Papers

A set of $m$ distinct positive integers $\{a_{1},\dots a_{m}\}$ is called a $D(q)$-$m$-tuple for nonzero integer $q$ if the product of any two increased by $q$, $a_{i}a_{j}+q$, $i\neq j$ is a perfect square. Due to certain properties of the sequence, there are many $D(q)$-Diophantine triples related to the Fibonacci numbers. A result of Baćić and Filipin characterizes the solutions of Pellian equations that correspond to $D(4)$-Diophantine triples of a certain form. We generalize this result in order to characterize the solutions of Pellian equations that correspond to $D(l^2)$-Diophantine triples satisfying particular divisibility conditions. % Subsequently, we employ this result and bounds on linear forms in logarithms of algebraic numbers in order to classify all $D(9)$ and $D(64)$-Diophantine triples of the form $\{F_{2n+8},9F_{2n+4},F_{k}\}$ and $\{F_{2n+12},16F_{2n+6},F_{k}\}$, where $F_{i}$ denotes the $i$th Fibonacci number.

We show that several categories of hyperstructures, including the mosaics of Nakamura and Reyes and the hypergroups of Marty, are realizable as categories of objects with extra structure within the category of complete atomic Boolean algebras (CABAs) and atom-preserving morphisms.

Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.

For a positive integer $d\geq 1$, we consider the sequences $(A_{n}^{(d)})_n$ and $(x_{n}^{(d)})_n$ given by $$ A_{n}^{(d)} =\sum_{n_1+\dots+n_d=n} \frac{(2n)!}{(n_1!)^2 (n_2!)^2 \dots (n_d!)^2} \quad \text{ and } \quad x_{n}^{(d)} = \frac{A_{n}^{(d)}}{\binom{2n}{n}}. $$ They have rich combinatorial interpretations, but we focus on the analytical properties of their generating functions $A_d$ and $F_d$. We use a modified Borel transform, and algebraic and combinatorial considerations to prove that $F_d$ is annihilated by an irreducible Fuchsian differentiable operator $L_{d-1,F}$ of order $d-1$. We determine the structure of $F_d$ as a global analytic function (analytic continuations from the original disk of definition, branches, finite singularities, and the structure of $F_d$ near the finite singularities). Additionally, we show that the sequence $(x_n^{(d)})_n$ satisfies a minimal recurrence of width $r=\lfloor (d+1)/2 \rfloor$ with polynomial coefficients $$ Q_r(n+r)\,x_{n+r}+\cdots + Q_0(n)\,x_n=0, \; n \ge 0. $$ These polynomials are shown to have very specific symmetries and we compute explicitly $Q_0$, $Q_1$, and $Q_r$. Similar results about the functions $A_d$ are obtained.

We develop a profinite theory for bounded commutative integral residuated lattices around three related questions. First, we prove a Stone–profinite theorem: a topological residuated lattice is profinite exactly when its underlying space is Stone. The proof refines clopen congruences of the semiring reduct into clopen congruences in the full residuated-lattice signature via the 1-class filter. Second, we solve the classwise Grätzer-type spectral representation problem: for every class K of residuated lattices, the K-representable prime-filter spectra are precisely the Priestley spectra of the bounded distributive lattices in the reticulation image Ret(K). This reduction is sharp, and we compute Ret(K) for several natural finite and logical classes, including finite MTL-, BL-, MV-, and n-potent/EDPC varieties. Third, we study profinite completions. We give an intrinsic criterion in terms of dense subalgebras and clopen congruences, prove a positive completion theorem under finite generation of finite-index congruences in n-potent varieties, and establish a finite-tag obstruction theorem. In particular, using the Heyting non-completion examples of Bezhanishvili–Bezhanishvili–Moraschini–Stronkowski and the three-element Łukasiewicz chain L3, we obtain genuinely non-idempotent profinite residuated lattices which are not profinite completions.

