Papers

In this article, we study the Bridgeland stability of the cotangent bundle on K3 surfaces and surfaces with ample canonical class. On a surface $X$ with ample canonical class $K$, we find values $t_0>0$ such that the cotangent bundle is semistable with respect to the Bridgeland stability $\sigma_{tK}$ for all $t > t_0$. We also study special cases such as the fake projective planes and the fake quadrics.

We consider a variation of the Cohomological Rigidity Problem: whether two quasitoric manifolds whose cohomology rings are isomorphic are homotopy equivalent. We show that if two quasitoric manifolds have isomorphic cohomology rings and the associated polytope of one of them is a product of simplices, then they are homotopy equivalent after localizing away from an explicit list of primes.

We study iterated cotangent integrals with cyclotomic shifts and express their values along specified reference paths in terms of cyclotomic multiple zeta values. Differences of cotangent forms extend to a punctured projective line over a totally real cyclotomic field, giving a family of "real cyclotomic multiple zeta values." With rational tangential endpoints, these difference integrals are periods of mixed Tate motives over that field; integral endpoint data yield periods over its ring of integers after inverting primes dividing the level. The associated framed motivic classes generate a Hopf subalgebra under Goncharov's coproduct.

We determine the leading logarithmic asymptotics of the fixed-maturity right tail of the SABR model with $\beta\in(0,1)$ and absorption at zero, for every correlation $\rho\in(-1,1)$. If $P(k)$ is the probability that the terminal forward is at least $f_0e^k$, then $-k^{-2}\ln P(k)\to(1-\beta)^2/(2\nu^2T(1-(\rho\wedge0)^2))$ as $k\to\infty$. For $\rho\ge0$ this is the rate predicted by the unrestricted hyperbolic geometry. For $\rho<0$ it is strictly larger: along the volatility excursion that produces the tail, survival forces the noise orthogonal to volatility to stay above a moving square-root barrier, at an additional cost of the same order $k^2$. Consequently the Black-Scholes implied volatility tends to $\nu\sqrt{1-(\rho\wedge0)^2}/(1-\beta)$: Henry-Labordère's conjectured wing limit $\nu/(1-\beta)$ holds for $\rho\ge0$ and fails for every $\rho<0$. For his second conjecture, whose unrestricted-distance Gaussian estimate does not hold for $\rho<0$, we also identify the correct replacement for this model: at the level of tail probabilities, the rate is given by the squared intrinsic Riemannian distance, within the survival domain, from the initial state to the terminal target set. The proofs are direct, combining a stopped Lamperti transform, a conditional time change, Gaussian moving-barrier bounds and change-of-measure constructions, and use no heat-kernel estimates.

We study the slow motion of phase transition layers for a time-fractional Allen–Cahn equation with a Caputo time derivative of order $\alpha\in(0,1)$. We show that the fractional memory preserves the distinction between exponentially and algebraically slow dynamics associated with non-degenerate and degenerate potentials, respectively. In the non-degenerate case, layered solutions persist on an exponentially long time scale of order $\exp(A/(\alpha\varepsilon))$, $A>0$, while in the degenerate case the corresponding algebraic time scale is modified by the factor $1/\alpha$. Finally, we extend the analysis to nonlinear diffusion operators, considering both the $p$-Laplacian and a Perona–Malik type diffusion, showing that the same fractional slow-motion mechanism persists in these more general settings.

We study the largest size $C(n)$ of a subset of the $n\times n$ integer grid containing no isosceles triple, including equally spaced collinear triples. We prove $C(n)=\Omega\left(n\sqrt{\log\log n/\log n}\right)$ by applying the sparse hypergraph coloring theorem of Cooper and Mubayi. Elementary geometry and primitive-direction counts give maximum degree $O(n^2\log n)$ and pair-codegree at most $5n$ for the forbidden-triple hypergraph. The resulting bound improves the explicit guarantee in PatternBoost by a factor of order $\sqrt{\log\log n}$. Recovered interaction records document how a Codex literature proposal, human route selection, finite verification, and author-relayed review informed proof synthesis and repair. We connect these actions to successive proof artifacts and extract two case-derived checking practices: tracking objects, parameters and conclusions in theorem applications, and seeding a definitional omission to test a finite verifier. Separately implemented enumerators agree on complete edge sets for every $2\le n\le12$; suppressing the degenerate-case branch loses exactly the equally spaced collinear triples. These records document assistance under human direction, and finite checks support implementation consistency rather than the asymptotic theorem itself.

