AI for Science

We extend the algebraic diversity (AD) framework from classical signal processing to quantum measurement theory. The Quantum Algebraic Diversity (QAD) Theorem establishes that a group-structured positive operator-valued measure (POVM) applied to a single copy of a quantum state produces a full-rank, group-averaged density matrix estimator whose eigenbasis and eigenvalue ordering track those of the true density matrix, with a bias toward the symmetrized state, analogous to the classical recovery of covariance eigenstructure from a single observation. We establish a Classical-Quantum Duality Map connecting classical covariance estimation to quantum state tomography, and an Optimality Inheritance Theorem showing that classical group optimality transfers to quantum settings via the Born map within the group-averaged family. SIC-POVMs are identified as AD with the Heisenberg-Weyl group and mutually unbiased bases as AD with the Clifford group, revealing the hierarchy $\mathrm{HW}(d) \subseteq \mathcal{C}(d) \subseteq S_d$ that mirrors the classical $\mathbb{Z}_M \subseteq G_{\min} \subseteq S_M$. The double-commutator eigenvalue theorem gives polynomial-time adaptive POVM selection. A worked qubit example shows the group-averaged estimator from a single computational-basis measurement, averaged over a matched $\mathbb{Z}_2$ group, reaching fidelity 0.99 where standard single-basis tomography gives a rank-1 estimate of fidelity 0.80. Monte Carlo simulations for $d = 2$ to $13$ confirm fidelity above 0.90 from a single outcome while standard fidelity degrades as $\sim 1/d$. The growing ratio reflects collapse of the rank-1 standard estimator, not fewer copies per parameter: the biased single-copy estimator reduces the number of distinct measurement settings, not the per-parameter sampling cost, and a genuine copy reduction holds only under exact symmetry.

Optical thermodynamics has recently emerged as a theoretical framework describing a Rayleigh-Jeans (RJ) modal power distribution of multimoded nonlinear photonic circuits. However, its applicability is constrained to systems exhibiting weak nonlinear mode-mode interactions. Here, by employing a Transfer Integral Operator, we circumvent this limitation and establish a steady-state interacting RJ modal distribution -- referred to as non-ideal RJ (NIRJ) -- with renormalized temperature and optical chemical potential. This also builds a natural bridge with earlier work on grand-canonical statistical-mechanical formulations of discrete nonlinear systems. The theory derives the optical analogue of the compressibility factor, which controls the transition from an ideal, non-interacting equation of state (EoS) to a van der Waals-like interacting EoS.

Identifying materials hosting an excitonic insulator ground state has been one of the major pursuits in condensed matter physics in recent years. Promising candidates in transition metal chalcogenide compounds (TMC), including $1T-\mathrm{TiSe_2}$, $\mathrm{Ta_2Pd_3Te_5}$, and $\mathrm{Ta_2NiSe_5}$, share a crucial common characteristic: their low-energy physics is governed by electrons in $d-$ orbitals subject to strong on-site Coulomb interactions. In this work, we investigate spatially indirect exciton condensation in two-dimensional semimetals on triangular lattice. Using a combination of dynamical mean-field theory and the determinant quantum Monte Carlo method, we study two- and three-orbital Hubbard models incorporating strong on-site ($U$) and inter-orbital interactions ($V$). Our results demonstrate that on-site Hubbard $U$ can strongly suppress the condensation temperature $T_c$, an effect that is particularly pronounced at higher electron-hole pair densities. This behavior contrasts sharply with the case without on-site $U$, where $T_c$ grows with pair density at fixed $V$. Moreover, we uncover competition among multiple electron-hole pairing channels in the three-orbital model, which also acts to suppress $T_c$ of exciton condensation. An orbital-selective electron-hole pairing state is identified. These findings may help explain the large discrepancy between strong binding-energy and relative low transition temperature for indirect excitons in TMCs materials, offering important insights for understanding and engineering exciton condensation in materials with strongly correlated $d-$ shell electrons.