We extend the recent work of Reis and Rothvoss on sparsifying sums of $\ell_1$ norms to the more general task of sparsifying (Minkowski) sums of centrally symmetric, convex sets. As our main result, we prove that for any $\varepsilon > 0$ and centrally symmetric, convex sets $C_1, \ldots, C_m\subseteq\mathbb{R}^n$ there is a choice of weights $\lambda_1, \dots , \lambda_m \in \mathbb{R}_{\geq 0}$ such that at most $O(n / \varepsilon^2)$ of the weights are non-zero, and \[(1 - \varepsilon)\cdot C\subseteq\sum_{i = 1}^m\lambda_i\cdot C_i\subseteq(1 + \varepsilon)\cdot C,\] where $C:= C_1 + \cdots + C_m$ refers to the Minkowski sums of the sets $C_1, \ldots, C_m$, and $\lambda\cdot C$ refers to the dilation of the set $C$. As immediate applications of this result, we obtain sparsifiers of size $O(n / \varepsilon^2)$ for sparsifying sums of seminorms in $n$-dimensional space, improving on the $O\left ( \frac{n \log(n/\varepsilon) \cdot \log^{2.5}(n)}{\varepsilon^2} \right )$ size sparsifiers from the work of Jambulapati, Lee, Liu, and Sidford (FOCS 2023). This further yields optimal size hypergraph cut sparsifiers with $O(n / \varepsilon^2)$ hyperedges, improving on the $O(n \log(n) / \varepsilon^2)$ size sparsifiers from the work of Chen, Khanna, and Nagda (FOCS 2020). More generally, this also gives optimal size sparsifiers for sums of symmetric submodular functions.

The homomorphism threshold $\delta_{\text{hom}}(H)$ of a graph $H$ asks how large the minimum degree of an $H$-free graph has to be in order to force a homomorphism to a bounded $H$-free graph. Determining this threshold is in general very difficult, and the odd cycles $C_{2k-1}$ are among the most important open cases for $k\ge 3$. Ebsen and Schacht proved the general upper bound $\delta_{\text{hom}}(C_{2k-1})\le\frac{1}{2k-1}$, while a breakthrough of Sankar, using topological methods and a graph-theoretic analogue of homotopy equivalence, gave the first positive lower bound. The value $\frac{1}{2k-1}$ appeared particularly compelling: Ebsen and Schacht obtained the same exact threshold when all odd cycles of length at most $2k-1$ are forbidden, Huang, Liu, Rong and Xu later proved that it is the exact blowup threshold of $C_{2k-1}$, and Letzter and Snyder also explicitly asked whether $\delta_{\text{hom}}(C_{5})=\frac{1}{5}$. Surprisingly, we show that the upper bound can be improved. More precisely, for every integer $k\ge3$, we prove $$ \frac{1}{2\left((k-1)^{4k-5}(2k-1)+\frac{(k-1)^{4k-5}-1}{k-2}\right)} \le \delta_{\text{hom}}(C_{2k-1}) \le \frac{4(k-1)}{4(k-1)(2k-1)+1} < \frac{1}{2k-1}. $$ The new lower bound comes from a new graph-theoretic construction based on a sparse homomorphism theorem of Nešetřil and Zhu, and it improves Sankar's quantitative bound. The improved upper bound follows from a new structural argument that controls common neighborhoods along short odd paths. Our results have various consequences, in particular, every odd cycle of length at least five has pairwise distinct chromatic, homomorphism, polynomial removal, and linear removal thresholds, resolving two conjectures of Fox and Wigderson.

Let $\boldsymbol{X} = (X_1,\ldots,X_n)$ be a centered Gaussian vector, not necessarily nondegenerate. It is proved that, for every $\alpha_1,\ldots,\alpha_n > 0$, \[ \mathsf{E}\left[\prod_{i=1}^n |X_i|^{\alpha_i}\right] \geq \prod_{i=1}^n \mathsf{E}\left[|X_i|^{\alpha_i}\right]. \] When all marginal variances are positive, equality holds if and only if the coordinates are independent. This settles Frenkel's 18-year-old Gaussian product inequality (GPI) conjecture and, in fact, its later strengthening to arbitrary positive exponents. Through Frenkel's hafnian formulation, this result also provides a short proof of the 28-year-old real linear polarization constant conjecture, which was settled this year as a consequence of the strong polarization inequality. The main result leads to the exact real linear polarization constant, a sharp weighted product inequality for real linear functionals, spherical moment bounds, hafnian inequalities for positive-semidefinite matrices together with complete characterizations of their equality cases, logarithmic variance and covariance inequalities, a Rényi total-correlation certificate, and unconditional versions of mixed-sign Gaussian moment bounds previously conditional on the positive-exponent GPI.