We establish statistical consistency for the penalized maximum-likelihood estimator underlying CPLASS, a method for detecting changes in velocity in $d$-dimensional time-series data. The signal is modeled as a continuous piecewise-linear trajectory observed with independent Gaussian noise. Unlike classical change-in-mean models, continuity across adjacent segments couples their parameters and prevents direct application of standard segmentation arguments. Under compact parameter spaces, minimum segment-length and velocity-jump conditions, and a strengthened Schwarz information criterion penalty $\rho_k(\log n)^\gamma$ with $\gamma>1$, we prove joint consistency of the estimated number of segments and all changepoint locations in the high-frequency regime $n\to\infty$. The maximum changepoint-location error is $O_{\mathbb{P}}\{(\log n/n)^{1/2}\}$. The proof first uses empirical-process theory to establish convergence of the fitted signal, variance, and likelihood for each fixed model size. It then combines overfitting and underfitting arguments with a two-stage geometric localization analysis, yielding an initial $(\log n/n)^{1/3}$ rate that is sharpened by exploiting the local two-segment continuous piecewise-linear structure.

Let $Q$ be a bipartite Dynkin quiver of finite simply-laced type and $\mathscr C_Q$ the corresponding Hernandez–Leclerc category. We study a two-parameter specialization of the Fedele–Hernandez toroidal Grothendieck ring and identify its generic positive part with a Cartan-type multiparameter quantum group. The structural input is the identity \[ 2D^{(2)}-D_0=\omega_Q\circ(\deg_Q,\deg_Q), \] where $D_0,D^{(2)}$ are the two effective toroidal commutation forms and $\omega_Q$ is the antisymmetric Euler form of \(Q\). It realizes the specialized character algebra, after an auxiliary scalar extension, as a graded bicharacter twist of the one-parameter algebra at $v=t_0t^{1/2}$. This yields the required root grading and PBW graded dimensions. For every source–sink edge $r\to s$, these dimensions combine with the local \(A\)–\(Y\) commutation formula to give a quadratic annihilator for \(\operatorname{Ad}_{Z_r}\). Consequently \[ \operatorname{Spec}_{Z_r}(Z_s)=\{t_0t,t_0^{-1}\}, \] without thinness or type-specific character formulas. The two spectral values yield the oriented multiparameter Serre relations. With \[ q_{ii}=t_0^2t,\qquad q_{ij}=t_0^{-1},\qquad q_{ji}=t_0^{-1}t^{-1}\quad(i\to j), \] and \(q_{ij}=1\) on nonedges, we obtain \[ U_{\mathbf q}^{+}(\mathfrak g) \simeq \mathscr K_\infty(\mathscr C_Q)\otimes_RK \] for $\mathfrak g$ of type $A_n,D_n,E_6,E_7,E_8$.

Multi-pass underfill dispensing couples cycle time with process reliability because the sequence determines the inter-pass flow times. This paper formulates a tri-objective multiset-permutation routing model that minimises nominal cycle time and additive duration variance while maximising reliability under hard- and soft-dwell policies; reliability combines pass, schedule-dependent flow and travel terms. Under independent durations and deterministic dwell thresholds, the variance measure equals the cycle-time variance under soft dwell and bounds it from above under hard dwell. A multiset-permutation binary-addition-tree algorithm enumerates the feasible routes without duplication and prunes prefixes by dominance. Six instances, including three BeagleBone Black benchmarks, yield reference Pareto fronts, the largest search space of $1.47 \times 10^{12}$ routes in 38 minutes on 20 CPU cores. Among simulated candidates, the nominally cost-selected routes incur maximum estimated excess costs of USD 0.0014 per board at the modelled duration variability and USD 0.0035 with it doubled. Round-robin dispatch is within 3.7 % of the reference optimum at the tested costs; NSGA-II and simplified swarm optimisation show at most 0.53 % observed cost regret while recovering about 30 % of the largest hard-dwell front. The framework quantifies sequencing trade-offs and heuristic decision regret; physical reliability prediction requires inspection-based calibration.

