Papers

We study an approximate system for the compressible Navier-Stokes-Korteweg equations in a bounded domain periodically perforated by small obstacles. As the size of the obstacles decrease much faster to zero than their mutual distances, we show in spatial dimension two and three that the limiting system remains unchanged. Our result applies for a large class of monotone and non-monotone pressure functions. In particular, it holds in the physically relevant case when the approximate system is used to describe the dynamics of a compressible viscous two-phase fluid extending the corresponding homogenization results known for the compressible Navier-Stokes equations in a single-phase setting.

Let $K$ be a compact Hausdorff space and let $E$ be an infinite-dimensional real Banach space. We prove that there is no continuous bijection $h\colon C_p(K)\to E_w$ whose inverse is continuous at $h(0)$. Consequently, $C_p(K)$ and $C_w(L)$ are not homeomorphic for any infinite compact Hausdorff spaces $K$ and $L$. This settles Krupski's problem and its two-space version due to Krupski and Marciszewski, and answers a question of Kąkol, Leiderman, and Michalak concerning $C_p([0,1])$ and weak Banach spaces.

We prove the absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-$1$ chains with nearest-neighbor interactions in which some of the quadrupolar couplings vanish, a class that is not covered by previous studies. Applying the technique of Shiraishi to these systems, we show that, for every model in this class on a periodic chain of $N$ sites, there is no $k$-local conserved quantity for any $3\le k\le N/2$. In particular, for a frustration-free spin-$1$ chain that exhibits spontaneous $U(1)$ symmetry breaking at zero temperature in one spatial dimension, we prove that every local conserved quantity with support up to half of the system size is a linear combination of the identity, the total magnetization $S^z$, and the Hamiltonian itself. This rigorously establishes that, unlike the Heisenberg ferromagnet, the model admits no local order parameter commuting with the Hamiltonian, so that its continuous symmetry breaking is enabled by the frustration-free structure rather than by a conserved order parameter. We also prove the absence of $k$-local conserved quantities for $3\le k\le N/2$ in the periodic Motzkin chain, a frustration-free spin-$1$ chain closely related to the original Motzkin chain, for which spontaneous $U(1)$ symmetry breaking at zero temperature has also been reported.

Starting from a model of $\mathrm{ZFC}$, we force the simultaneous failure of the Very Weak Club Guessing and the $\mho$ principles at $S_\kappa^{\kappa^+}$, for any regular cardinal $\kappa$. Our results are obtained by means of a forcing iteration technique, due to Krueger, that incorporates models as side conditions. At $\omega_1$, the failure of these principles is a well-known consequence of $\mathrm{PFA}$.

For $n\ge1$ and $\operatorname{Re}(x_j)>0$, we prove an alternating analogue of Mizuno's product formula: \begin{equation*} \prod\nolimits_{m=0}^{\infty} \left(\prod_{j=1}^{n}(m+x_j)\right)^{(-1)^m} = \frac{(\sqrt{\pi/2})^n} {\prod_{j=1}^{n}\widetilde{\Gamma}(x_j)}, \end{equation*} where $\widetilde{\Gamma}$ is the modified gamma function associated with the alternating Hurwitz zeta function. The polynomial factors are regularized using factorwise logarithms, and the formula establishes compatibility between this regularization and polynomial factorization. Applications include Gaussian and cyclotomic (Kurokawa–Wakayama type) evaluations; the single-factor case recovers the alternating Lerch formula, and a normalized ratio yields Wallis' product. A second group of results concerns an alternating double sine-type function. At nonexceptional positive periods, its positive-integer special values admit finite tangent-product representations. These products define locally meromorphic continuations $F_m$ in the period parameter. For $m\ge2$ and nonzero algebraic parameters, the continued values belong to $\mathbb P^1(\overline{\mathbb Q})$ at rational parameters and are finite, nonzero, and transcendental at nonrational parameters. We determine their orders at rational parameters and distinguish parameter continuation from fixed-period specialization at exceptional parameters. Additional results include Barnes factorizations of multiple alternating gamma functions, a Glaisher–Kinkelin special value, finite-product Stirling-type asymptotics, a symmetric two-parameter form of the known odd-integer multiplication formula, and an odd-scaling distribution relation for the alternating double sine-type function.

