Papers

Empirical risk minimization on massive datasets naturally exhibits a nested double finite-sum structure, where $N=nm$ total samples are logically or physically partitioned into $n$ blocks of size $m$ (e.g., in pooled data silos, out-of-core learning, or deliberate stratification). While variance-reduced methods achieve optimal oracle complexities for nonconvex objectives, they suffer from severe scaling bottlenecks in this centralized regime. Recursive estimators, such as PAGE, require periodic global full-gradient refreshes over all $nm$ samples, which are computationally expensive. Conversely, single-loop methods, such as SILVER, avoid such refreshes but require an impractical $\mathcal{O}(nm)$ memory footprint to store a control variate for every sample. In this paper, we propose SILAGE, a variance-reduced algorithm that addresses this trade-off. By actively exploiting the double-sum structure, SILAGE eliminates periodic global full-gradient refreshes over all $nm$ components (evaluating at most one local group gradient per iteration) while requiring only $\mathcal{O}(n)$ memory. Furthermore, we provide a tight convergence analysis that avoids pessimistic worst-case Lipschitz constants. Instead, SILAGE's complexity natively adapts to the underlying data geometry via nested functional similarities: across-group ($\delta_1$) and within-group ($\delta_2$) heterogeneity. Our results improve existing state-of-the-art bounds in several practically relevant regimes.

CMA-ES behaves, per restart, primarily as a local optimizer; multimodal search relies on restart strategies such as IPOP and BIPOP, which draw every restart uniformly and reuse no information from previous evaluations. Multi-Start Clustering CMA-ES (MSC-CMA-ES) makes restarts structure-aware: in alternating cycles, a Sobol pre-sample is partitioned into approximate basins of attraction by nearest-better clustering, restarts are seeded basin by basin with locally scaled step-sizes and population sizes, redundant basin visits are detected and excluded, and the remaining budget is spent on a budget-bounded, tolerance-disabled local refinement of the best-so-far solution. We evaluate the method on the four CEC suites (2014, 2017, 2020, and 2022) at their official budgets, across ten (suite, dimension) cells with dimensions 5–30, with 51 runs per function, and compare it with BIPOP-CMA-ES and five differential-evolution algorithms (ARRDE, jSO, j2020, NL-SHADE-RSP, and L-SRTDE). Read per function class over the $D\leq 20$ cells, the results split three ways. On composition functions MSC-CMA-ES attains the best value on all aggregate measures, with $2.7\times$ the fixed-budget target coverage of BIPOP-CMA-ES – the highest composition coverage of any algorithm evaluated – and leads in most suites, dimensions, and budgets. On unimodal and simple-multimodal functions it attains the best median error in aggregate but the lowest deep-target coverage, with orderings varying by suite and dimension. On hybrid functions both CMA algorithms trail the leading DE algorithms. All results and scripts are publicly available.

Let $S = (D, \sigma)$ be a signed digraph, where $D=(\mathcal{V},\mathcal{A})$ is the underlying digraph of $S$ and $\sigma:\mathcal{A}\rightarrow\{-1,+1\}$ is a sign function. In this paper, we study the adjacency singular values of signed digraphs. We provide a necessary and sufficient condition for the singular values of a digraph to remain invariant under signing. We study rank of signed digraphs and determine signed digraphs with rank one. As an application of this result, we obtain a lower bound for the trace norm of signed digraphs and characterize the extremal signed digraphs.

This paper establishes novel rigidity estimates for hyperbolic shells (surfaces with negative Gaussian curvature) and applies them to derive the \(\Gamma\)-limit of thin elastic shells. We prove a nonlinear rigidity estimate for \(H^1\) deformations on the mid-surface, and a nonlinear rigidity estimate for hyperbolic shells with clamped lateral boundary. The latter yields the optimal exponent \(h^{-4/3}\). As the main application, we characterize the \(\Gamma\)-limit of the nonlinear elastic energy for clamped hyperbolic shells across all scaling regimes \(\beta \in [0,2) \cup (8/3,\infty)\).

This paper proposes a geometrically constrained decentralized independent vector analysis (GC-Dec-IVA) method for distributed microphone arrays. Recently proposed Dec-IVA method enables source separation by exchanging only power-related statistics to exploit cross-array information. However, this initial attempt often provides negligible improvement over applying IVA locally at each array, mainly due to the potential permutation inconsistency among arrays and the strong cross-array dependency implied by its source model. To address these limitations, we incorporate direction-of-arrival (DOA) information to derive GC-Dec-IVA, which mitigates permutation mismatch across arrays and enhances source alignment. Furthermore, a new source model is introduced to weaken cross-array dependency, improving robustness against permutation inconsistency in noisy environments. Experiments show the proposed method improves both the separation performance and cross-array permutation consistency.

