# All Questions

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### Help me in showing that there is only 2 solutions for $f(x)=(x-1)e^{2x}+x^2-x-1=0$?

I have a problem with part (b) in the following question Question a) Function: $f(x)=(x-1)e^{2x}+x^2-x-1$, show that $f(x)$ has a zero between −1 and 1/2 and between 1/2 and 2. b) Show that there ...
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### Kernel of a group homomorphism

Is it true that for a group homomorphism $\phi: G\to H$, $\phi(e_G)$ necessarily equals $e_H$?
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### Real analysis problem: Rolle's theorem, Darboux's theorem, induction

I am taking a course in real analysis (undergraduate) and I'm supposed to solve a problem and present in class. I've just started, but I'm already stuck. This is the problem: Let a function $f$ be ...
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### Uncertainty propagation of a derivative

$y = \frac{a}{(x - b)^3}$ $y' = \frac{-3a}{(x - b)^4}$ a = 77.1 ± 15.2 b = -1.78 ± 1.18 x = 21 ± 1 How do I find the uncertainty of y'? Thanks.
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### Solving a bessel equation by substitution

For a problem solving class I need to find the general solution of ODE $y''+(e^{-2x}-\frac{1}{16})y=0$ in terms of $J_{\nu}$ and $J_{-\nu}$, if possible. $\nu$ represents the Bessel parameter. A hint ...
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this is a random math question I saw and it peaked my interest. I like cake. Start with a circular cake and cut it with five straight slices. What is the largest number of pieces that you can create ...
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### Is “polynomials in $x$” a monad?

The construction of polynomials $R \mapsto R[x]$ gives a functor $P: \mathbf{Ring} \to \mathbf{Ring}$ on the category of possibly noncommutative rings. Choosing a ring $R$ for the moment, there is a ...
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### $ABCD$ is a rectangle and $EFGH$ are any points on sides $AB,BC,CD,DA$, respectively. Show that $EF+FG+GH+HE>\sqrt{2} AC$.

$ABCD$ is a rectangle and $EFGH$ are any points on sides $AB,BC,CD,DA$, respectively. Show that $EF+FG+GH+HE>\sqrt{2} AC$. Is it true for all quadrilaterals $ABCD$?
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### Determine parameter for two quadratic equations having common root

The two quadratic equations: $10x^2+kx-2=0$ and $5x^2+14x-k=0$ have a common root,find the value of $k$.
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### The left and right side of an orientation in complex

The definition of the right side of an orientation $(z_1,z_2,z_3)$ is $$\{z: Im(z,z_1,z_2,z_3)>0\}.$$ Why is it defined this way? Note: ...
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### C*-algebraic intrinsic definition for compactness of an operator?

Some properties of operators (normal, self adjoint, hermitian) have intrinsic definitions for any element of a $C^*$ algebra. Is there such definition for compact operators? Equivalently: Let ...
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### Permutations and fundamental principle of counting.

find the number of different 8 letter arrangements that can be made from the letters of the word DAUGHTER (8 letters) so that --All vowels occur together. I thought of a solution which is as follows-- ...
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### Let $G$ be a group, if $H=\{b\in G\ |\ bab^{-1} \in \langle a \rangle\}$ is H a subgroup of G?

I've seen this question around when $G\wedge\langle a \rangle$ are finite, but what if $G$ is infinite and $\langle a \rangle$ is finite? My approach: Show $H$ is closed, which just follows ...
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### List of Example Equations For Problem Solving Software?

I'm currently working on an application for a class that will calculate: Intercepts Critical points Inflection points Asymptotes for an equation entered in by a user. So far, the ...
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### Proof on Fibonacci sequence: $F(1) + F(3) + \cdots + F(2n-1) = F(2n)$ using induction and recursion

The problem is: Use induction and the recursive formula to prove that: $$F(1) + F(3) + \cdots + F(2n-1) = F(2n)$$ For the base case I let $n=1$ which gave $$F(1) = F(2(1))$$ $$1=1$$ Which is ...
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### Simplify and rewrite in term of i

The problem click here I believe the answer is (b) i tried it in my calculator and it's by far the closest answer from the rest of the other options but it could also be (a) and that's where i need ...
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### Making sense of a definition - The Happy Number

Note, we are only using two colors in the definitions below (say red and blue). A n-graph (complete, edge-colored, and has n vertices) is called Happy if there exists a vertex coloring such that ...
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### Simple proof for a summation

Let $k=(1+\frac{1}{3} +\frac{1}{5} +\cdots+\frac{1}{2n+1}$) now prove for every $n\in \Bbb{N}$, $k$ is not nature number.
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### Recurrence relations closed form solutions

I am looking at various recurrence relations of the form: $x_{n+2} + bx_{n+1} +cx_{n} = f(n)$ In the book I use, I have been given an algorithm on how to solve these kinds of recurrence relations ...
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### Why does this ideal contain a power of that ideal?

