Papers

The Riemann-Hilbert problem for the focusing coupled nonlinear Schrödinger (CNLS) equation is formulated on the basis of the corresponding $3\times3$ matrix spectral problem. We remove the discrete spectrum of initial RHP with the aid of Darboux transformations. Based on the $\bar{\partial}$-steepest descent method, we establish the long-time asymptotic behavior of solutions to the CNLS equation for initial condition in the weighted Sobolev space. Compared to the improved nonlinear steepest descent method, we improve the error estimate up to order $\mathcal{O}\left(t^{-3/4}\right)$. Furthermore, we obtain the asymptotic stability of multi-soliton solutions for CNLS equation and analyze the law of multi-soliton collision in the view-point of Yang-Baxter map.

We study two-dimensional networks of FitzHugh-Nagumo oscillators in regimes that mimic cardiac tissue. The networks include small layers of cells that have fast-recovery characteristics and adjacent layers of cells that have slow-recovery. Numerical simulations show that the boundary between the fast- and slow-recovery cells can initiate fibrillatory-like behavior when activated by a single, healthy wave in the network. We discuss possible connections between these structures and the onset of atrial fibrillation, with the goal of improving our understanding and treatment of this costly condition. Our main conclusion is that a single, normal excitation of small areas of heterogeneity embedded in healthy tissue can initiate arrhythmias and the two dimensional network can then sustain irregular firing patterns in the form of double swirling-back curves and spiral waves. This occurs with a very simple model of excitable cells and we give a clear mechanistic description of the arrhythmia. The model illustrates how increasing heterogeneity in heart tissue can progressively cause first intermittent and then persistent arrhythmic behavior. Another conclusion is that the center of problematic firings can drift away from the heterogeneity into healthy tissue, thus presenting phantom targets for treatment.

We propose an augmented high-order compact finite difference method for biharmonic equations with clamped boundary conditions and variable coefficients. Standard mixed-type formulations introduce an auxiliary variable, but its boundary values are unavailable, leaving the resulting discrete systems globally coupled and difficult to solve at large scales. Our key contribution is the development of a new augmented formulation that treats these unavailable boundary values as additional unknowns, reduces the global coupling to a lower-dimensional Schur complement system, and yields decoupled second-order subproblems. The Schur complement is solved by matrix-free GMRES, while the subproblems are handled by FFT-based fast solvers. The method achieves fourth-order accuracy using compact stencils, and has $O(n\log n)$ computational complexity, enabling the solution of the biharmonic equation with $1024^3$ degrees of freedom within several minutes. To the best of our knowledge, this level of computational efficiency has not previously been achieved in either the literature or practice. Using energy estimates and Fourier analysis, we derive a new $L^2$-estimate for Poisson equations with inexact Dirichlet boundary and then prove the convergence of the proposed scheme. We provide ample numerical experiments to confirm the accuracy, efficiency, and further apply the fast and accurate solver to triharmonic equations, high-wavenumber problems, Stokes flow, and plate bending problems.

By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on $\mathbb T^2$ with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force $F\in C^\infty([0,1]\times\mathbb T^2)$, and a classical solution $\rho$ on $[0,1)$ whose density gradient and spatial velocity gradient diverge in $L^\infty$ as $t\uparrow 1$. Nevertheless, $\rho(t)$ converges in $C^\eta$ for every $0\leq\eta<1$.

We study finite-time blowup for the parabolic–elliptic Keller–Segel system on $\mathbb{R}^d$ in dimensions $d\geq11$, where the problem is mass supercritical. For every integer $l\geq2$, we construct smooth radially symmetric solutions whose radial mass variable concentrates the normalized stationary state $Q$ at a quantized scale. Each blowup regime can be realized by solutions with nonnegative population density throughout their classical lifespan. More precisely, near the blowup time $T$, \[ u(t,r)=\frac{1}{\lambda^2(t)}\left[Q\left(\frac{r}{\lambda(t)}\right) +\epsilon\left(t,\frac{r}{\lambda(t)}\right)\right], \qquad \lambda(t)=c(T-t)^{\frac{l}{\gamma(d)}}(1+o(1)), \] where $c>0$ and $\gamma(d)=\frac12\bigl(d-2-\sqrt{(d-2)(d-10)}\bigr)$. The remainder $\epsilon$ converges to zero in local $L^\infty$ norms and in a range of high-order homogeneous Sobolev norms. Since $2l>\gamma(d)$, the concentration scale is strictly smaller than the parabolic scale $\sqrt{T-t}$, and the resulting blowup is of type II. The $l$-th regime has exactly $l-1$ unstable radial modulation directions and is stable within a codimension-$(l-1)$ class of suitably regular radial initial data. The proof combines a generalized-kernel expansion driven by the algebraic tail of $Q$, modulation analysis, coercive weighted high-order energy estimates, and a finite-dimensional topological argument. This yields a quantized hierarchy of stationary-state concentration rates for the high-dimensional Keller–Segel flow.