This article addresses the problem of proving the nonlinear instability of an equilibrium starting from its linear instability when there is no existence theory for the corresponding equations. We design a general method based on the use of analytic functions to overcome this difficulty and apply it to the classical problem of the stability of a plasma column in magneto-hydrodynamics (MHD) as an illustration.

We study Blaschke-Santaló diagrams for the inradius, circumradius, and diameter in arbitrary-dimensional Minkowski spaces. Their boundary parts are regularly described by four linear inequalities and one more complex inequality, and we analyze which bodies fill the boundaries and for which (types of) norms that boundaries collapse to a single point. In this context, the diagram with respect to the 1-norm in 3-space plays an exceptional role. We complete the description of the according diagram by tightening Bohnenblust's inequality through involvement of the inradius.

We study the geometric regularization of positive closed currents by the Kähler-Ricci flow on compact Kähler manifolds. In a previous work of ours, it was shown that the Kähler-Ricci flow immediately smoothes out such a current when it has zero Lelong numbers. We study here the case when $T_0$ has divisorial singularities, showing that the flow gradually replaces the latter by Poincaré type ones, providing an approximation of $T_0$ by complete Kähler metrics with bounded curvature in a Zariski open set.

This paper investigates the rigidity of strain tensors on surfaces with sign-changing Gaussian curvature (mixed-type surfaces) and applies the results to determine the optimal thickness exponent in the first Korn inequality for thin shells. Using tools from Riemannian geometry and generalized tensor analysis, we derive an infinitesimal rigidity lemma for strain tensors, which establishes \(L^2\) regularity estimates for displacements decomposed into tangential and normal components. Specifically, we show that for a mixed-type shell with a middle surface \(S = S^+ \cup \Gamma_0 \cup S^-\) (where \(S^+\), \(S^-\) have positive and negative curvature, respectively, and \(\Gamma_0\) is a parabolic interface), the optimal constant in Korn's inequality scales as \(h^{4/3}\), matching the behavior previously established for hyperbolic shells. This result is obtained via a combination of geometric decomposition, Fredholm theory for linear operators, and compactness arguments to handle the curvature transition across \(\Gamma_0\). The findings bridge the gap between elliptic and hyperbolic shell theories, providing a unified framework for understanding rigidity in complex geometries with mixed curvature. The derived estimates are shown to be sharp, offering critical insights for the mechanical design of thin-walled structures with non-uniform curvature.

We explore the movement of a thin, circular quantum vortex filament within an infinite cylindrical pipe. The fluid surrounding the vortex ring moves through the pipe at a non-zero velocity denoted by $v$. Our study examines the energy spectrum $E = E(p)$, where $p$ represents the total momentum of a vortex ring. We have demonstrated that the function $E(p)$ significantly depends on the velocity $v$. The discovered spectrum $E(p)$ reveals the existence of states with both negative and extremely large effective masses. We also explored the hypothesis regarding the existence of coupled vortex pairs possessing finite summary effective masses. Every pair consists of vortices that possess both positive and negative masses, with the magnitude of these masses being unrestricted. In our model, the criterion for the appearance of these states is based on comparing two numbers. The first is seen as a quantum counterpart to the Reynolds number, while the second represents its critical value for a flow with a single vortex. We also explore how this studied effect might contribute to the emergence of quantum turbulence. This study discusses a method for determining the critical Reynolds number in quantum turbulence, using the proposed model as a framework. Here, we use a new quantization technique for classical closed vortex filaments developed by the author earlier.

We classify any entire area-minimizing surface $M^2\subset\mathbb{R}^n$ with density $2$ at infinity as either planar, or the zero set of a quadratic holomorphic polynomial inside some affine copy of $\mathbb{C}^2$. We also prove algebraicity of $M^2$ when the tangent cone at infinity has multiplicity one with general density. In our proof we introduce a sheeting-type theorem for minimizing surfaces near a multiplicity-two plane at infinity, and a new proof of holomorphicity (different from \cite{micallef}) based on a general “calibrated at infinity” principle.