We prove quantitative pathwise stability estimates for one-dimensional stochastic differential equations driven by a common Brownian motion and a common symmetric $\alpha$-stable process, where $\alpha\in(1,2)$. The comparison is made under a synchronous coupling and is measured by $\sup_{0\le t\le T}\mathbb E|X_t-\wt X_t|^{\alpha-1}$. The perturbed drift and Brownian diffusion coefficients are spatially Lipschitz, while the perturbed stable jump coefficient may be Hölder continuous down to the critical exponent $1/\alpha$. The estimate is expressed in terms of the initial error and three law-weighted coefficient errors: the drift error $B$, the Brownian diffusion error $A$, and the stable jump-coefficient error $S$. The Brownian component produces a second-order correction term in Itô's formula. This term is controlled by a second-derivative estimate for a Komatsu-type mollification of $|x|^{\alpha-1}$. For stable jump coefficients with Hölder exponent $\wt\eta>1/\alpha$, the estimate gives explicit power rates in $B,A,S$; at $\wt\eta=1/\alpha$, it gives a logarithmic rate. The same method yields estimates with predictable forcing errors and time-uniform tail bounds via stopped quasi-martingales.

An implicit Euler finite-volume scheme for degenerate pseudo-parabolic cross-diffusion equations is proposed and analyzed. The system describes the dynamics of an unsaturated two-phase flow mixture with dynamic capillary pressure in a porous medium. The numerical scheme is based on a two-point flux approximation that preserves the energy structure, ensures the conservation of total mass, and guarantees strict positivity and boundedness of the water saturation. These properties rely on carefully selected mean functions for the nonlinear components. The existence and uniqueness of a discrete solution and additional mesh-uniform bounds are proved. Numerical experiments in two space dimensions illustrate the effects of the dynamic capillary pressure.

Inverse learning within a statistical framework has a wide range of applications. It has garnered significant attention in machine learning, artificial intelligence, and related fields, where the goal is to infer unknown parameters from indirect and noisy observations. This work investigates the stable approximation of $u^{\dagger}$ which solves the equation $Au=g$, with $A$ being a linear operator between appropriate vector spaces. We will consider the domain to be a non-reflexive Banach Space and the co-domain to be a space of real-valued functions on a metric space $X$. The function $g$ is characterized by a finite number of independently and identically distributed data points, which are assumed to follow some unknown probability measure $\rho$. We employ Tikhonov regularization with an arbitrary convex functional to obtain the regularized solution corresponding to the given data point. The convergence analysis is carried out with respect to the Bregman distance, and an upper bound for the error is derived in probability terms. The theoretical findings are then supported by numerical experiments.

We give a systematic account of the shape theory of $\infty$-topoi, viewing the shape of an $\infty$-topos as its generalized homotopy type. We establish the basic functorial properties of the shape, including preservation of colimits, descent, and homotopy invariance. We then prove that shape preserves cofiltered limits under compactness and perfectness hypotheses and establish Künneth-type formulas for products. Finally, we give conditions under which the shape of an $\infty$-topos determines its cohomology and homology.

We propose a data-driven method for designing state-feedback gains that achieve stabilization, $H_2$-control, and $H_\infty$-control for continuous-time systems. The state-input data are assumed to be corrupted by process noise, measurement noise, and input disturbances. We first characterize the set of all systems consistent with the noisy data using operator-based data embedding. This characterization yields necessary and sufficient conditions for data informativity under a certain class of noise. These conditions are formulated as linear matrix inequalities, and the feedback gains are constructed from their solutions. To enable direct controller design from noisy sampled data for continuous-time systems, we also obtain an upper bound on the reconstruction error of continuous-time signals.

The Tchoukaillon array is an infinite array of the positive integers, arising from a one-row Mancala solitaire, in which each positive integer occurs exactly once. Its zeroth column is the Flavius Josephus sieve and its zeroth row is the sequence of Tchoukaillon numbers; the asymptotics of these two edges are classical results of Andersson and of Broline and Loeb. On the basis of numerical evidence, N. Beluhov conjectured (as relayed by Knuth) that the general entry $T_{i,j}$ satisfies $T_{i,j} \approx (\pi i+2j)^2/(4\pi)$ as $i,j \to \infty$. We prove this conjecture. In fact we establish the stronger uniform estimate $T_{i,j} = (\pi i+2j+2)^2/(4\pi) + O((i+j+1)^{4/3})$, in which both constants $\pi$ and $2$ are produced by the array's own recursion through a Wallis product, independently of the two edge theorems. Equivalently, the square root of the entry is asymptotically linear, $\sqrt{T_{i,j}} = (\sqrt{\pi}/2)\, i + (1/\sqrt{\pi})(j+1) + O((i+j+1)^{1/3})$, the linear blend of the two edge growth-rates. As corollaries we obtain that the level regions $\{T_{i,j} \le V\}$ are triangles up to a boundary of width $O(V^{1/6})$, and an $O(\sqrt{M})$ algorithm that locates the row and column of a given integer $M$.