Let \(\cthreshold(n)\) be the largest integer \(q\) such that every loopless digraph on \(n\) vertices with at most \(q\) arcs is isomorphic to a spanning subdigraph of a self-complementary digraph of order \(n\). We prove that \(\cthreshold(6)=7\). The upper bound is witnessed by \[ \bK{3}\dunion (x\longrightarrow y\longrightarrow z), \] and follows from a direct argument with a self-complementing permutation. We also determine the complete eight-arc obstruction layer: it consists of five isomorphism classes, or three after converse digraphs are identified. All five are arc-minimal. Each nevertheless packs with an isomorphic copy of itself, so ordinary packing is strictly weaker than same-order self-complementary completion already at this first failure layer.

Let $R$ be a nonzero commutative unital ring and let $\Gamma$ be a row-finite, strongly aperiodic higher-rank graph with no sources and a nonempty finite vertex set. We determine the prime ideals that occur as annihilators of modules induced from simple isotropy modules. They are parametrised by maximal tails of $\Gamma$ and prime ideals $\mathfrak p$ of $R$ for which the fraction field of $R/\mathfrak p$ is obtained by inverting a single element. Under the product description of the prime spectrum, these are precisely its locally closed points. The realisation argument uses a dense periodic orbit in every finite maximal-tail quotient. We also construct an explicit faithful simple module for the Leavitt algebra $L_R(1,2)$ whenever $R$ is a Goldman domain. Consequently, $L_{k[t]_{(t)}}(1,2)$ is primitive although its coefficient ring is not a field. In this example the zero ideal is locally closed but not completely irreducible. We obtain full prime-to-primitive coincidence for coefficient rings with finite spectrum and for semilocal rings of Krull dimension at most one, and transfer the results to split pullback Kumjian–Pask algebras. For pullbacks over an uncountable field, the full primitive spectrum is described without assuming algebraic closedness.

Let $X\sim\text{Mult}(N;p_1,\dots,p_n)$ with $p_i>0$ and $\sum_ip_i=1$. We study $\Pr\{X=m\}$ for lattice points near $Np$ from two complementary viewpoints. First, we give an exact integer-mode criterion: the multinomial mode is the Jefferson–D'Hondt divisor apportionment, given when the mode is unique by $m_i=\lfloor (N+d)p_i\rfloor$, $d\in[0,n)$. For $n\ge3$ the mode need not be obtained by rounding each $Np_i$ to a neighbouring integer; the excess $m_i-\lceil Np_i\rceil$ can reach $n-2$, so the two-sided localisation $|m_i-Np_i|<1$ stated in a standard reference is not valid in general. Second, by writing the mass as a gamma quotient with unequal scalings, we derive the complete local expansion of $\Pr\{X=m\}$ in integer powers of $N^{-1}$, with all coefficients in closed Bernoulli-polynomial form. The expansion contains the known $O(N^{-1})$ local limit theorem, reduces to the binomial local mass for $n=2$ and to the central multinomial coefficient in the symmetric case, and admits Cesàro averages of the oscillating coefficients in closed $\zeta$-value form.

For a field extension $E/k$ and a Chow motive $M$, let $I_E(M)=\ker\bigl(\operatorname{End}_k(M)\longrightarrow \operatorname{End}_E(M_E)\bigr)$ be the base-change ideal. We establish explicit nilpotence bounds for these ideals in several geometric settings. We also prove effective generic descent: if $M$ is a summand of $h(X)(a)$ and $M_{k(X)}$ has uniform Rost exponent $s$, then $M$ has exponent $s(\dim X+1)$. This yields the integral exponent $2n-1$ for twisted Milnor hyperplane sections of dimension $2n-2$. In characteristic zero, a weighted local-block refinement of the multilinear Rost filtration improves the uniform bounds of Gille and upgrades the elementwise estimates of Rosenschon–Sawant to uniform bounds at the ideal level. Using the Kahn–Sujatha description of pure birational motives, we obtain quantitative lifting results from birational to ordinary Chow motives, with applications to threefolds, including varieties birational to toric models and varieties admitting a decomposition of the diagonal supported on a surface. Finally, we extend the Kok–Zhou detector from individual correspondences to entire base-change ideals, obtaining an explicit ideal nilpotence bound from uniform annihilation of the critical refined unramified cohomology groups.