I'm reading the proof of Prop. I-18 in Eisenbud-Harris The Geometry of Schemes and I'm unsure of a claim they make. Let me just give you some necessary info: Let $R$ be a commutative unital ring and ...
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### Cycle index and graph

Determine the group $G$ that consist of all permutations $g$ of the nodes in the graph below so that if there is an edge between nodes $a$ and $b$ (i.e., $a$ and $b$ are neighbors), then there is an ...
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### Demystifying definition of the Random Variable

The intro on Random Variables says that it is a variable (in bold), whose value depends on a chance. IMO, it sounds like a random value generator, whose value depends on a chance, just as random ...
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### Sum for all element of the sum of x for every element

Let $i$ be an element of the set $S$. Suppose that you need to calculate a sum of many rates $\rho$ for every element $i$, and then the sum these sums for all elements of the set $S$. How do I write ...
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### Computing $4^{m+1} \cdot 9^{n-1}$ in terms of $2^m \cdot 3^n$

So I got this math question that I have to do. Unfortunately I don't understand a thing. The question is: If $2^m \cdot 3^n = a$, what is $4^{m+1} \cdot 9^{n-1}$? I will be grateful for any and ...
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### Why the disk algebra is not a C* algebra.

I'm trying to figure out why the set of analytic functions on the unit disk, A(D), is not a C* algebra. The norm is the sup norm and the involution is $f(z) \to \overline{f(\bar z)}$. I want to know ...
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### Is there convergence in the following series?

I am unable to figure out $\displaystyle \lim_{\epsilon \to 0} \sum_{n=1}^\infty \frac{1}{\epsilon}$ . Any help would be appreciated,Thanks
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### What is epsilon-delta defination of limit?What is it's geometrical meaning?

I know epsilon-delta defination but I can't compare it with the concept of right hand limit and left hand limit.Also a question arises in my mind which is "If a function is bounded and has a limit l ...
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### Uniqueness of an integer representation in an integer base

I'm trying to prove the uniqueness of representations of integers in integer bases and I'm doing so by contradictions. Let $a,d$ be natural numbers where $d>1$. A representation of $a$ in base $b$ ...
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### Viewing Semigroups as Categories?

I am wondering how to view semigroups as categories. For example, we can easily view monoids as categories with a single object. Unfortunately, semigroups do not necessarily have identities, so the ...
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### How to represent Pattern using a function?

Patterns are sets of multi-variable functions that express some physical entity, object, system, etc. The following is the Domain of patterns: $Ω = \{ f_r ( x ) \mid r = 1, 2 , \ldots \} ⊂ U$ What ...
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### A question about choosing certain covers in topological manifolds

Let $M$, $N$ be topological manifolds, and $f: M \to N$ a map. In the proof of theorem 1.5. in "Differential Topology" (M. Hirsch, Springer-Verlag, 1994) it is claimed the following: There is a ...
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### Construct the formal proof of validity for the given argument

This is the construction that I need to validate. 1. $\neg(E \vee \neg F)$ 2. $F \Rightarrow G$ $\vdash G \vee \neg E$
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### System of non-linear differential equation - crit. points

I have a system of non-linear coupled differential equations as follows: $x'=-9\sin(x)-0.1y$ and $yy'=-9\sin(x)-0.1y$ which I obtained from my original problem of finding and classifying the ...
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### Using the Cauchy integral to construct holomorphic functions

I was going through David C. Ullrich's wonderful text Complex Made Simple, when I came to Theorem 10.3.1. The Theorem states If $f\in C(\partial\mathbb{D})$ then $C[f]\in H(\mathbb{D})$, where ...
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### Find the maximum possible cardinality of collection of set

Suppose $B = \{B_1, B_2, \dots, B_k\}$ is an arbitrary collection of 3-element subsets of $n$-element set such that $|B_i \cap B_j| \neq 1$ for each pair of indices $1 \leq i, j \leq k$. Find the ...
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### Mashing eigenvalues together.

So we know all the elements of a matrix $\bf A$, and we want to tie it's eigenvalues (which are already close to each other) even closer to each other (preferrably without having to explicitly ...
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### Help with functions and relations?

I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to Relations and Functions. To be specific, I'm stuck on ...
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### Probability of at least two daughters in three-child family, given a daughter Mary

Bob and Jane have three children. Given that one child is their daughter Mary, what is the probability that Bob and Jane have at least two daughters? (I am also interested in wordings of this ...
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### How to evaluate this integral? Tried changing limits and adding , also tried using complex numbers but failed.

To Find : $$\int_0^\pi(\ln(\sin(x)))^2 \, dx$$
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### l’Hôpital’s rule to prove that $f \in \omega(g)$

Can anyone give me a hint on how to start this please? For $n \in R >1$ let $f(n) = n^{4/3}$ and $g(n) = n · (log$5$n)^2$. Use l’Hôpital’s rule to prove that $f \in \omega(g)$.
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### How to give a combinatorial proof for this forumula

I need to give a combinatorial argument that $$S(n,m) = \sum_{i =0} ^{n-1} {n -1 \choose i} S(i,m-1)$$ Where $S(n,m)$ is the Stirling numbers of the second kind. Here is my attempt. Well first ...
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### Calculating PDF of consequent events

My question is how to calculate the probability density function (pdf) of consequent, but otherwise independent events, defined as follows (in case of 3 events): $A$ is an independent event with pdf ...
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### Two Different Ways to Convert Latitude in Degrees to Meters in Mercator Space and their Inverses

The first way, using the mercator tangent half angle formula. $$y = \log(\tan(\frac{\pi}{4} + \frac{\lambda}{2}))$$ The inverse of this function being. $$\lambda = \arctan(e^{y*2\pi})$$ ...
Is it true that $f$ is uniformly continuous if $f(z) \to f(z_0)$ uniformly as $z \to z_0$? And I am a bit confused why all the definitions treat only the uniform convergence of sequences, why can't ...