Papers

We propose a Bayesian framework to quantify discretization errors in numerical solutions of ODE models based on observational data. The discretization error is modeled as a random variable, and its mean-referred to as the discretization error mean-is inferred from the observations. By introducing a Markov prior on the temporal evolution of the discretization error mean, we formulate the problem as a state-space model with a linear Gaussian observation process, which enables efficient inference via the Ensemble Kalman Filter. We also propose a specific form of a Markov prior motivated by classical discretization error analysis, in which global errors accumulate from local errors. The proposed prior depends on the step size of a numerical solver, and we establish its convergence rate in probability as the step size tends to zero. Numerical experiments on the pendulum system and the FitzHugh-Nagumo model demonstrate the effectiveness of the proposed approach.

Remarkable breakthroughs [25,43] established the so-called strong asymptotic freeness between classical Gaussian random ensembles and their semicircular free counterparts. Along the same lines, the Lehner formula [31], associated with the free counterpart, precisely determines the spectral edges of Kronecker-Gaussian matrices. We here revisit and study this formula without the utilization of random matrix theory and spectral methods. In particular, relying on concepts utilized within Random Duality Theory (RDT) [45,46,51], we reconfirm Lehner's formula and effectively reprove the key asymptotic freeness results obtained via spectral methods in [25,43].

We construct an explicit negative plurisubharmonic function on the unit polydisk in $\mathbb{C}^n, n\geq 2$ with an isolated singularity at the origin. Its Monge-Ampère measure is the Dirac mass at the origin, but its Lelong number vanishes. This gives a negative answer to the zero-mass conjecture of Guedj-Rashkovskii in every complex dimension at least two.

Using tools from topological dynamics, H.~Furstenberg introduced the notion of central sets and established the celebrated Central Sets Theorem. The sets which satisfy the conclusion of the Central Sets Theorem are called $C$-sets. Hindman, Maleki, and Strauss first brought the concept of an important type of $C$-sets called the quasi-central sets which are not central sets. In 2017, A. Ghosh gave the combinatorial treatment of $C$-sets in commutative adequate partial semigroups, where $C$-sets are the sets which satisfy the conclusion of Central Sets Theorem for commutative adequate partial semigroups. In this work, we discuss the Quasi-central sets algebraically and dynamically for commutative adequate partial semigroups. We give dynamical characterization of members of idempotent ultrafilters for commutative adequate partial semigroups, also we study the minimal dynamical systems for an adequate partial semigroup.

This thesis consists of two distinct projects situated in the areas of smooth dynamics and spectral theory, respectively. They are united by a common interest in mechanisms of chaos and statistical behavior in classical and quantum dynamical systems. The first concerns smooth dynamics. We prove that all ergodic linear automorphisms of the N-dimensional torus with two-dimensional center are stably ergodic, including all ergodic automorphisms in dimensions $N \leq 5$ and $N = 7$ . This generalizes a previous result of Rodriguez-Hertz, which required an additional algebraic condition on the characteristic polynomial of the linear automorphism. The second project deals with spectral theory of Schrödinger operators. We prove that delocalization of most eigenvectors is topologically common in the space of deterministic Schrödinger Operators on a given large finite graph, provided that the IDS satisfies a suitable regularity condition. This result generalizes a recent theorem of Avila and Damanik. We also describe a flexible family of graphs satisfying our criterion, by proving a variant of the Thouless formula.

The main result of the paper is that an inductive argument is established to prove the Fuglede's conjecture for an infinite sequence of square-free order cyclic groups. The tile-to-spectral direction of the conjecture holds for all square-free cyclic groups, whereas we prove the spectral-tiling direction only for those groups among them whose prime factors grow rapidly. To achieve this, we develop an inductive technique with roots in a previous paper by the author co-authored with Fallon, Kiss, and Mayeli. Following recent preprints, Fuglede's conjecture remains open only for finite cyclic groups, and until now, there was no cyclic group for which the conjecture was known that possessed an arbitrary number of distinct divisors.

