Papers

Multi-task learning (MTL) requires navigating unavoidable trade-offs among competing objectives. This paradigm is frequently formulated as multi-objective optimization (MOO), where the scalarization is favored to reduce an MOO problem to a single objective. We empirically find that existing merit-function-based scalarization approaches are sensitive to the relative scales of different objectives in practical MTL, where task losses commonly differ by orders of magnitude. The optimization process often favors objectives with larger scales even though the underlying Pareto optimal solutions remains invariant to rescaling (i.e., multiplying an objective by a positive constant). To address this issue, we propose Scale-Invariant Merit-function-based Scalarization (SIMS) for MTL. Specifically, SIMS adopts a transformation-induced merit function to convert the MOO problem of MTL to a single objective that renders optimization invariant to the magnitudes of losses. Theoretically, we prove that the requirement for scale invariance uniquely determines this transformation to be logarithmic. We further show that this general transformation-induced merit function preserves weak Pareto optimality and admits a smooth surrogate with controllable approximation error. Extensive experiments on representative multi-task benchmarks demonstrate that SIMS consistently outperforms existing scalarization methods and achieves state-of-the-art performance.

We establish the stochastic-gradient query complexity of sampling smooth strongly log-concave distributions in every fixed dimension $d\geq1$. For $\mu$-strongly convex, $L$-smooth potentials with unknown minimizers $x_f^\star$ in the ball $\mathbb{B}(0,\mu^{-1/2})$, and unbiased gradient oracles with variance at most $\sigma^2$, the minimax worst-case expected query complexity is $\Theta\left(\log(1+\kappa)+\frac{\sigma^2}{\mu\varepsilon}\right)$, jointly optimal for the condition number $\kappa:=L/\mu$, variance $\sigma^2\ge 0$, and TV accuracy $0<\varepsilon\leq 1/10$. In the noiseless setting $\sigma=0$, this tight complexity $\Theta\left(\log(1+\kappa)\right)$ is independent of $\varepsilon$. Moreover, a gradient-only sampler generates an exact sample with ${O}(\log(1+\kappa))$ worst-case expected queries. Without previous ball $\mathbb{B}(0,\mu^{-1/2})$, exact sampling from any initial point $x_0$ can be implemented with expected cost ${O}\left(\log(1+\kappa)+\log(1+\sqrt{\mu}\|x_0-x_f^\star\|)\right)$, without knowing the initial distance. We also show that dependence on initial distance is generally unavoidable.

A $\mu$-constant deformation of isolated complex hypersurface singularities of dimensions different from 2 preserves the diffeomorphism class of the Milnor fiber. If it would be known that $\mu$-constant deformations have no vanishing folds as defined in by O'Shea, this result is even true in all dimensions since the members in the family would have a common Milnor ball-radius. In this note a symplectic version of this is shown which in particular applies to weighted homogeneous polynomial singularities.

We establish a weighted restriction estimate in $\mathbb{R}^2$. Our proof combines new two-ends Furstenberg estimates with refined decoupling.As an application, our result yields decay estimates for the circular $L^p$-means of the Fourier transforms of fractal measures in $\mathbb{R}^2$.

In this paper, we introduce some weighted reduced quaternionic modules induced by a reduced quaternionic Cauchy-Riemann operator associated to arbitrary orthogonal bases of R3. A weighted conformal invariant property of the modules is given.

In this paper, we study the well-posedness and large-time behavior of a pseudo-parabolic problem associated with a generalized total variation energy in the $BV$-framework. A main mathematical issue arises from the mismatch between the Sobolev regularity of solutions at finite times and the $BV$-structure of the corresponding steady-state problem. We prove the existence and uniqueness of solutions and investigate the relationship between their large-time behavior and solutions to the steady-state problem. Moreover, our results include the convergence of the solution trajectory to the unique steady-state solution under a typical setting arising in image processing.

We exhibit a well-ordering principle that is equivalent to a theory of partial impredicativity. The latter goes back to Towsner and relates to recent work of Suzuki and Yokoyama. Together with part I of the present paper, this yields a combinatorial upper bound on the axiomatic strength of Fraïssé's conjecture.

