Papers

Let $n\equiv 3\ (\mathrm{mod}\ 4)$ and set $p=\frac{n-1}{2}$. On an oriented Riemannian $n$-manifold we consider the (middle-degree) curl operator, $\mathrm{curl}:*\mathrm{d}:\Omega^{p}\rightarrow\Omega^{p}$, and the associated conformally invariant Sobolev quotients on $(\mathbb{S}^n,g_{\mathrm{st}})$, \[ J_1(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\int\langle\mathrm{curl}\alpha,\alpha\rangle\,\mathrm{dV}}, \qquad J_2(\alpha)=\frac{\big(\int|\mathrm{curl}\alpha|^{\frac{2n}{n+1}}\,\mathrm{dV}\big)^{\frac{n+1}{n}}}{\inf_{\phi}\big(\int|\alpha-\mathrm{d}\phi|^{\frac{2n}{n-1}}\,\mathrm{dV}\big)^{\frac{n-1}{n}}}. \] Killing $p$-forms and their conformal images form a natural family of critical points for both functionals, analogous to the Aubin-Talenti family in the classical Sobolev inequality. We prove a quantitative local stability estimate for $J_1$ around this family, which in particular implies that every such form is a strict local minimizer in the conformally invariant space $W^{1,\frac{2n}{n+1}}$. In contrast, we show that these critical points are unstable for $J_2$ (and for related conformally invariant quotients), yielding a strict upper bound for the sharp constant of the $J_2$ inequality. By conformal invariance, the results on $\mathbb{S}^n$ transfer naturally to $\mathbb{R}^n$.

Considering that the family of bivariate Archimax copulas contains both the Archimedean and the extreme-value class, Archimax copulas constitute a flexible family allowing to model extreme and moderate levels of dependence. Despite their appeal, no fully nonparametric, consistent estimator that is itself an element of the Archimax family $\mathcal{C}_{am}$ has been established yet, mainly because Archimedean generators and Pickands dependence functions alone do not identify Archimax copulas. We resolve this identifiability issue by working with transformed generators and transformed Pickands dependence functions, and show that these functions do identify the Archimax copula uniquely. Building upon this result, we prove that uniform convergence of Archimax copulas is equivalent to uniform convergence of the corresponding transformed generators and Pickands dependence functions. Moreover, as for Archimedean and extreme-value copulas, uniform convergence in $\mathcal{C}_{am}$ is equivalent to weak convergence of almost all conditional distributions. Exploiting these equivalences, we construct two nonparametric estimators for Archimax copulas (a Pickands and a CFG type estimator, both elements of $\mathcal{C}_{am}$) and show that they are strongly consistent under mild regularity conditions. As a further consequence of the aforementioned weak conditional convergence, we obtain strongly consistent plug-in estimators for measures of directed dependence such as Chatterjee's $\xi$ and Trutschnig's $\zeta_1$. A large-scale simulation study shows that the proposed CFG type estimator outperforms both the standard empirical copula estimator and the Pickands type estimator; an application to precipitation data from Bregenz and Dornbirn (Austria) illustrates the practical use of our estimators on real data.

The Student-Teacher game is an extension of Herbrand's theorem. In the game, Student and Teacher take turns giving values for existentially and universally quantified variables, and Student is allowed to backtrack to propose other values for existentially quantified variables. The game has become increasingly important for proving lower bounds on provability in theories of bounded arithmetic. In those applications, the game is played in an arithmetical theory; however, this paper studies the Student-Teacher game in the setting of pure first-order logic, so it is more closely related to Herbrand's theorem and the midsequent theorem. When played in pure first-order logic, a prenex formula $\phi$ is logically valid if and only if there is a Student-Teacher game with a winning strategy for Student for establishing $\phi$. We present a refined version of the Student-Teacher game in arbitrary first-order universal theories and include a proof of the validity of Student-Teacher games from the sequent calculus midsequent theorem in an appendix. The game is presented as a finite tree with vertices and edges labeled by terms, and with a total order on the nodes. The totally ordered tree represents the players' interaction. Our main results show that minimal trees in the Student-Teacher game can be arbitrarily complex. Specifically, for every totally ordered tree $T$, we construct a valid prenex formula $\phi$ such that every winning Student-Teacher game for $\phi$ contains $T$ as a substructure. Using formulas parameterized by numerals, we prove similar results for computable and partial computable sequences of totally ordered trees. It follows not only that there is no computable bound on the size of Herbrand disjunctions, which is a well-known fact, but also that there is no computable bound on their complexity.