We study the structure of cyclic Riemannian nilmanifolds, that is, connected nilpotent Lie groups $N$ endowed with a left-invariant cyclic metric $\langle\cdot,\cdot\rangle$. We prove that a Riemannian nilmanifold $(N,\langle\cdot,\cdot\rangle)$ is cyclic if and only if its Lie algebra $\mathfrak{n}$ is at most two-step nilpotent and the associated family of skew-symmetric endomorphisms $J_{\mathfrak{z}}=\{J_Z : Z\in \mathfrak{z}\}\subset so(\mathfrak{a})$ is Abelian. As a direct consequence, every cyclic Riemannian nilmanifold is naturally reductive. We also determine the full isometry group of a connected and simply connected cyclic Riemannian nilmanifold without Euclidean factor.

We solve the four-dimensional Hamilton-Jacobi-Bellman (HJB) equation for two diffusively coupled Stuart-Landau-like oscillators to obtain full-state optimal feedback control. A sweep over coupling strength reveals a sharp change in the numerically optimal strategy: below a threshold coupling value, the controller steers the phase difference toward anti-phase while keeping both oscillators near the limit cycle, as reduced-order models would suggest. Above this threshold, the HJB solution changes qualitatively; the controller transiently collapses one oscillator's amplitude to near zero, thereby enabling large phase repositioning near the origin before rebuilding its amplitude. Direct gradient-based and stochastic optimization do not recover this lower-cost collapse trajectory from the initializations considered, suggesting that it occupies a region of the control landscape that is difficult to access by direct search. A joint sweep over nonisochronicity and coupling shows that collapse can occur even for an isochronous oscillator: phase repositioning near the origin can favor an off-cycle strategy. Nonisochronicity provides an additional energetic benefit through a phase-velocity surplus at small amplitude, quantitatively accounting for the observed reduction in control cost. Comparisons with uncoupled and coupled phase-reduced baselines show that phase models become increasingly inaccurate and cost significantly more energy for strong coupling. Results for coupled Van der Pol oscillators further demonstrate that exploitation of off-cycle dynamics is not specific to the Stuart-Landau-like oscillators.

Let ${\mathbb K}$ be an arbitrary field. We study tensor degenerations for square matrix multiplication over the field ${\mathbb K}$, and mainly prove for any integer $n\ge5$ the uniform dyadic estimate \[ \underline{\mathrm R}_{\mathbb K}({\mathsf M}_{2^n}) \le 2^{2n+1}\] which implies that $\omega(\mathbb K)\le 2+1/n$, where the border rank $\underline{\mathrm R}_{\mathbb K}(T)$ is the least $r$ for which the scalar extension of $T$ to an algebraic closure of $\mathbb K$ lies in the Zariski closure of the tensors of rank at most $r$, and $\omega(\mathbb K)$ is the matrix multiplication exponent over the field ${\mathbb K}$. This improves the current best record $\omega(\mathbb K)<2.371177$ greatly, and answers affirmatively the long-standing open question whether the value $\omega({\mathbb K})$ is exactly $2$. The implementation disclosure and appendices in this paper give fixed source maps, interpolation data, terminal incidence ledgers, and deterministic check counts for independent reconstruction.

We study the minion homomorphism poset of multisorted Boolean clones. Our main result completely determines the subposet induced by products of Boolean clones: every inequality between prod- ucts is induced by inequalities between their Boolean factors. Furthermore, we determine the minion cores of all products of Boolean clones, thereby identifying canonical representatives of their minion homomorphic equivalence classes.