Climate models often feature nonlinear responses to changes in model parameters or boundary conditions. Factor separation asks how much of the resulting change should be assigned to altered factors and their interactions. Lunt et al. (2010) showed that two factor attribution methods, the linear-sum and shared-interaction methods, coincide for $n\le 4$ factors and conjectured this holds for arbitrary $n$. We show that the linear-sum and shared-interaction formulas are respectively the permutation and Harsanyi-dividend representations of the Shapley value. Their equality therefore follows from a classical theorem in cooperative game theory. To our knowledge, this is the first explicit identification of the Stein–Alpert and Lunt climate-model factor-separation methods with Möbius/Harsanyi coefficients and Shapley values. The result imports many established and efficient Shapley sampling methods into climate-model experimental design, and extends to vector-valued responses.

We study time-uniform propagation and generation of chaos for a class of McKean-Vlasov equations with polynomial growth potentials. Using an entropy dissipation method based on time-uniform logarithmic Sobolev inequalities, we derive uniform-in-time bounds depending on the regularity of the potentials and the assumptions on the initial conditions. We thereby not only establish a result stated but unproven in [Mal01], but also extend it to non-chaotic and non-exchangeable initial conditions, moving beyond the classical Lipschitz setting at the expense of a logarithmic loss in the convergence rate.

We prove two families of distinguished cases of the heat flow conjecture for random matrices of Hall-Ho: elliptic Gaussian sources with planar targets, and deterministic Hermitian sources with targets on the boundary of the covariance disk. In particular, the empirical zero measure of the heat-evolved characteristic polynomial of a complex Ginibre matrix converges almost surely to the semicircle law. The strategy of the proof is to transfer estimates for the corresponding well-understood Gaussian matrix ensembles, through exact heat identities, into control of the zeros of the heat-evolved characteristic polynomials.

In a graph $G=(V,E)$, let $S$ be a non empty subset of $V$. For any $x\in S$, the vertex $x$ and its neighbors inside $S$ are defenders of $x$, whereas those lying outside $S$ are attackers on $x$. An attack on $S$ is an $|S|$-tuple of pairwise disjoint sets of attackers on vertices of $S$, whereas a defense is that of defenders. An attack on $S$ is defendable if $S$ has a defense, which provides at least as many defenders as attackers for each of its vertices. The set $S$ is a secure set if every attack on $S$ is defendable. Further, $S$ is an ultra secure set if $S$ has a defense which successfully defends $S$ against every attack. For an integer $k\geq 0$, a $k$-secure set $S$ is a secure set in which for any attack on $S$, there is a defense of $S$ with $k$ additional defenders for every vertex of $S$ having neighbors outside $S$. As a generalization of ultra security, in this paper ultra $k$-secure sets and ultra $k$-security number of a graph are introduced. A characterization of ultra $k$-secure sets is obtained and using which ultra $k$-security numbers of complete multipartite graphs and grid-like graphs are computed.

For a quantum Gibbs sampler whose mixing bottleneck is a weakly broken symmetry, I show that the correct initialization is determined by representation theory. I first prove a general speedup-versus-prefactor dichotomy: by exactly eliminating slow-mode overlap, I convert a nominal prefactor reduction into a fundamental, asymptotic acceleration of the system's mixing time. I then show that when the bottleneck is a weakly broken symmetry, the otherwise exponentially expensive bottleneck eigenvector is the symmetry charge.For a non-Abelian group $G$, I prove that the correct initialization target is the group-averaging asymmetry. Projecting this asymmetry out requires a $G$-invariant input. Partial, subgroup-invariant inputs produce a cascade of mixing-time speedups indexed by the subgroup lattice. Conversely, matching first moments alone is provably insufficient.The asymmetry is a directly measurable initialization target. I verify these results analytically and numerically in an $SU(2)$ Davies sampler where total-spin multiplets constitute the slow modes. The predicted speedup cascade emerges with relaxation rates scaling linearly with the symmetry-breaking parameter. Finally, the model maps the boundaries of the asymmetry target, illustrating where the separate matching of conserved logical data becomes necessary.

