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research_proposal_1 | research_proposal | Critical Lengths and Controllability for the Kawahara Equation | We study the (linearized) Kawahara equation on $(0,L)$:
$$u_t + u_x + u_{xxx} - u_{xxxxx} = 0,$$
with boundary conditions
$$u(t,0)=u(t,L)=u_x(t,0)=u_x(t,L)=0,$$
and boundary control acting at $x=L$ through
$$u_{xx}(t,L)=h(t).$$
A length $L>0$ is called \emph{critical} if there exists a nontrivial solution of the statio... | null | null | https://hal.science/hal-05111010v1/file/ADR-HAL.pdf |
research_proposal_2 | research_proposal | Critical Lengths and Controllability for the Kawahara Equation | For the Kawahara equation, the characteristic equation
$$\mu^5-\mu^3-\mu + i p = 0$$
has five roots, and criticality is determined by transcendental compatibility conditions involving exponentials $e^{\mu_j L}$. Unlike the KdV case, there is no obvious reduction to a simple resonance formula.
Classify possible configu... | null | null | https://hal.science/hal-05111010v1/file/ADR-HAL.pdf |
research_proposal_3 | research_proposal | Stabilizing quantities $\rho_p$ for boundary control of PDE | In the study of stability of nonlinear PDE of the form
$$\partial_t u + A(u)\,\partial_x u + B(u)=0\quad\text{on }[0,L],$$
with boundary conditions
$$\begin{pmatrix}u_+(t,0)\\u_-(t,L)\end{pmatrix}=G\begin{pmatrix}u_+(t,L)\\u_-(t,0)\end{pmatrix},$$
a natural quantity is
$$\rho_p(M)=\inf\{\|\Delta M\Delta^{-1}\|_p\;|\;\D... | null | null | null |
research_proposal_4 | research_proposal | Stabilizing quantities $\rho_p$ for boundary control of PDE | With $\rho_p$ defined by diagonal scaling as
$$\rho_p(M)=\inf_{\Delta=\mathrm{diag}(\delta_1,\dots,\delta_n),\ \delta_i>0}\|\Delta M\Delta^{-1}\|_p,$$
it is known that for any $n\in\mathbb N$ and any $M\in\mathbb R^{n\times n}$,
$$\rho_1(M)\ge \rho_2(M),\qquad \rho_1(M)=\rho_\infty(M),$$
and for $n\ge 2$ there exists $... | null | null | null |
research_proposal_5 | research_proposal | Boundary stability of the viscous Saint-Venant equations | Consider the viscous Saint-Venant system
$$\partial_t H + \partial_x(HV)=0,$$
$$\partial_t V + V\,\partial_x V + g\,\partial_x H + \frac{f(H,V)}{H} -4\mu\,\frac{\partial_x(H\partial_x V)}{H}=0,$$
with boundary conditions
$$V(t,0)=G_1(H(t,0)),\qquad V(t,L)=G_2(H(t,L),\partial_x V(t,L)),\qquad \partial_x V(t,0)=0.$$
(Co... | null | null | null |
Q764 | RMS | null | a) We consider a set $X = \{x_1 < x_2 < \cdots < x_n\}$ of real numbers, and $k \leq n$. For any subset $F$ of $X$, we denote $\delta(X, F) = \max_{x \in X} \min_{y \in F} |x - y|$. Give an efficient algorithm to calculate the subset $F$ with cardinality $k$ such that $\delta(X, F)$ is minimal.
b) Resume the previous ... | Discrete Mathematics - Computer Science | difficile | null |
Q825 | RMS | null | Let $\Omega_n$ be the set of binary trees with $n$ leaves.
\begin{enumerate}[a)]
\item Find a recurrence relation for the cardinalities of $\Omega_n$.
\item Describe an algorithm to simulate a random binary tree with a uniform distribution on $\Omega_n$.