Schimpf's connected reformulation of Pandharipande's pole conjecture for stable pairs predicts the following restriction: in class \(\beta\), connected descendent coefficients have nonzero \(p\)-poles only where \(-p\) is an \(m\)-th root of unity with \(1\leq m\leq\divis(\beta)\). We give a toric counterexample. The threefold is a smooth quasi-projective toric Calabi–Yau threefold whose compact torus-invariant curves form a chain \(C_1\cup C_2\). For the primitive class \(\beta=2[C_1]+[C_2]\), the coefficient of \(Q^\beta z^3\) in the connected series for the insertion \(\ch_z([v]_T)\), where \([v]_T\) is the equivariant class of \(v=C_1\cap C_2\), has Laurent expansion \[ {\frac{9}{4}}\frac{1}{p-1}+O(1) \] after restricting to the one-parameter subtorus with tangent weights \((1,2,-3)\) at \(v\). Since \(\divis(\beta)=1\), Schimpf's condition allows only the nonzero pole \(p=-1\). The pole at \(p=1\) is therefore forbidden.

We prove global strong well-posedness for the normalized isothermal, isobaric ternary Maxwell-Stefan diffusion system on bounded $C^{2+\alpha_0}$ domains, for arbitrary positive constant binary frictions and compatible nonnegative initial data, without a smallness condition. For distinct frictions, trace–determinant variables produce two separated spectral roots satisfying scalar parabolic equations with antisymmetric quadratic coupling. Euler-Poisson-Darboux one-sided truncations, stabilized by the mixing entropy, yield boundary De Giorgi oscillation decay and positive-time Hölder regularity. The mixing-entropy Hessian then provides a local symmetrizer; conormal Schauder estimates give uniform $C^{1+a}$ control, and frozen maximal $L_p$-regularity yields a uniform terminal trace bound excluding finite-time breakdown. Repeated frictions reduce to triangular scalar parabolic systems. Every component with positive conserved mass is strictly positive for positive time.

We study reachability and exact controllability for systems driven by a nondecreasing left-continuous Stieltjes derivator whose measure may have continuous, singular-continuous and atomic parts, while flat intervals represent inactivity. We construct an evolution family and post-jump variation-of-constants formula, and characterize the reachable subspace as the essential span of input directions. For constant coefficients, the Kalman condition is necessary for every derivator and, under regressivity, sufficient when the derivator is continuous and nonconstant on a subinterval. If the measure is purely atomic with exactly $N$ positive atoms, the reachable subspace is the truncated Krylov space generated by $B,AB,\ldots,A^{N-1}B$; for a Kalman-controllable pair, the minimum atom count among purely atomic derivators equals the controllability index. At the threshold count, one vanishing mass with the others converging to positive finite limits causes Gramian degeneration and worst-case energy blow-up; at or above threshold, compact positive mass ranges yield uniform coercivity. A mixed double-integrator example illustrates the results.

Let $X=\mathbb{P}^3$ and let \[ \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\mathcal{I}_Z$ for a plane $P\subset X$ and a length-$n$ subscheme $Z\subset P$. Hence the coarse stable-sheaf moduli space is the relative Hilbert scheme of points on the universal plane over the dual projective space $X^{\vee}$. Using the standard perfect obstruction theory for stable sheaves on a Fano threefold, we identify the obstruction bundle with a relative Carlsson–Okounkov twisted tangent bundle and obtain a closed product formula \[ \sum_{n\ge 0} \Gamma_n q^n=\prod_{m\ge 1}(1-q^m)^{-7} \] for the natural point-inserted two-dimensional Donaldson–Thomas invariants on $\mathbb{P}^3$. We then develop the analogous divisor-supported theory for a smooth divisor $D$ in a smooth projective Fano threefold, distinguishing moving and rigid divisors. For a rigid divisor satisfying $H^1(D,\mathcal{O}_D)=0$, the divisor-supported moduli component is a Hilbert scheme of points on $D$, its obstruction bundle is the twisted tangent bundle $\mathsf{T}_D^{[n]}(N_{D/X})$, and its generating series is \[ \prod_{m\ge 1}(1-q^m)^{-(c_2(X)\cdot D+D^3)}. \] We work out the examples of the exceptional divisor in $\operatorname{Bl}_p\mathbb{P}^3$ and of a rigid section in a Fano $\mathbb{P}^1$-bundle over $\mathbb{P}^1\times\mathbb{P}^1$.

We show that a positive answer to the centralizer problem for tracial matrix ultraproducts implies the existence of a finitely generated nonhyperlinear group.

