Papers

We determine the minimax cumulative log-loss regret of a finite-alphabet, exogenously driven source with genuinely infinite input memory: independent Rademacher inputs $(U_t)$ are observed sequentially and the next binary mark has logit $\sum_{j\ge1}\theta_jU_{t+1-j}$, the unknown coefficients obeying a summable envelope $|\theta_j|\le r_j$, $\sum_jr_j\le B$. At horizon $T$, lag $j$ can move the logit by at most $r_j$ and is exercised in only $n_{T,j}=(T-j+1)_+$ rounds, and the two limitations combine into the sum $\Gamma_T(r)=\sum_{j\le T}\log(1+n_{T,j}r_j^{2})$. One coordinate-localised Bayesian mixture achieves $R_T(r)\le C\Gamma_T(r)$ for every summable envelope with $C$ universal. Our main result is a matching nonasymptotic converse for the canonical exponential and polynomial envelopes; its new ingredients are a modular finite-sample information bound for logistic experiments with an exogenous random design, and a conditioning estimate for the overlapping Toeplitz lag matrix obtained by exhibiting each off-diagonal Gram sum as a sum of independent Rademacher variables indexed by the edges of a forest, needing neither local asymptotic normality nor any spectral theorem for random Toeplitz matrices. So $\Gamma_T(r)$ is the minimax regret scale here, giving $\Theta(\alpha^{-1}\log^{2}T)$ for $r_j=Ae^{-\alpha j}$ and $\Theta(T^{1/(2s)})$ for $r_j=Aj^{-s}$, $s>1$ — the latter without the extra $(\log T)^{1-1/(2s)}$ factor any window-truncation analysis pays. We also show memory decay cannot determine regret, and that a profile-scaled online Newton predictor attains $O_B(\Gamma_T(r))$ in polynomial time per round.

We construct period polynomial relations for finite systems of theta products stable under modular transformations and use them to derive identities among critical $L$-values. Our main example is a three-component theta system of weight $5$, whose coupled period polynomials are determined explicitly and yield new cross-form relations among critical values of the associated eta products. These relations are not consequences of the functional equations of the individual forms. We also develop a weight-$4$ system arising from a quadratic twist of conductor $3$ and obtain relations linking critical values of the original and twisted modular forms. Through modular parametrizations by complete elliptic integrals, these $L$-value identities yield corresponding moment identities. The method gives a unified modular-symbol explanation of several previously known elliptic integral moment relations while producing new critical-value relations between distinct modular forms.

We study Kirchhoff-Boussinesq-type equations on closed Riemannian manifolds $(M,g)$ of the form \[ \sum_{j=0}^m a_i(-\Delta_g)^m u \pm \text{div}_g(\vert\nabla u\vert_g^{p-2}\nabla u) = f(u),\qquad \text{ on }\ M, \] where $m<\dim M/2$, $(a_0,a_1,\ldots a_m)\in C^\infty_+(M)\times [0,\infty)^{m-1}\times(0,\infty)$, $2<p\leq\frac{2\dim M}{\dim M- 2}$ and $f:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous nonlinearity of superlinear type with subcritical or critical Sobolev growth. We briefly motivate the above equation in the case of $m=2$, as an extension of the stationary Kirchhoff-Boussinesq equation when modeling the dynamics of curved elastic plates. Under some symmetry assumptions, we prove the existence of multiple equivariant solutions when considering critical Sobolev nonlinearities and, for subcritical ones, we also prove the existence of ground-state solutions. As a byproduct, we prove a Gagliardo-Nirenberg interpolation inequality and give several equivalent norms on the higher order Sobolev space $H_g^m(M)$.

Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $\mu$ for Riemannian volume, and set $V(o,r)=\mu(B(o,r))$ for geodesic balls centered at $o$. For $0<\alpha<2$, let $X^{(\alpha)}$ be the process obtained by subordinating Brownian motion with an independent $\alpha/2$-stable subordinator; its $L^2$-generator is $-(-\Delta)^{\alpha/2}$. We prove that \[ \int^\infty\frac{dt}{V(o,t^{1/\alpha})}=\infty \] implies that \(X^{(\alpha)}\) is recurrent. The proof uses a spectral trace estimate and radial cutoffs on $M\times(0,\infty)$, where the auxiliary measure is $y^{1-\alpha}\,d\mu\,dy$. It proves the sufficient implication in Grigor'yan's Problem~26.

We prove that a strongly nondegenerate Reeb vector field on a closed contact three-manifold has infinitely many simple positive hyperbolic periodic orbits if it has at least three simple periodic orbits. The main ingredient is a result for area-preserving maps on compact surfaces, possibly with boundary. We prove that, under a natural boundary index condition, an area-preserving nondegenerate map with infinitely many periodic points has infinitely many positive hyperbolic periodic points in the interior. The proof combines this surface result with the broken book decomposition theorem of Colin–Dehornoy–Rechtman.

We prove interior $C^2$ regularity for convex viscosity solutions of the $3$-Hessian equation $\sigma_3(D^2u)=f(x)$ with $f\in C^{0,1}, \inf f>0$, under a strict $3$-convexity condition on $u$.

We study the compactness of two sets of positive radial solutions to “zero mass” quasi-linear Schrödinger equation \[ -\Delta u-u\Delta(|u|^2)=g(u) \quad\text{in }\mathbb R^N,\quad N\ge5. \] For nonlinearities that are either strictly subcritical or asymptotically critical at infinity, we prove that the set of least energy positive radial solutions is nonempty and compact in the natural space. Moreover, in the strictly subcritical case and under additional assumptions, we show that the set of all finite $L^2$-mass positive radial solutions is compact in \(L^2(\mathbb R^N)\). The proof relies on uniform decay estimates for the corresponding positive radial solutions of the transformed semilinear equation, which yield the required uniform \(L^2\)-tail control.

We study the entropy of random graphs generated by piecewise Hölder continuous graphons. We first present a result on the rate of convergence of the normalized entropy as the size of the graph grows. The core ideas of the proof are described, with the detailed proof provided in the appendix. From this result, we then derive quantitative bounds on the entropy for the stochastic block model and random geometric graph model. These bounds provide explicit formulae rather than asymptotic statements which have been found previously.

Let E be a polar compact set in $\mathbb C$. Let $f_\infty$ be a germ at $\infty$ that can be analytically continued along an arbitrary path $\gamma$ lying in $\widehat{\mathbb C}\setminus E$ and starting at the point $\infty$. In 1985–1986 Herbert Stahl presented his proof of a fundamental theorem on the convergence of diagonal Padé approximants constructed from such germ $f_\infty$. Since then, this theorem has borne his name. A crucial role in his proof is played by the existence of a compact set $S_{f_\infty}$ of minimal logarithmic capacity among all compact sets $K$ such that the germ $f_\infty$ extends as a single-valued meromorphic function to $\widehat{\mathbb C}\setminus K$. Unfortunately, the proof of this fact presented by H. Stahl in 1985 contains a crucial mistake. It is surprising that this mistake was made in the original Stahl's paper in 1985 and repeated in his last preprint in 2012, and, as far as we know, no one has pointed it out before! In this paper we explain this serious Stahl's mistake and present a correct proof of the existence of a compact set $S_{f_\infty}$. We emphasize that our proof will use ideas completely different from Stahl's ideas. Also we discuss the more general Stahl's conjecture about the existence of a compact set $S_{f_\infty}$ for an arbitrary germ $f_\infty$ without any assumption on the paths along that the germ $f_\infty$ can be continued.