We study diffeomorphisms of Banach spaces satisfying generalized hyperbolicity: the stable and unstable subspaces are invariant only up to inclusion, need not depend continuously on the point, and expansivity may fail. On closed manifolds, this condition is equivalent to Axiom~A with strong transversality. We introduce a continuous-selection method that converts uniform orbitwise estimates into bounded continuous solutions of cohomological equations for generalized-hyperbolic cocycles. For diffeomorphisms in the uniform global \(C^1\) category, this yields continuous semiconjugacies in both directions under sufficiently small bounded Lipschitz perturbations, and topological stability in the sense of Walters. For the infinite product of one-dimensional Morse-Smale systems on \(\ell^p(\mathbb Z)\), \(1\leq p\leq\infty\), we construct compatible forward and reverse cohomological solution operators and prove structural stability, including for perturbations that couple coordinates, although the product is not expansive. For \(p=\infty\) this gives a structurally stable diffeomorphism with uncountably many hyperbolic fixed points, in contrast with the finitely many basic sets of Axiom~A diffeomorphisms. The cocycle analysis yields Lipschitz shadowing for the induced operator on bounded continuous vector fields.

Euler's number $e$ is often introduced through compound interest, an infinite series, an area, or the slope of an exponential function. The last viewpoint is especially attractive: choose the base $b\in (2,3)$ so that the graph of $y=b^x$ has tangent slope $1$ at $(0,1)$. However, a curious student may raise several questions that a brief introduction necessarily leaves implicit. What does $b^x$ mean when $x$ is irrational? Why is $b^x$ differentiable at $x = 0$? How can one show that the slopes corresponding to bases $2$ and $3$ lie on opposite sides of slope $1$? Finally, why is there a unique base whose slope is exactly $1$? The paper provides an answer using tools from a first course in real analysis, including limits, continuity, derivatives, and the Intermediate Value Theorem. Its primary audience is undergraduate students familiar with standard concepts from an introductory analysis course who wish to apply what they have learned to explore questions left open in earlier courses. A secondary audience is instructors who may use selected portions as supplementary material in an introduction-to-proofs course. The mathematical ingredients are classical; our aim is to organize them into a self-contained exposition and to make explicit the logical steps that are often compressed in standard treatments.

Using a dioid algebraic structure, we show that there exist spherically symmetric trees $T(2^m 1^n)$ whose independence polynomials exhibit multiple breaks in log-concavity, a result established by estimating the asymptotic growth of the coefficients of these polynomials. Provided the parameter $n$ is a sufficiently large odd integer, the number of breaks is bounded below by the Jacobsthal numbers. This result affirmatively answers a question raised by F. Bencs [D. Galvin, Trees with non log-concave independent set sequences, arXiv 2502.10654.v2].

We consider the Wigner–Smith time-delay matrix of a chaotic cavity with broken time-reversal symmetry, a finite number $M$ of equivalent open channels, and a uniform tunnel barrier of reflection probability $R$. Exact random-matrix results for nonideal coupling are known at the level of the joint matrix distribution, whereas explicit finite-channel formulas for Schur moments are much less developed. We show that every Schur moment of the nonideal Wigner–Smith matrix factorizes into its ideal-coupling inverse-Laguerre value and a barrier factor given by a circular unitary ensemble (CUE) average of a Schur polynomial. A dual-Cauchy generating function reduces the latter to a Toeplitz determinant with a rational symbol, placing the barrier problem within the classical theory of exact CUE characteristic-polynomial ratios. For one-row partitions, specializing the exact CUE ratio formula gives a closed generating function for all barrier coefficients and therefore all finite delay-time moments in the allowed range $n\le M$. The exact ratio formula decomposes into a sector visible to the semiclassical expansion and an additional contribution proportional to $R^{M+1}$, which is invisible to every algebraic order in $1/M$. For the second moment, this contribution exactly reproduces the exponentially small term in the known random-matrix expression for $\langle\tau_W^2\rangle$ that is absent from the semiclassical expansion. We also derive an exact finite sum for all one-column partitions and an exact complement symmetry for Schur averages inside the $M\times M$ square. The new contribution therefore lies not in the underlying CUE ratio identities, which are classical, but in combining them with the exact nonideal Wigner–Smith mapping and extracting the resulting finite-channel time-delay formulas and corrections to semiclassical Schur-moment conjectures.