An isolated small system consisting of a movable shell and a finite number of internal particles is considered. Exact distributions of the shell coordinates and momenta are obtained for various types of internal particle motion under the conditions of conservation of the total energy and momentum of the system. The coordinate distribution consists of a finite number of branches and may contain plateau regions. The momentum distribution corresponds to a non-uniform distribution of energy among the degrees of freedom of the system. It is shown that the coordinate distribution depends on the number of internal particles and remains unchanged when the dimensionality of their motion varies, whereas the momentum distribution depends on the number of degrees of freedom. Consequently, comparison of the coordinate and momentum distributions of the shell makes it possible to draw conclusions about the internal structure of the system. The chaotic motion of the shell caused by impacts from the internal particles is analyzed; unlike Brownian motion, it persists even in the absence of an external medium. A universal expression for the root-mean-square deviation of the shell is obtained explicitly, depending on the number of internal particles and their masses.

In 1943, Hadwiger formulated his celebrated conjecture, connecting the chromatic number $\chi(G)$ of a finite, simple, undirected graph with the cardinality of the largest complete minor, $\eta(G)$. The disjoint union of all finite complete graphs shows that Hadwiger's conjecture fails for infinite, but a slightly weaker version is true in these graphs, and open for finite graphs. In this note we generalize that weaker version to hypergraphs and provide a simple, general, and purely set-theoretical formulation of Hadwiger's conjecture.

We introduce a reflection-substitution technique for sine inequalities that yields, via Hölder's inequality and its reverse, $2^{n-1}$ distinct upper bounds for products and lower bounds for ratios of weighted sines. The comparison between bounds reduces to simple sign conditions on linear combinations of the variables. We fully classify the two- and three-dimensional cases, providing explicit dominance criteria with detailed examples for each signature.

We characterize a noisy variant of a Quantum public encryption protocol recently introduced by Malavolta and Walter which demonstrated that the notion of everlasting security can be rigorously formulated for Quantum key distribution after two rounds of interaction between Alice and Bob. To address one possible direction of research that is related to injecting noise in the cryptographic protocol related to Quantum key distribution we formulate arguments for further examining the notion of everlasting security through malleability assumptions on transformations of the ciphertext. Assumptions surrounding malleability were introduced by Maurer and Tackmann for the purposes of comparing how authenticate then encrypt, and encrypt then authenticate, protocols behave through a variety of expressions for the forwarding error, deleting error, and reconstruction probabilities. Such probabilities are put to further use for obtaining connections between the indistinguishability and security threshold for a cryptographic protocol of interests. To further build upon such associations we demonstrate, through an adaptation of the Gentle Measurement Lemma from Quantum information theory, how upper bounds on the trace distance can be used to generalize the negligibility function obtained by Malavolta and Walter in the noiseless setting. Besides the fact that the negligibility function in the noisy setting is related to a higher security threshold it continues to remain of interest to determine whether computations provided in this work for upper bounding the trace distance can be related to other settings that are centered more on game-theoretic approaches.

We study traveling waves in high-dimensional plant–consumer reaction–diffusion systems with $N$ competing plants and $N$ associated consumers. We focus on the degenerate case in which consumer growth vanishes when plant populations are zero, so the standard Fisher–KPP linearization does not determine the leading-edge behavior. Under a weak-interaction condition, we identify a plant-driven lower threshold $s_{\mathrm P}$, a constructive upper threshold $s_{\mathrm C}^{\mathrm{ex}}$, and a universal necessary upper threshold $s_{\mathrm C}^{\mathrm{nec}}$, and prove $(s_{\mathrm P},s_{\mathrm C}^{\mathrm{ex}})\subseteq \mathcal S\subseteq[s_{\mathrm P},s_{\mathrm C}^{\mathrm{nec}})$, where $\mathcal S$ is the set of speeds admitting positive extinction-to-coexistence waves. Existence follows from new lower solutions that remove previously imposed diffusion and compatibility restrictions. Nonexistence below $s_{\mathrm P}$ follows from a Sturm argument, while center-manifold analysis and a Riccati crossing argument yield the finite upper-speed obstruction. An explicit two-species wave beyond $s_{\mathrm C}^{\mathrm{ex}}$ shows that this constructive threshold is not the true maximal speed.

