# All Questions

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### Trouble with converting the negation of a formula to CNF

I'm trying to convert the negation of the following formula to CNF: (p → (q → r)) → ((p → q) → (p → r)) These are the steps I am following: ¬((p → (q → r)) → ((p → q) → (p → r))) ¬(¬(¬p ∨ (¬q ∨ r)) ∨ (...
632 views

### Does there exist a bijective, continuous map from the irrationals onto the reals?

Let $\mathbb{P}$ be the irrational numbers as a subspace of the real numbers. $\mathbb{P}$ is homeomorphic to $\mathbb{N}^\mathbb{N}$, which is also called the Baire space. It is well known, and ...
156 views

### Name for integer "quotient" rounded up (ceiling) instead of down (floor), and its negative or complementary "remainder"

If $168$ cookies (dividend) are shared between $17$ people (divisor), that's almost $10$ cookies each but we're $2$ cookies "short"; alternatively we have slightly more than $9$ cookies each ...
1 vote
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### The linear system associated to a blow-up(of surfaces)

Take a point $p\in\mathbb{P}^2_k$ and blow it up. Then we get a morphism $f:X \rightarrow\mathbb{P}^2_k$ from the blow-up. This should then be describe by a linear system on $X$. What is that linear ...
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### Evaluate area bounded by curve S=0, its Latus Rectum and X-axis

Several equilateral triangles with respective sides 1, 2 ,3 ...n (n is a natural number) are placed end to end, starting from origin, one side of each lying on X-axis in the first quadrant. If the ...
1 vote
5 views

### Doubly transitive action on upper half plane.

Can I transform the triangle in the right hand side to a triangle in the left hand side by a Mobius transform? If $PSL(2, \mathbb{R})$ acts doubly transitive on $\mathbb{R} \cup \infty$, then do we ...
12 views

### Non-convex optimization problem transformation

I'm trying to solve a non-convex optimization problem, trying to figure if their are any tricks to transform it into an approximate convex form. here's a simpler form of the problem I'm trying to ...
46 views
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### Basis for a lattice

Let $p\in\mathbb{Z}^+$ and $\mathbf{n}\in(\mathbb{Z}^+)^n$. Consider the set $$\Lambda_{\mathbf{n},p}:=\left\{\dfrac{a}{p}\mathbf{n}+\mathbf{b}:a\in\mathbb{Z},\mathbf{b}\in\mathbb{Z}^n\right\}$$ Then, ...
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### If $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are b and c equal?

Is it true that, if $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are $b$ and $c$ equal? For coprime $(a,b)$ and $(a,c)$, it is trivial. It is also easy when $a|b$ and $a|c$. I am unable to construct ...
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### Find the minimum length

Given rectangle $ABCD$ where $AB=CD=4$ and $AC=BD=2$. Point $E$ is in middle of $AC$. Point $X$ is somewhere in the line $AB$. Find polygonal chain $EXC$ of minimum length. How do you find minimum ...
8k views

### Non-Metrizable Topological Spaces

What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. ...
14 views

### Rate of change of depth of a liquid in a spherical container

I'm just wondering about how you would do this question but I am unable to type it using LaTeX as for some reason it is not properly covnerting my code to the correct equation but here it is as an ...
1 vote
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### Intersection of perpendicular tangent lines - generalization of directrix?

This is a funny little problem that I came up with. For a differentiable function $f$, define a locus of points $P$ as follows: Let $m$ be an arbitrary tangent line to $f$, and let $n$ be another ...
14 views

### Let $n\geq 3$. Is there a connected, planar, bipartite graph with $n$ regions and $n$ vertices?

The answer given is that according to a Corollary of Euler’s formula (Corollary 3 Section 10.7), such a graph has at most $2n − 4$ edges. Applying this to Euler’s formula ($r = m − n + 2$), there are ...
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### Triply transitive action on upper half plane

I am calculating the area of a triangle in the upper half plane. Consider the following triangle in the upper half plane with the Poincare metric. Can I transform this triangle to the following ...
167 views

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### Arcwise connected but not connected?

In his book "Geometry, Topology and Physics", Nakahara makes the following claim with regard to topological spaces: With a few pathological exceptions, arcwise connectedness is practically ...
157 views

### If $X$ is a totally disconnected space, then is $\beta(X)$ totally disconnected?

I know that when $X$ is a normal and totally disconnected space, the Stone-Cech compactification $\beta(X)$ is totally disconnected. But I can't find a counterexample when considering $X$ totally ...
318 views

### cardinality of a basis for a topology

Suppose X is a space of cardinality $\le \kappa$. I would like to claim that any topology on X has a basis of cardinality $\le \kappa$. Intuitively it's true since even the discrete topology has such ...
1 vote
### The perimeter is equal to the area, i.e. $2a+2b=ab$.
The measurements on the sides of a rectangle are distinct integers. The perimeter and area of ​​the rectangle are expressed by the same number. Determine this number. Answer: 18 It could be $4*4$ = \$4+...