Papers

In this paper, the concept about a $k$-exponential separation of a linear cocycle $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$ is extended for a general linear coclye whose base space $\tilde{K}$ and fibre-map-value map $\mathcal{G}$ become a nonempty set and a continuous map from $\tilde{K}$ to $L(X)$ respectvely, by removing the prior assumptions in the classical sense that $\tilde{K}$ is a compact set contained in the Banach space $X$, and $\mathcal{G}(x)$ is compact for any $x\in K$. We prove that $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$ admits a $k$-exponential separation if the cocycle $(\tilde{F},\mathcal{G})$ is generated from a linear cocycle $(F,\mathcal{T})$ on $K\times X$ adimtting a $k$-exponential separation with $K$ being compact in the classical sense via a small perturbation. We also obtain some consequent results with their needed concepts spinning off from the one of a $k$-exponential separation of $(\tilde{F},\mathcal{G})$ on $\tilde{K}\times X$, as well as the unified terminology system around $k$-exponential separation is normalized. We apply our results to analyze the linearized structure near by an invariant set of a system generated from a dissipative system via a small perturbation, where the small perturbation is without the restriction of compactness.

For $s \in (1/4,1)$ and any degree the only $W^{s,\frac{1}{s}}$-minimizers for $\mathbb{S}^1 \to \mathbb{S}^1$ maps are Blaschke products. This gives a resolution of Open Problems 23 and 24 in Brezis-Mironescu's mappings to the circle book, as well as Brezis' Favorite Open Problem 5.4 in this $s$-range. Previous results of this type were partial and restricted only to a small neighborhood of $s=\frac{1}{2}$. In particular, Brezis' Favorite Open Problem 5.1 is completely settled. Moreover, as a consequence of the argument, one also obtains linearized stability results.

Let $N = N_{1} \times \dotsm \times N_{k}$, where each $N_{i} \subset \mathbb{R}^{n_i+1}$ is a closed strictly convex hypersurface. Let $M$ be a Riemannian spin manifold of dimension $n = \dim(N)$, and let $f \colon M \to N$ be an area non-increasing smooth map of non-zero degree. We show that $\mathrm{scal}_M \geq \mathrm{scal}_N \circ f$ implies $\mathrm{scal}_M = \mathrm{scal}_N \circ f$. Moreover, if $n \geq 3$ and $N$ has no circle factors, then every such map is a Riemannian isometry. In the presence of circle factors, we obtain the corresponding optimal splitting theorem for $M$ and $f$. Our results are based on an approach to the index-theoretic part of Llarull's scalar curvature rigidity theorem via Clifford-linear family index theory, which works independently of the parity of the dimension and extends naturally to products. This includes a proof of the Geroch conjecture for spin manifolds as the edge case with only circle factors.

In this paper, we study pseudo-Riemannian Hom-Lie algebras. We establish Hom-versions of the classical Bochner theorem in both the Riemannian and Lorentzian settings. Finally, we prove that the space of Killing vectors on a pseudo-Riemannian Hom-Lie algebra forms a totally geodesic Hom-Lie subalgebra.

This paper introduces second- and third-order integrating factor strong stability preserving Runge-Kutta methods for solving differential Lyapunov equations. The proposed schemes break the traditional order barrier while rigorously preserving the symmetry and positive semidefiniteness (SPSD) of numerical solutions-an essential property for stability analysis and control-theoretic applications. Furthermore, we provide a rigorous error estimate for the second-order scheme. Numerical experiments validate the accuracy of the methods and their ability to maintain SPSD properties.

Modern deep learning optimization features heterogeneous parameter structures, noisy gradients, and highly nonconvex landscapes, posing significant challenges for both algorithm design and theoretical analysis. Motivated by the limitations of SGD and the success of adaptive optimizers, we propose Schattor, a family of adaptive first-order methods based on Schatten norms. Schattor unifies SGD and the recently proposed matrix-variate adaptive optimizer Muon within a single Schatten-norm-based framework. We establish dimension-free stationarity guarantees for methods in the Schattor family for stochastic matrix optimization problems via a novel matrix martingale moment bound. We also develop multi-block extensions that adaptively balance block-wise optimization progress and prove dimension-free stationarity guarantees in this more general setting.

