Papers

Extended-domain spectral collocation expands the solution of a boundary value problem in the Laplacian eigenbasis of an enclosing box and collocates at as many nodes as modes. The square system is exponentially ill-conditioned, and the impossibility theorem for equispaced data rules out stable geometric convergence; yet such solvers converge geometrically to a few orders of magnitude above the unit roundoff. We explain why in one dimension. The collocation solution is the solution operator applied to the interpolant of the data in an operator-dependent test space whose Lebesgue constant must grow exponentially for geometric convergence, so exact solutions can amplify data errors exponentially. In floating-point arithmetic the error of any coefficient vector is bounded by its nodal residual, its distance to a competitor and two frame constants, a leakage and a restricted Marcinkiewicz–Zygmund constant, bounded uniformly in the number of modes at a fixed positive threshold; for a backward-stable solve, under a condition on the perturbed matrix, the bound is uniform in the conditioning for suitably analytic data and, with computed constants, within a factor of about ten of the error floor of a direct solver. For the Poisson operator the rate of the two constants along the tail of the spectrum is given by a constrained equilibrium problem of logarithmic potential theory. For an interval of length $L$ extended by $\delta$ at each end, the problems for $\delta$ and $L^2/(4\delta)$ are dual, the rate is linear for $\delta = L/2$, and it starts with slope minus the fraction of the extended domain occupied by the extension, so for $\delta < L/2$ the leakage constant obeys no power law with the exponent of the end of the spectrum. In experiments, convection–diffusion operators behave like the Poisson operator at moderate Péclet numbers.

We classify all nonconstant smooth harmonic maps $\mathbb{S}^2\to \mathbb{CP}^n$ and $\mathbb{S}^3\to \mathbb{CP}^n$ whose 0-homogeneous extensions are minimizing: they are, respectively, isometric embeddings or holomorphic isometric embeddings composed with a Hopf map.

Let $\hbar$ be a non-zero real number and put $q= e^{-\pi i \hbar}$. Let $G$ be a semisimple split real Lie group of classical type, with Lie algebra $\mathfrak{g}$ and associated Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$. We show that there exists a locally compact quantum group $(L_{\hbar}(B),\Delta)$ which integrates the Borel part $U_q(\mathfrak{b})$ of the Drinfeld–Jimbo quantized enveloping algebra $U_q(\mathfrak{g})$, viewed as a Hopf $*$-algebra. We further compute the associated Haar weights and modular data for $(L_{\hbar}(B),\Delta)$. Up to an excess Cartan part, the Drinfeld double of $(L_{\hbar}(B),\Delta)$ can be seen as an analytic quantization of the function algebra of the totally positive semigroup $G^+\subseteq G$.

We construct an action of the positive half of the Witt algebra on colored $\mathfrak{gl}_N$-Khovanov-Rozansky homology and prove that this action is compatible with the functoriality of colored link homology. To establish this result, we use Bar-Natan's framework of canopolies to formulate and prove the relevant statements at the level of tangles.

We derive the noncommutative open WDVV equations for the open Gromov-Witten superpotential introduced in joint work of the author and Solomon, extending the Solomon-Tukachinsky equations to noncommutative differential graded algebras. The resulting recursive formulas facilitate the computation of open Gromov-Witten invariants with multiple boundary point constraints for even-dimensional Lagrangians. We apply the open WDVV equations to compute all open Gromov-Witten invariants for the real loci of even-dimensional complex quadrics of dimension greater than two. These computations show that open Gromov-Witten invariants with multiple boundary point constraints can be nontrivial in arbitrary dimensions. We also construct a character for a natural subalgebra of big quantum cohomology associated to a Lagrangian endowed with a point-like bounding cochain. The character is in the spirit of the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono. Applying the character construction to the even-dimensional quadric illustrates how point-like bounding cochains refine the information about closed quantum cohomology obtained from weak bounding cochains of Floer theory. Open Gromov-Witten invariants with three boundary point constraints determine the two extensions of the character from the subalgebra to big quantum cohomology.

