We propose a data-driven linear modeling framework for controlled nonlinear hereditary systems that combines Koopman lifting with a truncated Grunwald-Letnikov memory term. The key idea is to model nonlinear state dependence through a lifted observable representation while imposing history dependence directly in the lifted coordinates through fixed fractional-difference weights. This preserves linearity in the lifted state-transition and input matrices, yielding a memory-compensated regression that can be identified from input-state data by least squares and extending standard Koopman-based identification beyond the Markovian setting. We further derive an equivalent augmented Markovian realization by stacking a finite window of lifted states, thereby rewriting the finite-memory recursion as a standard discrete-time linear state-space model. Numerical experiments on a nonlinear hereditary benchmark with a non-Grunwald-Letnikov Prony-series ground-truth kernel demonstrate improved multi-step open-loop prediction accuracy relative to memoryless Koopman and non-lifted state-space baselines.

Massively parallel hardware (GPUs) and long sequence data have made parallel algorithms essential for machine learning at scale. Yet dynamical systems, like recurrent neural networks and Markov chain Monte Carlo, were thought to suffer from sequential bottlenecks. Recent work showed that dynamical systems can in fact be parallelized across the sequence length by reframing their evaluation as a system of nonlinear equations, which can be solved with Newton's method using a parallel associative scan. However, these parallel Newton methods struggled with limitations, primarily inefficiency, instability, and lack of convergence guarantees. This thesis addresses these limitations with methodological and theoretical contributions, drawing particularly from optimization. Methodologically, we develop scalable and stable parallel Newton methods, based on quasi-Newton and trust-region approaches. The quasi-Newton methods are faster and more memory efficient, while the trust-region approaches are significantly more stable. Theoretically, we unify many fixed-point methods into our parallel Newton framework, including Picard and Jacobi iterations. We establish a linear convergence rate for these techniques that depends on the method's approximation accuracy and stability. Moreover, we give a precise condition, rooted in dynamical stability, that characterizes when parallelization provably accelerates a dynamical system and when it cannot. Specifically, the sign of the Largest Lyapunov Exponent of a dynamical system determines whether or not parallel Newton methods converge quickly. In sum, this thesis unlocks scalable and stable methods for parallelizing sequential computation, and provides a firm theoretical basis for when such techniques will and will not work. This thesis also serves as a guide to parallel Newton methods for researchers who want to write the next chapter in this ongoing story.

Accurate probabilistic modeling of the power system restoration process is essential for resilience planning, operational decision-making, and realistic simulation of resilience events. In this work, we develop data-driven probabilistic models of the restoration process using outage data from four distribution utilities. We decompose restoration into three components: normalized restore time progression, total restoration duration, and the time to first restore. The Beta distribution provides the best-pooled fit for restore time progression, and the Uniform distribution is a defensible, parsimonious approximation for many events. Total duration is modeled as a heteroskedastic Lognormal process that scales superlinearly with event size. The time to first restore is well described by a Gamma model for moderate and large events. Together, these models provide an end-to-end stochastic model for Monte Carlo simulation, probabilistic duration forecasting, and resilience planning that moves beyond summary statistics, enabling uncertainty-aware decision support grounded in utility data.

The primary objective of this paper is to establish several sharp results concerning the Bohr inequality, the refined Bohr inequality, and the improved Bohr inequality for the classes of analytic functions and harmonic mappings defined on the shifted disks \[ Ω_γ=\left\{z\in\mathbb{C}:\left|z+\fracγ{1-γ}\right|<\frac{1}{1-γ}\right\}\quad\text{for}\quadγ\in[0,1).\]

We develop LENORI, a Large Event Number of Outages Resilience Index measuring distribution system resilience with the number of forced line outages observed in large extreme events. LENORI is calculated from standard utility outage data. The statistical accuracy of LENORI is ensured by taking the logarithm of the outage data. A related Average Large Event Number of Outages metric ALENO is also developed, and both metrics are applied to a distribution system to quantify the power grid strength relative to the extreme events stressing the grid. The metrics can be used to track resilience and quantify the contributions of various types of hazards to the overall resilience.

