grandes-ecoles

Papers (191)

2025

centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38

2024

centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26

2023

centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15

2022

centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23

2021

centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44

2020

centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18

2019

centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26

2018

centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24

2017

centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26

2016

centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20

2015

centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22

2014

centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24

2013

centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9

2012

centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18

2011

centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10

2010

centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27

2015 centrale-maths1__mp

42 maths questions

QI.A.1 Groups Algebraic Structure Identification View

Determine a pair $(A, \vec{b})$ in $\mathrm{SO}(2) \times \mathbb{R}^2$ such that $M(A, \vec{b}) = I_3$.

QI.A.2 Groups Group Homomorphisms and Isomorphisms View

Let $(A, \vec{b})$ and $(A^{\prime}, \vec{b}^{\prime})$ be in $\mathrm{SO}(2) \times \mathbb{R}^2$. Show that $M(A, \vec{b}) M\left(A^{\prime}, \vec{b}^{\prime}\right) = M\left(A A^{\prime}, A \vec{b}^{\prime} + \vec{b}\right)$.

QI.A.3 Groups Subgroup and Normal Subgroup Properties View

Show that the elements of $G$ are invertible and explicitly determine the inverse of $M(A, \vec{b})$.

QI.A.4 Groups Subgroup and Normal Subgroup Properties View

Prove that $G$ is a subgroup of $\mathrm{GL}_3(\mathbb{R})$.

QI.A.5 Groups Group Actions and Surjectivity/Injectivity of Maps View

Is the application $\Phi : \left\{ \begin{array}{cll} G & \rightarrow & \mathbb{R}^2 \\ M(A, \vec{b}) & \mapsto & \vec{b} \end{array} \right.$ surjective? Is it injective?

QI.B.1 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

Draw a graph of $\Delta\left(0, \vec{e}_1\right)$ and $\Delta\left(2, \frac{\vec{e}_1 + \vec{e}_2}{\sqrt{2}}\right)$.

QI.B.2 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

Determine a Cartesian equation of $\Delta\left(q, \vec{u}_\theta\right)$.

QI.B.3 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View

Show that a parametrization of $\Delta\left(q, \vec{u}_\theta\right)$ is given by $\left\{ \begin{array}{l} x(t) = q\cos\theta - t\sin\theta \\ y(t) = q\sin\theta + t\cos\theta \end{array} \right.$ when $t$ ranges over $\mathbb{R}$.

QI.B.4 Straight Lines & Coordinate Geometry Collinearity and Concurrency View

Under what condition are the lines $\Delta(q, \vec{u})$ and $\Delta(r, \vec{v})$ identical?

QI.C.1 Straight Lines & Coordinate Geometry Matrix and Linear Algebra Approach to Lines View

QI.C.2 Groups Group Homomorphisms and Isomorphisms View

QI.C.3 Groups Group Homomorphisms and Isomorphisms View

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Verify that $\Psi\left(M\left(R_\theta, q\vec{u}_\theta\right)\right) = \Delta\left(q, \vec{u}_\theta\right)$; deduce that $\Psi$ is surjective.

QI.C.4 Groups Subgroup and Normal Subgroup Properties View

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Let $H$ be the set of matrices $M(A, \vec{b})$ of $G$ such that $\Psi(M(A, \vec{b})) = \Delta\left(0, \vec{e}_1\right)$.
a) Describe the elements of $H$.
b) Show that $H$ is a subgroup of $G$.
c) Show that for all $g$ in $G$ and all $h$ in $H$, we have $\Psi(gh) = \Psi(g)$.

QII.A.1 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Establish that $f$ is in $\mathcal{B}_1$.

QII.A.2 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Show that $\hat{f}$ is defined on $\mathbb{R}^2$ with $\hat{f}(q,\theta) = \frac{\pi}{\sqrt{1+q^2}}$.