In this paper, we investigate the local boundary behaviour of a recently developed hyperbolic-type metric $m_D$. First, employing a boundary-flattening technique and local behaviour of $m_D$-geodesics, we establish its asymptotic formula near any $C^1$-smooth boundary point. Next, we introduce a metric quantity analogous to the Nikolov–Andreev metric and show that $m_D$ is the inner metric associated with it. Finally, by establishing a sharp two-sided comparison inequality, we obtain an improved lower bound for the $m_D$-metric.

We study (bounded) finitely additive measures on the group of integers $\mathbb Z$, as elements of the Banach algebra $\mathrm{ba}(\mathbb Z)$, viewed as a natural generalization of ultrafilters. The algebraic structure of $\mathrm{ba}(\mathbb Z)$ extends the semigroup structure of the Čech–Stone compactification, allowing methods from ultrafilter theory to be applied in a broader measure-theoretic setting. We investigate idempotent finitely additive measures and establish additive properties of subsets of $\mathbb Z$ having positive measure. We then proceed to study almost translation-invariant and translation-invariant finitely additive measures, showing that these stronger notions yield correspondingly stronger additive conclusions. In particular, we prove that every subset of $\mathbb Z$ whose measure exceeds a certain explicit threshold necessarily is an $\mathsf{IP}_{n}$-set; with stronger properties and lower thresholds depending on the properties of the relevant measures. Several examples illustrating the sharpness and limitations of the results are also presented, together with a discussion of open problems and directions for future research.

We propose a novel facial reduction algorithm tailored to semidefinite programming relaxations of combinatorial optimization problems with quadratic objective functions. Our method leverages the specific structure of these relaxations, particularly the availability of feasible solutions that can often be generated efficiently in practice. By incorporating such solutions into the facial reduction process, we substantially simplify the reduction steps. On average, our facial reduction algorithm is four times faster than the standard implementation on the considered benchmark sets, providing significantly improved preprocessing for SDP relaxations in combinatorial optimization.

In this article, we first introduce the notion of BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash biproduct over a BiHom-Hopf algebra, denoted by $D\natural H$, where $m,n,p,q,s,t,u,v\in \mathbb{Z}$, and give the sufficient condition for $D\natural H$ to be a BiHom-bialgebra. Furthermore, we describe the concept of BiHom-$\binom{m,n,p,q}{s,t,u,v}$-Yetter-Drinfel'd-Long bimodule via BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash biproduct bialgebra, and prove that the category $\mathcal{LR}(H)(m,n,p,q)$ of BiHom-$\binom{m,n,p,q}{s,t,u,v}$-Yetter-Drinfel'd-Long bimodule is a strict braided monoidal category. Finally, for a finite-dimensional BiHom-Hopf algebra H, \(\mathcal{LR}(H)\binom{m,n,p,q}{s,t,u,v}\) is isomorphic to the BiHom-$\binom{s,t}{p,q}$-Yetter-Drinfel'd category \({}_{H\otimes H^*}^{H\otimes H^*}\mathcal{YD}\binom{s,t}{p,q}\) as braided monoidal categories.

Quantifying the association between a real-valued variable and a categorical variable is a fundamental task in data analysis. Existing methods often rely on parametric assumptions or arbitrary integer encoding, which may lead to unstable results. We propose a label-invariant population measure of association, $\xi'$, specifically designed for the mixed real-valued-categorical setting. The proposed measure is normalized between 0 and 1; it equals 0 if and only if the variables are independent and 1 if and only if the categorical variable is a measurable function of the real-valued one. We also introduce a corresponding sample estimator, $\xi_n'$, computable in $O(n \log n)$ time. These measures are invariant to permutations of category labels and strictly monotone transformations of the real-valued variable. We establish the strong consistency and asymptotic normality of the estimator $\xi_n'$, enabling a computationally efficient, permutation-free Wald test for independence, and an asymptotic confidence interval for the population measure $\xi'$. Extensive simulations and an application to The Cancer Genome Atlas (TCGA) data demonstrate that the proposed method provides coding stability, competitive power, and substantial computational advantages in nominal mixed-type settings.

