Papers

We propose a novel framework for modeling binary opinions (0 or 1) of individuals connected through a weighted directed network, where edge weights quantify interpersonal influence. Unlike classical models that assume complete access to previously expressed opinions, our framework allows individuals to update their biases using structured memory sets that capture limited and delayed information exchange. To analyze these opinion differences, we introduce a mathematically tractable notion of relative bias between pairs of individuals. The relative biases evolve according to a linear update rule involving past expressed opinions specified by the memory sets. We define the belief of an individual as the probability of expressing opinion 1 and derive a time-delayed dynamical system governing the evolution of network beliefs. We establish its asymptotic behavior and characterize its properties. The framework is further extended to networks containing bots, which maintain fixed biases while influencing neighboring individuals. We quantify the effect of bots by comparing the fixed points of the dynamics in their presence and absence. Finally, simulations illustrate the influence of memory, network structure, and bot interactions on the resulting opinion dynamics.

Let $X_1,\ldots,X_n$ be independent Gaussian tensors in $\mathbb{R}^{d_1}\otimes\cdots\otimes\mathbb{R}^{d_k}$ with a common covariance matrix given by the Kronecker product of $k$ unknown positive-definite factors, and let $D=\prod_{a=1}^k d_a$ and $d_{\max}=\max_a d_a$. Franks et al. (2026) established condition-number-free guarantees for the tensor-normal maximum likelihood estimator under the sample-size condition $nD\gtrsim k^2 d_{\max}^3$ and asked whether the cubic dependence on $d_{\max}$ could be reduced to a quadratic one. We answer this question affirmatively. For $t\geq 1$, if $nD\geq C k^2 d_{\max}^2 t^2$, then with high probability the maximum likelihood estimator exists, is unique, and satisfies $d_{\rm FR}(\widehat\Theta,\Theta)\leq C t \sqrt{k} d_{\max}/\sqrt{n}$ and $d_{\rm FR}(\widehat\Theta_a,\Theta_a)\leq C t\sqrt{k d_a} d_{\max}/\sqrt{nD}$ for every mode $a$. For every mode $a$ with $d_a=d_{\max}$, we further establish the sharp Thompson-metric bound $d_{\rm op}(\widehat\Theta_a,\Theta_a)\leq C t d_{\max}/\sqrt{nD}$. These guarantees are uniform over the unknown covariance factors and require neither condition-number bounds nor sparsity assumptions. Gaussian submodel lower bounds match the full and largest-factor Fisher–Rao rates up to a factor of $\sqrt{k}$ and the largest-factor Thompson rate up to universal constants. Consequently, for fixed $k$, the quadratic dependence of the sample-size threshold on $d_{\max}$ is optimal. GPT-5.6 Sol and Claude Fable 5 were used to assist with proof development, verification, and manuscript preparation.

In this paper, we develop a high-order, energy-stable and charge-conservative Lagrangian finite element method for variable-density incompressible inductionless magnetohydrodynamic (MHD) equations. The method utilizes the moving high-order curved tetrahedral mesh to track the material interface. Second-order Backward Differentiation Formula (BDF2) is used for the temporal discretization of material derivative, together with the second-order Adams–Bashforth method (AB2) for the update of control points of the meshes. High-order isoparametric Taylor-Hood elements with grad-div stabilization are used for the velocity-pressure pair. While, to ensure the discrete charge conservation, high-order parametric $\BH(\Div)$-conforming element is adopted for the current density. In the absence of external force, the unconditional energy-stability of the fully discrete scheme is proven. Finally, 3D numerical experiments are conducted to confirm the expected high-order accuracy for smooth solutions, the energy stability property and the capability of the proposed method.