We prove that the multiplicities of every symmetric $Q$-polynomial association scheme form a log-concave sequence. This resolves the 1984 unimodality conjecture of Bannai and Ito and establishes both related inequalities conjectured by Stanton. Our proof is a novel application of strong subadditivity and weak monotonicity inequalities for quantum entropy.

We introduce and study the rotating-memory Fibonacci numbers, a periodic variable-order analogue of the Fibonacci sequence in which the number of preceding terms used in the recurrence changes cyclically with the index. Despite this varying memory, the resulting sequences exhibit a remarkably rigid structure. We derive closed forms, rational generating functions, arithmetic properties, and exact growth behavior, and show that the sequence decomposes naturally into geometric subsequences. We also develop combinatorial interpretations in terms of periodically constrained tilings, and restricted compositions, including bijective explanations for the multiplicative structure of the sequence. In addition, the first two nonclassical periods admit natural geometry-driven realizations: the period-2 sequence arises from monomer–dimer tilings of a triangular chain, while the period-3 sequence is related to tilings of a double hexagon strip by single and double hexagons. These connections provide geometric interpretations of the rotating recurrence in which the periodic behavior is induced by the underlying structures themselves, and suggest a broader problem of constructing analogous models for higher periods.

Motivated by a within-host framework of immunity-modulated parasitic disease, we formulate a multitype Markovian branching process with reproductive parameters dependent on the number of individuals of a single type only. We impose a soft carrying capacity $K$ with respect to the controlling type, serving as a threshold between subcritical and supercritical dynamics. We prove that extinction occurs almost surely when the offspring batch of each non-controlling type includes an individual of the controlling type with positive probability, with the time to extinction possessing finite moments of all orders. We then construct a sequence of density-dependent processes indexed by $K$ and study scaling limits of the process using the canonical functional law of large numbers (FLLN) and central limit theorem, with a view towards characterising the emergence of metastable behaviour in the vicinity of asymptotically stable equilibria, or stable limit cycles of the FLLN.

We give a self-contained proof of Bourgain's finite $\Lambda(p)$ selection theorem with an explicit probability bound. For every $p>2$ and $N\ge2$, given $N$ pairwise orthogonal functions bounded by one on a probability space, a uniformly chosen subsystem of cardinality $\lceil N^{\frac{2}{p}}\rceil$ satisfies the $\Lambda(p)$ inequality with probability at least $1-\frac{1}{N}$, with a constant depending only on $p$. The inequality holds simultaneously for all complex coefficient vectors. More generally, for every fixed $A>0$, the success probability is at least $1-\frac{1}{N^A}$ with a constant depending only on $p$ and $A$, and independent of $N$. The proof uses a greedy approximation in $L^r$, with $r>2$, whose potential decreases throughout all approximation scales. A single weighted bound on the cumulative number of updates controls a union bound over discrete update histories, and increasing the weight assigned to each history gives the prescribed failure exponent without changing the cardinality exponent.

We construct equivalent norms on $L_1[0,1]$, arbitrarily close to the usual $L_1$-norm, which are weakly midpoint locally uniformly rotund and Gâteaux smooth, and whose dual norms have the Daugavet property. In particular, we obtain a strictly convex and smooth Banach space with the Daugavet property, thereby solving a longstanding open problem in the area.

In this paper, we investigate the asymptotic behavior of minimizers $u_\Lambda$ of the Bernoulli functional $\int_D |\nabla v|^2 \, dx +\Lambda|\Omega_v|$ for large $\Lambda$ with a boundary datum $g \ge 0$. In particular, in the case $g>0$, we show graphicality of the free boundary over $\partial D$ and obtain a second order asymptotic expansion in $\Lambda$ of the graph.