A long-lasting conjecture in non-Kähler geometry says that any compact Hermitian manifold with constant holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but remains open in dimensions 3 or higher, except in several special cases. In this article we consider compact Hermitian manifolds that are Bismut torsion-parallel (BTP for brevity), which means that the Bismut connection has parallel torsion. For BTP manifolds that are not balanced, the conjecture was proved by Chen and Zheng. They also confirmed the conjecture for balanced BTP manifolds in dimension 3, utilizing a classification result for such threefolds by Zhao and Zheng. In a recent work, we confirmed the conjecture for balanced BTP manifolds in dimension 4 through a detailed case-by-case analysis. In this article, we establish the conjecture for balanced BTP manifolds in all dimensions. We also disprove the full-rank $B$ conjecture by constructing a compact balanced BTP fivefold with full-rank $B$ that is neither Kähler nor Chern flat; its Chern holomorphic sectional curvature is nonconstant.

Let a random permutation $W\in S_n$ represent the order of $n$ distinct cards. After each guess, the player learns the actual card, and the objective is to maximize the expected total number of correct guesses. Diaconis, Fulman, and Holmes conjectured that, after a single uniform, unbiased shelf shuffle, the direction-tracking strategy is optimal: first guess the smallest card, and thereafter choose the nearest remaining card in the direction of the most recent rise or fall. We prove a more general theorem: if \[ \mathbb P(W=w)=\frac{f(v(w))}{\sum_{\sigma\in S_n}f(v(\sigma))}, \] where $v(w)$ is the number of interior valleys of the permutation, $f$ is nonnegative and nonincreasing, and the denominator is positive, then direction tracking maximizes the conditional probability of correctly guessing the next card after every history of positive probability. It therefore maximizes the expected total number of correct guesses among all complete-feedback strategies. Here an interior valley is a position $2\le i\le n-1$ satisfying $w_{i-1}>w_i<w_{i+1}$. Combined with the known shelf-shuffling probability formula, this theorem proves the conjecture for every deck size and number of shelves. The key to the proof is a swap of adjacent remaining values that preserves the revealed prefix: starting from the preferred candidate, the swap leaves the valley count unchanged or increases it by one, thereby turning monotonicity of the weights into a conditional probability ordering after every history.

We study the minimum and maximum expected scores over all adaptive strategies in sequential card guessing with yes/no feedback. A deck contains $n$ labels, each appearing $m$ times, and is shuffled uniformly. The cards are guessed one at a time, and after each guess the player is told only whether it was correct. Always guessing the same label gives exactly $m$ points. How far can the expected score depart from $m$ if all previous feedback is used either to seek or to avoid correct guesses? Diaconis, Graham, and Spiro conjectured that the minimum expectation is $m-o(m)$ as $m\to\infty$ with $n$ sufficiently large in terms of $m$, and that, for each fixed $m$, the limit superior of the maximum expectation as $n\to\infty$ is at most $(e-1)m$. Writing these extrema as $P^-_{m,n}$ and $P^+_{m,n}$, we prove \[ 0\le m-P^-_{m,n}\le3m^{3/4} \qquad(n\ge m\ge1), \] and the finite-parameter bound \[ \frac{P^+_{m,n}}m\le\sum_{k=1}^{n}\frac1{k!}<e-1 \] for all positive integers $m,n$. The first bound gives a first-order asymptotic uniform in $n\ge m$; the universal constant $e-1$ in the second cannot be reduced. Thus both conjectures follow. The lower bound combines a switching comparison for restricted permutations with martingale second-moment estimates. For the upper bound, we represent the repeated cards as a random interleaving of independent decks of distinct cards and prove that the optimal expectation is additive when the component-deck labels are revealed.

In this article, we prove two conjectures of Sun concerning the evaluation of two series involving the reciprocal of the binomial coefficient $\binom{4k}{k}$. The values of these series are related to $\pi$, $\log3$, and the Gieseking constant $\text{Cl}_2(\pi/3)$. Our proofs rely on the Beta function and the Clausen function.