We study two entities that have proved to be of interest and importance in their own right viz. the $H^p_{\alpha, \beta}$ spaces and the classical Gleason-Kahane-Żelazko (GKZ) theorem. We establish a GKZ-type theorem on the $H^p_{\alpha, \beta}$ spaces by identifying a natural class of functions serving as the counterpart of invertible elements, and proving that every continuous linear functional nonvanishing on this class is a point evaluation. As an application, we characterize weighted composition operators on these spaces. It must be noted that the $H^p_{\alpha, \beta}$ spaces do not possess the various structural advantages of the classical Hardy spaces where the GKZ theorem already exists and thus we have had to modify and establish methods that rely on suitable extensions and refinements of the known techniques. Additionally, we provide examples that illustrate the natural class of functions arising in our GKZ-type theorem and demonstrate the sharpness of our characterization of weighted composition operators under the assumption of surjectivity.

In this paper, we investigate Almost Rational Finsler (AR-Finsler) metrics, a class of Finsler metrics whose fundamental tensor admits a decomposition $g_{ij}(x,y)=\eta(x,y)a_{ij}(x,y),$ where $\eta$ is a positive smooth function and $a_{ij}$ is rational in the fiber variables. We derive necessary and sufficient conditions for local dual flatness and local projective flatness of AR-Finsler metrics. Furthermore, we obtain a compatibility relation characterizing AR-Finsler metrics that are simultaneously locally dually flat and locally projectively flat. As an application, we establish a rigidity result for AR-Finsler. Motivated by these results and several known rigidity phenomena in special classes of Finsler metrics, we formulate a conjecture concerning the local Minkowskianity of AR-Finsler metrics that are both locally dually flat and locally projectively flat. Several examples are presented to illustrate the theory.

We study the Bateman–Horn conjecture for generalised von Mangoldt functions. As an application, we prove that for $k \in \{2, 3\}$, almost all Bouniakowsky polynomials represent integers that are a product of exactly $k$ primes.

Data from simulations and experiments are rarely noise-free and often exhibit heterogeneous levels of fidelity. Measurement uncertainty may vary across repeated observations, sensing devices, or even within a single experiment. This work addresses the problem of discovering nonlinear dynamical systems from such inhomogeneous data. We extend the Sparse Identification of Nonlinear Dynamical Systems (SINDy) framework to account for variable noise levels by combining Ensemble SINDy and Weak SINDy within a weighted regression formulation derived from generalized least squares. A statistical justification for the weighting strategy is also provided. The methodology is validated on several benchmark systems, including ordinary and partial differential equations. In addition, we show the benefit of multi-fidelity integration for forecasting the dynamics of a double pendulum system. The results confirm that the proposed approach mitigates the adverse effects of heteroscedastic noise and that repeated, low-cost, low-quality measurements can improve model recovery, in some cases matching or outperforming reconstructions obtained using only high-fidelity data.

Digital twin (DT) networks require tight integration with wireless sensing, yet the fundamental limits of such coupling in cell-free integrated sensing and communication (ISAC) systems remain largely unexplored, particularly in the presence of fluid intelligent metasurfaces (FIM). This paper establishes a joint position-velocity Cramer-Rao bound (CRB) framework, operationalized through a twin-in-the-loop architecture. By leveraging a scatter-matrix decomposition of the velocity Fisher information, we show that single-base-station systems are inherently rank-deficient for two-dimensional velocity estimation, whereas cell-free deployments with multiple access-point pairs achieve full observability. The resulting CRB reveals a spatio-temporal decoupling: FIM shape optimization significantly improves position accuracy but does not affect the velocity CRB under isotropic waveforms while Doppler coupling asymmetrically enhances position estimation accuracy. Building on this analysis, we develop a closed-loop DT framework, deriving the critical mismatch angle in closed form and showing that angular diversity in cell-free systems mitigates DT prediction errors. We further characterize the optimal synchronization period and propose a confidence-aware scheduling strategy that reduces the DT update rate. Numerical results demonstrate substantial performance gains over single-base-station systems, with improvements attributed to angular diversity, Doppler-position coupling, and FIM adaptation.