In this paper we present an integrable model in two dimensions. It is a deformation of the sine(sinh)-Gordon model. We give its Lax connection. We also obtain its (classical) $r$-matrix. It satisfies the classical Yang-Baxter equation. Thus, the model is a classical integrable field theory in two dimensions. The underlying algebra is a ${\cal Z}$-graded non-Lie alternative Akivis algebra. An alternative Akivis algebra is the tangent space at the identity of an analytic alternative loop as a Lie algebra is the tangent space at the identity of a Lie group. We expect the model to be integrable at the quantum level. We also give a family of classically integrable models which are related to the Poisson-Boltzmann equation in two dimensions.

Random non-hermitian matrix ensembles with double-sided rotation invariance obey, in the limit of large matrix size, the Single Ring Theorem, which states that the support of the mean eigenvalue distribution in the complex plane is either a disk or an annulus. In contrast, rotational-invariant random normal matrix ensembles can have mean eigenvalue densities supported over any number of concentric annuli in the complex plane. In this paper we introduce and investigate, both analytically and numerically, a non-hermitian matrix model which flows from a generic matrix distribution obeying the Single Ring Theorem to a distribution of normal matrices by tuning a parameter which penalizes non-normality. We observe numerically breakdown of the Single Ring Theorem as the model flows towards normality, and determine the critical value of the parameter at which the transition occurs. We also study in detail the behavior of the singular values of these matrices under the flow. These singular values form a Fermi gas confined to the positive half-line. In particular, we find that at small values of the flow parameter, the interparticle spacings in the gas exhibit Wigner-Dyson repulsion, whereas for asymptotically large values of the flow parameter, at the normal matrix endpoint of the flow, the spacing statistics is Poissonian. The flow interpolates continuously between these two types of statistics. However, this change in statistics is not related directly to breaking of the Single Ring Theorem, which occurs very early-on along the flow, in the regime of Wigner-Dyson statistics. Finally, we introduce a certain ensemble of random permutations associated with the gas, and make a conjecture on how to use it in order to reconstruct approximately the average density of complex eigenvalues from that of the singular values in the large-$N$ limit.

In this paper, we study the existence of symphonic maps on compact or complet non-compact Riemannian manifold into Riemannian manifolds admitting a conformal vector field or a non-trivial Ricci solitons.

We consider a vectorial energy functional proposed in the physical literature as a simplified model for thin films of ferronematics – composite materials formed by dispersing magnetic nanoparticles into a nematic liquid crystals host. The model features two order parameters: a reduced $\Q$-tensor field $\Q$, describing the nematic liquid crystal component, and a vector field $\M$, accounting for the average polarisation vector field generated by the included particles. The energy contains a coupling term promoting alignment between $\Q$ and $\M$. It has been shown in~\cite{CDS1,CDS2} that, as a small parameter $\eps$ tends to zero, the energy of critical pairs $(\Q_\eps,\,\M_\eps)$ concentrates on distinct singular sets: a finite set of points for the $\Q_\eps$-component and the support of a $\H^1$-rectifiable varifold for the $\M_\eps$-component, with first variation supported on the singular set for the $\Q_\eps$-component. We show in this paper that, if the coupling constant is above a certain (explicit) threshold, then, up to rescaling its density by a material constant, such a limiting varifold has integer multiplicity.

This paper addresses coefficient problems for the apple-like convex subclass $\mathcal{C}_{\mathcal{AP}}$, defined by the subordination relation $1+zf''(z)/f'(z) \prec e^z\sqrt{1+z}$. We determine sharp bounds for the initial inverse logarithmic coefficients $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, and the consecutive modulus difference $|\Gamma_2|-|\Gamma_1|$. We also obtain the sharp upper bound for the second-order inverse logarithmic Hankel determinant, characterize the generalized Fekete–Szegő functional for all real parameters, and compute sharp bounds for the third-order Hermitian–Toeplitz determinant. Extremal functions are given for each case.

We study families of smooth, embedded, regular planar curves $ \alpha : \left [-1,1 \right ]\times \left [0,T \right )\to \mathbb{R}^{2}$ with generalised Neumann boundary conditions inside cones, satisfying three variants of the fourth-order nonlinear curve diffusion flow: (1) curve diffusion flow with a length penalisation, (2, 3) two forms of constrained curve diffusion flow with fixed length. We prove in cases (2) and (3) for cone angle less than $\pi$, if the initial curve has small oscillation of curvature and the initial curve is sufficiently far from the cone tip, then the solution exists for all time and converges exponentially in the $C^\infty$-topology to a circular arc with the same length as the initial curve. In case (1), a similar result holds under suitable rescaling. In all cases, the limiting arc is centred at the cone tip.