We prove that any subset $A \subseteq \{1, \dots, N\}$ lacking non-trivial polynomial progressions $(x, x + y, x + y^2)$ satisfies $\lvert A \rvert \le CN \exp\bigl(-c(\log \log N)^\beta\bigr)$ for effective absolute constants $C, c, \beta > 0$. The argument develops a density increment framework localized to anisotropic quadratic Bohr sets. To handle mixed linear and quadratic frequencies, we establish a multilinear transference principle coupled with a relative quadratic inverse theorem based on Balog–Szemerédi–Gowers additive energy rigidity. Boundary stability across increments is maintained via semi-algebraic model regularization using cylindrical algebraic decomposition and Cauchy–Crofton estimates, while dimension growth is controlled by dissociated basis reduction and multidimensional Erdős–Turán–Koksma equidistribution.

Heyer's left adjoint to derived parabolic induction gives rise to higher mod $p$ Satake transforms \[ \mathcal S_V^i: \mathcal H_G(V) \longrightarrow \mathcal H_M\bigl(H^i(K_U,V)\bigr), \] whose extreme degrees recover Herzig's invariant and coinvariant transforms. For $G=\mathbf G(F)$, where $\mathbf G$ is a connected reductive group defined over a finite extension $F/\mathbb{Q}_p$, we describe these maps explicitly using Mackey decompositions and corestriction in group cohomology. For $G=\mathrm{GL}_n(F)$, we evaluate them on the standard Hecke generators, showing that their support lies in a single Weyl orbit and that their values on the individual support cosets are concrete corestriction operators. For $G=\mathrm{GL}_2(F)$, we determine the image in every cohomological degree for each non-Steinberg Serre weight. The endpoint images are the classical polynomial Satake images, while every interior image is of the form \[ k[x,y^{\pm1}]/(x^r) \] for some positive integer $r$. Thus, every interior image is a finite nilpotent thickening of the Laurent polynomial algebra $k[y^{\pm1}]$. We give explicit bounds for its thickness, prove that they are sharp, and show that in the unramified case every interior image is either reduced or a square-zero thickening.

By our previous work, the category of unipotent monodromic mixed Hodge modules on $\mathbb{C}$ is equivalent to the category of can-Var diagrams of mixed Hodge structures. In this paper, we show that the associated graded diagram with respect to the weight filtrations decomposes as a finite direct sum of four basic types tensored with pure Hodge structures, which we call a split graded model. We then show that every unipotent monodromic mixed Hodge module is represented by a split graded model together with a finite family of $\mathbb{C}$-linear maps. We also obtain an analogous description in the case where the monodromy has no eigenvalue $1$.

In 1952, Pólya proved the inequality $\mu_2 < \lambda_1$ for the eigenvalues of Dirichlet and Neumann problems in domain $\Omega \subset \mathbb{R}^d$, $d \ge 2$. We show that $\mu_3 < \lambda_1$ if $d\ge 3$. In the two-dimensional case, in 2025, Rohleder proved that $\mu_3 < \lambda_1$ for simply connected domains. We prove that this inequality holds also true for double connected domains symmetric with respect to the origin.

We investigate the boundedness problem for log Calabi-Yau fibrations whose bases and general fibers are bounded. We improve the results in arXiv:2504.05243 from birational boundedness to actual boundedness, which confirms a conjecture of Birkar and Hacon when the boundary coefficients belong to a fixed finite rational set. We also get a new proof for the birational boundedness result in arXiv:2504.05243 which is substantially more conceptual.

We investigate restrictions on the dimensions and coordinate fields of real Weyl-Heisenberg fiducial vectors. Our main result is that such vectors can exist only in dimensions $d\equiv 3\pmod{4}$. This result is motivated by evidence showing that real Weyl-Heisenberg fiducial vectors can exist only in dimensions $d = 4m^2 + 3, m \in \mathbb{Z}_{\geq 0}$. We further restrict the possible dimensions when the coordinate field is a totally real number field, an even or odd degree extension of $\mathbb{Q}$, a real quadratic field, or a maximal real subfield of a cyclotomic field. Lastly, we establish non-existence over odd-degree number fields under either positivity of all non-trivial cyclic-shift overlaps or a conjectural containment of a distinguished quadratic field in the SIC field.