Stochastic trace estimation is a standard tool for approximating the trace of a large-scale matrix available only through matrix-vector products. However, in tensor-structured settings, unstructured Gaussian or Rademacher test vectors may be prohibitively expensive to store and compute with, while cheaper rank-one tensor-product vectors can require sample complexities that grow exponentially with the tensor order. This work studies Gaussian random tensor train vectors as a structured alternative for stochastic trace estimation. We show that, with a suitable choice of the tensor train rank, random tensor train vectors recover dimension-independent guarantees for the Girard–Hutchinson estimator. In particular, a median-of-means variant with tensor train rank $r \geq d-1$ achieves the same dependence on the accuracy $\varepsilon$ and failure probability $\delta$ as the classical estimator based on unstructured Gaussian vectors. We further prove an oblivious subspace injection result for sketches formed from independent Gaussian random tensor train vectors: tensor train rank $r\geq d-1$ and $\mathcal{O}(\varepsilon^{-2}(k+\log(1/\delta)))$ samples suffice for a $k$-dimensional target subspace. Finally, we investigate the use of such sketches within the Nyström++ framework. We show that the resulting estimator can achieve the desired $\mathcal{O}(\varepsilon^{-1})$ sample complexity under an additional spectral-tail condition. These results provide clarififcation on both the potential and the limitations of random tensor train vectors in stochastic trace estimation.

From any operadic category O we construct a simplicial groupoid X (slightly pseudo in a specific way), called the operadic nerve. It integrates all the structure of chosen-local-terminals, fibre functor, and cardinality functor into a single simplicial groupoid, which can be seen as an undecking of the ordinary nerve of O in the Kleisli category for the symmetric-monoidal-groupoid monad S: we have the equation DX = SNO, where D is upper decalage. The construction leads to a new characterisation of operadic categories, in which all the axioms end up as simplicial identities, and where the notion of operad over an operadic category takes the form of a simplicial map subject to well-known pullback conditions (the notion of IKEO map).

We prove an approximate fixed point theorem for maps of vanishing mean oscillation from the unit ball to itself. This gives a short proof of the Open Problem 5.10 of Brezis [Rend. Lincei Mat. Appl. 34 (2023), 307–335].

We revisit the critical Hardy–Rellich inequalities recently established by Castro in "A critical Hardy–Rellich inequality (preprint, arXiv:2511.16537, 2025)". Using the classical one-dimensional weighted Hardy inequality in Emden–Fowler variables, we prove that the weighted inequality holds exactly for $a\neq N$, with $a=N$ as the unique critical value. We derive explicit constants for $N\ge2$, obtain the sharp constant in dimension one, and extend the $\Delta u$ formulation to the Muckenhoupt range $-N<a<N(N-1)$, $a\neq N$. This provides a partial solution to an open problem raised in Castro's work.

This paper studies the information gap between mixture detection and label recovery in binomial logistic mixtures. Standard likelihood-based criteria such as the Bayesian information criterion (BIC) can detect the presence of two components, but this does not guarantee that the corresponding labels are recoverable. We show that this gap is intrinsic to binomial logistic mixtures with a fixed number of trials: observed-data evidence for mixture structure and per-observation information for label recovery have different local orders in the component separation, and only the former accumulates with the sample size. As a result, there exists a detectable-but-unrecoverable regime in which BIC selects two components while the posterior labels remain essentially uninformative. To address this issue, we propose two feasibility-aware inference procedures: a recoverability-aware BIC with a posterior-entropy penalty and an entropy-regularized estimator that mitigates the tendency of the maximum likelihood estimator to produce overly separated components and overly concentrated posterior responsibilities. Numerical experiments confirm the predicted gap and demonstrate that the proposed methods avoid misleading component selections and improve the calibration of posterior label probabilities.

We study the stability of transition fronts for the one-dimensional Cahn-Hilliard equation. More precisely, we prove that for any Hölder initial datum sufficiently close to a front in the $L^2$ norm, the corresponding solution of the Cahn-Hilliard equation exists globally and converges, up to a dynamical shift, to the front as time tends to infinity. The result holds for a class of double-well potentials in a $C^4$-neighborhood of the standard quartic potential. For the proof, we develop an $L^2$-stability framework adapted to Cahn-Hilliard fronts. The key ingredient is a nontrivial second-order Poincaré-type inequality that reveals a coercive structure in the indefinite quadratic energy form associated with the linearized operator about the front, once the dynamical shift is taken into account. This yields an $L^2$-contraction estimate and, via a far-field semigroup argument, asymptotic orbital stability of the front in the unweighted $L^2$ topology. The stability analysis requires only the $L^2$-smallness of the initial perturbation; no higher-order smallness, spatial localization, or moment assumptions are imposed.