Wu and Santhanam asked whether one can determine, from an increasing i.i.d. sample of binary random matrices, whether the unknown mean matrix is diagonalizable while making only finitely many errors almost surely. We answer this question affirmatively, for diagonalizability over either $\mathbb{R}$ or $\mathbb{C}$. The main observation is a general principle: every semialgebraic property of a fixed-dimensional bounded mean parameter is eventually almost surely predictable. We give a self-contained shrinking-confidence-set proof and an explicit predictor obtained from polynomial sign tests. Tarski–Seidenberg quantifier elimination shows that both the real- and complex-diagonalizable loci are semialgebraic, despite being neither closed nor open. We further extend the positive result to unbounded observations with any fixed finite moment of order $r>1$, using the Marcinkiewicz–Zygmund strong law. Combined with the Dembo–Peres topological criterion, this yields a sharp contrast: over the class of all merely integrable matrix laws, diagonalizability is not eventually almost surely predictable when the dimension is at least two. The construction is effective for fixed dimension, although no practical complexity bound is claimed.

For every prime $p$ and every perfectoid field $K$ of characteristic $p$, we construct a geometrically normal sousperfectoid affinoid adic space with a $C_p$-action whose quotient in the category of $v$-ringed spaces is not adic. The source is stably uniform and sheafy, while a cofinal sequence of rational neighborhoods at a fixed point acquires new degree-one invariant sections that do not descend by completed rational localization. Consequently, finite quotient stability for affinoid perfectoid spaces does not extend to sousperfectoid spaces.

For a $1$-rectifiable set $E \subset \mathbb{R}^d$ of positive length, a theorem of Federer shows that, among any $d$ linearly independent orthogonal projections of $E$, at least one has positive length. We develop a version of this finite-witness principle for generalized curve projections. Given scalar-valued mappings $\varphi_1,\ldots,\varphi_d:\mathbb{R}^d\longrightarrow\mathbb{R}$, we introduce the canonical encoding map $\mathsf{H}:=(\varphi_1,\ldots,\varphi_d)$. Where $D\mathsf{H}$ is invertible, a local bilipschitz change of variables and Federer's theorem show that $\varphi_j(E)$ has positive length for some $j$. If a positive-length portion of $E$ lies in the critical set, we instead argue intrinsically on a $C^1$ hypersurface containing it, provided that $\mathsf{H}$ retains full tangential rank there. This yields an abundance of deterministic finite witnesses, as well as structural bounds for those exceptional parameters where the good witness property fails. A new fold non-degeneracy condition makes our constructions stable under perturbations of the underlying parameters. At this rectifiable endpoint, these conclusions complement work of Peres–Schlag, which developed exceptional set estimates for fractal sets of Hausdorff dimension strictly greater than one. We then apply our framework to several nonlinear projection families. Affinely independent pinned squared-distance maps satisfy the tangential and fold conditions, whereas two planar radial projections lose tangential rank along their critical line. We also verify the hypotheses for more exotic examples, including nonlinear anisotropic distances, Bregman functionals arising from smooth approximations of polyhedral norms, and families whose critical hypersurfaces have prescribed $C^2$ geometry.

This paper introduces and studies \(\ell\)-rank additive intersection pairs (RAIP) of codes over finite fields for a given positive integer \(\ell\). The notion of \(\ell\)-RAIP provides a common framework that generalizes additive complementary dual (ACD) codes, additive complementary pairs (ACP) of codes, and the hull of an additive code. We establish necessary and sufficient conditions characterizing when a pair of additive codes forms an \(\ell\)-RAIP. Furthermore, for (q>2), we prove that any pair of additive codes is monomially equivalent to an ACP of codes. As a consequence, for (q>3), every additive code is monomially equivalent to an ACD code. { A key contribution of the paper is a general construction method for \(\ell\)-RAIP of codes derived from self-orthogonal additive codes, which yields families of \((\ell+1)\)-RAIP of codes under suitable conditions.} In addition, several explicit constructions of \(\ell\)-RAIP of codes are presented.

Let O be the ring of integers of a non-Archimedean local field with finite residue field, let p be its maximal ideal, and let O_2 = O/p^2. We show that the class equation of GL_n(O_2) depends on O only through the cardinality of its residue field: for two such rings O and O' with isomorphic finite residue fields, there is a canonical bijection between the conjugacy classes of GL_n(O_2) and of GL_n(O'_2) which preserves the size of every class. The argument constructs a section of the reduction map GL_n(O_2) -> GL_n(O_1) which is multiplicative on the centralizer of any element in its block Jordan canonical form, and uses it to transport the classification of conjugacy classes lying above a fixed class of GL_n(O_1) from one ring to the other. This note records the original argument for this result, developed in the author's 2010 doctoral thesis, which predates and is independent of two later proofs of closely related statements: the Ext-theoretic classification of similarity classes for n <= 4 by Prasad, Singla and Spallone, and the Hom-theoretic approach of Jambor and Plesken for general uniserial rings of length two. We record the centralizer-section argument here since it seems to be of independent interest and several colleagues have asked to see it in print.