In this paper we present a concrete example of a substochastic discrete-time Markov chain on a countable state space producing a spectrum of infinitely many quasi-stationary distributions (QSDs) for generic parameter values, with each QSD supporting a distinct escape rate. Our system is motivated by an open variant of the Pelikan dynamical system, a random map introduced in the 1980s. These QSDs, and their stability to perturbative random noise, are tested in numerical simulations. The existence of unique QEDs is also established for some parameter values.

We investigate time-periodic dynamics in a forager-exploiter system with a taxis cascade under both homogeneous and heterogeneous resource environments. The model describes the interactions among foragers, exploiters, and environmental resources, where foragers move toward higher resource densities while exploiters aggregate toward regions with higher forager densities. Our results show that different resource renewal mechanisms shape periodic dynamics in fundamentally different ways. Precisely, for time-periodic resource renewal rates, we establish the existence of positive time-periodic solutions for any positive renewal rate and further prove their global stability under suitable conditions on the parameters. In contrast, for homogeneous environments, only large resource renewal rates can destabilize the constant steady state through the Hopf bifurcation, thereby generating non-constant time-periodic solutions. Interestingly, for spatially heterogeneous and temporally homogeneous environments, our numerical simulations indicate that spatial heterogeneity exhibits opposing effects on periodic dynamics depending on total resource availability. When resources are sufficiently abundant, spatial heterogeneity tends to suppress the emergence of temporal oscillations, whereas when resources are relatively scarce, it may instead promote oscillatory behaviors, and sufficiently concentrated local resource supplies can trigger local or even global temporal oscillations. These findings reveal a delicate interplay among resource renewal mechanisms, resource availability, and spatial heterogeneity in shaping dynamics behavior.

Transport-based quantiles extend univariate quantiles to multivariate distributions via optimal transport. We study the influence function of the transport quantile map $\mathbf{Q}_P$, defined as the optimal transport map pushing a fixed reference measure $\mu$ forward to a target distribution $P$. For the Huber contamination $P_t=(1-t)P+t\delta_{x_0}$, we prove that the first-order limit $\mathbf{I}(x_0;\mathbf{Q}_P(z)) := \lim_{t\downarrow 0} [\mathbf{Q}_{(1-t)P+t\delta_{x_0}}(z)-\mathbf{Q}_P(z)]/t$ exists whenever $x_0\ne \mathbf{Q}_P(z)$ and characterize it uniquely. Specifically, $\mathbf{I}(x_0;\mathbf{Q}_P(z))=\nabla G_{x_0}(z)$, where $G_{x_0}$ is characterized by a uniformly elliptic equation with a Dirac source and a Neumann boundary condition. In every dimension $d\ge 2$, this influence function has a pole-type singularity. For fixed $z\in\operatorname{int}(\Omega_\mu)$, it remains bounded when $\mathbf{F}_P(x_0)$ stays away from $z$, where $\mathbf{F}_P=\mathbf{Q}_P^{-1}$ is the transport-based distribution function, but diverges as $x_0\to\mathbf{Q}_P(z)$, equivalently as $\mathbf{F}_P(x_0)\to z$. In fact, $\|\mathbf{I}(x_0;\mathbf{Q}_P(z))\|\asymp\|z-\mathbf{F}_P(x_0)\|^{-(d-1)}$. This contrasts with the bounded influence function of univariate quantiles and implies that $\mathbf{I}(X;\mathbf{Q}_P(z))$, for $X\sim P$, has infinite second moment. Numerical experiments further suggest that empirical transport quantiles may exhibit stable-type non-Gaussian fluctuations.