This PhD thesis studies the classification of group extensions, whose main result is the classification, up to isomorphism, of finite nonsolvable groups of order less than 23,040. To make this classification computationally effective, we develop reduction methods that replace extension computations involving a finite nonsolvable group by computations in suitable smaller solvable subgroups. This reduction from nonsolvable to solvable computations led to a major decrease in computing time and made it possible, on a standard personal computer available in 2002, to classify around 8.4 million nonsolvable groups of order less than 23,040. A nonsolvable group E is constructed from a given perfect kernel P, its perfect residual, extended by a given solvable quotient H=E/P. The isomorphism problem for the resulting nonsolvable groups E is reduced to isomorphism computations between extensions involving a small nilpotent subgroup U<P and the quotient H. We introduce and use (Z\phi)-extensions and supplement methods to obtain explicit classifications in a range of finite cases. These methods are applied to the classification of finite nonsolvable groups of order less than 23,040 and provide algorithms for constructing and identifying extensions up to isomorphism.

We develop a whole-trajectory relaxation framework for dissipative evolutions with non-convex, state- and history-dependent kinetics. The framework arises naturally from a game-theoretical formulation of energy–dissipation evolution. State–rate consistency is eliminated by causal Volterra reconstruction, reducing the problem to a diagonal equilibrium condition on complete rate trajectories. We establish a direct method based on vanishing-defect approximability, compactness, and complementary semicontinuity. The relaxation mechanism is computed exactly for a reduced simple-shear model with a non-convex kinetic potential and a local memory variable. The enriched state is a Young measure on complete local histories. We identify explicitly the sequential lower-semicontinuous envelope of the pure diagonal defect, characterize its zero set, construct pure recovery sequences for every relaxed zero, and prove existence through a frozen-best-response time discretization. The resulting coexistence of distinct rate populations with distinct memories provides a reduced mechanism for kinetic shear banding and connects naturally with negative strain-rate sensitivity and dynamic strain aging such as underlie the Portevin–Le Chatelier effect.

We study decoding procedures for a family of MDS array codes previously constructed from the Kronecker product of a superregular matrix and a non-singular matrix over a finite field. By exploiting the particular structure of their parity-check matrices, we develop decoding algorithms for different channel models. For the erasure channel, we provide an algorithm capable of recovering any pattern of up to $n-k$ symbol erasures. For the $q$-ary symmetric channel, we investigate the decoding of one and two symbol errors and give explicit procedures for determining their locations and values. We also consider the particular case in which the superregular matrix is a Vandermonde matrix, showing how its additional algebraic structure can be exploited in the decoding process. Explicit examples over different finite fields are provided to illustrate the proposed procedures.

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus $\mathbb T^d$, $d=2,3$, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.

Bubeck and Sellke's universal law of robustness says that fitting $n$ noisy samples below the noise level with a class of $p$ real parameters forces a Lipschitz constant of order $\sqrt{nd/p}$, when the covariate distribution on $\mathbb{R}^d$ is isoperimetric: smooth interpolation costs a factor $d$ of overparametrization. Its hypotheses are Gaussian-type concentration, a homogeneous covariate law, scalar outputs, and independent samples; we relax each one. Exponential concentration suffices, at the unchanged rate and absolute constant and at the price of a capacity burn-in, some form of which we show is necessary; this gives the law for normalized log-concave covariates, with a $\log d$ loss from the current KLS bound. For heterogeneous mixtures the worst component constant is replaced by the weighted average of the component constants, under a typicality condition on the mixture composition. For vector-valued predictors the bound at the Frobenius Lipschitz scale is free of the output dimension $K$, and the Euclidean bound loses only $K^{-1/2}$, a loss that covariate-independent noise removes. Stationary Markov covariates retain the i.i.d. rate, the pseudo-spectral gap entering only the sample-size conditions. One master theorem carries the shared argument, and constants are explicit throughout.