\item If $X$ is a random variable with a uniform distribution on ... | Discrete Mathematics - Computer Science | moyen | null |
Q1038 | RMS | null | Following the corrected oral exercise 227 (RMS 111-9) and the response R712 published in RMS 132-2. A matrix $A = (a_{i,j})$ is said to satisfy $P(n, h, k)$ if
\begin{itemize}
\item $A$ is square of order $n$;
\item $a_{i,j} \in \{0, 1\}$ for all $i, j$ ;
\item there exists, for each $j$, exactly $h + k$ indices $i$ su... | Discrete Mathematics - Computer Science | moyen | null |
Q1062 | RMS | null | In the response R1001 present in this journal, the solutions of the diophantine equation $(E_n)$ are studied:
\[ x_1 x_2 \cdots x_n = 2(x_1 - 2)(x_2 - 2) \cdots (x_n - 2), \]
where $5 \leq x_1 \leq x_2 \leq \cdots \leq x_n$. Here are three unresolved questions.
a) It is established at the end of the response that the ... | Discrete Mathematics - Computer Science | moyen | null |
Q604 | RMS | null | Let $A$, $B$ be polynomials in $K[X]$, coprime, and $U$ and $V$ be polynomials in $K[X]$ such that $\deg U < \deg B$ and $AU + BV = 1$. Let $m$ and $n$ be non-zero natural integers. Explicitly find the pairs $(U_{m,n}, V_{m,n})$ of polynomials in $K[X]$ such that $\deg U_{m,n} < \deg B^n$ and $A^m U_{m,n} + B^n V_{m,n}... | General Algebra - Arithmetic | facile | null |
Q831 | RMS | null | We denote $\Sigma_1, \Sigma_2, \ldots, \Sigma_n$ as the elementary symmetric polynomials of $X_1, X_2, \ldots, X_n$. For $m = 2, 3, \ldots$, can we make explicit the $\Sigma_i(X_1^m, X_2^m, \ldots, X_n^m)$? | General Algebra - Arithmetic | difficile | null |
Q834 | RMS | null | In oral exam exercise 5 (118-4) it is shown that there exists a real sequence $(a_n)_{n \in \mathbb{N}}$ such that for every natural number $n \in \mathbb{N}$, the polynomial $a_n X^n + \cdots + a_1 X + a_0$ has exactly $n$ distinct real roots, but without providing an explicit formula. Does there exist a sequence of n... | General Algebra - Arithmetic | difficile | null |
Q938 | RMS | null | Let $n$ be an integer at least equal to two. In the group $S_n$ of permutations of order $n$, we denote by $f(n, k)$ the number of elements that are powers $k^e$. Let $p$ and $q$ be prime numbers divisors of $n!$ such that $p < q$. Is it true that $f(n, p) < f(n, q)$? | General Algebra - Arithmetic | difficile | null |
Q947 | RMS | null | Characterize the pairs $(G, f)$ where $G$ is a finite group of odd order and $f$, an involutive automorphism of $G$. | General Algebra - Arithmetic | facile | null |
Q995 | RMS | null | a) Can we characterize the monic polynomials $A$ with integer coefficients such that for all prime $p$ there exists an integer $a$ such that $p$ divides $P(a)$? b) Can we characterize the polynomials $A$ with integer coefficients such that for all integer $n$ there exists an integer $a$ such that $n$ divides $P(a)$? | General Algebra - Arithmetic | facile | null |
Q1003 | RMS | null | One classically shows that a polynomial $P$ with real coefficients taking only positive values can be written in the form $A^2 + B^2$ where $A$ and $B$ are in $\mathbb{R}[X]$. Find necessary and sufficient conditions for a polynomial $Q$ with rational coefficients to be written in the form $A^2 + B^2$ where $A$ and $B$... | General Algebra - Arithmetic | difficile | null |
Q1021 | RMS | null | Let $G$ be a non-abelian finite group. What are the elements of $G$ that can be written as products of all elements of $G$ in any order, each element appearing exactly once? In particular, handle the case $G = S_n$.
Oral exercise 6, the solution of which appears in RMS 131-3, deals with the case $G = S_3$. | General Algebra - Arithmetic | moyen | null |
Q1037 | RMS | null | Let $K$ be a field. We denote by $(K^*)[2]$ the set of squares of the non-zero elements of $K$. A square root on $K^*$ is a group homomorphism $\varphi$ from $(K^*)[2]$ to $K^*$ such that $\varphi(x)^2 = x$ for all $x \in (K^*)[2]$. In the corrected oral exam problem 13 (RMS 128) we study for $K = \mathbb{Q}, \mathbb{R... | General Algebra - Arithmetic | facile | null |
Q1043 | RMS | null | An integer $n > 1$ is said to be a powerful number if, for every prime divisor $p$ of $n$, $p^2$ also divides $n$.
\begin{enumerate}
\item Show that there exists a sequence of $n$ consecutive natural integers containing no powerful numbers.