Let k be an integer at least 2, and let D_{N,k} be the set of permutations of {1,...,N} whose descent set is exactly {k, 2k, ..., k*floor((N-1)/k)}. We enumerate the elements of D_{N,k} avoiding each classical and each vincular pattern of length three. For classical patterns, we give recursive bijections from the 132- and 231-avoiding classes to ordered forests of complete k-ary trees, obtaining the Raney number. The 213- and 312-avoiding classes are obtained from these forest bijections by completing the last block and applying reverse-complement symmetry. The remaining classical case 321 is expressed by RSK. For vincular patterns, we enumerate the six fully consecutive patterns and the twelve patterns with exactly one adjacency. The closed-form results are accompanied by bijective models: the Catalan, Raney, Fuss-Catalan, and RSK cases are natural k-ary or fixed-descent extensions of classical bijections, while the product and poset cases arise from block-insertion and record/tree-poset encodings forced by the adjacency conditions.

We study the Superiorization Methodology (SM) in the context of the General Dynamic String-Averaging (GDSA) method in the inconsistent case (that is, where the input operators don't have a common fixed point) which primarily aims at achieving convex feasibility while simultaneously reducing an objective function. In many scientific and real-world problems modeled as constrained minimization tasks, striving for the exact constrained optimum can be costly in terms of time, energy, and resources. Therefore, applying the SM can offer a practical and efficient alternative. In particular, we present a new "theorem of alternatives" for the superiorization method which leads to investigation of theoretical conditions under which the superiorized version of the GDSA algorithm converges to a "superior" feasible point, i.e., one with an objective function value that is smaller or equal to that produced by the unperturbed feasibility-seeking algorithm. While this question has only been partially addressed in the existing literature, we present new sufficient conditions that guarantee that the SM attains such a superior outcome.

We construct segregated bubbling solutions for a two-component critical Schrödinger system of Brezis–Nirenberg type in a smooth bounded domain $\Omega\subset\mathbb R^N$, $N\geq5$, with any fixed competitive coupling $\beta<0$. Suppose that the Robin function has two distinct prescribed critical points, each satisfying a local degree condition. For every sufficiently small $\epsilon>0$, the system admits a nonnegative weak solution with both components nontrivial. Each component has a single-bubble profile and concentrates at one of the prescribed points. Each component also vanishes identically in a ball centered at the other concentration point; after rescaling by the natural bubble length, the radius of this ball tends to infinity. The main obstruction is that $p=N/(N-2)\in(1,2)$, so the gradient of the interaction potential $(s,t)\mapsto |s|^p|t|^p$ is not differentiable when one component vanishes and the other is nonzero. Hence the usual global Lyapunov–Schmidt reduction cannot be applied directly. We first solve a nonlinear exterior problem variationally. The resulting dead cores remove the cross-component coupling from the inner localization regions, where a projected critical reduction can then be carried out. The remaining scale and center equations are solved by Brouwer degree theory.

We introduce and study the category of d-spectral spaces, a bitopological analogue of the classical spectral spaces of Stone and Hochster. A d-spectral space is a compact, d-sober bitopological space such that both open set lattices are coherent frames, where d-sobriety is the bitopological notion of sobriety due to Jung and Moshier. We show that the category of spectral spaces embeds into the category of d-spectral spaces as a simultaneously reflective and coreflective full subcategory. Moreover, we prove that d-spectral spaces are precisely the spectra of d-lattices. Key to this result is the d-lattice of compact open sets associated to a d-spectral space and the spectrum construction for d-lattices. We also show that the patch space of a d-spectral space is d-Boolean and that the de Groot dual of a d-spectral space is again d-spectral, mirroring the corresponding classical properties of spectral spaces. Our results demonstrate that d-spectral spaces form a natural and well-behaved bitopological extension of the spectral space framework.