The terahertz (THz) band offers abundant spectrum resources for high-throughput communication and ultra high-precision localization. This paper investigates secure communication in cooperative THz orthogonal frequency-division multiplexing (OFDM) bistatic integrated sensing and communications (ISAC) systems, where multiple base stations (BSs) equipped with extremely large-scale antenna arrays (ELAAs) collaboratively serve downlink users while concurrently locating multiple targets. Malicious targets are assumed to act as potential eavesdroppers attempting to intercept confidential information intended for legitimate users. To mitigate these threats, we formulate a joint optimization problem for analog beamforming, digital precoding, true-time delayers (TTDs), and sensing signal covariance matrix design. The objective is to maximize the minimum secrecy rate subject to Cramer-Rao bound (CRB) constraints that ensure localization accuracy. This problem is highly challenging due to the non-convex CRB constraint, strongly coupled variables, high computational complexity from ELAA, and near-field channel modeling. To address these challenges, we propose a novel data-driven framework that integrates graph neural networks (GNNs) with the Mamba architecture. Our proposed framework first encodes the interactions among users, targets, and BSs into a heterogeneous graph and then employs message passing to optimize vertex features. The Mamba blocks further enhance this process through their selection mechanism and state space modeling capabilities, enabling dynamic and context-aware optimization of beamforming, TTD configurations, and sensing parameters. Numerical simulations validate that the proposed method outperforms both conventional and learning-based baselines, while offering high computational efficiency and strong generalization across different network conditions.

This paper establishes Turán-type inequalities for quaternionic canonical generalized polynomials with all zeros in the closed unit ball of radius $k \geq 1$. We extend the classical inequality due to Govil from the complex setting to the quaternionic framework. We prove that for certain classes of polynomials, the inequality \[\|P'\| \geq \frac{n}{1+k^n}\|P\|\] holds for all $k \geq 1$, where $n$ is the degree of the polynomial. Our main results provide sharp derivative estimates for quaternionic canonical generalized polynomials under specific algebraic conditions on their zeros. These findings contribute to the ongoing research of extending classical polynomial inequalities to the noncommutative setting of quaternions.

We study in this note the decomposition theorem for forgetful maps between moduli spaces of genus $g$ stable pointed curves $\overline{\mathcal{M}}_{g,m}\to \overline{\mathcal{M}}_{g,n}$. We prove that as $m$ goes to infinity, every stratum closure in $\overline{\mathcal{M}}_{g,n}$ appears as the support of a direct summand in the decomposition theorem. The proof uses Hassett's moduli spaces of stable weighted pointed curves and a wall-crossing argument.

In this paper, we define topological and measure-theoretic entropy scales in dynamical systems for the scaling introduced by Helfter. Firstly, we establish a variational principle between Bowen and lower entropy scales on compact subsets. Secondly, we prove factor inequalities for Bowen and upper capacity entropy scales on arbitrary subsets; unlike the classical additive case, the fiber contribution is described by a transfer function and need not be additive. We also study entropy scales and sequence entropy scales under the induced action. Under suitable conditions, zero values are preserved in both directions, while positive values become infinite; counterexamples show that additional assumptions are generally necessary. As an application, we use entropy scales to distinguish zero-entropy systems sharing the same entropy dimension but differ in finer orbit-growth rates.

Classical sumset inequalities originating in additive combinatorics admit geometric analogues for convex bodies in finite-dimensional real vector spaces, as recently developed by Fradelizi and two of the authors. We develop integral analogues for geometric $\alpha$-concave functions under $\alpha$-convolutions, an operation that arises naturally in the “geometrization of probability” program. In particular, we establish a Plünnecke–Ruzsa-type inequality, as well as sharp analogues of sum-difference and Ruzsa triangle inequalities, for $\alpha$-convolutions of $\alpha$-concave functions. We also prove a sharp Rogers–Shephard-type inequality and characterize its equality cases for $\alpha$-concave functions, bridging the log-concave case studied by Alonso-Gutiérrez, González-Merino, Jiménez, and Villa and the quasi-concave case studied by Colesanti.