A stochastic representation and sampling algorithm for Morillas-type copula-to-copula transformations and related distortions of multivariate distribution functions is derived, resulting as a byproduct in a novel sampling scheme for Archimedean and Archimax copulas. This closes a methodological gap and facilitates simulation-based applications of distorted copulas. For stable tail dependence functions (stdfs), a Morillas-type distortion framework is introduced, where monomial distortions with exponents below 1 are shown to preserve stdfs via a domain-restricted Pexider equation analysis. This characterization is leveraged to identify distortions preserving extreme value copulas, and convex combinations of distorted stdfs are proposed to increase flexibility in extremal dependence modeling. The impact of distortions on maximum domain of attraction limits is also analyzed. Explicit limiting EVC distortions are identified under non-restrictive regular variation assumptions. Examples of absolutely monotone distortions allowing to fine-tune the extreme value behavior after distortion, but also of non-regularly varying 2-absolutely monotone distortions are given.

We investigate Hartree-type equations driven by a nonlocal operator $\mathcal{L}_\mu$, defined as a superposition of fractional Laplacians through a signed Borel measure $\mu$. Under Berestycki-Lions type assumptions, we prove the existence of a Mountain Pass solution and show that its energy level coincides with the minimum on the Pohozaev manifold. We also establish the boundedness of non-negative solutions. The proof of this fact requires a careful use of the Sobolev embedding in the iterative argument and a delicate treatment of the integrals involved in the estimates, as well as a Kato-type inequality in our general setting. Finally, we establish a general Pohozaev identity for solutions under a suitable summability assumption.

The rapid growth of the global aging population presents severe challenges to healthcare systems, necessitating efficient, equitable, and patient-centered care models. While Industrial Engineering and Operations Research (OR) provide robust optimization and decision-support tools to address these multidimensional complexities, current applications often remain fragmented. This paper presents a thematic review of 30 seminal studies at the intersection of OR and elderly care, categorizing the literature into home healthcare operations, polypharmacy management, and clinical chronotherapy. Our analysis highlights a significant methodological evolution from static, deterministic models toward dynamic and stochastic frameworks integrated with artificial intelligence (AI). Despite these advancements, a critical translational gap persists: the current OR literature is heavily dominated by process-level optimizations, such as staff routing, and struggles to translate these operational efficiencies into measurable clinical outcomes. Furthermore, holistic models bridging the transition between hospital and community care remain critically underexplored. To develop resilient and smart healthcare systems, this study proposes a conceptual framework that shifts the research focus from isolated operational tasks to integrated, multi-level decision-making. We emphasize the critical need for robust systems analysis, human-inclusive design, and the smartification of care through emerging digital technologies - including digital twins and large language models - to successfully bridge the gap between theoretical operational metrics and tangible patient-level health outcomes.

Distributed optimization offers significant advantages over centralized methods in terms of scalability and robustness when solving large-scale problems. In this paper, we propose a novel decentralized optimization algorithm that integrates Inexact Consensus ADMM (IC-ADMM) with a finite-time decentralized quantized communication algorithm. The proposed method enjoys three main benefits: (i) it operates on directed communication graphs, (ii) it requires only quantized local information instead of exact values, and (iii) it does not rely on solving the local subproblems exactly. Under the assumption that each node's local objective is strongly convex and L-smooth, the algorithm is guaranteed to achieve global linear convergence to a neighborhood of the optimal solution. Extensive numerical experiments demonstrate its advantages over existing methods in terms of convergence speed and communication efficiency.