This paper introduces a novel approach to the General Assembly Line Balancing Problem (GALBP) by utilizing a receding horizon optimal control framework. The proposed discrete model for the assembly line offers a flexible representation, avoiding assumptions about specific line configurations. The control actions aim to optimize the industrial assembly line by minimizing the completion time while adhering to constraints such as task precedence, workstation capacity, and resource requirements. Control actions are represented through task assignment and resource allocation matrices, assigning tasks to specific workstations and assigning resources to workstations, respectively. The optimization problem is formulated as a Mixed-Integer Nonlinear Programming (MINLP) problem. The inherent robustness of the receding horizon approach ensures optimal solutions for the assembly line, effectively adapting to sudden changes. Numerical experiments demonstrate the robustness and effectiveness of the proposed control synthesis in efficiently distributing tasks and resources, minimizing the overall completion time.

Magnetic mirrors are among the conceptually simplest plasma confinement configurations and remain promising candidates for thermonuclear fusion. Their design requires shaping an externally applied magnetic field to confine plasma within an open-ended cylindrical device. In contrast to toroidally closed devices such as tokamaks and stellarators, confinement in magnetic mirrors depends intrinsically on kinetic mechanisms, particularly velocity-space trapping and particle loss through the open ends. We formulate magnetic mirror design as a PDE-constrained optimization problem governed by a reduced multispecies drift-kinetic-Poisson model. The resulting optimization reveals two physical effects not captured by the classical loss-cone argument. First, the self-consistent electric field generated through Poisson coupling acts as a secondary confinement barrier and substantially alters particle retention in the nonlinear regime. Second, the optimized magnetic-field configuration depends qualitatively on the underlying kinetic model: an electron-only model favors an unconventional centrally peaked field, whereas the fully coupled electron-ion model recovers the classical boundary-peaked mirror configuration. These results demonstrate that optimal magnetic mirror design cannot be determined solely from loss-cone considerations, but must account for the self-consistent nonlinear kinetic dynamics of the plasma.

We construct two fixed disjoint families $H_0,H_1$ of stationary ergodic binary process distributions for which a weakly asymptotically consistent test exists, but no strongly asymptotically consistent test exists. The construction combines a synchronizing binary suspension code, an independent i.i.d marker process, and countably many independent slowly switching two-state Markov chains. In particular, this disproves the asymptotic-consistency branch of a conjecture of Ryabko.

We continue the study of Phase Retrieval in the Quantum Information setting in the style of \cite{liu2023phase}, in a manner which is more general in two primary ways: (a) Instead of only pure states we consider also states (and positive semidefinite operators) of bounded rank, as is done in Quantum Tomography; (b) We consider general super operators instead of only quantum channels. We show that in order to have phase retrieval with respect to positive semidefinite operators of bounded rank, it suffices to discriminate between perfectly distinguishable pairs (i.e. those with orthogonal supports). From this we prove that phase retrievability is always Lipschitz stable with respect to the trace norm, and then we use the Fuchs-van de Graaf inequalities to deduce stability with respect to the Bures–Wasserstein distance (thus generalizing the known stability results for phase retrieval in Frame Theory). For Hermitian-preserving super operators taking values in block-diagonal matrices, we characterize phase retrievability in terms of conditions inspired by the classical complement property from Frame Theory.

The Jacobi triple product identity provides closed-form expressions for many infinite-product generating functions that arise naturally in combinatorics and number theory. A particularly important application is to Dedekind's eta function {\eta}({\tau}), which it expresses as a theta function. Using the theory of modular forms, Oliver [5] classified all eta quotients that are theta functions. In this paper, we present elementary proofs of the identities established by Oliver.

In this paper, we investigate the evolutionary processes of hypersurfaces within half of the $(n+1)$-dimensional light-cone. Depending on the evolutionary processes, our focus extends to exploring variational problems associated with a smooth function $f(S_1,\cdots,S_n)$, where each $S_r$ denotes the $r$-th elementary symmetric polynomial, defined as the sum of all possible products of $r$ distinct principal curvatures. We present several fundamental properties related to these variational problems. Furthermore, we examine a curvature-type flow defined locally within the light-cone, establishing its perpetual existence and smooth convergence to a circle whose length is preserved and equal to that of the initial curve.

