Estimating the division rate and kernel in the fragmentation equation

We consider the fragmentation equation

$\frac{?f}{?t}(t,x)=?B(x)f(t,x)+\underset{y=x}{\overset{y=\infty }{\int }}k(y,x)B(y)f(t,y)dy,$

and address the question of estimating the fragmentation parameters – i.e. the division rate $B(x)$ and the fragmentation kernel $k(y,x)$ – from measurements of the size distribution $f(t,?)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance Xue and Radford (2013) [26] for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x)=\alpha x^{\gamma }$ and a self-similar fragmentation kernel $k(y,x)=\frac{1}{y}{k}_0(\frac{x}{y})$, we use the asymptotic behavior proved in Escobedo et al. (2004) [11] to obtain uniqueness of the triplet $(\alpha ,\gamma ,k_0)$ and a representation formula for $k_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.     

Imprecise random variables,random sets,and Monte Carlo simulation

The paper addresses the evaluation of upper and lower probabilities induced by functions of an imprecise random variable. Given a function g and a family $X_{\lambda }$ of random variables, where the parameter λ ranges in an index set Λ, one may ask for the upper/lower probability that $g(X_{\lambda })$ belongs to some Borel set B. Two interpretations are investigated. In the first case, the upper probability is computed as the supremum of the probabilities that $g(X_{\lambda })$ lies in B. In the second case, one considers the random set generated by all $g(X_{\lambda })$, $\lambda \in \mathrm{\Lambda }$, e.g. by transforming $X_{\lambda }$ to standard normal as a common probability space, and computes the corresponding upper probability. The two results are different, in general. We analyze this situation and highlight the implications for Monte Carlo simulation. Attention is given to efficient simulation procedures and an engineering application is presented.     

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): $\dot{x}=y?12x^2+16y^2$, $\dot{y}=?x?16xy$, and (r20): $\dot{x}=y+4x^2$, $\dot{y}=?x+16xy$, and the periodic orbits of the quadratic isochronous centers $(S_1):\dot{x}=?y+x^2?y^2$, $\dot{y}=x+2xy$, and $(S_2):\dot{x}=?y+x^2$, $\dot{y}=x+xy$. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system $(S_1)$ and $(S_2)$ are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line $y=0$. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively $4n?3(n\ge 4)$ and $4n+3(n\ge 3)$ counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers $(S_1)$ and $(S_2)$ are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.     

A new type of approximation for the radical quintic functional equation in non-Archimedean (2,β)-Banach spaces

Let $\mathbb{R}$ be the set of real numbers. In this paper, we first introduce the notions of non-Archimedean $(2,\beta )$-normed spaces $(X,{6?,?6}_{?,\beta })$ and we will reformulate the fixed point theorem [10, Theorem 1] in this space, after it, we introduce and solve the radical quintic functional equation

$f\left(\sqrt[5]{x^5+y^5}\right)=f(x)+f(y),x,y\in \mathbb{R}.$

Also, under some weak natural assumptions on the function $\gamma :\mathbb{R}\times \mathbb{R}\times X\to [0,\infty )$, we show that this theorem is a very efficient and convenient tool for proving the hyperstability results when $f:\mathbb{R}\to X$ satisfy the following radical quintic inequality

${6f\left(\sqrt[5]{x^5+y^5}\right)?f(x)?f(y),z6}_{?,\beta }\le \gamma (x,y,z),x,y\in \mathbb{R}?\{0\},z\in X,$

with $x\ne ?y$.     

Majoration of the dimension of the space of concatenated solutions to a specific pantograph equation

For each $\lambda \in {\mathbb{N}}^{?}$, we consider the integral equation:

$\underset{\lambda y}{\overset{\lambda x}{\int }}f(t)\mathrm{d}t=f(x)?f(y)\text{ for every }(x,y)\in {\mathbb{R}}_{+}^2\text{,}$

where f is the concatenation of two continuous functions $f_a,f_b:[0,\lambda ]\to \mathbb{R}$ along a word $u=u_0{u}_1?\in {\{a,b\}}^{\mathbb{N}}$ such that $u=\sigma (u)$, where σ is a λ-uniform substitution satisfying some combinatorial conditions.There exists some non-trivial solutions ([1]). We show in this work that the dimension of the set of solutions is at most two.     

Lehmer's totient problem over Fq[x]

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x]$, where ${\mathbb{F}}_q$ is a finite field with q elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}?1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3$, $p(x)$ is the product of any 2 non-associate irreducibles of degree 1; or (iii) $q=2$, $p(x)$ is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.     

Bivariate identities related to Chebyshev and Dickson polynomials over Fq

Let ${\mathbb{F}}_q$ be a field of q elements, where q is a power of an odd prime p. The polynomial $f(y)\in {\mathbb{F}}_q[y]$ defined by$f(y):={(1+\sqrt{y})}^{(q+1)/2}+{(1-\sqrt{y})}^{(q+1)/2}$ has the property that$f(1-y)=\rho (2)f(y),$ where ρ is the quadratic character on ${\mathbb{F}}_q$. This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.