We study exponential-scale fluctuations of the Hardy-Szegő zero process, investigated by Peres and Virág in the disk setting, in its upper-half-plane realization, which reveals a different probabilistic geometry. This conformally invariant determinantal zero process is equivalent to its disk realization, but the upper-half-plane coordinates make real-translation invariance explicit and single out long horizontal windows as natural observables. For the vertical window from height one to height $a>1$, let $N_a(L)$ denote the number of zeros in the corresponding horizontal window of length $L$. We identify the limiting scaled log-moment generating function explicitly. As a consequence, we prove a large deviations principle for $N_a(L)/L$, with rate function given by the Legendre transform of this limit. We also prove a strong Szegő expansion for the log-moment generating function, including an explicit order-one correction, locally uniformly in the natural complex strip. The proof uses the planar determinantal structure before projection, reduces the problem to a one-dimensional Fredholm determinant, and combines fixed-power trace asymptotics with a Wiener-Hopf comparison.

For every finite-rank crystallographic affine Coxeter system $(W,S)$ and Coxeter element $c$, we construct a minimal linear graded free resolution of the trivial module over $k[M([1,c]_T)]$ supported on a rectified exceptional cluster complex. Hence the affine dual braid monoid algebra is Koszul over every field $k$. The exceptional complex is introduced to recover the principal-fibre topology missing from the direct Reading–Stella labelling. Half-orbit rectification replaces the transjective root labels by ordinary exceptional modules, so that a face $F$ determines an exceptional wide subcategory and the intrinsic weight \[ \omega(F)=\operatorname{cox}(\operatorname{wide}\langle F\rangle). \] The resulting principal fibres are induced subcomplexes and split canonically as joins of subcomplexes attached to connected Dynkin and affine blocks; these subcomplexes are contractible. Affine non-lattice divisibility creates the genuinely nonprincipal case. The McCammond–Sulway completion shows that whenever no greatest interval right divisor exists, all maximal interval right divisors share a common nontrivial complete finite Coxeter component. In the associated exceptional-wide decompositions, this common Coxeter component is the Coxeter element of a Dynkin block, and the subcomplex attached to that block occurs as a common contractible join factor. Thus every nonidentity fibre is contractible, and the weighted-face complex is exact, minimal and linear. In particular, $\operatorname{Tor}^{A_c}_q(k,k)$ is indexed by $q$-vertex exceptional cluster faces in internal degree $q$, and $\operatorname{pd}_{A_c}k=|S|$.

By means of the theory of compound (non-homogenoeus) Poisson processes we define a general framework which includes most of the generalizations of the Poisson processes, both in the Skellam sense and in the space-fractional sense. We prove that Bernstein subordination is the only time-changing leading to a compound Poisson process. We also consider the case of subordination with inverse Bernstein subordinators. Then, we focus on the integrals of compound processes. We give an explicit representation of the fractional integral of compound Poisson process, showing that for fixed t it is distributed as a compound Poisson random variable. We then extend this result to more general integral forms. Furthermore, we obtain some limit results, explicit forms of the iterated integrals and their governing equation. Finally, we study the integral of compound renewal processes in the Fourier-Laplace domain.

We show that the $ \mathrm{K}_1 $-functor modeled on locally isotropic reductive groups is hypoabelian if the base ring has finite Bass–Serre dimension and, for the Tits index $ \mathsf{E}_{8, 2}^{78} $, contains a field. For classical or globally isotropic reductive groups schemes $ \mathrm{K}_1 $ is actually solvable. This implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

In this paper, we derive some bounds on numerical radius of the weighted shift operator $T$ with weights $(1,sq,q^2,tq^3,q^4,sq^5,q^6,tq^7,\ldots)$ where $s,t >0$ and $0<q<1$. Furthermore, we provide an entire function $F_T(z)$. The reciprocal of the minimal positive root of $F_T(z)=0$ gives the numerical radius of $T$. These results generalize several previously known results on the numerical radius of weighted shift operators discussed in \cite{chakraborty2025numerical}.