Towards practical applications of statistical sequence matching, we propose low complexity tests that are exponentially consistent and bound the exponential decay rates for error probabilities of our proposed tests. In statistical sequence matching, one is given two databases of multiple sequences, where in each database, each sequence is generated i.i.d. from a distinct distribution. A pair of sequences, one from each database, is said matched if they are generated from the same distribution. The number of matched pairs of sequences, a.k.a, the number of matches, can be either zero or positive. We consider both cases of known and unknown number of matches. When the number of matches is known and positive, the task is to identify all matched pairs of sequences. To construct fixed-length tests, we calculate the scoring function values for all pairs of sequences across the two databases and identify the matched pairs as those having small enough pairwise scoring function values. We show that our proposed test is exponentially consistent, which strikes a better tradeoff between complexity and performance than existing fixed-length tests that use exhaustive search. To construct sequential tests, we need an additional positive threshold to determine the stopping time, and we show that our sequential test achieves strictly better performance than our fixed-length test. Subsequently, we generalize our results to the case with unknown number of matches.

Let $r\ge2$, let $T_r$ denote the transitive tournament on $r$ vertices, and write $d_G^*(v):=\max\{d_G^+(v),d_G^-(v)\}$. We prove that if $r\mid n$ and an $n$-vertex digraph $G$ satisfies $d_G^*(x)+d_G^*(y)\ge 2(1-1/r)n-1$ for every $x\ne y \in V(G)$ with $xy \notin E(G)$, then $G$ has a $T_r$-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree $d^*_G(x) \ge (1-1/r)n$ and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n - 1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.

We prove a finite-point version of the Schwarz-Pick inequality for holomorphic self-maps of the disk. The estimate compares Bergman Gram matrices built from several points and their images, and the sharp constant depends on both the number of points and, for finite Blaschke products, the degree of the map. These matrices also arise from the Szegő-kernel metric on the symmetrized polydisc, where the estimate becomes a sharp Schwarz-Pick inequality for the induced holomorphic maps.

We explore the classical question of enumerating the homotopy types of gauge groups of principal $\mathrm{SU}(n)$-bundles over $S^{2m+1}$ in the unstable range $m\geq n$. Using computations of certain $p$-local Samelson products, we obtain lower bounds on the number of such homotopy types. We also present a discussion of the last stable case.

Let $(S^1,\delta)$ be the unit circle endowed with the arc length metric. This paper concerns the question of which subsets of $(S^1,\delta)$ can be embedded isometrically into the sequence space $\ell_1$. It is well known that every finite subset of $S^1$ admits such an embedding, but the situation for infinite subsets, even including the whole circle, has perhaps been obscured by conflicting terminology in the literature. It is worth noting that the circle avoids the classical obstructions to isometric embeddability, since it is both of negative type and hypermetric. In this paper we show that if $X \subseteq S^{1}$ is a closed set and the Lebesgue measure of $X \cap X^{\ast}$ is positive, where $X^{\ast}$ is the antipodal set of $X$, then it is impossible to isometrically embed $(X, \delta)$ in $\ell_{1}$. As a result, no subset of $S^{1}$ with Lebesgue measure greater than $\pi$ can be isometrically embedded in $\ell_1$. Conversely, we show that any closed subset of $S^1$ whose intersection with any half-circle has measure zero can be isometrically embedded in $\ell_1$. These results have several consequences. They imply that the classical Banach space $L_1[0, 1]$, considered purely as a metric space, does not isometrically embed in $\ell_{1}$. Secondly, they yield a simple proof that a metric graph $(M, d)$ embeds isometrically in $\ell_1$ if and only if it is a tree. In contrast to $S^{1}$, we show that some related metric spaces, such as the cylinder and the flat torus, contain finite subsets that cannot be isometrically embedded in $\ell_1$.