\item Even better: show that there exists a sequence of $n$ consecutive natu... | General Algebra - Arithmetic | difficile | null |
Q1070 | RMS | null | Here is problem 3, day 1 of the IMC 2023 Bulgaria International Mathematics Competition.
Determine the real polynomials $P(x, y)$ satisfying for all real numbers $x, y, z, t$ :
\[
P(x, y) P(z, t) = P(xz - yt, xt + yz).
\]
Now here is a proposition with four variables. Determine all real polynomials $P(x_1, x_2, x_3, ... | General Algebra - Arithmetic | facile | null |
Q655 | RMS | null | Following R626. Let $K$ be any field and $E$ a $K$-vector space of finite dimension $n \geq 2$. Let $C(f)$ denote the commutant of $f \in L(E)$. When $f$ and $g$ are not homotheties, what is the maximum dimension of $C(f)+C(g)$ and which pairs $(f, g)$ achieve the equality? | Linear and Bilinear Algebra | difficile | null |
Q706 | RMS | null | We denote $S_n([a, b])$ as the set of real symmetric matrices whose coefficients are all in $[a, b]$. For any real symmetric matrix $A$, we denote $\lambda_k(A)$ as the k extsuperscript{th} eigenvalue of $A$ in decreasing order. Study as a function of $n$, $a$, $b$ the bounds of the set of $\lambda_k(A)$ where $A$ runs... | Linear and Bilinear Algebra | facile | null |
Q857 | RMS | null | The corrected oral exercise 73 (102-10) involved studying, for $A$ in $M_n(\mathbb{C})$, the infimum of $\|A - M\|$, where $M$ ranges over the set of matrices $M$ such that $\det M = 0$, $\|\cdot\|$ being the norm subordinate to the Hermitian norm.
Study in $M_n(\mathbb{C})$ (resp. $M_n(\mathbb{R})$) the infimum of $\... | Linear and Bilinear Algebra | difficile | null |
Q942 | RMS | null | This question follows on from R499 and the article “What is the maximum dimension of a vector subspace of \( M(n, \mathbb{R}) \) whose every non-zero element is invertible?”, (see RMS 127-4).
Let \( G(n) \) be the set consisting of \( \mathrm{GL}(n, \mathbb{R}) \) and the zero matrix of \( M(n, \mathbb{R}) \).
\begin{e... | Linear and Bilinear Algebra | difficile | null |
Q949 | RMS | null | Continuation of the corrected oral exam exercise 516 (124 4).
Let $E$ be a real normed vector space of finite dimension at least 2. In exercise 516, it is shown that for any non-zero element $a$ of $E$ and any element $b$ of $E$, the infimum of $\|\|\|f\|\|\|$, where $f$ is an endomorphism of $E$ such that $f(a) = b$, ... | Linear and Bilinear Algebra | facile | null |
Q1013 | RMS | null | Let $M \in \mathrm{M}_n(\mathbb{C})$. We say that $M$ satisfies (P) if the set of subalgebras of $\mathrm{M}_n(\mathbb{C})$ containing $M$ is finite. In the oral exercise 15 of the year 2014, corrected in RMS 125-4, it is proven that if $M$ has $n$ distinct eigenvalues then $M$ satisfies (P).
a) Show that, for $n = 2$... | Linear and Bilinear Algebra | facile | null |
Q1026 | RMS | null | Let $E$ be a finite-dimensional vector space; for each endomorphism $u$ of $E$, denote by $z(u)$ the algebra of endomorphisms commuting with $u$.
\begin{enumerate}[a)]
\item If $u = [v, w]$, where $v$ and $w$ are in $z(u)$, show that $u$ is nilpotent.
\item Conversely, if $E$ is of dimension 3 and if $u$ is a nilpotent... | Linear and Bilinear Algebra | difficile | null |
Q1034 | RMS | null | In this question, all matrices involved are complex square matrices of size $n$.
a) What are the real matrices $B$ such that for any real symmetric positive definite matrix $A$, $AB$ is diagonalizable in $M_n(\mathbb{R})$?
