Questions tagged [problem-solving]
The problem-solving tag has no summary.
58 questions
3
votes
1
answer
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Diophantine equation $a^2+b^4+c^6=d^8$ (parameterization)
How to get parametrization for the diophantine equation $a^2+b^4+c^6=d^8$ There is an infinite set of solutions to this equation (example $79^2+4^4+2^6=3^8$), but it is not easy to reduce it to a ...
- 33
0
votes
1
answer
134
views
Every region is made up of simple pieces in a suitable coordinate system
I’ve asked this question on mathstack, but did not get any answers, so I’ve decided to ask it here.
A region D in the plane is said to be of $\textit{type 1}$ if it can be written as $D = \{ (x, y) \...
- 171
0
votes
0
answers
119
views
Solving 2nd order partial differential equations
I have a study that includes the following system of 2nd order partial differential equations. The unknown variables in these equations are $y(k,t)$, $x(k,t)$, $z(k,t)$, and their derivatives.
My ...
- 117
2
votes
0
answers
131
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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 ...
- 21
3
votes
1
answer
2k
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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 ...
- 161
1
vote
2
answers
357
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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
2
votes
1
answer
399
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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 ...
- 21
7
votes
1
answer
870
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 ...
- 79
4
votes
0
answers
215
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}-\...
- 199
3
votes
0
answers
995
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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
vote
0
answers
84
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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
19
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, ...
- 341
1
vote
0
answers
120
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 ...
- 47
13
votes
1
answer
522
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). ...
- 1,461
17
votes
1
answer
643
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 ...
- 1,461
4
votes
0
answers
258
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:
Find ...
- 8,116
2
votes
0
answers
186
views
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
0
votes
1
answer
151
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$
- 13
4
votes
0
answers
249
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 ...
- 88.5k
-2
votes
1
answer
84
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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 ...
- 9
1
vote
2
answers
863
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), ...
15
votes
1
answer
593
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 ...
- 1,461
5
votes
1
answer
313
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 ...
- 2,626
16
votes
2
answers
1k
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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 ...
- 561
1
vote
1
answer
99
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 $...
- 145
13
votes
0
answers
2k
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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 ...
- 303
1
vote
5
answers
898
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$?
- 129
1
vote
0
answers
761
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$ ...
- 35
1
vote
0
answers
101
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}
$$
where $a,b,c,d$ are given positive integers. Could anyone give some references?
...
- 841
0
votes
2
answers
243
views
Characterize the Monotonicity of a root of a cubic equation
I have a cubic function:
\begin{equation*}
h(x)\triangleq \eta+x-\frac{V(\eta-x)^3}{c\eta}
\end{equation*}
we know that $x\in[0,\eta)$ and all letters are positive and $V>c/\eta$. Hence we know ...
- 191
2
votes
1
answer
351
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the sum of fractional parts times the ordinary powers
Is there any way to compute/express $\sum\limits^m_{i=0}\{\frac{q*i}{m}\}(\frac{i}{m})^n$ ? Here $q,m,n$ are natural numbers, one can assume $gcd(q,m)=1$. Furthermore, $n$ can be treated as a ...
- 2,282
16
votes
0
answers
464
views
Division of a square and value of a disk
[Full disclosure] I asked this question on math.stackexchange with little success : https://math.stackexchange.com/questions/1866295/division-of-a-square-and-value-of-a-disk
I cam across this problem ...
- 477
1
vote
0
answers
88
views
Competitive functions: uniqueness of solution [closed]
Let $f_i(x_1, x_2, ..., x_n)$ for $i=1,...,n$, be real-valued differentiable functions with the following properties:
1) $f_i(x_1, x_2, ..., x_n)=0$ if $x_i=0$.
2) $f_i(x_1, x_2, ..., x_n)=1$ if $...
- 41
15
votes
2
answers
874
views
(Non)existence of mirrors with more than two foci
Do there exist any mirrors $M$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ for which there exist three different points $x_1$, $x_2$, $x_3 \in \mathbb{R}^d$ such that if any ray of light passes ...
- 1,461
6
votes
1
answer
340
views
Strange problem about triplets of differential forms
Suppose we have the following map:
$$(\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+\...
- 2,131
9
votes
2
answers
1k
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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:
"...
- 159
1
vote
1
answer
260
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 $n\ge N$?
- 111
2
votes
3
answers
1k
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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 ...
- 123
0
votes
0
answers
543
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
204
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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 ...
- 13.6k
9
votes
2
answers
1k
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
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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 ...
27
votes
3
answers
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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 ...
1
vote
2
answers
323
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 ...
- 13
0
votes
1
answer
2k
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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 +...
- 209
0
votes
1
answer
294
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 ...
- 3
1
vote
1
answer
271
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 ...
- 23
0
votes
1
answer
218
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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 ...
- 103
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
4k
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 ...
- 11.6k