We prove that every biharmonic hypersurface in a space form of nonpositive sectional curvature is minimal in arbitrary dimension. This settles the hypersurface case of the generalized Chen's conjecture in hyperbolic space and gives a unified treatment of nonpositive space forms. For biharmonic hypersurfaces in the unit sphere, we derive quantitative restrictions on every possible nonconstant-mean-curvature solution. In particular, we obtain pointwise criteria forcing constant mean curvature, a strict scalar-curvature bound in dimensions at least five, and local classification results above the classical CMC gap threshold. These results provide further evidence for the BMO conjecture.

Let $G$ be a connected compact Lie group and $R$ a commutative ring over which $H_*(G;R)$ is an exterior algebra on odd primitive generators. We prove a rigidity theorem: the Batalin-Vilkovisky (BV) algebra $\mathbb{H}_*(LG;R)$ of Chas and Sullivan on the homology of the free loop space is that of the product of odd spheres with the classical exponents of $G$, tensored with the group ring of the torsion of $\pi_1(G)$. Conversely, over an integral domain the graded algebra $\mathbb{H}_*(LG;R)$ already determines the exponents and the group ring, hence the BV algebra. If moreover $G$ is simply connected, then $\mathbb{H}_*(LG;R)$, as a BV algebra, is the Hochschild cohomology of $H^*(G;R)$ with the BV operator induced by Poincaré duality. The isomorphism with the product of spheres is algebraic: for $\mathrm{Sp}(n)$, $n\geq2$, and $\mathrm{SU}(n)$, $n\geq3$, no map of spaces induces it over $\mathbb{Z}$. We compute, in Hepworth's presentation, the integral BV algebra $\mathbb{H}_*(L\mathrm{Sp}(n);\mathbb{Z})$ with its Gerstenhaber bracket, open for $n\geq2$, the running example of the rigidity theorem.

We discuss a new symmetric-key cipher for digital data encryption. Its implementation is fast, memory efficient, and resilient against classical, AI-assisted, and quantum attacks. Let $x$, $k$, and $c$ be elements of a finite abelian group $G$ with operation $+$ and neutral element $0$. Suppose one is given ciphertext $c=x+k$. Retrieving the plaintext $x=c-k$ from the ciphertext is trivial when the key $k$ is known. Without the key, however, recovering $x$ amounts to a blind search. Even if an efficient criterion were available for identifying the plaintext among all group elements, the average number of trials would remain prohibitively large when the group is sufficiently large. The challenge in constructing a practical cryptosystem is therefore to identify a sufficiently rich group $G$, together with efficient algorithms that enable an immersion of real digital data into $G$ and efficient evaluation of the group operations. The proposed cipher is based on generalized Dirichlet convolution over finite fields. We investigate its security against plaintext attacks from an information-theoretic perspective. The nonlocal structure of Dirichlet convolution naturally permits plaintext salting, providing protection against plaintext attacks while allowing controlled key reuse. We establish perfect masking properties over finite fields and prove that the resulting cipher achieves information-theoretic security under key reuse when the underlying arithmetic is carried out in $\mathbb{F}_p$ with $p>2$. For arithmetic over $\mathbb{F}_2$, some information leakage remains, leading to a finite key-reuse budget.

Let $P\subset\mathbb R^d$ be finite with $\operatorname{diam}(P)\le1$. For $0<\varepsilon<1$, let $\mathcal N_\varepsilon(P)$ denote the number of ordered pairs at distance at most $\varepsilon$, including the diagonal, and let $\mathcal A_\varepsilon(P)$ denote the number of ordered pairs at distance at least $1-\varepsilon$. We prove the sharp bound \[ \mathcal N_\varepsilon(P)\gtrsim_d \varepsilon^{(d-1)/2}\mathcal A_\varepsilon(P) \] conjectured by Steinerberger (2025). More generally, the same inequality holds for finite positive Borel measures. The main input is a uniform $L^2$ estimate, at cap thickness comparable to lattice spacing, for a supporting-direction incidence operator. In gnomonic coordinates, this operator is controlled by an incidence estimate for small Fenchel–Young gaps.