QII.A.3 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
We set $R(q) = \frac{1}{2\pi} \int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta$. Prove that $q \mapsto \frac{R^{\prime}(q)}{q}$ is integrable on $]0, +\infty[$ and that $$f(0,0) = -\frac{1}{\pi} \int_0^{+\infty} \frac{R^{\prime}(q)}{q}\,\mathrm{d}q$$ One may, to compute this last integral, use the change of variable $q = \operatorname{sh}(u)$.

QII.A.4 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We consider the function $f$ defined by: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{1}{1+x^2+y^2}$.
Is the function $\frac{\partial f}{\partial x}$ in $\mathcal{B}_2$?

QII.B.1 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
For $r \in \mathbb{R}^+$, compute $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.

QII.B.2 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Justify the convergence, for any real $q \geqslant 0$, of $\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2 - q^2}}\,\mathrm{d}r$.

QII.B.3 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Prove that the Radon transform of $f$ is defined on $\mathbb{R}^2$ and that $$\forall q \in \mathbb{R}^+,\quad \forall \theta \in \mathbb{R} \quad \hat{f}(q,\theta) = 2\int_q^{+\infty} \frac{r\varphi(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$

QII.B.4 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We assume that there exists a function $\varphi$ from $\mathbb{R}^+$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}^+$, such that: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \varphi\left(\sqrt{x^2+y^2}\right)$.
Deduce that $\forall q \in \mathbb{R}^+,\ \frac{1}{2\pi} \int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$.

QIII.A Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Verify that $\hat{f}$ is defined on $\mathbb{R}^2$.

QIII.B Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We consider a function $f$ belonging to $\mathcal{B}_1$ and we recall that $$\hat{f}(q,\theta) = \int_{-\infty}^{+\infty} f(q\cos\theta - t\sin\theta,\, q\sin\theta + t\cos\theta)\,\mathrm{d}t$$
Justify that for all $q$ and all $\theta$ we have $\hat{f}(-q, \theta+\pi) = \hat{f}(q,\theta)$.

QIII.C.1 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that $\bar{f}$ is of class $C^1$ on $\mathbb{R}$.

QIII.C.2 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that the function $r \mapsto r^2 \bar{f}(r)$ is bounded on $\mathbb{R}$.

QIII.C.3 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Show that if we further assume that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$, then $r \mapsto r^4 \bar{f}^{\prime}(r)$ is bounded on $\mathbb{R}$.

QIV.A.1 Integration using inverse trig and hyperbolic functions View

Justify the existence of the integral $\int_1^{+\infty} \frac{\mathrm{d}t}{t\sqrt{t^2-1}}$ and show that its value is $\frac{\pi}{2}$.

QIV.A.2 Integration by Substitution Substitution to Prove an Integral Identity or Equality View

Let $\varepsilon$ and $r$ be fixed such that $0 < \varepsilon < r$. With the change of variables $q = r\cos\theta$, establish that $$\int_\varepsilon^r \frac{\mathrm{d}q}{q^2\sqrt{r^2-q^2}} = \frac{\sqrt{r^2-\varepsilon^2}}{r^2\varepsilon}$$

QIV.B.1 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

Let $h$ be a function of class $C^1$ on $\mathbb{R}^+$. We assume that $r \mapsto r^2 h(r)$ is bounded and we set $H(q) = \int_1^{+\infty} \frac{t\, h(qt)}{\sqrt{t^2-1}}\,\mathrm{d}t$.
Show that $H$ is continuous on $]0, +\infty[$.

QIV.B.2 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

QIV.B.3 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

QIV.C.1 Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions View

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set, with the notations of part III: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
Justify that $F$ is of class $C^1$ on $]0, +\infty[$ and that near $+\infty$ we have $F(q) = O\left(\frac{1}{q}\right)$.