Motivated by an analogue of pancyclicity, we study minimum-degree conditions ensuring that a $2$-connected graph $G$ of order $n$ contains a $2$-connected subgraph of every order $\ell\in\{4,5,\ldots,n\}$. Yin and Wu [A minimum degree condition for a 2-connected graph containing all possible orders of 2-connected subgraphs, Discrete Appl. Math. 387 (2026), 129-136] initiated the study of this problem and showed that the condition $\delta(G)\ge \lceil n/3\rceil+1$ is sufficient. Kashima conjectured that the condition $\delta(G)\ge \sqrt{3n}$ is sufficient. In this paper, we prove that every $2$-connected graph $G$ of order $n$ with $\delta(G)\ge 2n^{2/3}+6n^{1/3}+2$ contains a $2$-connected subgraph of every order from $4$ to $n$. In particular, this gives the first sufficient minimum-degree condition of sublinear order in $n$.

This paper first investigates the propagation dynamics of solutions to the Cauchy problem for a one-dimensional reaction-diffusion equation in a spatially periodic environment consisting of two distinct patch types. The novelty of this work lies in the systematic analysis of a KPP-bistable heterogeneous framework. In this setting, the respective patch lengths, the linear stability of the zero solution and the positive periodic steady state, and the magnitude of the initial data play crucial roles in the long-time dynamics. We first establish persistence properties of the species, showing that uniform persistence holds when the zero steady state of the associated periodic patch model is unstable, while local persistence is obtained under additional suitable conditions. Using a dynamical systems approach, we further establish spreading properties and demonstrate the existence of pulsating traveling waves in two different cases, depending on whether the trivial solution is unstable or stable. Finally, we present two sets of sufficient conditions characterizing species extinction.

Regularized sparse regression has been extensively studied in the offline setting, but online formulation remains relatively under-explored. This gap stems from four key challenges: (i) the infeasibility of dynamically updating the regularization parameter in every online round, (ii) managing storage and memory complexity, (iii) enabling real-time computation via closed-form updates rather than solving full optimization problems at each round, and (iv) achieving optimal statistical guarantees under realistic assumptions. In this paper, we propose an online generalized-sparsity-constrained regression framework, focusing on online cardinality-constrained linear regression and low-rank matrix sensing. Unlike online regularized regression, our constrained formulation eliminates the need for dynamic parameter tuning. We introduce an efficient online hard-thresholding algorithm that performs closed-form updates and requires storing only summary statistics, making it computationally, memory, and storage efficient. Despite the inherent nonconvexity and combinatorial nature of the formulation, our algorithm achieves global convergence at the optimal statistical rate under realistic assumptions, provided that the projection set is properly overparameterized. Numerical experiments demonstrate that our method consistently outperforms state-of-the-art alternatives.

Sparse optimal control is considered in the infinite horizon. In the literature, sparse control has been considered mostly in a finite horizon for its formulation into a finite-dimensional optimization problem. It is shown in this paper that an optimal solution of the infinite-horizon sparse control problem can be obtained through a solution of some finite-horizon subproblem. This is due to sparsity of the optimal solution in the sense that the optimal control input is constantly equal to zero at its tail. An estimate is given on the horizon length required by this subproblem and its adaptive choice is also discussed. Implementation with a receding-horizon technique is considered and its optimality and sparsity are guaranteed.

We study the persistence probability of fractional Lévy fields, i.e. the analogue of fractional Brownian motion with generalised (multi-dimensional) index sets. First, we compute the persistence exponent of the hyperbolic fractional Lévy field. The result matches the rate obtained in Molchan (1999) for Euclidean space and the one in Aurzada/Helmer (2026) for the sphere. This enables us to study persistence for fractional Lévy fields indexed by a large class of Riemannian manifolds (whenever that process exists) through a local comparison argument with the spherical and hyperbolic case.

In this paper, we construct an infinite family of infinite order corks admitting Stein structures. This answers a question raised by Akbulut and Gompf.

This paper addresses the problem of efficiently estimating aggregate human preferences by dynamically adapting questions based on respondents'answers. To this end, we formulate and address two sub-problems: preference estimation and question design. First, regarding preference estimation, we model respondents' preferences and estimate them using Bayesian estimation, employing a particle filter as a computationally efficient approximation. The main theoretical contribution to this sub-problem is to analyze the preference estimation error using an information-theoretic approach, deriving a theoretical lower bound for the error. Second, regarding question design, we formulate the design problem as an Expected Information Gain maximization problem and employ an epsilon-greedy strategy to solve the problem in a computationally efficient way. We theoretically analyze the search efficiency of the approach, demonstrating that it achieves higher efficiency than a random search. Finally, we verify the effectiveness of the proposed method through numerical simulations.