This study presents a theoretical analysis of partitional clustering on networks, analyzing both hard and soft assignment schemes with different objective functions. Cluster centers are not restricted to vertices but can also be located along the edges. We examine four key models: P-Median (PMP) and Sum of Squares Clustering (SSC) under hard assignment, and Probabilistic Distance Clustering (PDC) and Fuzzy C-Means (FCM) under soft assignment. Through mathematical analysis, we uncover structural properties that differentiate these models, such as the significance of assignment bottleneck points and the role of vertex-restricted solutions in determining optimal cluster centers. Our findings reveal that, while SSC and FCM can yield optimal centers along edges, PMP and PDC inherently favor vertex placement, leading to insights into clustering behavior on networks. These insights offer new directions for designing efficient algorithms and have implications ranging from facility location and network design to clustering on the embedding graphs that power similarity search in modern retrieval systems.

This paper assesses the readiness of Türkiye's electric vehicle (EV) charging infrastructure to support cost-competitive intercity travel compared to internal combustion engine (ICE) vehicles. Utilizing a directed acyclic graph-based scheduling framework, charging stops are optimized for over 2 million synthetic intercity EV trips across the national highway network, evaluating 9,641 licensed charging stations. An eight-scenario case study examines the interplay between charger power (5–350 kW) and station capacity (5 and 50 plugs), supplemented by a 200-run parametric sensitivity analysis on EV range, discharge rate, charger power, and state-of-charge (SoC) operating bounds. A three-tier economic analysis isolates operational costs, total cost of ownership (TCO), and infrastructure investments. Economically, EVs demonstrate a 23–26% operational cost advantage driven by Türkiye's favorable fuel-to-electricity price ratio, maintaining a 10–12% TCO advantage despite higher purchase prices. However, full-system infrastructure savings heavily depend on station sizing, yielding positive returns at 5 plugs per station but negative returns at 50. Ultimately, vehicle discharge rate emerges as the dominant economic lever. This study provides the first countrywide, vehicle-level assessment of intercity EV charging readiness for an emerging market, offering actionable insights for policymakers on infrastructure gaps, station sizing tradeoffs, and adoption incentives.

Multiscale problems with evolving interfaces are ubiquitous in science and engineering. Phase-field models are a powerful tool for simulating interface-dominated phenomena in computational mechanics and materials modeling, but their application to large-scale problems is often constrained by the high computational cost of resolving thin diffuse interfaces over the entire domain. This paper presents an efficient strong-form phase-field solver that couples the Gradient Smoothing Method (GSM) with a hierarchical adaptive and moving structured mesh, enabling automatic localization of resolution within a narrow interfacial region while retaining coarse discretization in bulk domains. A layered refinement design is introduced to preserve locally uniform resolution across the interface, allowing the GSM discretization to maintain overall second-order accuracy despite strong mesh non-uniformity away from the interface. Although GSM incurs a higher per-degree-of-freedom cost than standard finite-difference schemes, the adaptive framework substantially reduces the total number of degrees of freedom, resulting in near-linear computational scaling compared with the quadratic scaling of uniform-grid approaches. Numerical examples based on the Allen-Cahn and Cahn-Hilliard equations demonstrate that the proposed adaptive GSM solver delivers desired accuracy for interface evolution while attaining more favorable computational complexity, O(N), than existing weak-form and strong-form solvers, becoming significantly more efficient for large-scale problems with thin interfaces or a small interfacial area fraction relative to the whole domain.