Recent work has shown that, for smooth convex optimization, plain gradient descent can be accelerated from its textbook convergence rate of $O(T^{-1})$ (where $T$ denotes the number of iterations) to $O\big(T^{-\log_2(1+\sqrt{2})}\big)$ using carefully designed stepsize schedules alone, without resorting to momentum or other algorithmic modifications. Despite this progress, however, little was known about lower bounds for such methods beyond the classical $\Omega(T^{-2})$ benchmark for general first-order methods. In this work, we present a new lower bound of $\Omega(T^{-1.9319})$ for the last-iterate convergence rate of gradient descent with predetermined nonnegative stepsize schedules. This result provides rigorous evidence that stepsize schedules alone cannot accelerate plain GD to the optimal $O(T^{-2})$ convergence rate. The proof was developed by GPT-5.6 Sol Pro under the authors' guidance.

Elliptical Lunar Frozen Orbits (ELFOs) are attractive candidates for lunar satellite constellations supporting lunar south pole exploration. Building on prior frequency-based orbit analysis methods, this investigation develops a torus-based, frequency-domain framework for constellation-level ELFO design and analysis. Within a doubly-averaged dynamical model, each ELFO is characterized by three frequency-angle pairs; a systematic comparison across progressively higher-fidelity models identifies the additional frequency components introduced at each level. The angular parametrization admits multiple symmetries that substantially reduce the design space, enabling a rapid, epoch-free global survey of constellation configurations. A higher-fidelity refinement and analysis layer then combines a (1) Frequency-Domain Differential Corrector (FDDC) that targets desired frequency properties and suppresses undesired long-period oscillations, (2) Fourier surrogate re-optimization for inter-satellite phasing without propagation in the loop, and (3) spectral attribution that separates design-refinable coverage losses from those inherent to the dynamics. Together, these elements supply a fidelity-bridging and diagnostic framework for ELFO constellation design and analysis.

This is an introduction to probability, written with a quantum idea in mind, namely that the semicircle law comes first. We first discuss discrete probability, notably with the binomial and hypergeometric laws, positive and negative, and the Poisson and compound Poisson laws. Then we get into the continuous case, with the basics of the theory explained, and with as starting examples the exponential, semicircle and beta distributions. Afterwards, we investigate the central limits and normal variables, both real and complex, and with a look at Rayleigh laws, and hyperspherical laws too. Finally, we discuss a number of more specialized distributions, and more specialized techniques too, and we end with an introduction to random matrices and freeness.

Powered landing with variable mass, free final time, and quadratic aerodynamic drag requires the repeated solution of local convex subproblems, whose main online cost lies in the long dynamics-equality chain and the inner iterations. This paper develops a condensed sequential convex approximation designed for low latency. Exact block elimination removes 217 intermediate-state components and 210 interval equations from a 31-node model, leaving 100 primal variables coupled by six terminal equalities. A low-weight energy term and fixed quadratic proximal regularization make the ideal surrogate strongly convex with predictable curvature. The inner solver is an extrapolated proportional–integral projected gradient (xPIPG) implemented with fixed-size arrays, $3\times3$ interval solves, and a fused one-pass node map. The one-pass map is a deliberate low-cost approximation, not the exact joint proximal operator. We therefore evaluate the timed code by nonlinear trajectory residuals and independent physical checks rather than by a claim of exact KKT convergence. The single-precision C implementation completes one plan in four outer updates and 336 xPIPG updates. On an Intel Core i7-10875H, the median end-to-end solve time is \SI{374}{\micro\second} and the P99 value is \SI{512}{\micro\second}. All 100 common initial-state perturbations pass validation, and the median remains below \SI{0.7}{\milli\second} for 15–51 nodes. An independent high-accuracy first-order-hold integration gives a terminal position error of \SI{0.183}{\meter}. Within the stated model, hardware, stopping rule, and timing boundary, this is, to the authors' knowledge, the first submillisecond end-to-end sequential-convex solve for a single powered-landing trajectory.