Classifying the spherical metrics on a torus $E_\tau$ with $4\pi$ conic angle at each half period point\, ${\omega_k}/{2}, k=0,1,2,3$\, is equivalent to classify solutions of the following curvature equation \begin{align}\label{eq0731093154} \Delta u+e^u=4\pi\sum_{k=0}^3\delta_{\frac{\omega_k}{2}}\text{\ on\ }E_\tau \end{align} where $\tau\in \mathbb{H}:=\{z\in \mathbb{C}\mid \mathrm{Im} \, z>0\}$ and $\delta_p$ is the Dirac measure at $p\in E_\tau$. By constructing a multiple Green function $$G_2(z_1, z_2;\tau):=G(z_1-z_2;\tau)-\frac{1}{2}\sum_{j=0}^3\left(G(z_1-\frac{\omega_j}{2};\tau)+G(z_2-\frac{\omega_j}{2};\tau)\right), $$ in terms of the Green function $G(z;\tau)$ on $E_\tau$, we classify the solutions of (\ref{eq0731093154}) into two types: special and non-special. Furthermore, we obtain the following conclusion about the solutions of (\ref{eq0731093154}): \begin{enumerate} \item any special solution is an even function and the set of special solutions is isomorphic to $SL(2,\mathbb{C})/SU(2)$ for all $\tau\in \mathbb{H}$. \item a non-special solution exists if and only if $\tau\in \mathcal{E}$. Moreover, if $\tau\in \mathcal{E}$, then there are six one-parameter families of nonspecial solutions. \end{enumerate} where $$\mathcal{E}:=\left\{\tau\in \mathbb{H}\mid G(z;\tau) \,\,\text{has exactly 5 critical points.}\right\}.$$ The set $\mathcal{E}$ is completely determined in \cite{CLW2018, Lin}, which is a union of countable many open triangular domains. As a byproduct, we completely determine and classify the critical points of $G_2$ and then obtain the degeneracy criterion of critical points for $G_2$, which may be of independent interest.

The hydrostatic balance is a fundamental equilibrium state in stratified fluids and plays a central role in geophysical fluid dynamics. Understanding its stability under incomplete dissipation is a longstanding challenge, since anisotropic diffusion alone is generally insufficient to control the nonlinear evolution and no robust stabilizing mechanism is known for the corresponding anisotropically dissipative Navier–Stokes equations. In this paper, we investigate the two-dimensional Boussinesq equations on $\mathbb{R}^2$ with only horizontal dissipation near the hydrostatic equilibrium $(U,\Theta)=(0,x_2)$. We show that the velocity–temperature coupling generates internal gravity waves whose dispersive decay, together with the horizontal dissipation, provides an effective stabilizing mechanism that compensates for the complete absence of vertical dissipation. This identifies a mechanism by which dispersive wave propagation restores stability in an incompletely dissipative fluid system. For sufficiently small initial perturbations in $H^k(\mathbb{R}^2)\cap W^{3,1}(\mathbb{R}^2)$ with $k\ge14$, we establish the global existence and uniqueness of classical solutions together with explicit anisotropic, componentwise large-time decay rates for the velocity and temperature, including faster decay of the vertical velocity.

Classically simulable measurements (CSMs) constitute an important class of restricted measurements in the odd-prime-dimensional magic resource theory, referring to those measurements with positive discrete Wigner functions. Since their discrimination power is weaker than that of global measurements, it is necessary to study how to improve the discrimination power of CSMs. In this paper, we consider three methods to improve the discrimination power of CSMs, including adding magic resources, using quantum catalysts, and using quantum memories. Specifically, we relate measurements with positive discrete Wigner functions to completely positive Wigner-preserving measurement channels, thereby transforming the problem of improving the discrimination power of CSMs into the problem of determining how many magic resources are required to simulate quantum channels using free operations. Based on this, we derive the lower and upper bounds of the simulation cost. Moreover, we provide a concrete example for which these bounds coincide and prove that consumable magic resources can enhance the discrimination power of CSMs. Finally, we establish a no-go theorem, which shows that for discriminating a pair of states with positive Wigner functions, neither finite-dimensional quantum catalysts nor finite-dimensional quantum memories can improve the optimal success probability of discrimination using CSMs.