b) What are the complex matrices $B$ such that for any real symmetric positive definite matrix ... | Linear and Bilinear Algebra | facile | null |
Q1051 | RMS | null | What is the maximum $a_n$ of the dimension of a subalgebra of $M_n(K)$ generated by two projectors? $K$ is an arbitrary field and $n$ is an integer at least equal to two. In the response R1020 published in the RMS 133-1, it is proven that $a_n = n^2$ if and only if $n = 2$ and $K$ has at least three elements. | Linear and Bilinear Algebra | facile | null |
Q1069 | RMS | null | We consider the space $S$ of real sequences, the subspace $B$ of bounded sequences, and we focus on operators of the form:
\[
\Phi : U \in S \mapsto V \in S \quad \text{where} \quad \forall n, \ v_n = a_1 u_{f_1(n)} + \ldots + a_d u_{f_d(n)}
\]
where the $a_i$ are strictly positive real numbers whose sum is $1$, an... | Linear and Bilinear Algebra | facile | null |
Q1072 | RMS | null | a) Let $A \in M_n(\mathbb{C})$ and $A = UH$ be its polar decomposition ($U$ unitary, $H$ Hermitian positive). Show that for all $k \in \mathbb{N}$, $| \operatorname{tr}(A^k)| \leq \operatorname{tr}(H^k)$.
b) More generally, if $U_1, \ldots, U_k$ are unitary matrices and $H$ a positive Hermitian matrix, is it true that... | Linear and Bilinear Algebra | facile | null |
Q756 | RMS | null | Following the corrected exercise 126 (121-4). Let $\beta$ be a real number and $\lambda$ a real number such that $|\lambda| > 1$. We denote by $F(\beta, \lambda)$ the set of functions $g$ in $C^1(\mathbb{R}, \mathbb{R})$ such that
\[ \forall t \in \mathbb{R}, \quad g'(t) = g(\lambda t) - \beta g(t). \]
There exists a... | Real and Functional Analysis | moyen | null |
Q758 | RMS | null | Let $v$ be a continuous function from $[0, 1]$ to $[0, 1]$. We denote by $E$ the vector space of continuous real functions on $[0, 1]$. For $f$ in $E$, we set :
$$Tf(x) = \int_0^1 f(xv(t))\,dt.$$
Find necessary and sufficient conditions for $T$ to be injective. | Real and Functional Analysis | difficile | null |
Q850 | RMS | null | It is known that any irrational algebraic number $\alpha \in \mathbb{R}$ satisfies a Diophantine inequality of the type
\[ \exists d \geq 2, \exists C > 0, \forall (p, q) \in \mathbb{Z} \times \mathbb{N}, \quad \left| \alpha - \frac{p}{q} \right| \geq \frac{C}{q^d} \cdot \]
A typical example of a number that does not s... | Real and Functional Analysis | facile | null |
Q909 | RMS | null | Let $\varphi : [0, 1] \rightarrow \mathbb{R}$ be a convex function and $I$ the interval $]0, +\infty[$. Determine a necessary and sufficient condition on $\varphi$ so that there exists a function $f \in L^1(I) \cap L^\infty(I)$ such that $\ln \|f\|_{L^p(0,+\infty)} = \varphi\left(\frac{1}{p}\right)$ for all $p \geq 1$. | Real and Functional Analysis | difficile | null |
Q912 | RMS | null | According to exercise 112 of the RMS 124-2. We denote by $E$ the set of $C^1$ functions, concave on $[-1, 1]$, vanishing at $-1$ and $1$, and not identically zero.
a) Show the existence of $c > 0$ such that for every $f$ in $E$,
\[
\int_{-1}^{1} f(t)\,dt \geq c \int_{-1}^{1} |f'(t)|\,dt.
\]
What is the best poss... | Real and Functional Analysis | difficile | null |
Q941 | RMS | null | Let $S$ denote the Schwarz space on $\mathbb{R}$, which is the vector space of functions $f \in C^{\infty}(\mathbb{R}, \mathbb{R})$ such that $\sup_{x \in \mathbb{R}^n} |x^n f^{(m)}(x)| < +\infty$ for all $(m, n) \in \mathbb{N}^2$. For every integer $p \geq 1$, we define
\[ J_p := \inf_{f \in S} \int_{\mathbb{R}} f'(x... | Real and Functional Analysis | difficile | null |
Q955 | RMS | null | Following oral exercise 93 corrected in RMS 128-3. For any natural integer $q$ and any real $x \in ]0, \pi[$, we consider the expression:
\[ S(q, x) = (q + 1) \sum_{k=q+1}^{\infty} \frac{\sin kx}{k} \cdot \frac{\sin x}{2}. \]
Determine its supremum. | Real and Functional Analysis | moyen | null |
Q983 | RMS | null | In the oral exercise 33 corrected in RMS 129-3, we are asked to study the limit values of the sequence $\{ \ln(n!) \}$ where $\{ x \}$ is the fractional part of $x$. For any integer $k > 0$, we define $u_k(n) = \{ \ln(\ln(\ldots(\ln(n!)) \ldots )) \}$ where the natural logarithm is iterated $k$ times.