We record lower bounds for permanent circuits with valid division over characteristic zero, counting nonscalar multiplications and divisions while additions and scalar operations are free. Chapter 5 of OpenAI's Ten Advances supplies the block construction and critical-locus estimate; these, with the parameter choice made here and the classical bounds of Strassen and Baur-Strassen, give liminf as n tends to infinity of L_div(per_n)/(n^2 log_2 log_2 n) >= 1/12. Our finite-parameter refinement for matching-minor polynomials observes that the coefficient vector at deletion size h lies in a space of dimension at most min{binom(t,h), binom(t,d-h)}. Twelve machine-checked declarations of the Lean development accompanying OpenAI's manuscript on border determinantal complexity of the permanent (September 24, 2026) imply, by a short written argument, the existence of a geometric slice; a further written, unformalized application of the same criterion gives L_div(per_n) >= (n^2/119790) log_2(n/44) for n >= 1936. Assuming Proposition 7.3 of that manuscript gives the bound (n^2/86400) log_2(n/48) for n >= 1408. These are order-of-growth results; the elementary Hessian bound n^2/2 is larger at practical orders.

The aim of this work is to provide a rigorous mathematical analysis of a well-known SIS epidemic model with a saturating contact rate. First, we establish the global asymptotic stability (GAS) of the model's equilibria by employing a suitable Lyapunov function in combination with the Poincaré–Bendixson theorem and the Bendixson–Dulac criterion. The resulting GAS results improve upon previous findings for this SIS model and may also be applicable to its extensions with more general saturating contact rates. Second, we construct families of first- and second-order nonstandard finite difference (NSFD) schemes that preserve the positivity and asymptotic stability properties of the SIS model for arbitrary step sizes. All these schemes are formulated within Mickens' framework; however, compared with their first-order counterparts, the second-order schemes employ a more elaborate construction that combines a weighted nonlocal approximation of the right-hand side with suitably renormalized denominator functions. The weighted nonlocal approximation guarantees dynamic consistency, whereas the denominator functions ensure second-order convergence. Finally, we conduct numerical experiments to validate the theoretical results and demonstrate the advantages of the proposed NSFD schemes. The numerical results are in good agreement with the theoretical results. The approach developed in this work is applicable not only to other epidemiological systems but also, more generally, to mathematical models arising in a wide range of real-world applications.

We prove a periodic long-time Boltzmann–Grad limit for hard spheres in every fixed spatial dimension $d\ge4$. Under the assumptions of the main theorem, the rescaled $s$-particle correlations converge in $L^1$ to the corresponding tensor-product Boltzmann profile at rate $\varepsilon^{1/(400d)}$, uniformly for $1\le s\le|\log\varepsilon|$ and $0\le t\le t_{\rm fin}$. In particular, when the activity and weighted solution bounds are fixed, $t_{\rm fin}=O(\log|\log\varepsilon|)$. The componentwise long-bond estimate used in dimensions two and three is insufficient in higher dimension. We replace it by a joint estimate for two connected time sublayers, selecting the first two lower collisions as landing roots. Disjoint landing edges yield a direct tangential frame. When the edges overlap, the newly appearing particle line contains a collision-free chord whose radial relative speed produces a rank-$(d-1)$ positive Schur factor. A positive Jacobi-network argument prevents focusing, while bounded-degree finite-fibre coarea globalizes the estimate for arbitrary nonnegative joint kernels. A first-failure construction repairs periodic double overlaps, and the resulting packet bound closes the exceptional top-layer contribution. A paid epoch restart for sealed complete joint kernels, using the fixed-word multi-landing operator for every fixed integer $k\ge1$, yields the stated time range. Within the regular connected full two-sublayer packet class, every packet with at least $2k-1$ physical lines admits a canonical $k$-birth flag and the operator factor $\varepsilon^{(d-1)(k-1)}\varepsilon_*^{-k(d-1)}$; $k=2$ is the first gain-producing case. A complementary full-chord estimate provides a coarse fallback.

We study graded arrays of highly dispersive subwavelength resonators through their volume-potential matrix reduction. The reduction gives a finite-dimensional nonlinear resonance problem whose diagonal entries encode the local dispersive response and whose off-diagonal entries encode volume-mediated coupling between distinct resonators. We show that monotone material grading turns this matrix into an analytically controllable frequency sorter. When graded detuning dominates coupling in the reduced characteristic matrix, the characteristic vectors localize near individual resonators; under an explicit half-lattice-site localization condition, this localization becomes a monotone frequency-position map. We also prove a constructive inverse-design statement: prescribed monotone target frequencies can be realized by choosing the local dispersive parameters, with robustness under sufficiently small grading errors. Numerical experiments, whose parameters and algorithms are recorded in an appendix, illustrate localized resonances, designed frequency-position maps, disorder stability, spacing dependence, and a resolved physical Lippmann-Schwinger validation beyond the leading logarithmic quadrature model.