Beyond conditional average treatment effects, treatments may impact the entire outcome distribution in covariate-dependent ways, for example, by altering the variance or tail risks for specific subpopulations. We propose a novel estimand to capture such conditional distributional treatment effects, and develop a doubly robust estimator that is minimax optimal in the local asymptotic sense. Using this, we develop a test for the global homogeneity of conditional potential outcome distributions that accommodates discrepancies beyond the maximum mean discrepancy (MMD), has provably valid type 1 error, and is consistent against fixed alternatives -- the first test, to our knowledge, with such guarantees in this setting. Furthermore, we derive exact closed-form expressions for two natural discrepancies (including the MMD), and provide a computationally efficient, permutation-free algorithm for our test.

We study for the first time the action of the Hilbert matrix $$\mathcal H=(c_{n,k})_{n,k\geq 0}, \quad c_{n,k}=\frac{1}{n+k+1}$$ on the analytic tent spaces $AT^q_p, 1<p,q <\infty,$ of the unit disc $\mathbb D$ of the complex plane. They were proposed by Triebel as the natural analytic version of the tent spaces of measurable functions defined by Coifman, Meyer and Stein. The $AT_p^q$ spaces are consisted of those analytic functions $f$ in $\mathbb D$ such that $$ \|f\|_{AT_{p}^{q}}= \left\{\int_{\mathbb T} \left(\int_{Γ_{1/2}(ξ)} |f(z)|^p \ \frac{dA(z)}{1-|z|^2} \right)^{q/p}\ |dξ|\right \}^{1/q}<+\infty, $$ where $$ Γ_{1/2}(ξ) =\bigl\{ z\in \mathbb{D} : |z|< 1/2 \bigr\} \cup \bigcup_{|z|<1/2}[z,ξ), $$ $dA(z)$ is the normalized area Lebesgue measure in $\mathbb D$ and $|dξ|$ is the arc length in the unit circle $\mathbb T$. The Bergman spaces $A^p, p>1,$ stand among the $AT_{p}^{q}$ and correspond to the case $p=q$. The multiplication of the Hilbert matrix with the column matrix with entries the Taylor coefficients of an $f(z)=\sum_{k\geq 0} a_k z^k $ analytic in $\mathbb D$ introduces the series $$ \mathcal H (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \frac{a_k}{n+k+1}\right)z^n\,, \quad z\in \mathbb D\,\, $$ known in the literature as Hilbert operator. We prove that it is a bounded operator on the $AT_{p}^{q}$ when $1/p + 1/q <1,\, p>2$. This is a natural range for the values of the indices $p,q$ compared to what is known in the special case of the Bergman spaces. We confront the question under discussion through a more general point of view by studying an associated integral operator defined with respect to a positive Borel measure $μ$ on $[0,1)$. Finally, we provide an estimation of the norm of the Hilbert operator. Our work extends in a non-trivially way previous results on the Bergman spaces to the analytic tent spaces.

We introduce a model for directed spatial networks. Starting from an age-based preferential attachment model in which all arcs point from younger to older vertices, we add \emph{reciprocal} connections whose probabilities depend on the age difference between their end-vertices. This yields a directed graph with reciprocal correlations, a power-law indegree distribution, and a tunable outdegree distribution. We consider two versions of the model: an infinite version embedded in $\mathbb{R}^d$, which can be constructed as a weight-dependent random connection model with a non-symmetric kernel, and a growing sequence of graphs on the unit torus that converges locally to the infinite model. Besides establishing the local limit result linking the two models, we investigate degree distributions, various directed clustering metrics, and directed percolation.

The automorphism group of a regular locally finite tree is shown to admit irreducible Banach representations that are not admissible. The dense subspace of smooth vectors contains no algebraically irreducible component.

We establish new examples and non-examples of pro-p limit groups among the class of Demushkin groups, that is, pro-p Poincaré duality groups of dimension 2.

Computational properties of the Hahn-Banach theorem have been studied in computable, constructive and reverse mathematics and in all these approaches the theorem is equivalent to weak Kőnig's lemma. Gherardi and Marcone proved that this is also true in the uniform sense of Weihrauch complexity. However, their result requires the underlying space to be variable. We prove that the Hahn-Banach theorem attains its full complexity already for the Banach space $\ell^1$. We also prove that the one-step Hahn-Banach theorem for this space is Weihrauch equivalent to the intermediate value theorem. This also yields a new and very simple proof of the reduction of the Hahn-Banach theorem to weak Kőnig's lemma using infinite products. Finally, we show that the Hahn-Banach theorem for $\ell^1$ in the two-dimensional case is Weihrauch equivalent to the lesser limited principle of omniscience.