We propose an artificial noise (AN)-aided, secure OFDM with index modulation (OFDM-IM) framework for integrated sensing and communication (ISAC). Consider a security-critical scenario, where the sensing target is also a potential eavesdropper. Instead of suppressing the signal power toward the target, the proposed method injects AN into inactive OFDM-IM subcarriers. The precoder spatially nulls the AN at the legitimate receiver, while projecting active-subcarrier energy levels toward the target to deceive its index detector. Since the monostatic ISAC transmitter knows the AN waveform, the same inactive subcarriers also contribute to sensing. We formulate this scenario as an optimization problem that maximizes transmit power toward the target while providing secure communication with a legitimate receiver. Numerical results show that the proposed scheme improves targeted transmit power compared to classical OFDM.

The Bender–Brody–Müller (BBM) Hamiltonian was proposed as a non-Hermitian Hilbert–Pólya operator. We analyze the Hilbert completion induced, on the standard half-line $L^2$ core, by BBM's candidate metric $\hat\eta=\sin^2(\hat p/2)=\Delta^\dagger\Delta/4$. The form $\eta_0=\Delta^\dagger\Delta$ is positive with trivial kernel but is not coercive. Completing $C_c^\infty(0,\infty)$ in the norm $\|\psi\|_{\eta_0}=\|\Delta\psi\|$ gives a Hilbert space canonically unitarily equivalent to $L^2(\mathbb R_+)$. Its free self-adjoint realization is the dilation generator, with simple, purely absolutely continuous spectrum $\mathbb R$. The analysis yields two spectral statements of interest beyond the BBM problem. First, no bounded sandwich $\Delta^\dagger h(D)\Delta$ is boundedly invertible. Second, the transported symmetric operator has deficiency indices $(\infty,\infty)$ and an adjoint with every real point as an eigenvalue of infinite multiplicity, while its free extension is purely continuous. The realization-independent BBM conclusion concerns the candidate eigenfunctions: $\Delta\psi_z=x^{-z}$, so for $\operatorname{Re}z=1/2$ they do not belong to the completed space. Thus the original BBM boundary-condition/eigenfunction mechanism cannot produce point-spectrum Riemann-zero states in this $L^2$-based metric completion.

This work is devoted to the study of traveling wave solutions of reaction-diffusion systems, where the diffusion rate of~$v$, the second component, depends on~$u$, the first component. Such systems arise in prey-predator models, where the predator~$v$ is actively hunting its prey~$u$, or in epidemiological models where the disease induces erratic behavior, for example rabies. This results in a quasilinear coupling in the highest-order term of the equation for~$v$. From the theoretical point of view, we fully classify traveling wave solutions when the first component does not diffuse, in which case the problem can be reduced to a nonlinear scalar equation. The case where both components diffuse is investigated numerically.

We study a local closure operation on the cover edges of a Boolean lattice: whenever the two lower edges or the two upper edges of a square face are present, all four edges of that square are added. We prove that every set of cover edges generating the full cover graph of $\mathcal{B}_n$ has cardinality at least $n$, and we classify all generators attaining this bound. For a graph $G$ on $[n]$, let $S_G=\{N_G(i)\to N_G(i)\cup\{i\}:i\in[n]\}$. Then $S_G$ diamond-generates the full cover graph if and only if $G$ is a cograph, and every minimum-cardinality generator arises uniquely in this way. Consequently, labeled minimum diamond-generating sets of $\mathcal{B}_n$ are in bijection with labeled cographs on $n$ vertices.

In satellite design, it is common practice to add safety margins to the constraints to achieve conservative solutions that are robust to uncertainties. This robustness often comes at the expense of performance and it may be more appropriate to include uncertainties in the definition of the design problem. This work addresses such a challenge by applying techniques of multidisciplinary design optimization under uncertainty to an industrial use case. The latter is a pair of telecommunication satellites launched together in a stacked configuration. Each satellite is a strongly coupled multidisciplinary system while the two satellites are not coupled at all. This use case can be extended to an arbitrary number of satellites. By considering uncertainty quantification techniques such as sensitivity analysis and reliability-based design optimization, this study demonstrates that accounting for uncertainties in the design problem results in a 66% reduction in performance loss compared to the adding of a predefined safety margins, while guaranteeing the feasibility of constraints with high probability.