We construct simply connected integral symplectic four-manifolds forming an exotic pair, with equal values of $c_1^2$, $c_2$, $c_1\mathbin{\cdot}[\omega]$, and $[\omega]^2$, but with symplectic Kodaira dimensions $-\infty$ and $2$. Since symplectic Kodaira dimension is invariant under Luttinger surgery, the two manifolds are not Luttinger-surgery equivalent. This gives a negative answer to Auroux's uniqueness question.

Complex hypergraphs (chygraphs) contain graphs, hypergraphs, multiplex and interacting networks as special cases, and the percolation threshold of all of them follows from one symbolic calculation: the spectrum of a tensor $A$ built from four matrices of first moments. That calculation stops at the threshold. Here I show that $A$ is the Jacobian, at its trivial fixed point, of a non-linear self-consistency map whose non-trivial fixed point is the giant component fraction, so threshold and order parameter are two orders of one object on one index set. Carrying the expansion one order further gives the critical amplitude $B$ in $S=B\Lambda+O(\Lambda^2)$ in closed form, and exposes a hierarchy: first moments fix the threshold, second moments the amplitude, and only the generating functions themselves the order parameter away from it. Dropping the assumption that a complex's participation in different layers is independent generalises the tensor again, replacing unconditional first moments by inclusion-biased second moments: distributions with identical marginals have different thresholds and different order parameters. Six constructions from the literature are then solved by substitution, among them AND- against OR-logic hypergraph percolation, which share a chygraph and differ in one generating function, and SIR epidemics with two levels of mixing, whose household reproduction number falls out of the tensor. A reported failure of mean-field theory on strongly clustered graphs is shown to be a failure of applying it to the wrong object: mapped onto the treelike backbone rather than the clustered graph, the same calculation reproduces the exact threshold, order parameter and critical exponents. Everything is implemented in one class, in which a chygraph is specified once by its generating functions and then returns all three, and validated against Monte Carlo simulation.

In this paper, we prove the weak and linear convergence of Tseng algorithm applied to the zero of sum of two operators problem where each operator is not necessarily monotone while existing researches in the literature require the monotonicity of both operators. Consequently it allows us to decompose more efficiently and can reduce the Lipschitz constant significantly, which is meaningful especially in big data. Numerical examples supporting the theoretical results are also provided.

In this paper, we mainly provide some methods for constructing antisymmetric infinitesimal (ASI) bialgebras and Lie bialgebras using Zinbiel bialgebras and diassociative bialgebras. We first show that there is an ASI bialgebra structure on the tensor product of a diassociative bialgebra and a quadratic Zinbiel algebra, while there is an infinite-dimensional ASI bialgebra structure on the tensor product of a Zinbiel bialgebra and a quadratic $\bz$-graded diassociative algebra. For a special quadratic $\bz$-graded Leibniz algebra, the property that its tensor product with a Zinbiel bialgebra forms an ASI bialgebra characterizes the Zinbiel bialgebra. By examining the relationship between solutions of the (classical) Yang-Baxter equation in a Zinbiel algebra and the induced associative algebra, we prove that the induced ASI bialgebra is quasi-triangular (resp. triangular, factorizable) whenever the original Zinbiel bialgebra is quasi-triangular (resp. triangular, factorizable). These conclusions enable us to provide a method for constructing Lie bialgebras from Zinbiel bialgebras and two approaches for constructing Lie bialgebras from diassociative bialgebras. We also provide specific descriptions of the connections between the solutions of the Yang-Baxter equations and the connections between the $O$-operators corresponding to these constructions.

