, x_2, ..., x_n)=0$ if $x_i=0$. 2) $f_i(x_1, x_2, ..., x_n)=1$ if $...
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14 votes
2 answers
738 views

(Non)existence of mirrors with more than two foci

Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...
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6 votes
1 answer
284 views

Strange problem about triplets of differential forms

Suppose we have the following map: $$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$ $$(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\...
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  • 2,051
8 votes
2 answers
1k views

Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?

My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling: "...
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1 vote
1 answer
199 views

Problem on the digits of $n!$

let $m$ be a natural number, does always exist a $N\in \mathbb{N}$ such that $m$ or more "$0$" digits (excluding the terminal ones) appears amongs the decimal digits of $n!$ if $n\ge N$?
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2 votes
3 answers
1k views

Expected value of swaps

Suppose you have a list of non negative numbers of size N. Now you calculate the maximum element in the list by scanning the list linearly and constantly updating a variable which has initial value of ...
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  • 123
0 votes
0 answers
461 views

How to simplify conditional probability of union of several events

I have an output binary scalar, $y∈B=\{0,1\}$, and an input binary vector $x=[x_1, x_2,…x_M]$ where $x_i∈B=\{0,1\}$. I know that the output $P(y)=1$ depends entirely on the input x. Thus, I want to ...
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4 votes
0 answers
180 views

A challenging non homogenous fractional inequality

I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...
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6 votes
1 answer
808 views

Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...
4 votes
4 answers
2k views

When did you "meet Polya"? [closed]

I guess most of us didn't meet Polya in person (this is the answer to the title)! Perhaps, it is much easier to guess that most of us have met one of his writings (or alike) on problem solving, and ...
27 votes
3 answers
3k views

Is “problem solving” a subject to be taught?

I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all grades. The peoples involved has sent me the textbook for seven graders (13 ...
1 vote
2 answers
259 views

Reducing system of polynomials with symbolic factors

Getting nowhere with maple using its triangularize and groebner decompositions for even moderate size systems with any symbolic factors. Any suggestions on how better to approach this would be ...
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0 votes
1 answer
1k views

number of non-negative integer solutions for a set of equations [closed]

How to find the exact number of non-negative integer solutions of the following set of equations : $$x_1 + x_2 + x_3 + x_4 + x_5 + x_6=6 $$ $$ 2x_1 + x_2 + x_3 = 4$$ $$ x_2 + 2x_4 + x_5 = 4$$ $$ x_3 +...
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  • 209
0 votes
1 answer
229 views

Problem solving equation of type x=a*b^x [closed]

I have the equation x = a*b^x and want to solve it for x. But every online solver I tried says that it is not possible. But when I choose a==8 and b==0.5 there is a solution for x==2 Is it not ...
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1 vote
1 answer
227 views

First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function: $f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...
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0 votes
1 answer
178 views

Transformation problem involving 2 random variables

Any help in this problem? Suppose U and V are independent random variables with density f(u) and g(v) respectively. The domain of U is the interval (0, 1) and the domain of V is v > 0. After the ...
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7 votes
4 answers
2k views

Help me find good math questions for my students [closed]

I am a teacher at 西铁一中。 I teach mathematics in English for students going abroad. Now this is my problem, there are few mathematics books written in English that are at the level of high school, ...
14 votes
3 answers
3k views

Truncated Exponential Series Modulo $p$: Deeper meaning for a Putnam Question.

Apparently B6 of the Putnam this year asked: Suppose $p$ is an odd prime. Prove that for $n\in \{0,1,2...p-1\}$, at least $\frac{p+1}{2}$ of the numbers $\sum^{p-1}_{k=0} k! n^{k}$ are not divisble ...
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80 votes
18 answers
11k views

Contest problems with connections to deeper mathematics

I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay. We all know that problems from, for ...
0 votes
0 answers
240 views

Fast removal of weighted edges in a graph in a way such that all shortest paths are preserved

This problem is analogous to fast removal of the minimum number of edges in a weighted graph such that if the graph were to be drawn on paper with edge lengths linear in proportion to their weights, ...
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15 votes
11 answers
5k views

Elementary mathematical books

I understand that this is a bit of offtopic but mathoverflow is my last resort, as google did not help. I am about to publish an English translation of my Russian book for high school students. The ...
2 votes
2 answers
2k views

Question about Banach's matchbox problem.

Hi, I've been struggling with this for awhile ( http://en.wikipedia.org/wiki/Banach%27s_matchbox_problem) and I put together this little bit of Python code ...
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54 votes
16 answers
13k views

Examples of using physical intuition to solve math problems

For the purposes of this question let a "physical intuition" be an intuition that is derived from your everyday experience of physical reality. Your intuitions about how the spin of a ball affects ...
0 votes
1 answer
665 views

Math Puzzle: calculating the dimensions of variable rectangles in a fixed square

I've got the following problem, I've got a fixed size square and within there are a fixed number of rectangles to be contained within it. I want the rectangles to cover the maximum amount of space ...
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