The paper aims to introduce a digital-topological ($DT$-, for brevity) $k$-ring and a $DT$-$k$-field. They are indeed endowed with both a digital image (or digital object) $(X, k)$ and a ring structure or a field structure $(X, \ast_1, \star)$, where $X \subset {\mathbb Z}^n$ and the $k$-adjacency is the digital $k$-connectivity of ${\mathbb Z}^n$. Besides, some properties of them are investigated. The ring $(SC_k^{n,l}, \ast_1, \star)$ is proved to be isomorphic to the ring $({\mathbb Z}_l, +, \cdot)$, where $SC_k^{n, l}$ is a simple $k$-cycle with $l$ elements in ${\mathbb Z}^n$, $n\in {\mathbb N}\setminus \{1\}$, and ${\mathbb N}$ is the set of natural numbers. However, $(SC_k^{n,l}, \ast_1, \star)$ is proved not to be a $DT$-$k$-ring. Meanwhile, we prove that for $l \in \mathcal{P} \setminus \{2,3\}$, while $(SC_k^{n, l}, \ast_1, \star)$ is a field, it cannot be a $DT$-$k$-field, where $\mathcal{P}$ indicates the set of prime numbers. Besides, the paper proves that the field $(X:=\{-1, 0, 1\}, \ast_1, \star)$ is a $DT$-$2$-field derived from the digital image $(X, 2)$ and the field $(X:=\{-1, 0, 1\}, \ast_1, \star)$, and further, $(Y:=\{0, 1\}, \ast_1, \star)$ is also a $DT$-$2$-field derived from the digital image $(Y, 2)$ and the field $(Y:=\{0, 1\}, \ast_1, \star)$.

Fluid antenna systems (FASs) have recently emerged as a promising reconfigurable antenna technology for future wireless networks, owing to their unique ability to exploit fine-grained spatial channel variations within a compact aperture. In this paper, a single-input multiple-output (SIMO) FAS employing maximum-ratio combining (MRC) is investigated under the block-diagonal correlation model, where the ports of FAS are partitioned into independent blocks and the strongest port within each block is selected for MRC combining. Exact outage probability (OP) expressions are first derived in both convolution and characteristic-function forms. To gain further insights, closed-form high-SNR asymptotic expressions are developed, from which the diversity order is shown to approximate the number of ports. This result reveals that block partitioning influences only the coding gain and can therefore be optimized without compromising the diversity performance. For the ergodic rate (ER), a Gamma-matching approximation together with a tighter Jensen-based approximation is derived in closed form. Simulation results corroborate the analytical framework and demonstrate that: i) increasing either the number of ports or the number of blocks improves the system performance; ii) the diversity order depends solely on the number of ports; and iii) the proposed SIMO-FAS achieves comparable or superior outage performance to conventional MRC receivers despite employing fewer combining branches.

We introduce anisotropic Banach spaces of distributional 1-forms on compact surfaces designed to capture the fine regularity properties of Morse gauge-fixed Yang–Mills random connections. Using singular connections supported on unstable curves of a Morse gradient flow, we provide a new description of the moduli space of flat connections and of its canonical Atiyah–Bott–Goldman symplectic form. When the group is the special unitary group, we prove that in the zero-area limit, the Yang–Mills random connection converges, within these anisotropic spaces, to a random distributional 1-form whose law coincides with our Morse-theoretic representative of the Atiyah–Bott–Goldman measure. Our approach extends the works of Witten, Forman, Liu, and Sengupta by establishing the zero-area limit of the Yang–Mills measure at the level of random distributional connections, rather than only at the level of holonomies. This answers a question of Thierry Lévy on the semiclassical limits of Yang–Mills random connections.