Entropy over chain-of-thought tokens decides which tokens receive the policy gradient, which get pruned, and whether a run has collapsed, yet each such statistic reads a next-token distribution mixing three choices: whether to emit connective scaffolding, which connective, and what the substantive continuation should be. Designating a scaffold vocabulary subset separates the three, exactly, for entropy, Kullback–Leibler divergence, and the first-order entropy velocity of a softmax policy. We prove the raw and content conventions disagree about which position is the larger fork on an explicit open region, and bound answer diversity by the content channel plus a leakage term a measured witness certifies. Across twenty-three configurations the scaffold side carries up to 41% of the raw high-entropy set; on a matched-tokenizer ladder, coupling changes only at the math-corpus step while the scaffold's entropy share keeps growing through distillation; a closed-form forecast from one channel correlation tracks selection retention over a 54-point range to five points, unfitted. On compression, the content convention beats raw surprisal in every cell; an answer-leakage audit then corrects our own headline control: re-fed chains earn a quarter to a half of their accuracy from restated answers, and once stripped, no token scorer beats a random contiguous block.

The two-dimensional Cutting Stock Problem (2D-CSP) is an NP-hard problem with direct economic and environmental impacts on manufacturing and logistics. We encode its fixed-plate variant, with free piece repetition and full non-overlap and containment constraints, as a Quadratic Unconstrained Binary Optimization (QUBO) problem for quantum annealing and compare two formulations from opposite encoding paradigms. The first was a coordinate-based set-packing model with one binary variable per candidate placement. Its variable count grows linearly with plate area and resolution, but its ground state is, by construction, a geometrically feasible maximum-area packing. The second is a coordinate-free sequence-pair model whose variable count is independent of plate resolution and size. We prove that this compactness has a structural limit: no coordinate-free QUBO of bounded interaction degree whose penalties vanish on every geometrically feasible layout can have a geometrically feasible ground state for 2D containment, because containment is a longest-path constraint that bounded-degree penalties cannot enforce on chains longer than their interaction order. We evaluate both formulations under multi-seed simulated annealing, simulated quantum annealing, and an exact integer-programming baseline. Hardware experiments include D-Wave minor embedding, a calibrated direct-QPU sweep, and Leap hybrid solvers in both penalty and constraint-native form, across a six-instance campaign with per-instance calibration. The hybrid solver returns our certificate configuration, tying its energy to thirteen decimal places while overflowing the plate. We claim no quantum speedup. Our contribution is an impossibility result characterizing the limits of compact packing QUBOs, and a practical rule for choosing between the two formulations.

A Gray code for $S_n$ is genlex when words sharing a suffix are consecutive. We determine the genlex Gray codes for $S_n$ that use the fewest possible operations: Zaks' recursion generalises to a superfactorial family of such codes, and no code outside this family attains the minimum. Every code in the family closes into a cycle, and the cycle is invariant under a group of left translations, cyclic of order $n$ or dihedral of order $2n$ according to an explicit criterion on the operations. In the pancake graph the family reduces to a single member, the classical Zaks order; with respect to the standard parabolic chain of $W(A_{n-1})$, that order attains one extreme of total Coxeter length.

In this paper, we study a slow-fast model of Intracellular Calcium Concentrations (ICC) within a stochastic framework, where stochastic perturbations are introduced to capture the intrinsic randomness of such biological systems. We first analyse the three-dimensional single-cell model, based on the FitzHugh-Nagumo structure, and subsequently extend the analysis to a coupled two-cell six-dimensional setting. We prove existence and uniqueness of solutions and establish positivity of the calcium variable. We obtain explicit mean-square stability estimates around deterministic equilibria and quantify how they depend on the noise intensity. Numerical simulations for both the single-cell and coupled systems illustrate relevant qualitative behaviours and reveal additional dynamical features arising in the stochastic framework.