Suppose $X=(X_t)_{t\ge0}$ is a supercritical superprocess with a non-local branching mechanism on a Lusin space $E$, whose mean semigroup admits a positive leading eigenvalue $\lambda_1$ with a bounded positive eigenfunction $\varphi$ and a finite eigenmeasure $\widetilde{\varphi}$. Let $Y_t:=\int_0^t X_s\mathrm{d}s$ be its occupation measure. We first prove a law of large numbers, that is, for every bounded measurable function $f$, $e^{-\lambda_1 t}\langle f,Y_t\rangle$ converges in $L^2$ to $\lambda_1^{-1}\langle f,\widetilde{\varphi}\rangle W^\varphi_\infty$, where $W^\varphi_\infty$ is the limit of the martingale $W^\varphi_t:=e^{-\lambda_1 t}\langle\varphi,X_t\rangle$. We then establish functional central limit theorems for the fluctuations of $\langle f,Y_t\rangle$ around this random centering. We show that the normalisation and the limit are determined by the spectral decay rate of $f$ compared with the critical value $\lambda_1/2$, and there are three possible regimes: after suitable normalisation, the fluctuation process converges in distribution in $\mathcal{C}[0,+\infty)$ either to a mixed Gaussian process $\sqrt{W^{\varphi}_{\infty}}G(f)$, or to a constant Gaussian process with random variance depending on $W^{\varphi}_{\infty}$, or to a non-Gaussian limit given by the limit of a martingale associated with the spectral projection of $f$. These results are the occupation-time counterparts of the fluctuation theorems for instantaneous linear functionals of supercritical non-local branching superprocesses.

We show that any bounded strictly convex domain in $\Bbb C^N$, endowed with the Kobayashi distance, is visible.

In this article we examine a problem stated in Grübaum and Shephard's classic book on tiling theory, Tilings and Patterns, which asks the reader to find all unilateral and equitransitive tilings by squares of sizes $1, 2, 3, \ldots, n$ where $n \in \mathbb{N}$. The unilaterality condition requires that two squares of the same size in such a tiling may not intersect along an entire side, and the equitransitivity requires that two tiles of the same size in such a tiling must be in the same transitivity class with respect to the symmetry group of the tiling. These two conditions on tilings by squares allows for surprisingly interesting and complex tilings by squares. In this work we provide the answer to Grübaum and Shephard's question for the case $n = 4$, as well as developing methods and computer algorithms that may be extended to the cases where $n \geq 5$.

We study stochastic flows with interaction through an infinite-dimensional semigroup approach. By means of a kernel lifting procedure, we represent the solution to measure-valued stochastic equation as a Hilbert-valued Markov process. We identify its generator, construct the associated Markov semigroup, and establish a dense range condition using measure polynomial functionals. The results provide a framework for the analysis of stability of stochastic equations with interaction.

We explore the dynamics of a Vlasov gas propagating in an external axisymmetric potential consisting of a central potential with an additional quadrupolar component which gives rise to two potential wells along the symmetry axis. Employing independently $N$-particle simulations and statistical methods, we show that an initially homogeneous and isotropic configuration evolves to an inhomogeneous final state with overdensities located at the wells. The quadrupolar component is chosen such that the equations of motion form an integrable Hamiltonian system, which allows one to compute the final state of the gas analytically. On the one hand, this allows one to compare the late-time behaviour of the simulations with an analytic prediction and on the other hand to understand the relaxation process through phase space mixing. Our model constitutes a Newtonian analogue of the particle motion in a Kerr spacetime, and we discuss possible applications to recently observed astrophysical phenomena.