We investigate the propagation profile of positive solutions to \begin{equation*} u_t-du_{xx}=f(t,u) \mbox{ for } t>0,\ x\in(g(t),h(t)), \end{equation*} where $f(t,u)$ is monostable in $u$ and $T$-periodic in $t$, and the free boundaries $x=g(t), \ x=h(t)$ are determined by the Stefan condition $g'(t)=-\mu u_x(t, g(t)),\ h'(t)=-\mu u_x(t,h(t))$, coupled with $u(t, g(t))=u(t, h(t))=0$. For a special nonlinearity satisfying the strong KPP condition, the long-time behavior and asymptotic spreading speed of this problem were considered by Du, Guo and Peng \cite{DGP}. In this paper, by employing new techniques, we extend the results of \cite{DGP} to general monostable nonlinearities beyond the KPP framework and at the same time we obtain more precise description of the propagation profile: we prove the existence and uniqueness of a semi-wave and show that the spreading solution converges to this semi-wave as time goes to infinity.

We show that $id:\mathbb{S}^1 \to \mathbb{S}^1$ is not a minimizing $W^{s,\frac{1}{s}}$-harmonic map for $s \in (0,\frac{1}{8}$). On the other hand, for $s \in (\frac{1}{3},1)$ it is a local minimizing map, and for $s\in [\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]$ it is a global minimizer. The usual extension or Fourier techniques being unavailable, our argument relies instead of stability analysis in $s$.

In this paper, we apply the long resonator method to exhibit large values at the central point of the family of quadratic Dirichlet $L$-functions of prime-related moduli under the generalized Riemann hypothesis.

In this paper, we address the channel estimation (CE) problem in SIM-based multi-user (MU) millimeter-wave (mmWave) near-field communication systems. To address the severe path loss and blockage in mmWave communication systems, many meta-atoms are typically integrated into each layer of the SIM. Then, the number of radio frequency (RF) chains at the base station (BS) is fewer than that of meta-atoms per layer, resulting in an underdetermined problem. Additionally, the increase in the number of meta-atoms in each layer expands the SIM's near-field region, leading to the user equipment (UEs) being mostly situated in this region, necessitating precise modeling of the channel under the spherical wavefront assumption. To address these issues, we introduce a compressed sensing (CS)-based CE protocol to tackle the underdetermined problem. In contrast to the traditional CS-based estimation framework, we investigate a polar-domain channel representation to tackle the severe energy spread effect of the classical angular-domain channel representation in near-field communication systems. Specifically, we design a novel polar-domain transform matrix for uniform planar arrays (UPAs), thereby transforming the CE problem into a sparse recovery task of the paths' support set and complex gains. To overcome the limitations of the sparse Bayesian learning (SBL) framework in tackling high-dimensional dictionaries, we propose a low-complexity polar-domain SBL (LCPD-SBL) algorithm, which significantly reduces computational complexity without compromising estimation accuracy.

We study the large-scale spatial fluctuations of excursion volumes for a class of smooth stationary Gaussian fields. In the case of Berry's random wave model in dimension $d \geq 2$, we show that the spatial fluctuations for fixed $u>0$ converge to the fractional Gaussian field $(-\Delta)^{-1/4}W$ in the space of tempered distributions $\mathcal S'(\mathbb{R}^d)$, where $W$ is the $d$-dimensional Gaussian white noise. This explains the long-range correlations in the apparent filament structure of the Random Plane Wave model. For a class of smooth planar Gaussian fields whose spectral density has a power-law singularity at the origin, we prove convergence to fractional Gaussian fields with an index determined by the singularity exponent. More generally, the results illustrate that, for stationary random measures, large-scale spatial fluctuations are determined by the behaviour of the spectral measure density exponent near zero.