Hydrodynamic ram (HRAM) loading remains a persistent challenge in impact mechanics, less due to exotic physics than to the difficulty of achieving convincing quantitative agreement in reduced-order representations. Prevailing studies typically employ a single mesh and boundary treatment validated against one experimental dataset, leaving unresolved whether agreement reflects genuine fidelity or compensating discretization errors. This work presents a planar 2D, GPU-accelerated WCSPH solver employing an Adami-type ghost-particle boundary condition, extended for the first time to a moving, decelerating disk penetrating a confined liquid channel at ballistic velocity. The boundary condition is examined against a purpose-derived potential-flow added-mass solution and the analytical acoustic reflection coefficient at the steel-water interface; this exploration reveals that the stiff equation of state and cavitation cutoff impose identifiable delineation of the solver's operating regime on quantitative agreement. A three-point resolution convergence study reveals a non-monotone near-probe pressure peak, attributed to a bounded ghost-fluid density inconsistency. A comprehensive coefficient sweep, conservation diagnostics, and O(N) GPU throughput scaling enable exhaustive characterization impractical in 3D. Qualitative comparison against literature evidence at 900 and 600m/s clarifies load bearing versus approximate elements. Notably, energy diagnostics reveal a modest total energy growth over the simulated window, with the fluid absorbing disproportionately more energy than the projectile relinquishes - an instructive signature of prescribed, non-back-reacting projectile kinematics that helps map the formulation's operating envelope. The resulting solver, while confined to planar geometry, delivers verified, convergence-checked performance well suited for rapid parametric design exploration.

It is known that $q$-Painlevé equations admit the symmetry of the extended affine Weyl groups. Kajiwara, Noumi and Yamada developed a unified description of the symmetry by introducing representations of the extended affine Weyl groups. On the other hand, a parallel translation associated with the extended affine Weyl groups describes the time evolution in Sakai's theory of discrete Painlevé equations. In this paper, we introduce the equivalences to the representations of Kajiwara, Noumi and Yamada. The equivalences imply reducibility of the representations, and they are related to the expressions of the parallel translations of the $q$-Painlevé equations.

Which distinctions in a present state must be retained to predict a specified future exactly? We study this question on a finite reachable set of a deterministic five-agent ABCW model in which binary agent actions and strategies coevolve with a weighted directed influence field. The dataset is generated from four initial-field families and all 4,096 action-strategy initial conditions, yielding 56,536 transitions and 2,562 distinct current fields. We retain the current action and seek the coarsest field-only partition that uniquely determines the complete one-step-ahead field on every observed transition. Nine natural feature families and all 511 of their nonempty combinations provide strong approximate predictors but no exact nontrivial field compression. We therefore construct an incompatibility graph whose vertices are current fields and whose edges join fields that produce different next fields under a shared observed action. A proper 692-coloring gives a constructive upper bound. An independent anchor-action decomposition, followed by exact solution of the 50 nontrivial induced subgraphs, gives the matching lower bound. Hence the minimum number of field classes is exactly 692. This number applies only to the specified finite reachable set, field-only compression, retained current actions, one-step target, and exact-prediction requirement; it is not a universal macrostate count for ABCW systems.

We determine by exhaustive enumeration all main-diagonal symmetric Costas arrays of orders 37 through 42. There are 4 arrays of order 37, none of order 38, 16 of order 39, 2 of order 40, 12 of order 41, and 4 of order 42. All belong to known finite-field constructions. The arrays at orders 37-40 are Lempel arrays over $\mathbb{F}_{41}$ or their corner deletions and augmentations; the twelve arrays at order 41 are the Lempel arrays over $\mathbb{F}_{43}$; and the order-42 census consists of two corner augmentations and a reverse-complement pair in the Rickard-Golomb family. Combined with the published census through order 36, these results show that no sporadic main-diagonal symmetric Costas array occurs at orders 24-42. We also develop a collision model for random involutions. The first two terms of its collision exponent are derived analytically, $n^2/18+n^{3/2}/360$, with an $O(n)$ remainder. The lower-order model and clumping correction are empirical. The six new censuses are consistent with the model's prediction of a rapidly declining expected sporadic population.