After removing suitable thin subsets, we count rational points of bounded height on some hypersurfaces of small bidegree in $\mathbb{P}^3\times\mathbb{P}^3$ and in $\mathbb{P}^3\times\mathbb{P}^2$. One of the ingredients is a new estimate for the number of rational points of bounded height on diagonal quadratic surfaces which improves as the coefficients grow.

We study the Hausdorff dimension of attracting sets of three-dimensional flows whose one-dimensional quotient dynamics is non-uniformly expanding. Our main assumption is that the quotient map admits a full-branch uniformly expanding Gibbs–Markov induced map. We first prove that every such attractor contains a hyperbolic horseshoe of Hausdorff dimension strictly larger than two. As a consequence, after an arbitrarily small $C^2$ perturbation, the associated dynamical Lagrange and Markov spectra have robustly non-empty interior for an open and dense set of $C^1$-observables. We then establish intermediate dimension properties. Every admissible intermediate Hausdorff dimension is realized by an ergodic invariant measure supported on the attractor. Moreover, every intermediate stable and unstable dimension is realized by a strictly ergodic hyperbolic subsystem. The approach extends the dimension theory previously developed for geometric Lorenz attractors to more general non-uniformly hyperbolic attractors including singular-hyperbolic attractors, Rovella attractors, and broad classes of skew-product suspension flows modeled by intermittent and Benedicks–Carleson maps.

We investigate the time-asymptotic stability of composite waves consisting of a non-classical Liu shock and a rarefaction wave for non-convex viscoelastic system. By choosing delicate time-dependent shift functions to the viscous shock and suitable weight functions for the relative entropy, we succeed in proving the time-asymptotic stability of the composite of a viscous Liu shock and a rarefaction wave from two different characteristic families for the one-dimensional viscoelastic system with a cubic non-convex stress-strain relation by $a$-contraction method. Here the non-classical Liu shock can be either degenerate or non-degenerate, which can be seen as the system version of the scalar Oleinik shock. Note that both the shift and the weight functions are chosen quite differently in the two cases of nearly degenerate and fully non-degenerate. The paper represents the first time-asymptotic stability result for the composite waves consisting of both non-classical Liu shock and rarefaction wave from two different families for the non-convex physical system.

Timely information dissemination is essential in modern communication systems, where low informational latency is critical for networked control, monitoring, and real-time decision-making. The Age of Information (AoI) has emerged as a fundamental metric for quantifying information freshness at the receiver. A key challenge in AoI optimization is selecting the update generation rate: infrequent updates lead to stale information, whereas overly frequent updates increase queueing delays. We develop a structure-aware, data-driven framework for adaptive AoI optimization modeled as single-server queues with unknown system statistics. Our key idea is to exploit analytical representations of the AoI gradient, which depend on a finite-dimensional collection of system quantities that can be estimated from observations collected during system operation. This reduces the online learning problem to estimating the quantities required by the gradient, rather than learning the AoI objective as a black box. We propose an online stochastic approximation algorithm that jointly estimates these quantities and optimizes the update generation rate. We prove almost sure convergence of the algorithm to the optimal update generation rate under mild regularity and moment conditions, despite the temporal dependence induced by the underlying queueing process. We demonstrate the framework in four representative incomplete-information settings, including FCFS queues with unknown service characteristics or hidden background traffic, a preemptive LCFS system with partial observations, and a generate-at-will system with threshold waiting.

In the literature, research on representing the non-dominated set has focused almost exclusively on multi-objective optimization problems with Pareto-dominance. However, general ordering cones are used to incorporate preferences of decision makers into the optimization process and to model other specific types of optimization problems, such as ordinal optimization problems. In this work, we consider several representation approaches based on representation relations in the context of multi-objective optimization problems with polyhedral ordering cones. Our main research question is whether existing theory and solution approaches developed for the Pareto case can be used to address the more general setting.

