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Questions tagged [examples-counterexamples]

Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

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7 views

Give an example of bounded linear functional

Give an example of bounded linear functional on L^{$\infty$} space. I taken T (x) = $\sum$ xn Is it true?
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0answers
18 views

inequality regarding Stirling numbers of second kind

I am looking to prove/disprove the following inequality: Let $n$ be even, then ${n\brace n/2-1}\leq \frac{n}{2}{n\brace n/2}$. Clearly, we have ${n\brace n/2}\leq {n\brace n/2-1}$ by the ...
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1answer
17 views

Example of a sequence of bounded functions $(f_n)$ on $[0, 1]$ which converges pointwise to an unbounded function $f$

Give an example of a sequence of bounded functions $(f_n)$ on $[0, 1]$ which converges pointwise to an unbounded function $f$ I was thinking of modifying the example (which runs into trouble at $x=...
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4answers
129 views

Find all matrices which satisfy $M^2-3M+3I = 0$

I am trying to find all matrices which solve the matrix equation $$M^2 -3M +3I=0$$ Since this doesn't factor I tried expanding this in terms of the coordinates of the matrix. It also occurs to me ...
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1answer
24 views

Choice of measure in $L^p$ spaces [on hold]

I need to find a measure $\mu$ and a $\sigma$-algebra $\mathcal{E}$ on the set $[0,\infty)$ such that for any $p$ and $q$ with $1 \leq p < q \leq \infty$, the sets $L^p \backslash L^q$ and $L^q \...
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0answers
38 views

Example of a one dimensional GCD domain which is not a UFD.

I know that every UFD is a GCD domain. But every GCD domain is not a UFD. I want to make sure that a one dimensional GCD domain is not necessarily a UFD, so I'm looking for an example to confirm ...
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1answer
48 views

If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15\;$ then which of the following equals $a×b×c$?

The problem is: If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15,\;$ then which of the following equals $a×b×c$ ? A) $\frac {1}{375}$ B) $\frac{1}{125}$ C) $\...
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0answers
35 views

Finding the harmonic conjugate of $T(x,y)= e^{-y} \sin x$?

I try the following two method on finding the harmonic conjugate of $T(x,y)= e^{-y} \sin x$ : Method 1 : by a method of the book Complex Variables and Applications by Brown and Churchill, Chapter 9 ...
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2answers
48 views

If $A\subseteq B$, then $A'\subseteq B'$

Proof or counterexample: If $A\subseteq B$, then $A'\subseteq B'$. I have no idea where to start. Only thing I know is the definition of limit points. $A'$ is the set of all limit points of $A$.
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2answers
66 views

Inequality in 5 variables

Let 5 positive real variables $(a,b,c,d,e)$. Prove or disprove: $$ \sum_{cyc} a^2 b d (c+e)\ge \sum_{cyc} a b c e (a+d) $$ where $\sum_{cyc}$ means all 5 cyclic shifts $(a,b,c,d,e) \to (b,c,d,e,a) \to$...
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1answer
28 views

How pathological can the boundary of an open, simply connected subset of $\mathbb{C}$ be?

In my complex analysis class we’re currently covering the Riemann Mapping Theorem, and as is well-known there are no conditions imposed on the actual shape of the region $\Omega$, except that it must ...
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2answers
54 views

Betweenness preserving implies monotonic?

For this question, we can assume that $f:\mathbb{R}\rightarrow\mathbb{R}$. However, I hope that an answer can generalize to arbitrary linearly ordered sets. I assume that everyone will know what I ...
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0answers
9 views

Example for intersection, multiplicity and tangents for projective plane curves

We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, ...
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2answers
364 views

An example of a sequence which does not have any subsequence with a finite limit.

Find and example of a sequence $\{x_n\}$ which does not contain any subsequences having a finite limit. I've been thinking of the following seqeunce: $$ x_n = \sin(n)\cdot\sin(\sqrt{3}n) $$ But is ...
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1answer
31 views

Tensor product of two irreducible representations is not irreducible

I have seen that if $\rho: G \longrightarrow \text{GL}_{\mathbb{C}}(V)$ and $\alpha: G \longrightarrow \text{GL}_{\mathbb{C}}(W)$ are two irreducible representations of a finite grup $G$, then its ...
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1answer
36 views

Homogeneity does not suffice for a map between vector spaces to be linear

The following problem is taken from Sheldon Axler's book Linear Algebra Done Right, more precisely Exercise 1. from Chapter 3: Problem: Give an example of a function $f : \mathbb{R}^2 \to \mathbb{R}$ ...
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1answer
12 views

Prove falsity of argument schemas in predicate logic

I have to give a counterexample for the following argument schema: ∃x (Px ∧ Qx) ⊨ ∀x (Px ∨ Qx) by definig its domain and the interpretation function, which is where I have some slight problems. On ...
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1answer
44 views

A counterexample of Banach Steinhaus Theorem

I was reading about a consequence of Banach-Steinhaus theorem which states that: Let $E$ be a Banach space and $F$ be a normed space, and let $\{T_n\}_{n\in \mathbb{N}}$ be a sequence of bounded ...
14
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2answers
767 views

Can an uncountable group have a countable number of subgroups? [closed]

Can an uncountable group have only a countable number of subgroups? Please give examples if any exist! Edit: I want a group having uncountable cardinality but having a countable number of ...
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2answers
38 views

Counterexamples about function discontinuity.