Let $q\geq3$ be odd and put $m_r=\lfloor r\log_2q\rfloor$. We count parity sequences with exactly $r$ odd entries for which the coefficient in the $qx+1$ iteration first drops below $1$ at step $m_r+1$. Their number $a_q(r)$ also counts the corresponding residue classes modulo $2^{m_r+1}$. We derive a first-violation recurrence which, for $q=3$, proves a recurrence conjectured in 2015. We establish the sharp inequalities \[ \frac1r\binom{m_r-1}{r-1}\leq a_q(r)\leq\frac1r\binom{m_r}{r-1}. \] Equality is characterized by the lower and upper record indices of $\{r\log_2q\}$, equivalently by denominators of convergents and semiconvergents from below and above. At these indices $r\geq2$, the values $a_q(r)$ are rational Catalan numbers. We also characterize the dependence on $q$ and prove, for each fixed odd $q\geq3$, that $a_q(r)=\Theta_q(r^{-3/2}B_q^r)$, where $B_q=(\log_2q)^{\log_2q}/(\log_2q-1)^{\log_2q-1}$. For $q=3$, this settles Hikawa's conjecture concerning the polynomial factor in a related counting function.

We prove blue percolation in the Chayes-Machta-Redner representation of the zero-field Edwards-Anderson spin glass in finite dimensions. For fixed symmetric iid couplings with $0<m=\mathbb{E}|J|<\infty$, the first blue-percolation inverse temperature $\beta_{\mathrm{b}}(d)$ satisfies $2dm\,\beta_{\mathrm{b}}(d)\to1$ as $d\to\infty$, in periodic joint limits. For symmetric $\pm1$ couplings, both overlap signs percolate at explicit temperatures in every dimension from 7 to 12. That proof reveals the overlap field, shows that it dominates a weakly coupled Ising field, and bounds a weighted second moment of open paths; in dimensions 7 to 9 the paths make short lateral excursions, and an interaction-matrix criterion controls the second moment. These proofs are computer-assisted: finitely many inequalities between rigorously bounded quantities are decided in exact rational arithmetic. For exponential-moment disorder, we also establish a percolating regime with exactly equal infinite-blue-sector densities and finite spin-glass susceptibility. In these periodic limits, persistent density imbalance would imply distinct spin-flip-related Gibbs states and ordinary spin-glass order: positive spatial-overlap variance, infinite spin-glass susceptibility and a cusp of the replica-coupling pressure, without requiring finite-blue cancellation. Proving imbalance remains open.

Let $h(n) = n + \tau(n)$, where $\tau$ counts divisors. Erdős and Graham asked, after Spiro, whether the orbits of any two positive integers under $h$ eventually share a point (Problem 414 on Bloom's list). The problem is open, and a finite computation cannot close it; nothing about it has been checked by a proof kernel, and the statement in the formal-conjectures repository carries a sorry. We replay what can be certified through the Lean 4 kernel under an axiom gate (axioms exactly propext, Classical.choice, Quot.sound; no sorry; no native_decide). Three elementary facts recorded by Li, namely that coalescence is an equivalence relation, $\tau(n)$ is odd exactly for squares, and $\tau(n)$ is bounded by $2\sqrt{n}$ so that orbits skip no square annulus $[k^2, (k+1)^2)$, are formalized, and his frontier lemma is formalized in a window form: every orbit started below a level $N$ has a point within $\lceil 2\sqrt{N} \rceil$ below $N$. The frontier lemma becomes a certificate: the $\lceil 2\sqrt{N} \rceil$ values $\tau(m)$ just below $N$ and the orbit points above $N$ until the orbits through the crossing points have merged. Its soundness is a kernel-checked theorem, and the certified rungs state that every pair of positive integers below $N$ coalesces for $N = 10^5, 10^6, 10^7, 10^8$; the certificate at $10^8$ has 44,530 entries rather than $10^8$. Every ledger and every quoted count was produced by two programs sharing no code that agree by hash. We record what is known about a non-coalescing pair and measure the exit-set sizes Li bounds. Nothing here is a proof of the conjecture.

Let $p$ be an odd prime and let $K_\infty/K$ be a $\mathbb{Z}_p$-extension. For each finite layer $K_n$, let $G_n$ be the Galois group of its maximal unramified pro-$p$ extension. We study the augmentation and Zassenhaus filtrations of the groups $G_n$ as $n$ tends to infinity. Assume that the classical Iwasawa $\mu$-invariant is positive. If $d_n$ denotes the minimal number of generators of $G_n$, then there is an integer $\delta>0$ such that $d_n=\delta p^n+O(1)$, while the minimal number of relations is $O(p^n)$. For each fixed $m\geq2$, as $n\to\infty$, we show that the $m$-th augmentation quotient has the same leading asymptotic as that of a free pro-$p$ group on $d_n$ generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.