We recall several derivations of temperature-dependent Cahn-Hilliard equations. The resulting equations are systems of mixed order. For a suitable subclass of equations we discuss its local-in-time well-posedness based on $L^p$-maximal regularity theory. We also discuss stability of equilibria.

We establish the strict Dirichlet Pólya inequality on every bounded simply connected planar domain. The proof uses a unitary boundary transport formulation of the Helmholtz equation and relates Dirichlet resonances to unitary eigenphase crossings through the Dirichlet-to-Neumann map. An exact area identity yields the sharp Weyl coefficient, while a Ky Fan estimate leaves a strictly positive remainder in the counting estimate. The general case follows by conformal exhaustion.

Evaluating 5G positioning on a physical testbed ties every measurement to one propagation environment and one antenna layout, so any question about an alternative deployment requires a new field campaign. We present a ray-traced digital twin of the GEO-5G testbed at EURECOM that drives an unmodified OpenAirInterface (OAI) 5G stack with per-antenna channels computed by Sionna-RT over a 3D model of the site. The distributed gNB antennas are modelled as independent radio devices rather than as elements of a compact array, which preserves the per-antenna absolute delays on which time-difference-of-arrival (TDoA) positioning depends. With them preserved, the twin drives OAI's SRS-based UL-TDoA pipeline to metre-level accuracy consistent with that reported for the physical deployment. We then use the twin to answer a question the physical testbed cannot. Widening the carrier truncates the tail of the horizontal error distribution without improving a typical position, and leaves the vertical error untouched, because the latter is limited by the array geometry rather than by the measurement. That limit is predictable from the antenna coordinates alone, and a height-diverse layout reduces the RMS height error by a factor of four.

We establish a principle that converts qualitative regularity into uniform interior estimates for degenerate equations. For the $p$-Laplace equation, $p>2$, we prove that local $C^{1,\gamma}$ regularity of every solution, without any uniform bound on the corresponding seminorms, implies uniform $C^{1,\beta}$ estimates for every $0<\beta<\gamma$. Thus, the optimal qualitative and quantitative gradient Hölder exponents coincide as suprema in every dimension. In the plane, this yields estimates controlled by the $L^p$ norm of the solution for every exponent below the sharp Iwaniec–Manfredi threshold. Applied to Xu's recent qualitative $C^{1,1/3}$ regularity for planar infinity-harmonic functions, we obtain uniform interior $C^{1,\beta}$ estimates for every $0<\beta<1/3$. For the nonlocal superconductivity equation $|Du|(-\Delta)^s u=0$, with $1/2<s<1$, we prove that pointwise differentiability of all solutions with integrable tails implies uniform interior $C^{1,\alpha}$ estimates for bounded solutions, for some $\alpha>0$. The proofs combine compactness for shifted equations with an improvement-of-flatness iteration at variable scales.

We investigate generalized Stieltjes constants attached to Hurwitz-type Dirichlet series of the form $$ L(s, f, x)=\sum_{n \geq 0} \frac{f(n)}{(n+x)^s} $$ where $f$ is a periodic arithmetic function and $x\in\mathbb C\backslash\{0,-1,-2,\dots\}.$ We prove that these series admit meromorphic continuation to the whole complex plane, with at most a simple pole at $s=1$. We further derive functional equations satisfied by $L(s,f,x).$ Our main result gives an explicit formulas of the Stieltjes constants $\gamma_n(f, x)$ appearing in the Laurent expansion at $s=1$. In particular, we obtain explicit evaluations in the case of rational $x$. As an application, we derive explicit closed-form expressions for the first three generalized Stieltjes constants associated with nonprincipal real primitive odd Dirichlet characters arising from quadratic fields. These expressions are given in terms of the class number of the corresponding quadratic number field. These results extend and unify recent work of Blagouchine and others on generalized Stieltjes constants and Hurwitz-type $L$-functions.