Simple and complex contagions differ mechanistically; multiple exposures act synergistically in the latter but independently in the former. Yet correlated mixtures of simple contagions may appear complex when inferring global contagion rules, a phenomenon we call "emergent complexity." We present a measure of contagion complexity and an inferential framework for estimating mixtures of nonparametric contagion rules from time-series data. Our work reframes past studies on complex contagion by offering heterogeneous mixtures of simple contagions as an alternative explanation.

Safety inflation can cause nearby obstacles to overlap, violating the disjoint-obstacle assumptions used by many modulation-based reactive planners. We investigate Star-World workspace reshaping for three-dimensional reactive control of a Franka Emika Panda manipulator. At each update, intersecting inflated obstacles are clustered and replaced by star-shaped proxies before a dynamical-system-based end-effector controller is evaluated. A null-space artificial-potential-field term provides complementary arm-body avoidance. We compare reshaped and unreshaped obstacle representations in six PyBullet scenarios using goal attainment, path-length ratio, and computation time. In this preliminary 12-trial evaluation, reshaping reaches the goal in five of six scenarios, compared with four of six for the unreshaped baseline. It resolves the canonical overlapping-wall case and requires 0.68–8.70\,ms per workspace update for scenes containing one to seven obstacles. However, it also increases path length, produces near-equilibria in two cases, and closes a navigable corridor through over-aggressive merging. These results show both the promise and the practical limitations of transferring Star-World guarantees from workspace geometry to a redundant manipulator controlled through inverse kinematics.

Matrices and involutions in/on the algebras $B(\mathbb{R})$ and $B(\mathbb{C})$ of real (resp., complex) row- and column-finite $\omega\times\omega$ matrices are studied. It is proved that any positive definite $\mathbb{R}$-algebra involution on $B(\mathbb{R})$ (resp., any positive definite conjugate-linear involution on $B(\mathbb{C})$) is given by conjugating the transpose involution (resp., the conjugate-transpose involution) with a positive definite matrix from the algebra in question. All positive definite matrices in these algebras have Cholesky factorizations within the algebra. Examples are constructed to show that such positive definite matrices need not have any eigenvalues for their canonical action on column-finite column vectors, and they need not have any square roots in $B(\mathbb{C})$.

We develop a geometric–spectral framework for the computation and stability analysis of quasi-normal modes (QNMs) in open optical cavities of compact support. The hyperboloidal approach, transferred from gravitational physics to electromagnetism [1], incorporates outgoing boundary conditions directly into the formulation of the optical resonance problem. Dispersive and absorbing media are described by a Lorentz-model permittivity through an auxiliary-field formulation, and the resulting non-Hermitian spectral problem is solved using Chebyshev spectral discretization.Pseudospectrum analysis is then extended to the optical setting following [2] and used to study the stability of optical resonances under perturbations. The pseudospectrum provides information on the sensitivity of the resonances that is not contained in the QNM spectrum alone. Particular attention is given to the role of the norm used to define the pseudospectrum, by comparing the stability properties obtained with different choices of scalar product. The results show that the assessment of spectral stability is closely related to the functional setting in which perturbations are measured. Combining the hyperboloidal formulation with pseudospectrum analysis therefore makes it possible to compute optical QNMs while also studying their spectral stability. The approach extends techniques developed for resonance problems in gravitational physics to dispersive electromagnetism and provides a basis for studying stability and sensitivity in open optical systems.

We establish twisted automorphic descent from $(\mathrm{GL}_{2n}, \mathrm{GSpin}_2^{\delta})$, where $\mathrm{GSpin}_2^{\delta}$ is anisotropic, to the odd $\mathrm{GSpin}$ group $\mathrm{GSpin}^{\delta,\alpha}_{2n+1}$, which may be split or non-split, generalizing a construction of Jiang–Liu–Xu–Zhang. As an application, we give a new explicit construction of the global Jacquet–Langlands correspondence between $\mathrm{GL}_2$ and $D^\times$, where $D$ is a quaternion division algebra over a number field. As another application, we prove a new case of the global Gan–Gross–Prasad conjecture for $(\mathrm{GSpin}_{2n+1}, \mathrm{GSpin}^{\delta}_2)$.