مسابقة دكتوراه 2026 — المدرسة الوطنية العليا للرياضيات

Let $\lambda \in ]0,1[$ be fixed, and let $f,g,h : [0,1] \to [0,+\infty[$ be continuous functions such that

$$\forall x,y \in [0,1], \qquad h(\lambda x+(1-\lambda)y) \geq f(x)^\lambda g(y)^{1-\lambda}.$$

Let

$$\alpha = \int_0^1 f(x)\,dx \qquad\text{and}\qquad \beta = \int_0^1 g(x)\,dx.$$

1. Show that $\alpha \neq 0$ and $\beta \neq 0$.

2. Consider the functions

$$\Phi : x \mapsto \frac{1}{\alpha}\int_0^x f(t)\,dt \qquad\text{and}\qquad \Psi : x \mapsto \frac{1}{\beta}\int_0^x g(t)\,dt.$$

Check that $\Phi$ and $\Psi$ are bijections from $[0,1]$ onto $[0,1]$.

3. Show that the function

$$U : x \mapsto \lambda \Phi^{-1}(x)+(1-\lambda)\Psi^{-1}(x)$$

is continuously differentiable on $[0,1]$ and is a bijection from $[0,1]$ onto $[0,1]$.

4. By using question (3), establish the following inequality:

$$\int_0^1 h(x)\,dx \geq \left(\int_0^1 f(x)\,dx\right)^\lambda \left(\int_0^1 g(x)\,dx\right)^{1-\lambda}.$$

Consider a multiple linear regression model with $k$ regressor variables and $n$ observations $x_{ij}$, $i=1,\ldots,n$, $j=1,\ldots,k$:

$$y_i = \beta_0 + \beta_1 x_{i1} + \cdots + \beta_k x_{ik} + \varepsilon_i,$$

where $x_{ij}$, $j=1,\ldots,k$, are assumed to be deterministic.

Let

$$\widehat{y}_i = \widehat{\beta}_0 + \widehat{\beta}_1 x_{i1} + \cdots + \widehat{\beta}_k x_{ik}$$

be its least squares estimated model.

Assuming that the columns of the matrix

$$M = (x_{ij})_{1 \leq i \leq n,\; 1 \leq j \leq k}$$

of the observed levels of the regressor variables are linearly independent, and using the matrix

$$X = [1,M], \qquad 1=(1,1,\ldots,1)^T,$$

1. Give a simple evaluation of

$$\sum_{i=1}^{n}\operatorname{Var}(\widehat{y}_i).$$

2. Find a simple function $\phi(\beta,\varepsilon)$ such that

$$\widehat{\beta} = \phi(\beta,\varepsilon).$$

Justify your answers.

Note: $\beta$ and $\varepsilon$ are the vectors of true coefficients of the model and the error vector, respectively; $\widehat{\beta}$ is the vector of coefficients of the estimated model by the ordinary least squares method.

Let $A$ be a finite field. We endow $A^{\mathbb{Z}}$ with the metric

$$d(x,y)= \begin{cases} 0, & \text{if } x=y,\\[4pt] 2^{-N}, & \text{where } N=\min\{|m| : x_m \neq y_m\}, \end{cases}$$

Recall that the shift map

$$\sigma : A^{\mathbb{Z}} \to A^{\mathbb{Z}}$$

is defined by

$$(\sigma(x))_m = x_{m-1}.$$

In this exercise, a convolutional encoder is defined as a map

$$T : A^{\mathbb{Z}} \longrightarrow A^{\mathbb{Z}}$$

for which there exist integers $L \leq R$ and a map

$$\Phi : A^{R-L+1} \longrightarrow A$$

such that

$$(T(x))_m = \Phi(x_{m+L},\ldots,x_{m+R})$$

for every $m \in \mathbb{Z}$.

1. Prove that every convolutional encoder is continuous.

2. Prove that every convolutional encoder commutes with the shift, i.e.,

$$T \circ \sigma = \sigma \circ T.$$

3. Does the converse hold? Justify your answer.