Graph matrices that combine structural and metric information have attracted considerable attention in spectral graph theory. In this paper, we introduce the Degree–Adjacency–Diametrical (DAD) matrix of a connected graph $G$. The DAD matrix is defined by \[ (DAD(G))_{ij}= \begin{cases} d(v_i), & \text{if } i=j, 1, & \text{if } v_i\sim v_j, \operatorname{diam}(G), & \text{if } d(v_i,v_j)=\operatorname{diam}(G), 0, & \text{otherwise}, \end{cases} \] where $d(v_i)$ denotes the degree of the vertex $v_i$. We establish fundamental spectral properties of the DAD matrix by deriving trace identities, characteristic polynomials, and spectra for several classes of graphs. Furthermore, we obtain a spectral expression for the Wiener index of graphs with diameter two, derive upper bounds for the largest DAD eigenvalue, and establish lower bounds for the DAD energy. Explicit spectral characterizations are also obtained for complete multipartite graphs. The proposed matrix provides a unified spectral framework that simultaneously incorporates degree, adjacency, and diametrical information, thereby enriching the study of graph spectra and introducing a new graph matrix for spectral graph theory.

Let $G$ be a finite abelian group, and let $c_0(G)$ be the least integer such that every regular sequence over $G$ of at least this length is an additive basis. Let $\mathsf N(G)$ be the least integer $l$ such that every weak-regular sequence $S$ over $G$ with $|S|\ge l$ has a nonempty zero-sum subsequence $T$ satisfying $\mathsf v_g(T)=\mathsf v_g(S)$ for some $g\mid S$. We obtain a common upper bound for $c_0(C_q\oplus C_{nq})$ for groups of odd order, where $q$ is prime, and prove $\mathsf N(G)\ge r(n_r-1)+1$ when $G=C_{n_1}\oplus\cdots\oplus C_{n_r}$ with $1<n_1\mid\cdots\mid n_r$. If $n\ge1$ and $p\ge3$ is the smallest prime divisor of $np$, we obtain $\mathsf N(C_p\oplus C_{np^2})=2np^2-1$ and give upper bounds for $\mathsf N(C_p\oplus C_{np})$ when $p\nmid n$. We also determine $\mathsf N(C_n)$ for two infinite families of cyclic groups arising in earlier work.

We prove that for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This settles Smale's sixth problem for five bodies in the plane. Albouy and Kaloshin proved it for masses outside a closed algebraic subset of codimension two. We follow their method. Infinitely many central configurations would give an algebraic curve of complex solutions. At the places at infinity of the curve the valuations of the mutual distances must balance. We determine the possible valuation vectors at all positive masses, together with the mass conditions under which each can occur. We treat limits in which four bodies form a cluster of zero potential energy by expanding the potential, which is constant along the curve, at a place. This excludes, at every positive mass, one of the two diagrams in which such clusters occur, and part of the other. The same expansion excludes one further diagram at every positive mass. What remains is an exact analysis, by computer algebra, of the sets of mass conditions that can hold together at positive masses. Exact rational certificates show that the balance fails under each of them, except on a few one-parameter families of masses, which we treat separately. On the rays through $(1,4,4,4,4)$ and $(1,1,4,4,4)$ the certificates do not apply. On the first ray they cannot apply: there the complex equations have a continuum of solutions, as Albouy and Kaloshin observed. On these rays we count the real central configurations: there are 306 and 294 up to orientation-preserving similarity. The certificates, the programs that check them and a proof of the theorem in the Lean proof assistant are available.

Following the celebrated breakthrough in Bénard-He-Zhang, we prove Khintchine dichotomy for push-forwards of self-similar measures on $\mathbb{R}^d$ under uniform $C^{1+\alpha}$ maps. For this, we also study almost random walks on $\mathrm{SL}(d+1,\mathbb{R})/\mathrm{SL}(d+1,\mathbb{Z})$ and prove their effective equidistribution.