Motivated by sampling and approximation schemes arising in nonsmooth optimization, we study the stability of parameterized set-valued integrals under weak perturbations of the underlying probability distribution. For a compact parameter set, we show that compact convex-valued and jointly outer semicontinuous integrands induce set-valued integral maps that converge graphically in excess distance along any weakly convergent sequence of probability measures. The result holds under a superlinear integrability condition and provides a unified stability principle for measure approximations of set-valued expectations. We discuss the sharpness of the assumptions through examples. In particular, we emphasize that joint outer semicontinuity is required in general and that the superlinear envelope condition is tight relative to the classical i.i.d. empirical setting. As a consequence, we obtain outer stability of solution sets for stochastic generalized equations. We illustrate the stability result in several settings, including stochastic nonsmooth optimization with Markovian sampling, smoothing by mollifiers, and parameter-dependent distributional dynamics.

We determine the minimum possible least common multiple of a distinct covering system whose minimum modulus is $7$. Klein previously constructed such a system with least common multiple $15120$ and conjectured that this value was minimal. We give a construction with least common multiple $10080$, and we prove that no smaller least common multiple can occur. The proof is organized as a successive filtering argument. Starting from the possible multiples of $7$ below $10080$, we first apply a reciprocal-sum filter, then a divisor-completed integer-programming filter, then a stronger partial-sum filter. The few remaining hard cases are finally certified by complete Gurobi computations.

We study the spreading dynamics for two-species diffusive competition systems in shifting environments caused by climate changes. Our main goal is to derive conditions for extinction and persistence of each species in the case of a strong-weak competition. Depending on the pace of the climate change, but also on whether the strong competitor is faster or slower, we will uncover dramatically different outcomes in the asymptotic behavior of solutions. For instance, it may be that a weak and fast competitor survives while a strong and slow one does not. Furthermore, we find some parameter regime where the outcome depends not only on the climate change speed, but also on both species' resilience to it, and even on the initial populations' distributions. Our results show how strikingly complex the effect of climate change may be on population dynamics as soon as several species are considered.

We generalize a construction of Yves Meyer of sparse crystalline measures arising from powers of a Blaschke factor. Starting from a recursion $f_n=\theta^n f_0$ on the unit circle where $\theta$ is an inner function, we project the Fourier coefficient array $\widehat{f_n}(k)$ to the real line by placing its entries at the frequencies $k+\alpha n$. We identify the role of model spaces in this construction: in Meyer's one-factor Blaschke recursion, the requirement that the coefficient array $\widehat{f_n}(k)$ vanish whenever $kn<0$ is equivalent to $f_0\in K_{zb_\lambda}$, and for general inner functions the condition $f_0\in K_{z\theta}$ yields a purely atomic Radon measure with locally finite support and polynomial growth on the Fourier side. We also show that, when $f_0$ is holomorphic in an annulus containing the unit circle, exponential Fourier decay is sufficient to obtain a purely atomic Radon measure of polynomial growth, though not necessarily locally finite support. For finite Blaschke products, the coefficient recursion gives an explicit annihilating exponential polynomial whose zero set controls the support and separation of the inverse Fourier transform. This yields Meyer-Blaschke-type crystalline measures and Poisson identities with sampling and finite-truncation consequences.

In this work, we take a topological approach and introduce a homotopy model of the orbit space of a maximal torus action on $\operatorname{Gr}(2, \mathbb{C}^{5})$. Using this homotopy model, we compute the integral homology groups and study the homotopy type of the orbit space. More precisely, we endow the homotopy model with a CW complex that contains one top cell and determine the homotopy class of the attaching map of the top cell of the orbit space.

This paper investigates the algebraic and differential geometric properties of smooth mappings between domains in $\mathbb{C}^2$, classifying them according to the constant ranks of their holomorphic and antiholomorphic derivative components. Particular emphasis is placed on strictly paired CR (SPCR) linear automorphisms of $\mathbb{R}^4$, where both block matrices $A$ and $B$ have rank 1. We characterise the set of SPCR linear automorphisms as a 12-dimensional submanifold of $\mathrm{GL}(2,\mathbb{C})^2$ and analyse its underlying CR structure, demonstrating its geometric and structural rigidity.