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a function with a point $\textbf{x}\in\mathbb{R}^n$ of discontinuity. Is it possible that the image $f(O_{x_i})$, the image of an open ball (containing $...
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1answer
33 views

Examples of functions characterized by sequence

Let $ \Psi$ be a set of functions $\chi :\mathbb{R}^+\rightarrow [0,1)$ satisfying $$\chi(t_n)\rightarrow 1\Rightarrow t_n\rightarrow 0$$ I want to find some functions that belongs to $ \Psi$, other ...
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1answer
40 views

Differentiable function with no second derivative at $0$?

What is an example of a function that is differentiable, concave up everywhere, and $f''(0)$ does not exist?
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1answer
27 views

Counter example of continous function such that there is Set S with $f(S)^\circ \subset (f(S^\circ))$

$S^\circ$ denotes the interior of a set $S$. Is there an example of a continuous function $f$ and a set S with $(f(X))^\circ \not\subset f(S^\circ)$ ? I know that$f(S^\circ)\subset (f(S))^\circ$ is ...
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1answer
87 views

does there exist a smooth function which is nowhere convex/concave?

Consider $g\in {\rm C}^1[0,1]$. We say that $g$ is nowhere convex (concave, resp.) on $[0,1]$ if there is no open interval $I\subseteq [0,1]$ on which $g$ is convex (concave, resp.) Is it possible to ...
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2answers
47 views

$\mathcal{f}$(A $\cap$ B) = $\mathcal{f}$(A) $\cap$ $\mathcal{f}$(B) $\iff$ $\mathcal{f}$ is injective. Numerical example and counterexample.

Need some help to check whether my understanding of a subject is right or not. $1$. Example of injective function. $\mathcal{f}$: $\mathbb{N}$ $\longrightarrow$ $\mathbb{N}$ $x$ $\longmapsto$ $2x$ ...
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1answer
22 views

Decimal representation of the set [0,1)

I have encountered the next statement in statistics lecture (translated from german): "From the analysis you know that all but a countable number of $$w ∈ [0, 1)$$ represent a unique decimal ...
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0answers
14 views

reciprocal of the Schwarz theorem

May you help me with a counterexample of the reciprocal of the Schwarz theorem? i.e. a $\mathbb R^2$-function $f$ verifying $\frac{\partial^2 f}{\partial x \partial y} (a) = \frac{\partial^2 f}{\...
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1answer
49 views

Is it possible for the sequence $\{\frac{x_{n+1}}{x_n}\}$ to be unbounded but have $\lim_{n\to\infty} x_n = x$, $x_n \ne 0$

Given: $$ \begin{cases} \lim_{n\to\infty} x_n = x \\ x_n \ne 0 \\ n \in \mathbb N \end{cases} $$ Is it possible for $\{\frac{x_{n+1}}{x_n}\}$ to be and unbounded sequence? This problem comes in ...
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0answers
43 views

Counterexample in Kolmogorov theorem about existence of almost surely continuous modification

I want to understand this Kolmogorov theorem about existence of almost surely continuous modification: A process $\{\xi_t, \in[0,T]\}$ admits an almost surely continuous modification if there exist ...
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1answer
37 views

Im not able to find a counter example ? [closed]

Let $(f_n)$ be a sequence of continuous functions on $\mathbb R$. If $(f_n)$ converges to $f$ uniformly on $\mathbb R$ then $$\lim\limits_{n\to \infty}\int^{\infty}_{-\infty}f_n(x)dx \neq \int^{\...
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1answer
14 views

bounded differentiable functions

I need some help with the following problem: Suppose $f\colon(0,\infty) \to \mathbb{R}$ is differentiable. If $f$ is bounded and $f'(x) \to 0$ as $x \to +\infty$, does this imply that $\lim_{x \to +\...
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2answers
60 views

Example of Series such that $\sum a_n$ converges but $\sum a_n^4$ diverges

I am interested in finding Example of Series such that $\sum a_n$ converges but $\sum a_n^4$ diverges I am able to find converse like if $\sum a_n^4$ converges but $\sum a_n$ diverges using ...
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1answer
34 views

Counterexample for $f$ is strictly increasing ,$ g$ and $f\circ g$ is continous but f is not continous

I wanted to find counterexample Counterexample for $f$ is strictly increasing,$ g$ and $f\circ g$ is continuous but $f$ is not continuous Where f and g are function form $[0,1]\to [0,1]$ How to ...
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2answers
26 views

Example of holomorphic function with no natural root

If $f$ is holomorphic over a simply connected set $\Omega\subset\Bbb C$ and $f(z)\ne0\ \forall z\in\Omega$ then it is known that for any $n\in\Bbb N^*$ there exists a holomorphic function $g$ such ...
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0answers
19 views