We study the SNR, aperture, and port count needed to meet a target outage in a single-user downlink with a fixed transmit antenna and a single-RF fluid antenna receiver. For known spatially correlated Rayleigh channels and ideal port selection, a sampling-free residual certificate controls reduced-rank outage error over a continuous SNR interval. Positive Bessel features support large arrays, and bounded importance sampling converts the certificate into simultaneous design intervals. At 1 bit/s/Hz and outage $10^{-5}$, a four-wavelength, ten-port receiver requires 4.40–4.55 dB. Reducing its port count from 256 to ten costs 0.70–1.25 dB under a joint 95% probability statement. At 3.5 wavelengths, ten ports require 4.90–5.05 dB. Integer-port screening across 3–8 dB shows how aperture and port requirements change with available SNR; an effective-SNR transformation extends these decisions to other rates. The reported intervals include model and sampling uncertainty under checked Gaussian covariance inputs. A constructive slow-growth rank bound and a 16–1024-port study support compact evaluation as port count grows.

Let $G$ be a connected Lie group equipped with a compatible right-invariant metric and let $G$ act continuously on a compact metrizable space $X$. We prove that $X$ admits a $G$-equivariant embedding into the right argument shift action on the space of one-Lipschitz maps from $G$ to $[0,1]$ if and only if the set of fixed points topologically embeds into $[0,1]$. This generalizes a theorem proved by Gutman, Huo and Tsukamoto for $\mathbb{R}^k$-flows.

We study pricing in a vehicle sharing network with $N$ locations and $C$ vehicles, motivated by on-demand bike, scooter, and car sharing, as well as robotaxi and ride sharing services. Customers request directed trips between locations, with travel times following general distributions, and leave without service if no vehicle is available. Pricing affects both demand for trips and the distribution of vehicles across locations. Computing an optimal dynamic pricing policy is challenging because the system state depends on the number of vehicles available at each location and in transit (including how long they have been traveling), motivating the use of simple static policies that offer a fixed price for each trip. For any number of vehicles and locations, we establish a tight performance guarantee for static pricing relative to optimal dynamic pricing. The guarantee holds for any concave objective functions (including revenue rate) and general travel time distributions. Our tight guarantee is bounded below by $ \frac{C}{C + \min \{ C -1, \sqrt{ CN} \} }$ (over 50%), achieving the first universal, non-asymptotic constant factor guarantee. We also prove that the optimal static policy can be efficiently computed by solving a convex optimization problem. Finally, we extend our performance guarantee to the setting with supply redirection.

We prove Pisanski's conjecture on quadrilateral embeddings of regular bipartite graphs. For integers $m\geq d\geq4$ with $4\mid md$, we construct both a connected simple $d$-regular bipartite graph $G$ with $m$ vertices in each bipartition class and an orientable quadrangulation. This graph's genus is \[ \ga(G)=1+\frac{m(d-4)}4. \] These quadrangulations attain the lower bound on the genus from Euler's formula. We give dihedral constructions for even $m$ and a cyclic construction for every $m\geq d$ with $4\mid d$. In each case, two cyclic permutations of a connection set of size $d$ determine all of the local rotations.

For every fixed real quintic polynomial $F$, we prove that the slow–fast Liénard system $\dot{x}=y-F(x),\quad \dot{y}=-\varepsilon x$ admits at most two limit cycles whenever $0<\varepsilon<\varepsilon_0(F)$ for a suitable $\varepsilon_0(F)>0$. Using scalar comparison near the equilibrium and a logarithmic fold-passage correction, we exclude \rev{canard singular limits}. Every sequence of limit cycles with $\varepsilon\to0^+$ has a subsequence converging in the Hausdorff metric to a common slow–fast path, whose regular slow arcs are either all normally attracting or all normally repelling. We also establish uniform Floquet separation for fixed $F$. The global counting argument uses shared regular arcs, nesting, and stability alternation. These results give a global small-parameter bound and describe the singular-limit structure, complementing the local cyclicity analysis of Li and Lu(J. Differential Equations, 2014).