Let $p$ be an odd prime and $H$ a finite $p$-group. Hertweck and Soriano parametrize the central Frattini extensions of $H$ by subspaces of a relation module $V$, matching isomorphism classes of extensions with orbits of $\mathrm{Out}(H)$ and isomorphism classes of the associated small group rings with orbits of $\mathrm{Out}(\mathbb{F}_p H)$. The discrepancy between the two is measured by the group $U(H)\leq GL(V)$ of displacements induced by the normalized unipotent automorphisms of $\mathbb{F}_p H$; it is the mechanism behind the counterexamples to the modular isomorphism problem in characteristic $2$. We prove that for $H$ of exponent $p$ and class at most $p-2$ every such automorphism acts on $V$ exactly as an automorphism of $H$ does, and that $U(H)=\exp\Omega\bigl(\mathrm{Der}_{>0}(\mathfrak{l}(H))\bigr)$, where $\mathfrak{l}(H)$ is the Lazard Lie algebra of $H$, $\mathrm{Der}_{>0}$ its derivations raising the lower central filtration, and $\Omega$ the natural action of derivations on $H_2(\mathfrak{l}(H))$. Hence $U(H)=1$ precisely when every derivation of positive degree acts trivially on $H_2(\mathfrak{l}(H))$. We compute $U(H)$ for all groups of class $2$ and exponent $p$, obtaining $p^{2n(n-1)}$ for the Heisenberg groups $\mathrm{Heis}(p^n)$.

Topological complexity, introduced by Farber, measures the number of continuous local motion-planning rules on a configuration space. For every simply connected space $X$ of finite rational homotopy type whose rational cohomology is a finite-dimensional Poincaré duality algebra, we prove $\mathrm{MTC}(X)=\mathrm{TC}_0(X)=\mathrm{TC}_0^M(X)$, where $\mathrm{TC}_0(X)=\mathrm{TC}(X_{\mathbb Q})$ and $M$ denotes the monoidal invariant. No formality or ellipticity assumption is required. The result extends to higher topological complexity and gives product additivity under duality. For every map $f$ between simply connected spaces of finite rational homotopy type whose rationalization admits a homotopy retraction, we establish the sharp bound $\mathrm{Msecat}(f_{\mathbb Q})\leq\mathrm{secat}(f_{\mathbb Q})\leq\mathrm{relcat}(f_{\mathbb Q})\leq\mathrm{Msecat}(f_{\mathbb Q})+1$. In particular, $\mathrm{MTC}(X)\leq\mathrm{TC}_0(X)\leq\mathrm{MTC}(X)+1$ holds without a duality assumption. The proofs use controlled Sullivan resolutions and an ideal-valued criterion for relative category. For sections of fibrations with Poincaré duality fibre, we prove that a derived self-intersection formula, vanishing of Thom multiplication on the augmentation ideal, and ideal-valued evaluation are equivalent. This criterion yields equality of the three rational sectional invariants for smooth sections of smooth bundles over simply connected smooth bases of finite rational homotopy type, with closed simply connected manifold fibre, including rationally hyperbolic fibres. Equality also holds when the fibre has finite-dimensional total rational homotopy, without a cohomological dimension bound on the base, and for suitable finite relative Frobenius models. Geometric applications include sphere, Grassmann, and projective bundles.

In this paper we prove the weak Hurwitz stability conjecture for the strong Turán expressions of Eulerian polynomials, as well as the corresponding conjecture for Narayana polynomials, which were conjectured by Liu and Yan. We generalize the notion of Turán expressions of polynomial sequences, and provide a criterion for the generalized stability of generalized Turán expressions. Based on our sufficient condition, we not only confirm Liu and Yan's conjectures but also obtain the generalized stability of generalized Turán expressions of a class of recurrent polynomial sequences and generalized Narayana polynomials.

We study the vanishing discount selection problem for critical Hamil\-ton–Jacobi equations posed on ${\mathbb R}^d$ with Hamiltonian $G=H-V$, where $H$ is continuous, periodic in space, convex and superlinear in the momentum, and $V$ is a compactly supported potential of constant sign. For $V\leqslant0$, we prove local uniform convergence of the discounted solutions to a distinguished critical solution, which we characterize in terms of Mather measures and of the solution selected by the unperturbed periodic problem. The proof relies on a decomposition of limiting discounted measures into a retained component and a periodic component accounting for the mass lost at infinity. For $V\geqslant0$, we establish convergence under the additional assumption that the unperturbed selected solution has zero average with respect to every periodic Mather measure. We also give explicit examples in which the discounted solutions converge, but natural formulas for the candidate limit fail to identify the selected solution.