\begin{itemize}
... | Real and Functional Analysis | difficile | null |
Q986 | RMS | null | Exercise 291 from RMS 119-2 asked if there exists a subset $A$ of $\mathbb{N}$ such that $f_A : x \mapsto \sum_{n \in A} \frac{x^n}{n!} \sim_{x \rightarrow +\infty} \frac{e^x}{x^2}$. A negative and fairly simple answer was given in RMS 119-4. This answer shows that the exponent 2 is not exceptional but leaves room for ... | Real and Functional Analysis | difficile | null |
Q1000 | RMS | null | Following R827, published in RMS 130-3.
For all integers $p$ and $q$ such that $0 < p < q$, we define $M(p, q) = \max_{x \in \mathbb{R}} |\sin(px) - \sin(qx)|$.
a) Show that the sequences
\[M(1, 4n)\]_{n \geq 1},
\[M(1, 4n + 1)\]_{n \geq 0},
\[M(1, 4n + 2)\]_{n \geq 0}
are strictly increasing.
b) Show that, for all $n... | Real and Functional Analysis | moyen | null |
Q1007 | RMS | null | Consider the functional equation: (E) $f'(t) = f(t - t^2)$. We know that the set of functions defined on $[0, 1]$, differentiable and solutions of (E) is a vector space of dimension 1. Let $f$ be the unique solution on $[0, 1]$ such that $f(0) = 1$.
\begin{enumerate}
\item Is there a solution $g$ of $E$ defined on an ... | Real and Functional Analysis | difficile | null |
Q1023 | RMS | null | It is well known that the series with general term $\sin(\pi e n!)$ converges. Study the set of real $x$ such that the series with general term $\sin(x n!)$ converges. | Real and Functional Analysis | difficile | null |
Q1032 | RMS | null | n X k=0 a_n X^n \in \mathbb{R}[X] of degree n, has a mode at M \leq n if we have a_0 \leq a_1 \leq \cdots \leq a_M \geq a_{M+1} \geq \cdots \geq a_n.
Let $P$ be a polynomial with strictly positive coefficients; we are interested in its powers. If $P$ represents the generating function of a random variable $X$, then $... | Real and Functional Analysis | facile | null |
Q565 | RMS | null | Let $T(n)$ be the Bell number of index $n : T(n) = e^{-1} \sum_{k=0}^{\infty} \frac{k^n}{k!}$. In \cite{R506} (116-4) it is established that $T(n) \sim f(n)$ where
\[ f(x) := \frac{1}{\sqrt{\ln x}} \frac{w(x)^x}{x} e^{w(x)-x-1}, \]
and $w : [0, +\infty[\to[1, +\infty[$ is the inverse function of $y \mapsto y \ln y$. ... | Asymptotics | difficile | null |
Q776 | RMS | null | Let the entire series $f_m(x) = \sum_{n=0}^{\infty} \frac{x^n}{(n!)^m}$ for $m = 2, 3, \ldots$. Find a simple equivalent in $-\infty$ for $f_m$.
Note. In the article by Jean-Christophe Feauveau “some simple equivalents of sums of entire series” (107-5), we find the following equivalent of $f_m$ for $+\infty$, as a pa... | Asymptotics | facile | null |
Q839 | RMS | null | Let $q$ be a real number strictly between $-1$ and $1$. Let $f$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that is a solution of the functional equation
\[ f(x) = (1 - qx)f(qx). \]
In oral exam 21 (103-5), it is asked to prove the existence of a power series expansion of $f$ on $\mathbb{R}$.
Here, it ... | Asymptotics | difficile | null |
Q855 | RMS | null | Consider the real sequences $(u_n)$ given by the second-order recurrence relation:
\[ u_{n+2} = u_{n+1} + \lambda u_n \]
In the answer R773 (124-1) it has been proven that the solution space has a basis
$\{(v_n), (w_n)\}$, where $(v_n)$ is equivalent to $n^\lambda$ and $(w_n)$ to
\[ (-1)^n \frac{n^\lambda}{n! n^{\la... | Asymptotics | facile | null |
Q865 | RMS | null | This question followed the corrected oral exercise 2 (113-4) and then the answer R469 (116-2).