A weighted dynamical version $\mathbf{Z}_f$ of the classical Ihara zeta function is presented on connected finite regular graphs, expressed in terms of weighted periodic orbit data. We establish its meromorphic continuation, with poles given by the resonances of the associated non-backtracking transfer operator acting on a suitable Banach space, and compute its residues. At simple spectral parameters, the residue of $\mathbf{Z}_f$ is identified with the invariant Ruelle distribution on the graph phase space. Combining this result with a relation obtained by Arends-Palmirotta, we further derive Patterson-Sullivan and Wigner residue formulae. This provides a finite-graph analogue of residue formulae for weighted dynamical zeta functions for geodesic flow on rank one locally symmetric spaces, in the spirit of Schütte-Barkhofen-Weich.

We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}\mu_{A,B}(a,b)$, where $\mu_{A,B}$ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve $t\mapsto \mathrm{tr}[A-tB]_+$ is piecewise affine exactly when $A$ and $B$ commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.

For a finite abelian group $\Gamma$, a map $f\colon V(G)\to\Gamma$ is a $\Gamma$-magic map if the sum of its values over the neighbors of a vertex is independent of the vertex. It is affinely generating if its pairwise differences generate $\Gamma$. When $|\Gamma|=|V(G)|$, a bijective $\Gamma$-magic map is a $\Gamma$-distance magic labeling. For a regular graph $G$ with adjacency matrix $A$, write $\mathbf1$ for the all-ones vector indexed by $V(G)$, and let $\overline A$ denote the endomorphism induced by $A$ on $\Lambda_{\mathbf1}=\mathbb Z^{V(G)}/\mathbb Z\mathbf1$. When $\overline A$ is nonsingular over $\mathbb Q$, define the reduced Smith group by $\mathsf S_{\mathrm{red}}(G)=\operatorname{coker}\overline A$. In previous work, we established that for every regular graph $G$ of positive degree with $\overline A$ nonsingular over $\mathbb Q$, one has \[ G\text{ admits an affinely generating }\Gamma\text{-magic map} \quad\Longleftrightarrow\quad \Gamma\hookrightarrow\mathsf S_{\mathrm{red}}(G). \] We determine this group explicitly for Kneser graphs. If $r\ge1$, $n\ge2r$, and $m_j=\binom nj-\binom n{j-1}$, then \[ \mathsf S_{\mathrm{red}}(K(n,r))\cong \bigoplus_{j=1}^{r} \left(\mathbb Z/\binom{n-r-j}{r-j}\mathbb Z\right)^{m_j}. \] When the labeling group and the Kneser graph have the same order, the resulting embedding criterion leaves only one case in which an affinely generating map exists. More precisely, if $\Gamma$ is an abelian group of order $\binom nr$, then $K(n,r)$ admits an affinely generating $\Gamma$-magic map if and only if $(n,r)=(9,2)$ and $\Gamma\cong\mathbb Z/6\mathbb Z\oplus\mathbb Z/6\mathbb Z$. A weak-Sidon-set bound shows that the affinely generating maps in the exceptional case cannot be bijective. Hence $K(n,r)$ admits no $\Gamma$-distance magic labeling throughout the range $r\ge1$ and $n\ge2r$.

In this work, we prove that any Boolean function of degree $d$ on $\mathbb{Z}_{q}^n$, $q\geq 3$, has at most $m_qq^d$ relevant variables, where $m_q=\frac{2q(q^2+4q+1)}{(q-1)^4}$. For $q\in \{3,4,5,6,7\}$, we improve this bound to $2.854\cdot 3^d$, $1.749\cdot 4^d$, $1.263\cdot 5^d$, $0.994\cdot 6^d$, and $0.814\cdot 7^d$, respectively.