We prove the existence of clopen strong marker sets in $F(2^{\mathbb{Z}^n})$ for arbitrary finite generating sets. Specifically, for any positive integers $n, d_0\geq 1$ and any finite generating set $S\subseteq \mathbb{Z}^n$, we construct a clopen set $M\subseteq F(2^{\mathbb{Z}^n})$ and a positive integer $\Delta$ such that (1) for any distinct $x,y\in M$ in the same orbit, $\rho(x,y)\geq d_0$; (2) for any $v\in S$ and any $x\in F(2^{\mathbb{Z}^n})$, there are non-negative integers $a,b\leq \Delta$ such that $av\cdot x\in M$ and $-bv\cdot x\in M$. Here $\rho$ denotes the Euclidean metric. The same result then holds for the standard supremum-norm metric $\rho_\infty$ (with an adjusted constant), by the equivalence of norms on $\mathbb{Z}^n$. The proof introduces polyhedral packages in $\mathbb{R}^n$ as a generalization of the rectangular packages used in earlier work, enabling the construction to handle generating vectors with arbitrary coordinate patterns. As an application, we obtain a continuous proper edge $(2|S|+1)$-coloring of the Schreier graph on $F(2^{\mathbb{Z}^n})$ with generating set $S$, recovering a result of Gao–Wang–Wang.

We establish matrix versions of the comparisons between the $\ell^p$-norms or quasi-norms for sequences of complex numbers. For instance, given $1\ge q>0$, and a family of $m$ normal $d\times d$ matrices $A_1,\ldots, A_m$, we show that $$ \left|\sum_{k=1}^m A_k\right| \le \frac{1}{d}\sum_{i=1}^d V_i\left\{\sum_{k=1}^m |A_k|^{q}\right\}^{1/q}\!\!\!\!V_i^* $$ for some unitary $d\times d$ matrices $V_1,\ldots, V_d$. We also give applications to Olson's spectral order and to the comparison between the symmetric modulus and the quadratic symmetric modulus. In particular we show that the sum $A+B$ of two positive matrices submajorizes their Kato supremum $A\vee B$, thereby completing majorization results due to Ando.

We prove a sharp product theorem for $r$-cross-intersecting families in the $p$-biased measure. If $r\ge2$, $0\le p\le \frac{r-1}{r}$, and $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq 2^{[n]}$ are $r$-cross-intersecting, then $$\prod_{i=1}^r \mu_p(\mathcal{F}_i)\le p^r.$$ The bound is attained by a common $1$-star, and the range of $p$ is best possible. In particular, this proves the equal-bias case of a conjecture of Frankl and Tokushige and, for $r=3$, confirms a conjecture of Tokushige. We also prove a stability theorem: for $r\ge3$, every near-extremal $r$-tuple is close, in $p$-biased measure, to a common $1$-star, with an optimal linear dependence on the product deficit. The extremal proof uses a coordinatewise coupling at the critical bias together with an isoperimetric inequality for increasing families. The stability proof uses dual families and random ordered partitions to obtain Fourier concentration, then applies biased Friedgut–Kalai–Naor theorem to force the star structure.

We study the numerical solution of electromagnetic scattering by an infinitely long impedance cylinder under oblique incidence. After separation of the axial phase factor, the axial electric and magnetic components satisfy a pair of coupled two-dimensional Helmholtz equations. The Leontovich impedance condition couples these components through tangential derivatives, and the associated boundary integral system contains both logarithmic kernels and principal-value tangential derivative terms. Building on existing coupled integral-equation formulations for oblique-incidence cylinder scattering, we construct a high-order Nystrom implementation based on Kress-type logarithmic kernel decomposition, periodic product quadrature, Fourier differentiation for the tangential derivative contribution, and a block diagonal preconditioner associated with the scalar impedance subproblem. Under uniqueness of the continuous scattering problem and a uniform discrete stability assumption, we formulate a high-order convergence framework for the boundary densities and far-field patterns. Numerical experiments include a manufactured Fourier-Bessel benchmark, a plane-wave circular-cylinder validation, a smooth non-circular boundary test, condition-number and GMRES comparisons, and a variable-impedance scattering-width reduction example in a prescribed backward angular sector. The results indicate that the method provides a stable high-accuracy forward solver for the coupled impedance system, rather than a new physical model.