For an integer $k\geq2$, let $\alpha_k$ be the supremum of the real numbers $a$ for which almost every integer $n\geq2$ has divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr).$ Let $\mathcal A\subseteq\N$ be the logarithmic random set in which the events $m\in\mathcal A$ are mutually independent and $\Pp(m\in\mathcal A)=1/m$ for every $m\geq1$. For a finite set $B\subseteq\N$, write $\Sigma(B)=\sum_{b\in B}b, \Sigma(\varnothing)=0$ and $m(B)=\max_{s\in\Z}\#\{C\subseteq B\mid\Sigma(C)=s\},$ and define \[ \beta_k=\sup\left\{c<1\,\middle|\,\lim_{D\to\infty}\Pp\bigl(m(\mathcal A\cap(D^c,D])\geq k\bigr)=1\right\}. \] Ford, Green and Koukoulopoulos proved $\alpha_k\geq\beta_k/(1-\beta_k)$ and conjectured that equality holds for every fixed $k\geq2$. In this paper, we confirm their conjecture. More precisely, for every fixed $a>\beta_k/(1-\beta_k)$, almost every integer $n\geq2$ has no divisors $d_1<\cdots<d_k\mid n$ satisfying $d_k\leq d_1\bigl(1+(\log n)^{-a}\bigr)$. We also correct local errors in their paper [Invent. Math. 232 (2023), 1027–1160], concerning the finite-subflag reduction, the residual-sum count, the moment estimate and the lattice adjustment. These corrections preserve the entropy-threshold comparison used in our proof.

Unmanned aerial vehicle (UAV) swarms deployed in mission-critical applications must simultaneously track a mobile aerial target and maintain reliable data links. However, active integrated sensing and communication (ISAC) operation imposes a dual energy burden on propulsion and transmission, threatening mission continuity through premature battery depletion. In this paper, we propose a two-tier UAV swarm architecture in which mission UAVs (MUAVs) execute cooperative ISAC for mobile aerial target tracking while dedicated charging UAVs (CUAVs), equipped with solar harvesting panels, replenish low-battery MUAVs via aerial UAV-to-UAV wireless power transfer (WPT). We formulate the joint minimization of the cooperative posterior Cramér-Rao bound (PCRB) over MUAV trajectories, per-slot sensing-communication time splits, WPT scheduling and admission, and CUAV rendezvous trajectories, subject to minimum uplink rate, dual-tier energy causality, WPT proximity, collision-avoidance, and speed constraints, yielding a non-convex mixed-integer program (MIP) that, to the best of our knowledge, is the first to jointly couple cooperative ISAC sensing quality with aerial WPT and dual-tier energy management. To solve it efficiently, we propose Receding-Horizon Alternating Optimization (RHAO), a four-block per-slot algorithm that decomposes the problem into: charging admission via the Hungarian algorithm, MUAV trajectory and time-split via successive convex approximation (SCA), CUAV rendezvous, and WPT power allocation, with monotone convergence guarantees. Simulation results demonstrate that RHAO reduces the mean PCRB by 16.8 times over a fixed-time-split baseline and 5.4 times over a static-trajectory scheme, while the aerial WPT subsystem sustains all MUAVs above the energy-critical threshold throughout the full mission horizon.

We prove that the Auslander–Reiten conjecture, the generalised Nakayama conjecture, the Nakayama conjecture, and each of the two Tachikawa conjectures are globally equivalent for Artin algebras over any fixed commutative artinian ring. For finite dimensional algebras over a fixed field, we also prove that the derived invariance of the Nakayama conjecture is equivalent to the truth of the Nakayama conjecture for all such algebras, answering a question raised by Chen and Xi.

Consider the ring of complex-valued continuous functions on a compact Hausdorff space $X$. Cortiñas–Thom computed its nonpositive algebraic $K$-theory, confirming Rosenberg's expectation. We give a motivic refinement of their result: For $i\geq 2j\geq 0$, the $i$th integral motivic cohomology group of weight $j$ is naturally isomorphic to the $i$th Betti cohomology group of $X$.