The deployment of a geoengineering program to cool Earth's climate may be imminent. It is crucial that tools be developed to ensure that such a program would achieve its objectives while minimizing disruption. Here we exploit recently developed differentiable atmospheric models to demonstrate a novel geoengineering control strategy. In the differentiable primitive-equation atmospheric model JAX-GCM we impose a uniform $+4$\,K ocean warming and ask what pattern of sea-surface temperature cooling – in five ocean-masked zonal bands of prescribed SST forcings whose amplitudes are free – returns land near-surface air temperature closest to the model's own unwarmed climatology. This idealized set-up represents a cooling pattern that could be delivered physically either by marine cloud brightening or stratospheric aerosol injection. Gradients through chaotic dynamics decorrelate from the true sensitivity beyond the Lyapunov horizon, so we optimize greedily over segments of 8 to 14 days, following receding-horizon control. The learned strategy removes $92.3 \pm 0.4\%$ of the realized land warming across a ten-member ensemble of two-year rollouts, and a three-year run sustains it. If we use the spatial pattern of land temperature as the optimization objective, the distributions of precipitation, evaporation, and specific humidity over land are restored as well, even though they are not included in the objective function. The learned strategy from JAX-GCM replayed in the AI emulators LUCIE and NeuralGCM without re-optimization is successful, suggesting robustness. These promising results demonstrate a strategy for designing optimal climate interventions that can be applied broadly for geoengineering scenarios under consideration.

We obtain the existence results of least energy sign-changing solutions and ground state solutions for a class of asymptotically cubic Kirchhoff equations with Dirichlet boundary value on a locally finite graph $G=(V,E)$, and obtain the least energy of sign-changing solutions is strictly larger than twice of the ground state energy.

In this work, a sharp Kähler spectral almost-rigidity theorem was established, resolving Conjecture 1.8 of Chu–Wang–Zhang. For compact Kähler manifolds satisfying $\Ric(\omega)\geq\omega$, pinching the first $n^2+3$ nonzero complex eigenvalues to one forces the manifolds to be Gromov Hausdorff close to normalized complex projective space and determines the biholomorphism type. The smaller index $n^2+1$ is the sharp threshold for noncollapsing. More generally, if a normalized measured limit has essential real dimension $r$, then the multiplicity of the critical eigenvalue is at most $r+\lfloor r/2\rfloor^2$, with equality attained in every dimension. A uniform energy estimate for kernel projections of complex gradient Gram matrices leads to a Lie algebra action on the whole spectral resolution.

We study the strategic design of a two-echelon spare-parts inventory network where evaluating each candidate topology requires an expensive inventory optimization model. The design partitions hundreds of sites into feasible clusters and selects a central replenishment site for each cluster to reduce costs while maintaining service levels. Because the optimizer favors candidates with high predicted savings, it can exploit optimistic surrogate errors. We develop a conservative framework combining a graph neural network ensemble, variable neighborhood search, and set-partitioning recombination. The surrogate is trained on exact cluster evaluations, while a lower quantile of ensemble-predicted savings guides the search to limit optimism. Clusters found during the search are recombined through set partitioning using surrogate-based objective coefficients. The resulting network is evaluated with the exact inventory model, and only this evaluation is used to report performance. In a case study of 246 fulfillment centers in Amazon's North American network, the framework improves combined savings by 30.5% over an optimization baseline based entirely on exact cluster evaluations, while maintaining approximately 99.8% service across six independent replications. Under equal computational budgets, graph-surrogate-guided search achieves higher mean exact savings than a tabular alternative under both scoring schemes. Conservative scoring improves mean savings for both surrogate classes and reduces the share of final-network clusters overestimated by the graph surrogate from 68% to 28%. Predictive and ranking accuracy deteriorate among search-generated candidates with high surrogate scores, indicating that random holdout performance can incompletely characterize surrogate quality during optimization.

Let $S_1$ and $S_2$ be rational elliptic surfaces with section over an algebraically closed field of characteristic zero, and let $X=S_1\times_{\mathbb{P}^1}S_2$. We determine the local equations produced by pairs of Kodaira fibers, including non-reduced fibers, and construct crepant resolutions for the admissible local models. For every construction we record whether it is projective and compute the number of irreducible components and the Euler characteristic of the resolved central fiber. Non-existence is asserted only in the normal Gorenstein cases covered by the terminal factorial obstruction; the remaining entries are stated as open for the methods of this paper. Under an explicit global compatibility hypothesis, the local data give formulas for the Hodge numbers of a smooth projective resolved fiber product. We also study Kummer quotients by fiberwise involutions. Their fixed curves are analyzed through the monodromy-orbit description of normalized fiber products of multisections. The final section records what the same constructions imply after reduction in characteristics $5$ and $7$, without interpreting characteristic-$p$ Euler and Picard data as complex Hodge numbers.