We prove the Feldman–Ilmanen–Knopf conjecture for finite time Type I singularities of the Kähler–Ricci flow. More precisely, for any compact Kähler manifold $Y$ and its blow-up $\pi:\operatorname{Bl}_pY\longrightarrow Y$, if $[\omega_0]-Tc_1(M)=\pi^*[\omega_Y]$, then any Type I parabolic blow-up limit of the Kähler Ricci flow along the exceptional divisor is the FIK shrinker $\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{n-1}}(-1))$.

We prove that, for a suitable small external force in $\dot H^{-2}(\mathbb R^2)$, the stationary Navier–Stokes equations on $\mathbb R^2$ admit multiple solutions in $L^2(\mathbb R^2)$.

Purcell's swimmer is a well-known planar model of a swimming microorganism, governed by low Reynolds number hydrodynamics, which is comprised of three rigid links connected by actuated rotary joints. This model has been analyzed as a robotic locomotion system governed by first-order nonlinear dynamics with a periodic input (gait) of the two joint angles. In this work, we present a robotic macro-scale realization of this three-link swimmer moving in a highly viscous fluid. We propose a simple variant of Purcell's theoretical model with non-slender links and a central rigid sphere which represents the added drag of the robot's central flotation block, and calibrate the model's parameters to fit experimental measurements. Next, we apply optimal control formulation based on Pontryagin's Maximum Principle (PMP) in order to find optimal gaits that maximize the displacement per cycle under bounds on the joint angles. Employing a differential geometric method that transforms the problem to area integral enclosed by the gait trajectory in the plane of joint angles, enables visual interpretation which explains topological changes in displacement-optimal gaits upon varying the bound on the joint angles. We then apply PMP formulation to the problem of maximizing Lighthill's energy efficiency in order to obtain a boundary value problem (BVP) whose solution gives efficiency-optimal gaits for Purcell's swimmer model, as well as its variant with a central sphere. Finally, we utilize numerical methods such as parameterizing the input gait as a truncated Fourier series, as well as GPOPS-II solver, to produce sufficient initial guess values for solving the BVPs and obtaining efficiency-optimal gaits.

We study a Brownian time change on the unit square whose speed measure is constructed from a Dirichlet eigenfunction expansion with independent non-Gaussian coefficients. For $0<\gamma<\sqrt2$, the measure is obtained by a second-moment martingale argument. Finite coefficient translations induce coherent exponential tilts of the speed measure, and conditioning on the complementary coefficients gives positive Lebesgue densities on every finite-dimensional orbit. Unitary transport along these orbits gives a common-domain analytic family and explicit first-order cluster derivatives. A first-order splitting argument proves almost-sure simplicity, while the local eigenfunction-square identity and a Vandermonde argument give joint densities for all finite vectors of ordered positive eigenvalues.

Moving least-squares approximation is a popular method for approximating multivariate functions from given discrete data. For higher accuracy higher degree polynomials have to be used, resulting also in higher computational cost and numerical instabilities. Recently, the combination of low-order moving least squares with a multilevel scheme showed superior numerical behavior. In this paper we will prove, amongst other things, that such a combination of moving least-squares with a multilevel scheme indeed leads to improved convergence results, at least if the data sites form a regular grid.

We study two open path problems with prescribed endpoints posed by Bang-Jensen and Gutin. The first asks for a longest $(x,y)$-path in a semicomplete digraph. The second asks whether a locally semicomplete digraph has a Hamiltonian $(x,y)$-path. For semicomplete digraphs, we solve the first problem when the endpoints lie in different strong components. We also prove that if a non-Hamiltonian longest $(x,y)$-path omits a set of vertices, then these vertices together with $x$ and $y$ have a Hamiltonian $(y,x)$-path. This gives an equivalent cycle problem. We then give an exact algorithm that runs in polynomial time when the number of omitted vertices is fixed. For locally semicomplete digraphs, we determine the possible endpoints in the connected nonstrong case. Known results then leave only strong, nonsemicomplete, non-$4$-strong digraphs unresolved. Every such digraph of order at least five has a strong vertex cut of size at most three. Two examples show that a spanning directed path together with a vertex-disjoint directed cycle is not sufficient, and that vertices of one strong component need not occur consecutively on a Hamiltonian path.

We study algebras with straightening laws on join- or meet-semidistributive lattices. We show that for a join-semidistributive (resp. meet-semidistributive) lattice, meet-distributivity (resp. join-distributivity) and Cohen–Macaulayness are equivalent, and that integrality implies these conditions. Thus, Hibi's conjecture that every integral lattice is Cohen–Macaulay holds for join- or meet-semidistributive lattices. For semidistributive lattices, distributivity, integrality and Cohen–Macaulayness are equivalent.