Questions tagged [problem-solving]
The problem-solving tag has no usage guidance.
55 questions
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Constructing a system of two cubic polynomial equations with exactly 9 real solutions in Maple [closed]
I am trying to construct a system of two cubic polynomial equations in two variables (x and y) with exactly 9 real solutions using Maple. However, I am having trouble finding the appropriate ...
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2
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1
answer
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Where can I access American Mathematical Monthly problems given an index?
I don't know if this is the appropriate website to ask, so I understand if this post gets closed. I want to explore (and maybe solve) some of the currently-unsolved problems submitted by readers on ...
- 151
1
vote
2
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Method to solve system of exponential sums of the form $a^x+b^x=c$ given more equations than variables [closed]
Cross post with mse
For example, let's say I have the following equations.
\begin{gather*}
a^{x-1}+b^{x-1}=337 \\
a^{x}+b^{x}=1267 \\
a^{x+1}+b^{x+1}=4825 \\
a^{x+2}+b^{x+2}=18751.
\end{gather*}
What ...
- 13
1
vote
1
answer
255
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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 ...
- 11
6
votes
1
answer
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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 ...
- 69
4
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0
answers
176
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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}-\...
- 199
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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 ...
- 161
1
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0
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62
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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 \...
- 47
20
votes
3
answers
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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, ...
- 351
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0
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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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1
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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). ...
- 1,461
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votes
1
answer
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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 ...
- 1,461
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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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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{...
- 31
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1
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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$
- 13
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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 ...
- 82.6k
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votes
1
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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
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2
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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), ...
15
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1
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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 ...
- 1,461
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votes
1
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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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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
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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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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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vote
5
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774
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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0
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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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0
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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}
$$
where $a,b,c,d$ are given positive integers. Could anyone give some references?
...
- 841
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2
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Characterize the Monotonicity of a root of a cubic equation
I have a cubic function:
\begin{equation*}
h(x)\triangleq \eta+x-\frac{V(\eta-x)^3}{c\eta}
\end{equation*}
we know that $x\in[0,\eta)$ and all letters are positive and $V>c/\eta$. Hence we know ...
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1
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the sum of fractional parts times the ordinary powers
Is there any way to compute/express $\sum\limits^m_{i=0}\{\frac{q*i}{m}\}(\frac{i}{m})^n$ ? Here $q,m,n$ are natural numbers, one can assume $gcd(q,m)=1$. Furthermore, $n$ can be treated as a ...
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Division of a square and value of a disk
[Full disclosure] I asked this question on math.stackexchange with little success : https://math.stackexchange.com/questions/1866295/division-of-a-square-and-value-of-a-disk
I cam across this problem ...
- 477
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Competitive functions: uniqueness of solution [closed]
Let $f_i(x_1, x_2, ..., x_n)$ for $i=1,...,n$, be real-valued differentiable functions with the following properties:
1) $f_i(x_1, x_2, ..., x_n)=0$ if $x_i=0$.
2) $f_i(x_1, x_2, ..., x_n)=1$ if $...
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(Non)existence of mirrors with more than two foci
Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...
- 1,461
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votes
1
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Strange problem about triplets of differential forms
Suppose we have the following map:
$$(\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+\...
- 2,091
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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
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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 $n\ge N$?
- 111
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3
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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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0
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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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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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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 ...
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4
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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 ...
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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 ...
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2
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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 ...
- 13
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votes
1
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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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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 ...
- 3
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vote
1
answer
242
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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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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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4
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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
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3
answers
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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 ...
- 11.4k
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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 ...
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0
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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, ...
- 1
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11
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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 ...