A B-coloring of a graph $G$ is a proper edge-coloring in which every $4$-cycle receives four distinct colors; let $q_B(G)$ be the minimum number of colors in such a coloring. Every graph of maximum degree $\Delta$ is $K_{2,\Delta+1}$-free; hence the known $2\Delta$ bound for planar graphs with $\Delta\ge38$ (Kong et al., 2026) motivates our study of $K_{2,t}$-free planar graphs, where $t\ge2$ is an integer. We prove $q_B(G)=\Delta(G)$ when $t=2$ and $\Delta(G)\ge7$, or when $t\ge3$ and $\Delta(G)\ge14(t-1)$. For $t\ge35$, the bound $q_B(G)\le\Delta(G)+t-1$ holds regardless of $\Delta(G)$; for every $t\ge2$, it also holds when $\Delta(G)>428$. Finally, for every integer $k\ge1$, every $k$-degenerate $K_{2,t}$-free graph satisfies $q_B(G)\le\Delta(G)+(k-1)\min\{t-1,\Delta(G)\}$, with equality for $K_{k,t-1}$ when $k\ge2$ and $t-1\ge k$.

We study long-time behavior of the solutions to the final state problem for the defocusing nonlinear Schrödinger equation (NLS) in one space dimension with the power nonlinearity $|u|^{2\sigma}u$ in the subcritical long-range regime $\frac{2}{\sqrt{7}}<\sigma<1$. Given a prescribed asymptotic profile in a weighted $L^2$-space, without size restriction, obtained by modifying the free solution with a nonlinear polynomial phase correction, we construct a unique global solution of the NLS that scatters to this profile, thereby proving the existence of modified wave operators. The proof relies on two new ingredients. Extending our previous work for the cubic case, we incorporate the leading part of the nonlinear term into the linear part as a linear potential by linearizing the NLS around the asymptotic profile and prove a global modified energy estimate for the linearized equation. We also exploit a specific structure of the nonlinearity arising from the linearization, which gives rise to a crucial cancellation when estimating the nonlinear terms in the modified energy space and enables us to control the polynomial growth of the nonlinear phase correction in the subcritical case.

Motivated by the wide applications of the quadratic phase integral transform in signal processing, optics, and time frequency analysis, in this paper we develop a new integral transform called the quaternion quadratic phase Dunkl transform (QQPDT). We present a detailed study of its fundamental properties. In particular, we establish continuity, linearity, scaling, modulation, Riemann Lebesgue result, inversion formula, and Parsevals identity. In addition, we estab lish Heisenberg and Donoho-Stark-type uncertainty principles for the QQPDT. Finally, we develop a signal recovery algorithm in the QQPDT domain. The proposed method enables the reconstruction of missing signal information from partial observations under suitable localization conditions.

This note presents new findings on the distribution of the volume and surface area of both the Poisson polyhedron and the Crofton polyhedron. It also considers the distribution of the area of the Poisson polygon and the Crofton polygon in the plane. This note motivates the modeling of the distributions of these convex body functionals as mixture distributions and the essential inclusion of the generalized gamma distribution therein as mixture components. Finally, model parameters are estimated and models are selected based on extensive stochastic simulations of these random polytopes. This includes that for the first time, this note gives very accurate estimates of the side number probabilities for both the Poisson polyhedron and the Crofton polyhedron, as these are used as mixing weights in the distribution models.

By studying wall crossings of Hassett moduli spaces of weighted pointed curves, we prove a recursive formula that reduces the computation of the cohomology of $\overline{\mathcal M}_{g,n}$ to that of the moduli space $\overline{\mathcal M}_{g,(0^+)^n}$ of stable pointed curves of minimal weights. For $g=1$, we further prove an explicit formula for the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,(0^+)^n}$. Combining these, we prove formulas that enable us to compute the cohomology of $\overline{\mathcal M}_{1,n}$ efficiently.