We study mean estimation for a Gaussian distribution with identity covariance in $\mathbb{R}^d$ under a missing data scheme termed realizable $ε$-contamination model. In this model an adversary can choose a function $r(x)$ between 0 and $ε$ and each sample $x$ goes missing with probability $r(x)$. Recent work Ma et al., 2024 proposed this model as an intermediate-strength setting between Missing Completely At Random (MCAR) -- where missingness is independent of the data -- and Missing Not At Random (MNAR) -- where missingness may depend arbitrarily on the sample values and can lead to non-identifiability issues. That work established information-theoretic upper and lower bounds for mean estimation in the realizable contamination model. Their proposed estimators incur runtime exponential in the dimension, leaving open the possibility of computationally efficient algorithms in high dimensions. In this work, we establish an information-computation gap in the Statistical Query model (and, as a corollary, for Low-Degree Polynomials and PTF tests), showing that algorithms must either use substantially more samples than information-theoretically necessary or incur exponential runtime. We complement our SQ lower bound with an algorithm whose sample-time tradeoff nearly matches our lower bound. Together, these results qualitatively characterize the complexity of Gaussian mean estimation under $ε$-realizable contamination.

Any quantum state of the radiation field, sliced in small non-overlapping space-time bins is a collection of single-rail qubits, each spanning the vacuum and single-photon Fock state of a mode. Quantum logic on these qubits would enable arbitrary measurements on information-bearing light, but is hard due to the lack of strong nonlinearities. With unentangled ancilla single-rail qubits, an $8$-port interferometer and photon detection, we show any single-rail qubit measurement in the $XY$ Bloch plane is realizable with success probability $147/256$, which beats the prior-known $1/2$ limit.

In his approach to Jones theorem on the interpolation of Hardy spaces on the torus, Pisier introduced an original method allowing the computation of complex interpolation spaces by means of real interpolation techniques. This approach has been successfully extended to noncommutative analytic Hardy spaces arising from subdiagonal algebras. In this paper, we formulate and prove an abstract version of Pisier s method in a more general setting. The method is then applied in the study of noncommutative martingale transforms.

We show stabilisation of solutions to one-dimensional advective Cahn-Hilliard equation modeling the Langmuir-Blodgett thin films. This problem has the structure of a gradient flow perturbed by a linear term $βu_x$. Through application of an abstract result by Carvalho-Langa-Robinson, we show that for small $β$ the equation has the structure of gradient flow in a weak sense. Combining this with the finite number of steady states implies stabilization of solutions.

This paper consider the LAN property for the mixed O-U process under high-frequency observation when H>3/4. As considered in mixed fractional Brownian motion, we will also use the projection step to get the non-diagonal rate matrix.

Quantum computing has been regarded as a promising approach to accelerate power system optimization. However, challenges such as limited qubits and inherent noise hinder their widespread adoption in power systems. In this paper, we propose a qubit-efficient framework for solving a crucial power system optimization problem, the probabilistic optimal power flow (POPF). We demonstrate that quantum noise, traditionally viewed as a drawback, can in fact be leveraged to provide a built-in differential privacy (DP) guarantee. Specifically, we first linearize POPF into a multi-parametric linear program (MP-LP) with renewable uncertainties being the parameters. This decomposes the parameter space into critical regions with precomputed solution maps. Second, a variational quantum circuit (VQC) classifies the critical region based on each uncertainty realization and then recovers the final solution. In this way, the required qubits scale with the uncertain parameters instead of the network size, with only 5 qubits versus 600+ for direct quantum OPF in a 69-bus system. Moreover, we prove the depolarizing noise of VQC provides DP guarantees and characterize the privacy-cost tradeoff. Case studies validate the proposed VQC achieves 2.1$\times$ smaller privacy budgets compared to its classical counterpart. At matched privacy levels, the VQC also maintains lower infeasibility and prediction error.

A procedure for the algebraization of a $CR$-manifold and its holomorphic automorphisms is described. Examples of the application of algebraization are considered. Questions arising in connection with the algebraization of a $CR$-manifold are formulated. The possibilities of extending this procedure to other branches of geometry and analysis are discussed.

