Conditional risk measures and their associated risk contribution measures are commonly employed in finance and actuarial science for evaluating systemic risk and quantifying the effects of risk interactions. This paper introduces various types of contribution ratio measures based on the multivariate conditional value-at-risk (MCoVaR), multivariate conditional expected shortfall (MCoES), and multivariate marginal mean excess (MMME) studied in [34] (Ortega-Jiménez, P., Sordo, M., & Suárez-Llorens, A. (2021). Stochastic orders and multivariate measures of risk contagion. Insurance: Mathematics and Economics, vol. 96, 199–207) and [11] (Das, B., & Fasen-Hartmann, V. (2018). Risk contagion under regular variation and asymptotic tail independence. Journal of Multivariate Analysis 165(1), 194–215) to assess the relative effects of a single risk when other risks in a group are in distress. The properties of these contribution risk measures are examined, and sufficient conditions for comparing these measures between two sets of random vectors are established using univariate and multivariate stochastic orders and statistically dependent notions. Numerical examples are presented to validate these conditions. Finally, a real dataset from the cryptocurrency market is used to analyze the spillover effects through our proposed contribution measures.

In this paper, we prove a new uncertainty principle for functions with radial symmetry by differentiating a radial version of the Stein–Weiss inequality. The difficulty is to prove the differentiability in the limit of the best constant that unlike the general case it is not known. We provide also an integral alternative formula for the logarithmic weight $(\log|\xi|)$ in Fourier domain.

Consider, for any integer $n\ge 3$, the set $\operatorname {\mathrm {Pos}}_n$ of all n-periodic tree patterns with positive topological entropy and the set $\operatorname {\mathrm {Irr}}_n\subset \operatorname {\mathrm {Pos}}_n$ of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\operatorname {\mathrm {Pos}}_n$, $\operatorname {\mathrm {Irr}}_n$ and $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$. Let $\unicode{x3bb} _n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty )$. We explicitly construct an irreducible n-periodic tree pattern $\mathcal {Q}_n$ whose entropy is $\log (\unicode{x3bb} _n)$. We prove that this entropy is minimum in $\operatorname {\mathrm {Pos}}_n$. Since the pattern $\mathcal {Q}_n$ is irreducible, $\mathcal {Q}_n$ also minimizes the entropy in the family $\operatorname {\mathrm {Irr}}_n$. We also prove that the minimum positive entropy in the set $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$ (which is non-empty only for composite integers $n\ge 6$) is $\log (\unicode{x3bb} _{n/p})/p$, where p is the least prime factor of n.

In this paper, we consider the defocusing nonlinear wave equation $-\partial _t^2u+\Delta u=|u|^{p-1}u$ in $\mathbb {R}\times \mathbb {R}^d$. Building on our companion work (Self-similar imploding solutions of the relativistic Euler equations, arXiv:2403.11471), we prove that for $d=4, p\geq 29$ and $d\geq 5, p\geq 17$, there exists a smooth complex-valued solution that blows up in finite time.

In this paper, we establish suitable characterisations for a pair of functions $(W(x),H(x))$ on a bounded, connected domain $\Omega \subset \mathbb{R}^n$ in order to have the following Hardy inequality:

\begin{equation*}\int_{\Omega} W(x) |\nabla u|_A^2 dx \geq \int_{\Omega} |\nabla d|^2_AH(x)|u|^2 dx, \,\,\, u \in C^{1}_0(\Omega),\end{equation*}

where d(x) is a suitable quasi-norm (gauge), $|\xi|^2_A = \langle A(x)\xi, \xi \rangle$ for $\xi \in \mathbb{R}^n$ and A(x) is an n × n symmetric, uniformly positive definite matrix defined on a bounded domain $\Omega \subset \mathbb{R}^n$. We also give its Lp analogue. As a consequence, we present examples for a standard Laplacian on $\mathbb{R}^n$, Baouendi–Grushin operator, and sub-Laplacians on the Heisenberg group, the Engel group and the Cartan group. Those kind of characterisations for a pair of functions $(W(x),H(x))$ are obtained also for the Rellich inequality. These results answer the open problems of Ghoussoub-Moradifam [16].

Let $(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation

\[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\]

$b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$

$A \in \mathbb{R}^{d \times d}$

$(Z_t)_{t\geqslant 0}$

$\alpha$

$\alpha \in (1,2)$

$x\in\mathbb{R}^{d}$

$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$

$(X_t)_{t \geqslant 0}$

$\alpha$

$\gamma^{\frac{1}{\alpha}}_n$

$\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$

$\kappa \in [1,\alpha)$

$\gamma^{\frac{2-\alpha}{\alpha}}_n$

$\alpha$

In this article, we study an optimization problem for a couple including two breadwinners with uncertain life times. Both breadwinners need to choose the optimal strategies for consumption, investment, housing, and life insurance purchasing to maximize the utility. In this article, the prices of housing assets and investment risky assets are assumed to be correlated. These two breadwinners are considered to have dependent mortality rates to include the breaking heart effect. The method of copula functions is used to construct the joint survival functions of two breadwinners. The analytical solutions of optimal strategies can be achieved, and numerical results are demonstrated.

We show that for all real biquadratic fields not containing $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$ and $\sqrt{13}$, the Pythagoras number of the ring of algebraic integers is at least 6. We also provide an upper bound on the norm and the minimal (codifferent) trace of additively indecomposable integers in some families of these fields.

We prove that every locally compact second countable group G arises as the outer automorphism group $\operatorname{Out} M$ of a II1 factor, which was so far only known for totally disconnected groups, compact groups, and a few isolated examples. We obtain this result by proving that every locally compact second countable group is a centralizer group, a class of Polish groups that arise naturally in ergodic theory and that may all be realized as $\operatorname{Out} M$.