We analyse fully discrete Streamline Diffusion finite element methods, also known as SUPG methods, for time-dependent first-order systems with skew-symmetric spatial operators. To the best of our knowledge, this is the first stability and a priori error analysis of strongly consistent Streamline Diffusion finite elements combined with linear multistep schemes of order greater than two for multidimensional first-order hyperbolic systems. Local-in-time $L^2$-residual minimisation singles out $\delta=b_0\tau$, where $\delta$ is the stabilisation parameter, $\tau$ the time step, and $b_0$ the current-time coefficient in the method average. For this choice, each implicit linear system is a symmetric positive definite normal equation in a graph norm. A direct dual-residual estimate yields errors of order $O(h^{k+1/2}+\tau^\nu)$ when $\delta$ is proportional to $h$ and $\tau=O(h)$, where $h$ is the mesh size, $k$ the polynomial degree, and $\nu$ the temporal order. Stability is established for the $\theta$-method and the Adams-Moulton methods AM3-AM5. The general-degree AM3 and AM4 estimates use strengthened CFL conditions; for continuous piecewise affine elements, an elementwise cancellation recovers the standard hyperbolic regime. AM5 and the explicit Adams-Bashforth methods AB3 and AB4 are stable under a standard hyperbolic CFL condition, though the explicit schemes do not have normal-equation structure. Order-matched acoustic tests recover orders two through five and material-residual rates one half-order lower. In a d'Alembert test with discontinuous initial data, normal-equation SUPG recovers second-order local $\mathbb P_1$ convergence away from the wave fronts, whereas the unstabilised error is close to half order. A two-dimensional compact-wave test shows improved localisation with modest dissipation.

Vision-based reinforcement learning holds strong potential for decision-making in controlled environment agriculture (CEA). However, its development is hindered by the scarcity of labelled crop images. To address this gap, LettuceVisSim, a lettuce growth simulator that generates labelled time series of crop images, was developed and validated. The simulator contains a process-based model (PBM) for shoot dry weight dynamics, a canopy layout algorithm for deriving canopy layout representations from shoot dry weight, and a Unity rendering engine for image generation. Five findings support the simulator. First, the PBM reproduced shoot dry weight under dynamic plant-density management with $\mathrm{R}^{2}=0.84$. Second, a piecewise cubic regression mapped shoot dry weight to potential projected area with $\mathrm{R}^{2}=0.94$. Third, the canopy layout representation was validated using 12 experimental datasets each having different dynamic environmental and spacing conditions. It reproduced the ground coverage ratio dynamics observed in measured images, achieving $\mathrm{R}^{2}=0.84$ when driven by measured shoot dry weight and $\mathrm{R}^{2}=0.40$ (0.76 excluding one outlier) when driven by PBM-simulated values. Fourth, the Unity rendering engine converted canopy layout representations into RGB and segmentation images at less than 10~ms. Fifth, a demonstration showed that a lighting-control policy can be learned and applied by observing only crop images that were generated with LettuceVisSim, providing a proof of concept of vision-based reinforcement learning in CEA using LettuceVisSim.

Sárközy's theorem says that if $A \subset \mathbb{Z}$ has positive upper asymptotic density, then there are distinct $a_1, a_2 \in A$ and $n \in \mathbb{Z}$ such that $a_1-a_2 = n^2$. The same is true if $n^2$ is replaced by $F(n)$ for any polynomial $F \in \mathbb{Z}[x]$ with constant term zero. Green proved an $\mathbb{F}_q[t]$-analog of Sárközy's theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial $F \in \mathbb{F}_q[x]$. This condition was recently removed by Li and Sauermann. In this paper, we generalize Green's argument to accommodate equations in more variables in $\mathbb{F}_q[t]$, while pointing out that the technical condition can be removed by means of a simple observation.