Deep spline superposition networks face a tension between approximation order and stability across depth. We study approximation under a hard layerwise Lipschitz budget, and organise it around two quantities: the factorisation stability complexity of a given deep factorisation, and the budget-compatible approximation complexity of a discretisation operator. First, we solve exactly the finite-depth diagonal balancing problem for a fixed chain of nonnegative envelope matrices: the optimal uniform layer budget equals $\|M_{L-1}\cdots M_0\|_{\infty\to\infty}^{1/L}$, attained by an explicit one-pass minimiser, for rectangular layers, with a complete treatment of degeneracies and non-attainment. The optimum can be arbitrarily larger than the Lipschitz constant of the network itself, because passing to envelopes destroys sign cancellation. Second, we give a constructive spline discretisation theorem preserving the budget up to a controlled slack, with an explicit grid threshold. Conversely, for linear spline-valued operators that preserve the budget exactly, we prove budget-compatible minimax lower bounds on classes constrained simultaneously in the first and third derivative norms – a constraint pair that is forced by the problem and that rules out the usual scaling escapes. Finally, we show that the corresponding layer errors need not cancel under composition: for every operator of the class there is a stable depth-$L$ tower realising a constant fraction of the accumulated error, so the linear-in-depth accumulation of the upper bound is not a proof artefact.

Let $q$ be an odd prime power and let $X$ be a maximal curve of genus at least two over $\F_{q^2}$. We prove that the automorphism group of $X$ acts transitively on its rational points if and only if $X$ is the Hermitian curve, or $q=5$ and $X$ is the unique $\F_{25}$-maximal curve of genus $3$, namely the maximal Klein quartic. Together with the result of Giulietti and Korchmáros in characteristic two, this gives the classification for every prime power $q$. In particular, the Hermitian curve is the only maximal curve of genus at least two whose automorphism group is doubly transitive on its rational points.

We study libraries of cross-sectional signals: at each date, a forecast vector over $d$ assets intended to predict the next period's cross-sectional return. Demeaned and unit-normalized, a signal is a point on a sphere and its $T$-date history a point on a product of $T$ spheres. A pairwise correlation cap on histories is a minimum angular separation on that product, so growing a library under such a cap is a packing problem. If a large library is not too pairwise correlated and is equally weighted, does the equally weighted sum (EWS) tend to a known principal-component quantity as it grows? We answer this under explicit assumptions on how the library is filled. Separation alone guarantees nothing: it fixes no limiting distribution; a saturated library covers the sphere yet can carry a biased count; and near-maximum packing on a fixed domain forces uniform volume, which on the unrestricted sphere gives zero mean and no distinguished principal component (PC1). Alignment depends on the admission rule and candidate distribution. Under the uniform product-volume benchmark, screening on positive average information coefficient (IC) yields a nonzero, target-aligned mean but an isotropic second moment, whereas a positive IC margin $\beta$ makat every finite $T$, with athree-level spectrum whose leading eigenvalue tends to $\beta^2$ while residual levels decay as $1/T$; margins of order $T^{-1/2}$ keep a positive admission rate but a vanishing eigengA finite residual-spectrum criteS-PC1 alignment.Gilbert-Varshamov codes show separation permits both outcomes: exponentially large positive-IC libraries exist whose EWS is PC1, and others whose EWS is orthogonal to PC1. $\log J=o(T)$ suffices for uniform estimation among $J$ candidates from $T$ iid dates. Derived results are proved and checked numerically; no market data are used.

We determine the Aharonov invariants of order at least two of every elliptic double cover by evaluating its projective kernel. In orders at least three, the formula separates the invariant into a constant Eisenstein term and an Eisenstein–Kronecker function evaluated under multiplication by two. The formula determines the ramification principal parts, torsion specializations, and isogeny traces, including the correction from two-torsion in an isogeny kernel. It also realizes the same intrinsic de Rham tensor through every degree-two projection: in higher orders this is the image of an Eisenstein section, while order two gives the classical Weierstrass complement to the Hodge line. A separate differential calculation shows that the Bernoulli-normalized invariant of order $p-1$ reduces to the quotient of the $p$th and first iterates of the defining derivation. For an elliptic invariant derivation this quotient is the Hasse invariant, independently of the rational function on its separable locus. The universal double cover provides a global specialization, integral away from two-torsion, whose reduction extends regularly across that locus. Classical Eisenstein zero theorems and the supersingular divisor congruence then describe the exactness loci and their reduction.