In this paper, we present a fifth-order finite difference divergence-free Hermite weighted essentially non-oscillatory (HWENO) scheme for the ideal magnetohydrodynamics (MHD) equations. In this framework, both the solution and its spatial partial derivatives are evolved in time and jointly employed in the spatial reconstruction procedure. A major challenge in MHD simulations is preserving the divergence-free constraint of the magnetic field, which is generally violated by standard numerical methods designed solely for hyperbolic conservation laws. To address this issue, we first solve the MHD equations within the HWENO framework for hyperbolic conservation laws, yielding a magnetic field divergence that remains zero up to high-order accuracy in smooth regions. Subsequently, we apply a correction that evenly distributes the divergence error among the partial derivatives involved in the divergence-free constraint, thereby rendering the magnetic field discretely divergence-free at the new time level. This approach offers several advantages. First, the scheme retains the conservation property, as only the partial derivatives of the numerical solution are corrected, leaving the conserved variables unchanged. Second, the correction applied to the partial derivatives of the magnetic field components introduces only a high-order perturbation, thereby preserving the overall accuracy. Third, the divergence-free treatment significantly enhances robustness, as most benchmark test cases cannot be run stably without such a correction. Fourth, the correction is a simple linear operation applied at each stage of the time integration, incurring negligible additional computational cost. Extensive numerical experiments demonstrate the accuracy, resolution, efficiency, effectiveness, and robustness of the proposed scheme.

In recent years, neural networks have significantly advanced numerical solutions of partial differential equations (PDEs). However, solving PDEs with discontinuous solutions, such as hyperbolic conservation laws, remains challenging for neural network-based methods such as physics-informed neural networks (PINNs). Existing methods often rely on strong prior assumptions such as knowledge of discontinuity locations, or they introduce artificial smoothing terms that degrade accuracy. However, accurately solving these conservation laws and predicting the formation and propagation of discontinuities in solutions is crucial in many practical applications, including gas dynamics and traffic flow modeling. In this paper, we introduce a novel Weak-Entropy PINN (WEPINN) framework for hyperbolic conservation laws with discontinuous solutions. The method enforces the governing equations in their weak (integral) formulation and incorporates the entropy condition to select the physically admissible solution, while employing the discrete fast Fourier transform (DFFT) for efficient numerical integration. Our method is tested through extensive numerical experiments on a variety of scalar conservation laws and systems of conservation laws in one and two dimensional spaces. These experiments demonstrate that our method can accurately resolve sharp discontinuities while effectively capturing interactions between multiple shock and rarefaction waves.

For a torsion-free abelian group $G$, let \[ K_{\mathbb{Q}}(G)=\operatorname{Frac} \mathbb{Q}[G] \] be the fraction field of its rational group algebra. We prove that this field determines the group up to isomorphism: \[ K_{\mathbb{Q}}(G) \cong K_{\mathbb{Q}}(H) \quad \Longleftrightarrow \quad G \cong H. \] The main structural input is that, over every field $k$ of characteristic zero, the monomial defect group \[ \Delta_k(G)=K_k(G)^\times/(k^\times X^G) \] is free abelian. We give a self-contained proof. It first treats the one-variable Puiseux field $F(t^{\mathbb{Q}})$: factorization from $F(t^{1/n!})$ to $F(t^{1/(n+1)!})$ gives split inclusions because the substituted irreducibles are square-free in characteristic zero. A transfinite decomposition of the divisible hull $\mathbb{Q} \otimes G$ then proves the general case. Given a field isomorphism, we compare the two monomial subgroups inside the common multiplicative group. Their intersection produces isomorphic subgroups $M\leq G$ and $N\leq H$, while the quotients $G/M$ and $H/N$ embed in free defect groups and are therefore free. Relative transcendence degree shows that these two free quotients have the same rank, completing the reconstruction. As consequences, rational-function stabilization of group fields exactly records free stabilization of groups, and Rickard's bounded sequence group yields a field $F$ with $F\cong F(x,y)$ but $F\not\cong F(x)$.