For $n \geq 2$, we consider the polynomial $P_n = X^n - \sum_{k=0}^{n-1} X^k$.
In R469, we showed the following results.
(i) The polynomial $P_n$ has a unique positive real root denoted $\rho_n$. The sequence $(\rho_n)$ is in... | Asymptotics | facile | null |
Q1039 | RMS | null | If $P = \sum_{k=0}^{n} a_k X^n$ is a real or complex polynomial, we denote
\[ \|P\|_\infty = \sup_{t \in [0,1]} |P(t)| \]
and
\[ N(P) = \max_{0 \leq k \leq n} |a_k|; \]
and, for $n \in \mathbb{N}$, we denote
\[ C_n = \sup_{P \in \mathbb{R}_n[X], \|P\|_\infty = 1} N(P). \]
What estimates of $C_n$ can we establish? By "e... | Asymptotics | moyen | null |
Q1046 | RMS | null | In oral exam 116 from the RMS list 132 2, we ask to study certain properties of the solutions of the differential equation, defined on $\mathbb{R}^+$ :
\[ y'' = \frac{8x^n}{(1 + x^2)^2} y. \]
Study, according to the natural integer $n$, the asymptotic behavior of the solutions when $x$ tends towards $+\infty$. | Asymptotics | difficile | null |
Q1066 | RMS | null | Voici l’exercice d’oral 83 corrigé dans RMS 133-3. For $x \in \mathbb{R}$, let $\lceil x\rceil$ be the smallest integer greater than or equal to $x$. We define a sequence $(u_n)_{n \in \mathbb{N}^*}$ by $u_0 = 1$, $u_n = 2u_{n-1}$ for every integer $n \geq 2$ which is a power of $2$, $u_n = \lceil \frac{u_{n-1}}{3} \rc... | Asymptotics | difficile | null |
Q701 | RMS | null | Let $a \in ]0, 1[$; we consider the set $C$ of functions from $[0, 1]$ to $\mathbb{R}$ which vanish at $0$ and are Hölder continuous of order $a$ with exponent $1$. Equipped with the uniform convergence on $[0, 1]$, $C$ is a convex compact set. Can we exhibit simple properties of the extreme points of $C$ and can we ev... | Topology | facile | null |
Q748 | RMS | null | Let $A$ be a subset of the Euclidean space $\mathbb{R}^n$, arcwise connected and $C^1$ piecewise. If $x$ and $y$ are any two points $x$ and $y$ of $A$, we define their geodesic distance $d(x, y)$ as the infimum of the set of lengths of arcs connecting them. We identify $\mathrm{M}_n(\mathbb{R})$ with the Euclidean spac... | Topology | difficile | null |
Q803 | RMS | null | Let $(a_n)$ be a real sequence, with strictly positive terms, decreasing and summable. We denote $\Sigma$ as the set of sums of extracted series (finite or infinite) from this sequence. Can we characterize the sequences of this type such that $\Sigma$ has an empty interior? | Topology | facile | null |
Q805 | RMS | null | Consider \( \mathbb{R}^2 \) with the usual dot product. For any norm \( N \) on \( \mathbb{R}^2 \), the dual norm \( N^\star \) is defined by: \[ \forall x \in \mathbb{R}^2, \quad N^\star(x) = \sup_{y \in \mathbb{R}^2\setminus\{0\}} \frac{\langle x, y \rangle}{N(y)}. \] Can we characterize the norms \( N \) such that t... | Topology | facile | null |
Q820 | RMS | null | \text{Following the corrected oral exercise 39 (113-4).} \\ \text{Let } C \text{ be a convex compact in } \mathbb{R}^2 \text{ with non-empty interior. We denote } \delta(C) \text{ the diameter of } C. \\ \text{For any point } P, \text{ we denote } \mu(P, C) \text{ the average of the distances from } P \text{ to the poi... | Topology | difficile | null |
Q963 | RMS | null | For each $\|\cdot\|$ on $\mathbb{R}^3$ we denote by $|||\cdot|||$ the associated subordinate norm. For any real $p$, we denote $A(p)$ the matrix
\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & \frac{1}{2} & p \\
0 & 0 & \frac{1}{2}
\end{pmatrix}
\]. In oral exam 997 of RMS 128-2, $\|\cdot\|$ being one of the three standard ... | Topology | difficile | null |
Q978 | RMS | null | Let $n \geq 1$. For all $P \in \mathbb{R}_n[X]$, we denote $\|P\|_1 = \int_0^1 |P(t)| \, dt$ and $\|P\|_\infty = \max_{0 \leq x \leq 1} |P(x)|$. Find $\inf_{P \in \mathbb{R}_n[X]} \frac{\|P\|_1}{\|P\|_\infty}$. | Topology | moyen | null |