Let $X$ be a normal complex algebraic variety. Let $\mathcal{G}^s_{\mathbb{Z}}(X)$ be the maximal torsion free nilpotent quotient of $\pi_1(X)$ of nilpotency class at most $s$. Let $F^{\bullet}\mathfrak{g}^s$ be the Morgan–Hain Hodge filtration on the Lie algebra of the $s$-th lower central quotient of the complex Malcev completion of $\pi_1(X)$. We show that the natural map $H^k(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z}) \to H^k(X, \mathbb{Z})$ vanishes for $k > \dim F^1\mathfrak{g}^s$. If $\mathcal{G}^s_{\mathbb{Z}}(X)$ is of nilpotency class greater than two, this includes the top nonvanishing degree of $H^{\bullet}(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z})$. We deduce that if the fundamental group of an aspherical normal variety is virtually nilpotent, it is virtually two-step nilpotent. This gives a positive answer to a question of Aguilar and Campana in this case of aspherical varieties. The ingredients of the proof are the $q$-convexity of higher Albanese manifolds and the definability of higher Albanese maps.

Let \[ {\mathcal I}(h)=\int_0^1\int_{\mathbb T}h(x,z)\,\frac{dz}{2\pi iz}\,dx \qquad \bigl(h\in{\mathbb C}[x,z,z^{-1}]\bigr). \] We give the three-term Laurent polynomial \[ f(x,z)=(1-z^{-1})\bigl((1-x)+xz\bigr) \] for which \[ {\mathcal I}(f^n)=0, \qquad {\mathcal I}(z^{-1}f^n)=\frac{(-1)^{n-1}}{n+1}\neq0 \qquad(n\geq1). \] Since ${\operatorname{Sp}}(f)=\{-1,0,1\}$, this disproves the $xz$-conjecture already with one interval variable and one torus variable, and it also shows that $\ker{\mathcal I}$ is not a Mathieu–Zhao subspace. Padding gives counterexamples to every mixed case of the $xz$-conjecture. Writing the coordinate functions on $SU(2)$ as \[ g=\begin{pmatrix}a&c\\ b&d\end{pmatrix}, \] the same example lifts, through the integration formula of Müger and Tuset, to the regular functions \[ F=(1+c)(ad+b),\qquad G=-c, \] which satisfy \[ \int_{SU(2)}F^n\,dg=0, \qquad \int_{SU(2)}F^nG\,dg=\frac{(-1)^{n-1}}{n+1}\neq0 \] for every $n\geq1$. Thus the Mathieu conjecture for $SU(2)$ is false.

This paper presents a unified cost-modeling and optimization framework designed to evaluate hybrid discrete-variable/continuous-variable quantum key distribution (DV/CV-QKD) infrastructures operating within the SAFE (Secure and Efficient) long-term storage (LTSS) protocol. Our methodology jointly addresses the information-theoretic and computational dimensions of cryptographic durability over multi-decade horizons by coupling a quantitative risk model with a stochastic integer linear programming (ILP) formulation. Going beyond idealized physics-informed limits, the framework incorporates realistic next-generation industrial implementations and multiplexed coexistence specifications, together with a sample-average approximation (SAA) pipeline, to determine cost-optimal and structurally feasible SAFE topologies under transmission-budget and security constraints. The proposed framework is evaluated on both randomized synthetic deployments and realistic metropolitan-scale QKD infrastructures, including the Paris Metro-Scale and Greater Paris networks, to characterize the interplay between network topology, QKD modality, and deployment cost while enforcing a target global compromise tolerance $\epsilon$. Numerical results show that the economically optimal architecture is highly scenario-dependent and that both hybrid and homogeneous modality allocations naturally emerge from the optimization. More importantly, the analysis reveals a non-monotonic relationship between the minimum required QKD link capacity and the infrastructure cost: contrary to intuition, increasing the minimum required per-link secret-key rate may reduce the overall deployment cost by enabling a global transition toward more economical CV-based solutions, whereas further capacity increases do not necessarily yield additional savings.