This paper is concerned with the spatial propagation dynamics of time-delayed nonlocal diffusion equations in shifting habitats. We employ the theory of viscosity solutions for Hamilton-Jacobi equations to provide a complete classification of the spreading speeds. In particular, we derive variational formulas that explicitly characterize how these speeds depend on the decay rate of the initial data, the habitat shifting speed, and the maturation delay across three regimes: locally determined, nonlocally selected, and locked. We reveal a distinct directional asymmetry in the nonlocal selection mechanism and a novel delay-insensitivity phenomenon. We also establish the threshold conditions under which the habitat locking effect eliminates the classical decelerating role of time delay.

Domain generalization (DG) aims to learn from multiple source domains and generalize to unseen target domains. Most DG methods pursue invariance: they seek a causal representation whose prediction rule is invariant across domains. This principle is effective when the causal mechanism is stable, but becomes restrictive when the domain itself modulates how causal content maps to the response. In this case, directly feeding domain style into the predictor can create misleading shortcuts, since style does not by itself cause the response. Yet the apparent chaos of multiple styles can become a ladder: style can locate the unseen target domain among source domains and guide which domain-dependent prediction rules should be trusted. We propose Latent Adaptive Domain Disentanglement and Environment Reweighting (LADDER), a fixed-model DG pipeline that learns causal/style representations, freezes the encoders, fits source-specific classifiers, and uses an unlabeled target-domain covariate set only at inference to compute weights over these fixed classifiers, with no target labels or model-state updates. We establish theoretical guarantees for source reweighting and validate LADDER on simulations, FMoW, and a location-grouped iWildCam protocol, with gains in overall and group-averaged accuracy.

This paper addresses infinite-horizon continuous-time stochastic linear quadratic optimal control problems with regime switching. We propose a paradigm shift from model-based design by adopting an adaptive dynamic programming approach, specifically developing on-policy and off-policy Q-learning algorithms that learn the optimal controller solely from online state trajectory data. The theoretical core of our work consists of a complete proof of the equivalence between the on- and off-policy architectures, alongside a rigorous analysis establishing the stability of the closed-loop system and the convergence of the algorithms to the optimal solution. For computational tractability, we implement these algorithms using vectorization and Kronecker product algebra. The theoretical results are corroborated by numerical case studies that clearly demonstrate the operational effectiveness and practical feasibility of the proposed model-free control strategy.

We develop a representation–symbol correspondence for $q$-Hamiltonian mechanics on the quantum plane. The coordinate algebra and its covariant differential calculus are realized simultaneously on a smooth commutative function space by multiplication, dilation, and Jackson operators. Normal ordering identifies the coordinate algebra with a polynomial symbol space and transports operator composition to an explicit associative star product. Under this correspondence, the induced covariant $q$-derivatives intertwine exactly with the Jackson operators, and the ordered Hamiltonian action descends to an exact star-Jackson action. For the coordinate observables, the relevant Jackson derivatives are constants, so the star factors reduce to the unit and the Jackson coordinate equations are exact symbol images of the formal quantum-plane equations. For nonlinear observables, pointwise multiplication produces a genuine commutativization error. We quantify this distinction through a first-order expansion of the operator quantization map, study the derivation, divergence, and energy defects of the induced pointwise Jackson dynamics, and prove convergence of the represented Hamiltonian, the symbol actions, the Jackson vector field, and its finite-time trajectories to their classical counterparts as $q\to1$. The resulting framework connects covariant quantum-plane differential calculus, deformation products, and Hamiltonian dynamics while keeping exact algebraic statements separate from commutative and classical approximations.

For $n\ge1$, let $F(n)$ be the least $H$ such that any $H$ consecutive integers contain $n$ pairwise distinct integers $a_1, a_2, \dots, a_n$ with $k \mid a_k$ for $1\le k\le n$, and define $h_{\mathbb P}(n)$ analogously for the primes at most $n$. We prove \[ F(n)\le n^{4/3}\exp\!\left(O\!\left(\frac{\log n}{\log\log n}\right)\right), \qquad h_{\mathbb P}(n)\ll \frac{n^{4/3}}{(\log n)^{1/3}}, \] and \[ F(n) \ge h_{\mathbb P}(n)\ge n\exp\!\left( \left(\frac{\log 2}{2}-o(1)\right) \frac{\log n}{\log\log n} \right). \] The upper bounds follow from a new estimate for unions of arithmetic progressions. The lower bound adapts a quadratic-residue compression construction of Green and Ruzsa.