We represent constructive integrals in spatially cut-off $\phi^4_2$ by sequences of exponential-motivic periods. For a nonzero spatial cutoff, a single Galois transformation makes the partition periods diverge in modulus along every sequence of approximations satisfying the constructive estimates. For a fixed cutoff presentation, we characterize the subgroup of transformations admitting a bounded realization on the free field's $L^1$ space and prove that all fixed-observable limits on this subgroup are a common scalar multiple of their physical values. A one-variable quartic example shows that all transformed fixed-observable sequences can converge without such a bounded realization, with limits that are not proportional to the physical values.

Modern training on a new task often starts from a previously trained model rather than from scratch, raising the question of how this initialization affects the subsequent training trajectory. Classical implicit-bias results characterize the direction selected by prolonged training, but this direction alone does not provide information regarding the intermediate behavior. We address this question through a geometric approximation of full-batch logistic gradient descent (GD) trajectories on strictly linearly separable data, with large initialization of scale $R$ motivated by prior training. From any limiting normalized initial position, we use minimum-norm projection rules to construct a unique continuous ideal path consisting of finitely many linear segments. The path has two stages: negative-margin correction followed by minimum-margin growth. We prove that, after an explicit two-stage time reparameterization, the fixed-step GD trajectory divided by $R$ converges uniformly to this path on every fixed parameter interval as $R\to\infty$. Further, our quantitative error bounds account for initialization perturbations and the transition between stages. This approximation provides asymptotic formulas for peak evaluation loss and cumulative training loss. In particular, peak evaluation loss can grow linearly in $R$ even when both endpoint losses tend to zero. The cumulative losses in the correction and margin-growth stages, normalized by $R^2$ and $R$, respectively, converge to explicit limits. Experiments on controlled geometries and fixed image features complement our theoretical results.

Mixture-of-experts (MoE) language models send each token to a few experts, so each expert is trained on a different part of the data, and this part changes during training. With a shared learning rate, Muon applies updates of roughly the same size to expert matrices of the same shape, even when an expert's update is poorly aligned with its current gradient. We here propose ExpertMuon-Compass (Compass), which multiplies the Muon step of each expert by two factors. A family factor compares the cosine between the expert's orthogonalized update and its gradient with the same cosine for the other experts in its layer. A scalar radius aggregates the alignment between corresponding rows of the update and gradient into one step-length multiplier. Compass keeps the update direction and the momentum buffer of Muon. In pretraining on FineWeb-Edu, Compass with Nesterov momentum on all matrices performs as well as or better than Muon, NorMuon, and other optimizers, with weight decay matched to NorMuon in the longer runs. Adding its factors to NorMuon gives the same or a lower loss than NorMuon. Compass is the most effective when the data seen by each expert varies during training, for example, as when the languages of a multilingual corpus arrive in separate blocks. With Compass, the expert load stays balanced, and the router assigns tokens to experts more decisively. We also prove a perturbation bound for the two factors.

We study central configurations for $n$ bodies interacting through pair potentials \[ U_\sigma = \frac{\kappa}{\sigma} \sum_{i<j}m_i m_j r_{ij}^{\sigma}, \] with logarithmic limit at $\sigma=0$. For the Newtonian exponent $\sigma=-1$, the classical Chazy–Wintner–Smale finiteness problem asks whether, for prescribed positive masses, there are only finitely many central configurations up to similarity. Finiteness is known in several important special cases and under additional dimensional or genericity hypotheses. It is therefore natural to ask whether an analogous finiteness principle continues to hold for other homogeneous exponents. We show that for every positive even exponent \[ \sigma=2k, \qquad k\ge2, \] the answer is emphatically negative. In fact, these exponents admit an abundance of continuous families of pairwise non-similar central configurations. The mechanism is provided by spherical designs: on a common sphere the force kernel for $\sigma=2k$ restricts to a polynomial of degree at most $k$, and consequently every positive weighted spherical $k$-design is a central configuration. We compute the corresponding central multiplier explicitly.