The notions of tree-like loop and Lipschitz tree-like loop were introduced by Hambly and Lyons in their 2010 Annals of Mathematics paper. They showed that the Lipschitz tree-like property determines an equivalence relation on the set of paths of bounded variation in a given metric space and then asked if this notion could be extended to paths without the Lipschitz requirement. We show that after eliminating the Lipschitz requirement, the resulting relation is no longer transitive and thus is not an equivalence relation. The counterexample is obtained by analyzing an explicit fractal construction in the plane.

We introduce a concept of the commuting probability of a skew left brace analogous to group theory. We establish upper and lower bounds for the commuting probability and prove that, for finite non-trivial skew left braces, it is always at most $\frac{3}{4}$. Interestingly, there is no skew left brace with commuting probability in the open interval $(5/8, 1)$, except $\frac{3}{4}$, for which we construct an explicit example. A characterization of skew left braces having commuting probability $\frac{3}{4}$ or $\frac{5}{8}$ is presented. We further show that the finite skew left braces with commuting probability larger than $\frac{65}{128}$ are necessarily nilpotent. We prove that the commuting probability remains invariant under isoclinism of skew braces. We introduce a concept of a compact Hausdorff topological skew left brace $B$, where we prove that the set of all elements of $B$ having finite centraliser index in $B$ is a Borel subgroup. For such infinite non-trivial skew left braces too $\frac{3}{4}$ is the upper bound for the commuting probability, and $\frac{3}{4}$ is the only rational number which occurs as commuting probability in the open interval $(5/8, 1)$.

Motivated by the recent development of noncommutative Novikov algebras and multi-Novikov algebras from the study of regularity structures of stochastic PDEs, this paper gives a general approach to study various multi-Novikov algebras and multi-differential algebras, with close connection with Poisson algebras. The construction of S. Gelfand of Novikov algebras from differential commutative algebras is generalized to this context. Free noncommuting multi-Novikov algebras are constructed from typed decorated rooted trees and from noncommuting multi-differential polynomials with populated conditions.

We prove that a finitely generated virtually RFRS group of cohomological dimension at most $2$ is coherent if and only if its second $L^{2}$-Betti number vanishes if and only if it is virtually free-by-cyclic. The non-vanishing of the second $L^{2}$-Betti number provides the first known global obstruction to coherence in any reasonably wide class of groups, allowing for proofs of incoherence without needing to exhibit explicit witnesses to incoherence. As applications of this result, we completely characterise coherence among two-dimensional Coxeter groups, confirming conjectures of Jankiewicz and Wise, and show that incoherence is generic in groups of nonpositive deficiency, confirming a conjecture of Wise. We also find that, among virtually compact special groups of virtual cohomological dimension two, coherence is algorithmically decidable and is a quasi-isometry, measure equivalence, and profinite invariant. In an appendix, Marco Linton applies one of the main results to prove that cubulated locally quasi-convex hyperbolic groups are virtually free-by-cyclic, solving problems of Abdenbi--Wise and Wise in the cubulated case.

Fermionic Gaussian states are a fundamental tool in many-body physics, faithfully representing non-interacting quantum systems and allowing for efficient numerical simulations. Given a many-body wave function, it is therefore interesting to ask how much it differs from that of a Gaussian state, as quantified by the notion of non-Gaussianity. In this work, we relate measures of non-Gaussianity with the Shannon entropy of the particle-number distribution, coinciding with the particle-number asymmetry for pure states. We derive a lower bound on the relative entropy of non-Gaussianity in terms of the exponential of the Shannon entropy, and study numerically its tightness for large system sizes. Our bound is non-trivial for large values of the asymmetry and relies on the concentration of the particle-number distribution of (mixed) fermionic Gaussian states. Since the Shannon entropy of the particle-number distribution is often efficient to compute or experimentally measure, our results can be viewed as a practical way to lower bound non-Gaussianity, highlighting a non-trivial interplay with particle-number asymmetry.

This note serves as an attempt towards the Jiang conjecture on the upper bound nilpotent orbits in the wavefront sets of representations in local Arthur packets of classical groups, which is a natural generalization of the well-known Shahidi conjecture, reflecting the relation between the structure of wavefront sets and the local Arthur parameters. Applying the character identities of local Arthur packets and the matching method of endoscopic liftings, we reduce the study of the upper bound to certain properties of the wavefront sets of the corresponding bi-torsor representations of general linear groups.