Let $H$ be a 3-partite 3-uniform hypergraph whose three vertex classes all have size $n$. For a vertex $v \in V(H)$, the link graph $N_H(v)$ is defined on $V(H)\setminus\{v\}$ with edge set $\{e\setminus\{v\}: v\in e\in E(H)\}$, and we denote by $\rho(N_H(v))$ its spectral radius. We prove that for every $\alpha>0$ there exists $n_0$ such that for all $n\ge n_0$ the following holds: if \[ \rho\bigl(N_H(v)\bigr) > \left(\frac{\sqrt{2}}{2}+\alpha\right)n \] for every vertex $v\in V(H)$, then $H$ contains a perfect matching. This spectral condition is asymptotically best possible.

The saturation number $\mu^*(G)$ of a graph $G$ is the minimum cardinality of a maximal matching, and $H(G)$ is its harmonic index. TxGraffiti conjectured in 2023 that $\mu^*(G) \le H(G)$ for every nontrivial connected graph $G$, and B{\i}y{\i}koğlu refuted this by showing that the ratio $\mu^*(G)/H(G)$ can be made arbitrarily large. Restricting to trees bounds the ratio sharply. Every nontrivial tree $T$ satisfies $\mu^*(T) < \frac{3}{2} H(T)$, with the constant $3/2$ best possible. A complementary bound $H(G) < 4\mu^*(G)$ holds for every graph with an edge, so on a nontrivial tree the saturation number is pinned to $\frac{1}{4} H(T) < \mu^*(T) < \frac{3}{2} H(T)$, both constants best possible. The friendship graph $F_4$ is a smallest counterexample to the conjecture, on nine vertices, and the smallest tree counterexample is the subdivided star on eleven vertices. For each positive integer $m$ a family of graphs with $m$ hubs has ratio approaching $m+1$, while the conjecture holds whenever all vertices have equal degree. Both invariants arise in applications, the harmonic index as a molecular descriptor and the saturation number as a measure of adsorption inefficiency, and the bounds estimate the latter, which is NP-hard to compute, by the former, which is computable in linear time.

For a polarized complex manifold with discrete automorphism group, it is known that if the first Chern class admits a cscK metric, then the balanced metrics, which are characterized in terms of the algebro-geometric notion of Chow stability, approximate this cscK metric. In this paper, we study a non-Archimedean analogue of this phenomenon. In particular, we prove that such an analogue holds for polarized totally degenerate abelian varieties. As an application, we also show that, for a totally degenerating family of polarized abelian varieties, the validity of this non-Archimedean analogue yields a uniform estimate for the Calabi–Yau metrics on fibers sufficiently close to the degenerate fiber.

Self-organized critical systems often exhibit three macroscopic features simultaneously: nonextensive thermodynamics (quantified by the Tsallis index $q$), structural fractality (measured by the Hausdorff dimension $D$), and non-Markovian dynamics (characterized by the memory exponent $\alpha$). Historically, these parameters have been treated as independent, to be empirically fitted case by case. Here we demonstrate that phase-space self-consistency imposes a unique algebraic closure: $\alpha=D/(2D-1)$. This relation, together with $q=1+1/D$ derived from the extensivity of Tsallis entropy on fractal supports, yields the known result $\alpha=1/(3-q)$ as a consequence, not as an independent assumption. The closure contains no free parameters and satisfies the physical boundary conditions $\alpha(1)=1$ (ballistic transport in Euclidean spaces) and $\alpha\to1/2$ as $D\to\infty$ (maximally subdiffusive regime). We validate the Troika relation across eight independent experimental systems, including seismicity, electromagnetic precursors, EEG, urban networks, botanical architectures, and space plasma. All measured values fall within error bars of the theoretical prediction, establishing the universality of the closure.

In this paper, we consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with a step-like initial data. Based on the Riemann-Hilbert problem associated with the mKdV equation, we derive the long-time asymptotic expansion of the solution to the defocusing mKdV equation in two transition regions using the nonlinear steepest descent method. It comes out that the leading term in the expansion is shown to match the corresponding background constants. The subleading term, however, decays at the order $\mathcal{O}(t^{-2/3})$, and its coefficient is derived from the associated Painlevé \uppercase\expandafter{\romannumeral34\relax} model.