We give an answer to a question of Xu and Ye (Disjointness with all minimal systems under group actions, to appear in Israel J. Math., arxiv:2212.07830) on the denseness of distal points in the Bernoulli shift $2^G$ for a countable discrete group $G$. For a related but stronger notion of almost automorphic points, we answer the similar question by showing that the corresponding collection of the groups coincides with maximal almost periodic ones. These characterizations allow us to construct 2-step nilpotent groups for which the answers to the Xu-Ye question differ. In search for an intrinsic answer to the Xu-Ye question, we introduce a notion of point-distal radical for a countable discrete group and show that a necessary condition is for the point-distal radical to be trivial. Finally, we consider some related questions, and show that the collection of all countable groups $G$ for which the set of distal points is dense in $2^G$ is closed under finite-index extension, and that the collection of countable groups $G$ for which the constant sequences are the only distal (almost automorphic) points coincides with the minimally almost periodic ones.

At parking speeds, the kinematic bicycle is the prevailing model for car-like vehicles. Yet, despite its wide use, stabilizing feedback laws for this system are scarce in the literature, and existing designs often do not reproduce realistic parking maneuvers. This limitation is inherent to the Cartesian coordinates, where Brockett's condition rules out smooth static feedback stabilization. We bypass this obstruction by transforming the system into polar coordinates together with additional range-normalized coordinates that encode the geometry of human-like parking maneuvers. In the transformed coordinates, the dynamics take a strict-feedback form, enabling a nonconventional backstepping design. We exploit the particular structure to develop smooth feedback laws that achieve global exponential stabilization in the transformed coordinates which in turn generates parking trajectories resembling the one performed by human drivers through feedback alone.

Given positive lengths $a$ and $b$ and a positive integer $k$, let $c$ be determined by $$ \frac{1}{c^k}=\frac{1}{a^k}+\frac{1}{b^k}. $$ We prove that a single finite unmarked-straightedge-and-compass construction producing $c$ for every pair $a,b$ exists if and only if $k$ is a power of two. Necessity follows by specializing to $a=b=1$ and applying the algebraic-degree obstruction to $2^{1/k}$; sufficiency is established by a recursive construction using third, fourth, and mean proportionals. We give an explicit construction for $k=4$ and its iteration for $k=8,16,\ldots$. We also describe a plane construction, intersecting a line with a Lamé curve, that realizes the same relation geometrically for every real $k>1$, although generally not by classical straightedge-and-compass operations.

Let $D$ be a bounded analytic Jordan domain and let $\gamma$ be chordal $\mathrm{SLE}_2$ in $D$, equipped with its $5/4$-dimensional natural-content measure $\mu_\gamma$. We retain the root in the standard integrated Brownian-bridge representation of Brownian loop measure and let $\mathcal M_\varepsilon^\gamma$ be the root-intensity measure of loops with duration in $[\varepsilon^2,t_0]$ whose traces hit $\gamma$. For every $f\in C_c(D)$, we prove that $\varepsilon^{5/4}\mathcal M_\varepsilon^\gamma(f)$ converges in $L^1$ to $\frac{4}{5\pi}\bar v_{\mathrm{BB}}\mu_\gamma(f)$, where $\bar v_{\mathrm{BB}}\in(0,\infty)$ is the mean specific area swept out by the Brownian-bridge offset of a natural-time two-sided whole-plane $\mathrm{SLE}_2$. In particular, the positive random measures converge vaguely in probability. The proof uses a duration-octave identity, a deterministic finite-$R$ reference coefficient obtained from a stopped two-arm Markov skeleton, a marked physical Palm tangent, an annular remote-return estimate, and a legal mesoscopic diagonal. As an application, the uniformly time-marked roots of an independent Brownian loop soup satisfy the corresponding vague law of large numbers. The analytic-boundary hypothesis enters only through a global finite-domain uniform-integrability estimate; its analogue for an arbitrary bounded Jordan domain remains open.