Transit Network Design is a well-studied problem in the field of transportation, typically addressed by solving optimization models under fixed demand assumptions. Considering the limitations of these assumptions, this paper proposes a new framework, namely the Two-Level Rider Choice Transit Network Design (2LRC-TND), that leverages machine learning and contextual stochastic optimization (CSO) through constraint programming (CP) to incorporate two layers of demand uncertainties into the network design process. The first level identifies travelers who rely on public transit (core demand), while the second level captures the conditional adoption behavior of those who do not (latent demand), based on the availability and quality of transit services. To capture these two types of uncertainties, 2LRC-TND relies on two travel mode choice models, that use multiple machine learning models. To design a network, 2LRC-TND integrates the resulting choice models into a CSO that is solved using a CP-SAT solver. 2LRC-TND is evaluated through a case study involving over 6,600 travel arcs and more than 38,000 trips in the Atlanta metropolitan area. The computational results demonstrate the effectiveness of the 2LRC-TND in designing transit networks that account for demand uncertainties and contextual information, offering a more realistic alternative to fixed-demand models.

In two-phase multiwave sampling, inexpensive measurements are collected on a large sample and expensive, more informative measurements are adaptively obtained on subsets of units across multiple waves. Adaptively collecting the expensive measurements can increase efficiency but complicates statistical inference. We give valid estimators and confidence intervals for M-estimation under adaptive two-phase multiwave sampling. We focus on the case where proxies for the expensive variables -- such as predictions from pretrained machine learning models -- are available for all units and propose a Multiwave Predict-Then-Debias estimator that combines proxy information with the expensive, higher-quality measurements to improve efficiency while removing bias. We establish asymptotic linearity and normality and propose asymptotically valid confidence intervals. We also develop an approximately greedy sampling strategy that improves efficiency relative to uniform sampling. Data-based simulation studies support the theoretical results and demonstrate efficiency gains.

For the regime-switching diffusion process with and without advection term we propose an integro-differential equation describing the densities of states continuously distributed over a segment. We demonstrate that there exists a constructive algorithm for solving the Cauchy problem. We then show that for some initial distributions of states, the solution can be found explicitly. We also discuss how a model with a discrete number of hidden states can be approximated by a model with continuously distributed states.

For any prime $p>0$, we prove that simple superspecial abelian surfaces over $\mathbb{F}_{p}$ admit CM liftings after base change at most to $\mathbb{F}_{p^2}$, by using the residual reflex condition (RRC) and Lie types. The CM-liftability of ordinary simple abelian surfaces is proved by Serre-Tate, and the CM-liftability of almost ordinary simple abelian surfaces is proved by Oswal-Shankar and Bergström-Karemaker-Marseglia, respectively. As there can only be ordinary, almost ordinary, or supersingular simple abelian surfaces over $\mathbb{F}_{p}$, our work is another step to complete the CM-liftability of simple abelian surfaces over $\mathbb{F}_{p}$.

Given a group $G = H_1 \ast_A H_2$ which is the free product of two finitely generated groups $H_1$ and $H_2$ with amalgamation over a cyclic subgroup $A$ which is malnormal in $G$, we study relations between the structure of its subgroups and the structure of the group $G$ itself. Firstly, we show that if $H_1$ and $H_2$ are 3-free products of cyclics of rank $\ge 3$ then $G$ is also a 3-free product of cyclics. Secondly, we prove that if $H_1$ and $H_2$ are 4-free products of cyclics of rank $\ge 4$ then every 4-generated subgroup of $G$ is a free product of $\le 4$ cyclics or a 1-relator quotient of a free product of four cyclic groups. Here a group is called an $n$-free product of cyclics if every $n$-generated subgroup is a free product of $\le n$ cyclic groups. These results are based on ubiquitous applications of the Nielsen method for amalgamated free products which we recall carefully. Lastly, given an infinite, finitely presented group which is not free, but all of its infinite index subgroups are free, a well-known conjecture says that it is isomorphic to a surface group. We revisit and elaborate on predominantly group theoretic proofs of this conjecture for cyclically amalgamated products as above, as well as for certain HNN extensions.

We prove that the homological and Balmer spectra in tensor-triangular geometry are functorial in certain definable functors, thereby providing an alternative perspective on functoriality in tensor-triangular geometry from the viewpoint of purity, and generalising current results in the literature.