Q1068 | RMS | null | Let $f : \mathbb{R}^n \to \mathbb{R}$ and $g : \mathbb{R}^n \to \mathbb{R}$ be two convex, differentiable, and lower-bounded functions. We equip $\mathbb{R}^n$ with the Euclidean norm $\| . \|$. Let $(\nabla f)(x)$ (resp. $(\nabla g)(x)$) denote the gradient vector of $f$ (resp. $g$) at $x$. Show that if $\| (\nabla f)... | Topology | facile | null |
Q793 | RMS | null | Following the corrected oral exercise 24 (121-4). We denote $E_n$ the set of square matrices of order $n$ where all coefficients are 1 or -1. In the mentioned exercise, it is shown that the average of the determinants of the elements of $E_n$ is zero and that the average of the squares of these determinants is $n!$. St... | Probability | moyen | null |
Q975 | RMS | null | This question is from the oral exercise 169 corrected in the present journal.
Let $Z$ be a random variable with values in $\mathbb{N}^*$.
\begin{itemize}
\item[a)] Give a necessary and sufficient condition on $Z$ for there to exist $X$ and $Y$, independent random variables that are not almost surely constant, such th... | Probability | moyen | null |
Q1048 | RMS | null | This question follows the answer R859 published in RMS 133-1.
Let $(X_n)$ be a sequence of independent random variables with the same Bernoulli distribution, that is, uniformly distributed over \{0, 1\}. We consider the random version of the Fibonacci sequence $(u_n)$ defined by the recurrence relation
\[ u_{n+1} = u_n... | Probability | facile | null |
Q1060 | RMS | null | In exercise 28 corrected in RMS 117-4 it is proven that any function $f : \mathbb{Z}^2 \to \mathbb{R}$ which is positive and harmonic on $\mathbb{Z}^2$, that is, satisfying
\[ f(m, n) = \frac{1}{4} \left(f(m, n -1) + f(m, n + 1) + f(m -1, n) + f(m + 1, n) \right) \]
for all integers $m$ and $n$, is constant.
The pr... | Probability | moyen | null |
Q1063 | RMS | null | Let $X$ be a real random variable such that $E(|X|)$ is finite. We assume that $X$ is symmetric, i.e., $X$ and $-X$ have the same distribution.
A point is placed on the real line according to the distribution of $X$. I am initially placed at $0$ and I set off blindly in search of the point, making a series of journeys ... | Probability | difficile | null |
Q804 | RMS | null | We denote by $S$ the unit sphere of $\mathbb{R}^3$. For every finite subset $X$ of $S$ with at least two elements, we denote $\delta(X)$ as the minimum distance between two points of $X$; we denote $d(n)$ as the maximal value of $\delta(X)$, where $X$ consists of $n$ points of $S$.
a) Suppose there exists a regular p... | Geometry | facile | null |
Q832 | RMS | null | Let us consider a regular dodecagon $P$ and form the twelve squares 'directed' towards the center of $P$ with sides that are sides of $P$. The twelve squares thus obtained verify the following two properties:
\begin{itemize}
\item two different squares do not share a common side;
\item a vertex belongs to exactly two ... | Geometry | difficile | null |
Q896 | RMS | null | In the oriented real affine plane, equipped with a direct coordinate system $(O, \vec{ı}, \vec{ȷ})$, we consider $n$ distinct rays all having $O$ as origin. They divide the plane into $n$ zones $Z_1, Z_2, \ldots, Z_n$ which are the connected components of the complement of the union of these rays. It is assumed that no... | Geometry | moyen | null |
Q971 | RMS | null | Construct the sets of four distinct, non-collinear points in the Euclidean plane such that every distance between two of these points is an integer. In exercise 163 of RMS 125-2, solved in RMS 125-3, it is shown that these distances cannot all be odd integers. Moreover, the book "Key Problems for Advanced Mathematics" ... | Geometry | facile | null |
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