A connectivity function on a finite set $E$ is a function $f\colon 2^E\to\mathbb Z$ that is submodular and symmetric, with $f(\varnothing)=0$. Given a connectivity function $f$ via a value oracle, terminals $t_1,\ldots,t_r\in E$, and an integer $k$, the Multiway $f$-Cut problem asks whether $E$ has a partition $(P_1,\ldots,P_r)$ with $t_i\in P_i$ for every $i$ and $\sum_{i=1}^r f(P_i)\le k$. We prove that Multiway $f$-Cut is fixed-parameter tractable parameterized by $k$. Cut functions of graphs are connectivity functions, so as a special case we recover the classical result that Edge Multiway Cut in graphs is fixed-parameter tractable. Our proof of correctness is completely elementary, and is arguably the simplest known proof of this fact.

We initiate the study of conjugacy languages in free inverse monoids. Motivated by the notion of conjugacy languages in groups and the study of conjugacy in semigroups, we introduce the language of shortest representatives of conjugacy classes and study it for free inverse monoids of rank at least $2$ under the natural notion of conjugacy. We show that, contrary to the free group case, where this language is regular, in a free inverse monoid it is neither context-free nor co-context-free. In the monogenic case, this language is context-free. We show that this non-context-freeness comes from elements with nontrivial conjugacy classes. We define an equivalence relation as follows: all elements with a nontrivial conjugacy class are related and elements with trivial conjugacy class are only related to themselves. We call this relation $\mathrm{UConj}$. We show that the language consisting of geodesics representing elements whose conjugacy class is trivial is context-free by providing an explicit context-free grammar generating it. For groups, we show that the language of minimal representatives of $\mathrm{UConj}$ classes is regular if the group is hyperbolic and for right-angled Artin groups with the standard generating set, it is piecewise testable. For virtually abelian groups, we show that there is a generating set for which this language is piecewise excluding and exhibit an example of a virtually abelian group admitting a generating set for which this language is not regular.

Recently, Lamzouri proved a lower bound for the least positive integer $n_f$ for which the Hecke eigenvalue $\lambda_f(n_f)<0$, showing it satisfies $n_f \ge (\log k)^{1-o(1)}$ for many holomorphic Hecke cusp forms of even integral weight $k$. In this paper, we extend Lamzouri's result to Hecke-Maass forms without assuming the Generalized Ramanujan Conjecture, and further prove that similar results also hold for the coefficients of symmetric power $L$-functions of Hecke-Maass forms.

A set of intervals $I = \{ I_1, I_2, \dots, I_n \}$ forms a simple chain if, for every $2\leq i \leq n-1$, interval $I_i$ overlaps only with $I_{i-1}$ and $I_{i+1}$. We show that a deterministic memoryless one-directional revoking algorithm achieves a competitive ratio of $2(1 - 1/\sqrt{e}) \approx 0.786$ on the simple chain in the random order model, hence performs worse than the basic greedy algorithm without revoking that has a competitive ratio of $(1 - 1/e^2) \approx 0.864$, but better than any deterministic revoking algorithm in the adversarial model that has a competitive ratio of at most $0.75$. The proof of the latter also leads to a lower bound of $n/4$ for the advice complexity.

We study a noncooperative resource-allocation game in which $m$ players distribute fixed resources among $n$ projects and the payoff of player $j$ is given by \[ F^j(x) = \sum_{i=1}^n \frac{a_i x_i^j}{b_i+\sum_{\ell=1}^m x_i^\ell}, \] where $a_i$ and $b_i$ are project parameters, while $x_i^j$ is the amount of resources player $j$ allocates to project $i$. This specification combines diminishing returns with congestion generated by competitors. We prove that the game has a unique Nash equilibrium and characterize it by an equimarginal principle. We show that, after the projects are ordered by $a_i/b_i$, each player invests in an initial segment of projects, and these segments are nested across players, so the equilibrium decomposes into consecutive activity zones; players with larger resources invest weakly more in every project. In the fully active regime, where every player invests in every project, we reduce the equilibrium to a single scalar nonlinear equation for the aggregate marginal-utility rate; all individual marginal rates, investments, and payoffs then follow from explicit formulas. We also provide a projected marginal-utility algorithm with global linear convergence under an explicit step-size condition, together with a structure-exploiting Block Pandora algorithm that reconstructs and certifies the equilibrium conditional on a proposed nested cutoff structure.