QIV.C.2 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

QIV.C.3 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We consider a function $f$ in $\mathcal{B}_1$ whose partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. We set: $$\forall q \in \mathbb{R}^+,\quad F(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(q,\theta)\,\mathrm{d}\theta = 2\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r$$
We admit that we can interchange the two integrals and therefore that $$\forall \varepsilon > 0 \quad \int_\varepsilon^{+\infty} \left(\frac{1}{q^2}\int_q^{+\infty} \frac{r\bar{f}(r)}{\sqrt{r^2-q^2}}\,\mathrm{d}r\right)\mathrm{d}q = \int_\varepsilon^{+\infty} \left(\int_\varepsilon^r \frac{r\bar{f}(r)}{q^2\sqrt{r^2-q^2}}\,\mathrm{d}q\right)\mathrm{d}r$$
Deduce that $\forall \varepsilon > 0,\ \int_\varepsilon^{+\infty} \frac{F^{\prime}(q)}{q}\,\mathrm{d}q = -2\varepsilon \int_\varepsilon^{+\infty} \frac{\bar{f}(r)}{r\sqrt{r^2-\varepsilon^2}}\,\mathrm{d}r$.

QIV.D.1 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$, where $R_{x,y}(q) = \frac{1}{2\pi}\int_0^{2\pi} \hat{f}(x\cos\theta + y\sin\theta + q, \theta)\,\mathrm{d}\theta$.
Establish the Radon inversion formula for this function $f$ at the point $(x,y) = (0,0)$.

QIV.D.2 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$.
Are the hypotheses made on $f$ necessary for the Radon inversion formula to be verified at the point $(x,y) = (0,0)$?

QIV.D.3 Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View

We consider a function $f$ in $\mathcal{B}_1$ such that $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are in $\mathcal{B}_2$. The Radon inversion formula states: $\forall (x,y) \in \mathbb{R}^2,\ f(x,y) = \frac{-1}{\pi} \int_0^{+\infty} \frac{R_{x,y}^{\prime}(q)}{q}\,\mathrm{d}q$.
Propose a method to obtain the Radon inversion formula at any pair $(x,y)$ from the formula at $(0,0)$.

QV.A.1 Groups Group Actions and Surjectivity/Injectivity of Maps View

If $f$ is a function defined on $\mathbb{R}^2$, we denote by $f^*$ the function $f \circ \Phi$, defined on $G$ by $f^*(g) = f(\Phi(g))$ where $\Phi : G \rightarrow \mathbb{R}^2$ is the function introduced in question I.A.5.
Prove that for all $g$ in $G$ and $r$ such that $\Phi(r) = \overrightarrow{0}$ we have $f^*(gr) = f^*(g)$.

QV.A.2 Groups Group Homomorphisms and Isomorphisms View

We now assume that $f$ satisfies the hypotheses allowing us to define its Radon transform.
Demonstrate that if two lines $\Delta\left(q_1, \vec{u}_{\theta_1}\right)$ and $\Delta\left(q_2, \vec{u}_{\theta_2}\right)$ coincide, then $\hat{f}\left(q_1, \theta_1\right) = \hat{f}\left(q_2, \theta_2\right)$.

QV.A.3 Groups Group Actions and Surjectivity/Injectivity of Maps View

We define $\hat{f}^\star$ on $G$ by composing $\hat{f}$ with $\Psi$: we set, for all $g \in G$, $\hat{f}^\star(g) = \hat{f}(\Psi(g))$.
Demonstrate that $\hat{f}^\star$ is $H$-invariant, that is, for all $g \in G$ and $h \in H$, $\hat{f}^\star(gh) = \hat{f}^\star(g)$.

QV.B.1 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

We model the density of tissues by an unknown function $f$ zero outside the zone to be studied. Assuming that each incident X-ray beam is carried by an affine line $\Delta$, and denoting by $I_e$ and $I_s$ its intensity measured on either side of the targeted zone: $$\ln\left(\frac{I_e}{I_s}\right) = \int_\Delta f$$
Propose a rigorous definition of the right-hand side of this equation in the case where $\Delta = \Delta\left(q, \vec{u}_\theta\right)$.

QV.B.2 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View

We model the density of tissues by an unknown function $f$ zero outside the zone to be studied. Assuming that each incident X-ray beam is carried by an affine line $\Delta$, and denoting by $I_e$ and $I_s$ its intensity measured on either side of the targeted zone: $$\ln\left(\frac{I_e}{I_s}\right) = \int_\Delta f$$
Explain how the Radon inversion formula allows us in principle to know the density of tissues in the radiographed zone.