An example of the fact that from measurability of a random process does not follow measurability of its integral

Let {$ \xi _t(\omega), t\in[0,\infty)$} be a random process and $ \xi _t(\omega)\in \{\mathfrak F_t\}$ (some filtration). If $ \xi _t(\omega) $ is $ \mathfrak F_t $ measurable then $\int_0^t\xi _s(\...
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2answers
17 views

Give an example of a skew–symmetric2×2–matrix B with entries in C for which I2+B is not invertible [closed]

Give an example of a skew-symmetric $2\times2$ matrix $B$ with entries in $\mathbb C$ for which $I_2+B$ is not invertible. I'm struggling with this Lin Algebra problem if you could help me with it ...
1
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1answer
28 views

Finite group of size n for each n > 1 example

So I am trying to think of an example of a finite group of size $n$ for each $ n \gt 1 $, but nothing is coming to mind. If it is a finite group denoted as $G$, then the order of G is is $|G|$, but I ...
6
votes
2answers
94 views

Is there a function $f(x)$ such that $\lim_{x\rightarrow x_0}f(x)=\infty$ for all $x_0$ in some interval? [closed]

Let $f(x)$ be a real valued function with its domain in $\mathbb{R}$. Is there an example of $f(x)$ such that $$\lim_{x\rightarrow x_0}f(x)=\infty$$ for all $x_0$ in some interval?
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1answer
160 views

How to give counterexample for given claim

Suppose $A_1,\dots,A_m$ be distinct $n\times n $ real matrices such that $A_iA_j=0$ for all $i\neq j$. Show that $m\leq n$. I think this true because i tried for $3\times 3$ and $2\times 2$ case I ...
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2answers
57 views

Determinant of the $n\times n$ matrix whose $jk$-th entry is $1/\min\{j,k\}$

So on my exam I got a True/False question that asked the following: For any $n\in\mathbb N$, compute the determinant of the matrix \begin{bmatrix} 1&1&1&\cdots&1 \\1&1/2&...
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0answers
16 views

An example of random functions that are stochastically equivalent but aren't modifications of each other.

I'm trying to show that stochastic equivalence of random functions (in the broad sense) doesn't imply that they are modifications of each other. For this, I'm looking for a counterexample. Perhaps ...
2
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4answers
38 views

Example showing $\lim\limits_{x \to x_0} xf(x) \neq x_0\lim\limits_{x \to x_0} f(x)$

I can looking for a simple example to illustrate $\lim\limits_{x \to x_0} xf(x) \neq x_0 \lim\limits_{x \to x_0} f(x)$ For example I have tried $f(x) = x-1, x_0 = 1$ hoping that I would get a zero on ...
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2answers
20 views

If a space is complete with respect to some metric is it complete with respect to any other metric?

In proving if a metric space is complete the defined metric on it is considered. For example $\mathbb{R^n}$ is complete with respect to the 'standard' Euclidean metric. I was wondering if being ...
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1answer
39 views

How to argue some of following maps form $\mathbb Z\to Q$ are not possible? [closed]

{$f:Z\to Q|$f is bijective and monotonically increasing} Actually I only now that there is function bijective form $N\to N\times N$ I can make it from $Z\to N\times N$. But after that I could not ...
2
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1answer
45 views

Is there a bounded domain on which Poincaré's inequality does not hold?

Suppose that $U$ is a bounded domain in $\mathbb{R}^n$. Poincaré's inequality states that (for $U$ sufficiently "nice") there exists a constant $C>0$ such that if $u\in H^1(U)$ satisfies $\int_U u =...
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2answers
24 views

Existence of nonnegative and nonconstant martingales

Are there martingales that are nonnegative and nonconstant? If so, are their any intuitive examples for such?
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1answer
20 views

Prove that fn is converge uniformly to 0

In the first section of this problem which about f and it's solution, I try to cheak it but I confused about who to prove uniformity to this given example ?
1
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0answers
38 views

Example of measure for some algebra over $\mathbb N$

$\mathcal F$ is a set of events ($\sigma$-algebra). Can you give an example for some algebra $\mathcal A$ over $\mathbb N$ a non-zero finite additive measure $\mu $ on this algebra, which has a ...
2
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2answers
40 views

Example of sequences $\{x_n\}$ and $\{y_n\}$ with matching ranges of values but different limits.

Find an example of two sequences $\{x_n\}$ and $\{y_n\}$ such that $R(x_n) = R(y_n)$ and: $x_n$ and $y_n$ are convergent, and $\lim_{n \to \infty} x_n \ne \lim_{n \to \infty} y_n$; $x_n$ ...
4
votes
1answer
58 views

A function whose averages lie in a set does not need to take values in this set almost everywhere

Let $ (X,\Sigma,\mu)$ be a finite measure space. ($\mu(X)<\infty$). Let $f \in L^1(\mu)$ be a complex-valued function, and let $S \subseteq C$ be a non-closed subset. Suppose that for every $E \in ...