This paper contains a survey of some of the results of the author related to unrectifiablity and is an extended version of the author's talk given at the Second Winter School Geometric Measure Theory Rectifiability vs. Pure Unrectifiability in Hanghzou, China. These results include irregular/purely unrectifiable $1$-sets on the graphs of continuous functions like the Takagi, the Weierstrass-Cellerier and the typical (in the sense of Baire) continuous function. It is also discussed that there exists $ {\alpha}_{0}<1$ such that the fractal percolation is almost surely purely $\alpha$-unrectifiable for all $\alpha>\alpha_0$. The background of the $1$-unrectifiability is discussed in more detail.

The aim of this note is to illustrate that the set of integrable functions on the torus $\mathbb{T}$ for which the Fourier series diverges almost everywhere is of the second Baire category in $L^1(\mathbb{T})$.

In this paper, we consider nonlinear Schrödinger equation. Duyckaerts–Roudenko [Comm. Math. Phys. 334 (2015), no. 3, 1573–1615] investigated time behavior of solutions to the equation, whose mass-energy is greater than that of the ground state. We revisit the results and show by using the action. Moreover, we apply it to other equations without scale invariance.

This paper revisits the functional equation of the symmetric-square $L$-function from the perspective of trace formulas. We show that, for level-one holomorphic cusp forms and even Hecke–Maass cusp forms, this symmetry can be recovered directly from the Petersson and Kuznetsov trace formulas. Although the functional equation itself is classical, our aim is to provide a concrete and transparent example of how the analytic structure of an $L$-function can emerge from a comparison of the spectral and geometric sides of a trace formula. The argument leads to an averaged reciprocity identity and treats the holomorphic and Maass settings within the same general framework. In this way, the paper offers a simple model for extending trace-formula methods beyond standard $L$-functions.

We study a class of IMplicit-EXplicit Runge–Kutta (IMEX-RK) schemes for the numerical approximation of reaction and diffusion-reaction problems arising in a variety of biological and physical applications. Such models may admit travelling wave solutions, with the Fisher–Kolmogorov equation representing a prototypical example. Motivated by this feature, the proposed time integration schemes are designed to accurately capture sharp propagating fronts. We also investigate a less standard use of IMEX-RK methods that circumvents a splitting of reaction terms into linear and nonlinear components, while still requiring the solution of linear systems at each stage. This semi-implicit formulation, referred to as SI-IMEX-RK, enables a targeted treatment of stiffness by isolating its relevant contributions. The time discretization is coupled with a high-order polygonal discontinuous Galerkin method for space discretization, resulting in a flexible and robust framework for the treatment of multiscale dynamics in complex geometries. A comprehensive validation strategy is presented to assess the accuracy and stability properties of the proposed schemes across a hierarchy of increasingly challenging test problems.

In this paper, we establish a sharp comparison between Carleson-cube and Bergman-metric-ball conditions on the open unit ball $\B$ and combine it with a Berezin-type characterization to prove embedding theorems for Besov spaces and Bergman spaces on $\B$ into logarithmic tent spaces in the Bergman metric. As applications, we characterize the boundedness, compactness, and essential norms of the Volterra-type integral operators $T_g$ and $I_g$ acting from the Besov space $B_t(\B)$ to the general function space $F(p,q,s)$.

We present an equivariant K-theoretic construction of affine quantum Schur algebras of type C with three parameters by taking advantage of Kato's exotic framework. This enables the derivation of an equivariant K-theoretic realization of affine iquantum groups of type AIII with three parameters. Additionally, in Appendix A, we provide an equivariant K-theoretic construction of affine quantum Schur algebras of types B/C/F/G with two parameters, following Antor's recent work on affine Hecke algebras with two parameters.

This paper investigates the algebraic structure of Krom logic programs, consisting only of facts and rules with at most one body atom. We show that sequential composition endows the class of Krom programs with a natural monoid structure and that this structure admits rich algebraic extensions to Krom seminearrings, Krom quemirings, Krom-Conway seminearrings, and Krom-Conway omegaseminearrings. Furthermore, we establish explicit generating sets and canonical decompositions, study the associated ${}^\omega$-operator, characterize the Kleene star in graph-theoretic terms, and relate finite Krom monoids to transformation monoids and finite-state automata. These results provide new connections between logic programming, algebraic automata theory, and algebraic graph theory.