It is well known that the reaction-diffusion equation $u_t=du_{xx}+f(u)$ with compactly supported nonnegative initial functions exhibits trichotomy dynamics for bistable and combustion type $f(u)$ \cite{DM, zlatos}. The same is true for the corresponding Stefan type free boundary problem \cite{DL}. In this paper, we reveal a rather different type of trichotomy for this reaction-diffusion equation under a new set of (free) boundary conditions, arising as a model for biological invasion with $u(t,x)$ representing the density of an invading species over the one dimensional spatial regin $[0, h(t)]$. The evolution of the invading front $x=h(t)$ is governed by $h'(t)=-\frac d\delta u_x(t, h(t))$ and $u(t, h(t))=\delta\in (\hat\theta_f, 1)$, with $\hat\theta_f \in [0, 1)$ uniquely determined by $f$; they allow $h(t)$ to advance as well as to retreat when time increases. At the fixed boundary $x=0$, the density is controlled by $u(t,0)=\delta_0\geq 0$. We completely classify the long-time dynamics of the model when $f(u)$ is a monostable, or bistable, or combustion type nonlinear function. In the biologically interesting case that $\delta_0<\delta$, we show that there are exactly three scenarios: (i) successful spreading, (ii) finite-time vanishing, (iii) a transition state characterized by $h(t)\to l_*\in (0, \infty)$ and $u(t,x)\to w_*(x)$ as $t\to\infty$, where $(u(t,x), h(t))\equiv (w_*(x), l_*)$ is the unique stationary solution of the free boundary problem. The model here does not have the usual order-preserving property enjoyed by those considered in \cite{DM, zlatos, DL} and elsewhere (i.e., $u(0,x)\leq v(0,x)$ implies $u(t,x)\leq v(t,x)$ for all $t>0$ if $u$ and $v$ are two solutions of the problem), which is intrinsically linked to the many novel features of the model.

The usual description of the superposition of two (pure quantum) states is ambiguous, since the binary operation of addition in a Hilbert space does not pass down to the quotient projective space. In the setting of complex projective space this paper gives a new geometrical definition of the superposition of two pure states, viewed as two distinct points in the projective space, as the unique (complex) line on which those two points lie. Using the Erlangen program the geometry appropriate for quantum theory of the projective space is precisely defined. Also this rigorous approach is compared with others.

There are two purposes of the present paper which are interrelated. The first goal is to construct the structure of a non-unital monoidal category $\mathfrak{Cont}$ of contact manifolds, not necessarily coorientable, by developing the contact topology without contact forms. The non-unital monoidal product is the functorial contact product $\star$, called star product, introduced in \cite{oh:shelukhin-conjecture}. We prove that the product $\star$ is associative and there exist a collection $\alpha = \{\alpha_{X,Y,Z}\}$ of the associator isomorphisms $\alpha_{X,Y,Z}: X \star (Y\star Z) \cong (X \star Y) \star Z$ for $X, \, Y, \, Z \in \mathfrak{Cont}$, that satisfy the pentagon axiom, i.e., that the triples $(\mathfrak{Cont}, \star, \alpha)$ form a nonunital monoidal category. The second goal is to develop the calculus of Legendrian correspondences, which are by definition embedded Legendrian submanifolds of the contact product $Q \star Q'$. Legendrian correspondences will play the role of 1-morphisms in the $2$-categorical structure to be equipped with $\mathfrak{Cont}$ whose two morphisms are contact instanton cohomologies $HI(R_{ab},R'_{ab})$ associated to a pair of Legendrian correspondences $R_{ab}, \, R'_{ab} \in \mathfrak{Leg}(Q_a,Q_b)$. With this future application in mind, we define the composition of Legendrian correspondences and prove that the composition of a generic pair is again embedded and hence canonically becomes a Legendrian correspondence.

In this paper we consider a multivariate risk model, with common counting process and common process of logarithmic returns for the investment portfolio. We assume that the claim-vectors, the counting process and the logarithmic returns of the investment portfolio satisfy a weak dependence structure. Further, we consider that the counting process represents an inhomogeneous renewal process, and the logarithmic returns represent a cadlag process with independent but not necessarily stationary increments. Under these conditions we provide an asymptotic expression for the infinite-time entrance probability of the discounted aggregate claims into some rare set xA, where A denotes a set from a general set family, crucial for the actuarial practice, when the common distribution of the claim vectors belong to a multivariate heavy-tailed distribution class. This result, is derived under a moment condition for the financial risks, and underlines the multivariate linear single big jump principle. When we restrict the distribution class of the claim-vectors to multivariate regular variation, we find more explicit asymptotic expressions, weakening the moment conditions on the financial risks. The asymptotic formulas, derived through double dependence solution, become more direct and practical in applications. With respect to the technical part, due to non Levy-Renewal framework, the classical Kesten-Goldie theorems are not applicable, nor their extensions. The way we make the discretization of the process of the discounted aggregate claims permits to derive uniform asymptotics with respect to the number of summands, that facilitate the approximation of the infinite sums of the main results.