The accurate characterisation of the 3D deformations of slender fibres and thin sheets in flow, is a key experimental challenge in the study of particle-laden flows. We propose a high-resolution, single-camera method to visualise non-intrusively the shape of a transparent crumpled sheet, as it translates, rotates and deforms. We perform periodic scans of the crumpled shape by illuminating it with a sequence of stacked light sheets at a rate much faster than its deformation and image the scattered light signal in a plane near-orthogonal to the plane of lighting. Processing of the data using a pinhole camera model yields a noisy spatio-temporal dataset of the strongly deformed time-evolving surface of the sheet, which we reconstruct in 3D using a neural autoencoder. We validate the robustness of the shape reconstruction algorithm to noise using synthetic data sets, and demonstrate the accurate reconstruction of laboratory sedimentation experiments with elastic disks. We find that the inclusion of isometricity-enforcing penalties into the cost function of the autoencoder enables us to robustly reconstruct highly folded shapes, where different regions of the sheet overlap.

The tropical semiring is a semiring of extended real numbers, where the operations of `max' and `+' replace the usual addition and multiplication, respectively. Difference equations obtained from the ultradiscrete limit of discrete dynamical systems are described in terms of the tropical semiring. We propose a tropical linearization approach for the stability analysis of difference equations, including those describing ulradiscrete dynamical systems. We show that the fixed point at the tropical origin is asymptotically stable if the maximum eigenvalue of the tropical Jacobian matrix is negative. On the other hand, it is unstable if the maximum eigenvalue of the tropical Jacobian matrix is positive. Since $0$ is the tropical multiplicative identity, these results are analogous to those in the usual linearization process.

We report the observation of charge-exchange collisions between trapped calcium monohydride molecular ions ($^{40}$CaH$^+$) and ultracold potassium atoms ($^{39}$K) in a hybrid ion-atom trap. The measured charge-exchange rate coefficient is significantly suppressed relative to the Langevin rate constant for the system. We use $\mathit{ab\ initio}$ quantum-chemical calculations to model the (CaH-K)$^+$ complex in the ground and excited electronic states and to identify possible charge-exchange mechanisms. Our calculations rule out a direct non-radiative charge-exchange reaction and instead point to a radiative mechanism, but do not quantitatively reproduce the measured rate, highlighting the need for a full-dimensional quantum dynamics treatment that includes vibrational motion and intermediate complex formation. Our work demonstrates that cold hybrid ion-atom platforms with molecular ions enable access to richer chemical complexity and collisional dynamics inaccessible in purely atomic systems.

We consider image registration as an optimal control problem using an optical flow formulation, i.e., we discuss an optimization problem that is governed by a linear hyperbolic transport equation. Requiring Lipschitz continuity of the vector fields that parametrize the transformation leads to an optimization problem in a non-reflexive Banach space. We introduce relaxations of the optimization problem involving smoothed maximum and minimum functions and appropriate Orlicz spaces. To derive well-posedness results for the relaxed optimization problem, we revisit and establish new existence and uniqueness results for the linear hyperbolic transport equations. We further discuss limit considerations with respect to the relaxation parameter and discretizations.

We propose a Bayesian framework for feedback boundary control of hyperbolic balance laws. The method propagates a probability distribution over feedback parameters using Lyapunov decay estimates as a likelihood. For linear models, it recovers available analytical stability results and extends to nonlinear regimes where theory is limited. Using first-order local Lax-Friedrichs (LLF) discretizations, we validate the approach on the decoupled wave system and the linearized Saint-Venant equations, reproducing known stability intervals and mixed boundary couplings. We then treat nonlinear and stochastic problems, including the nonlinear Saint-Venant system, one- and two-dimensional Burgers equations, Burgers equation with random initial data, and nonconservative perturbations with source terms, and show that the inferred stability domains are robust with respect to the indicator and the prior. Finally, we demonstrate transfer to a second-order semi-discrete LLF scheme and to a two-parameter feedback model for laser powder bed fusion with power regulation.

We consider the optimal control of large-scale systems using distributed controllers whose network topology mirrors the coupling graph between subsystems. In this work, we introduce spatial regret, a graph-informed metric measuring the worst-case performance gap between a distributed controller and an oracle with access to additional sensor information. The oracle's graph is a user-specified augmentation of the information graph, yielding a benchmark policy that penalizes disturbances for which additional sensing would improve performance. Minimizing spatial regret yields distributed controllers - respecting the nominal information graph - that emulate the oracle's response to disturbances characteristic of large-scale networks, such as localized perturbations. We show that minimizing spatial regret admits a convex reformulation as an infinite program with a finite-dimensional approximation. To scale to large networks, we derive an upper bound on the spatial regret that can be efficiently minimized in a distributed way. Numerical experiments on power-system models show that the resulting controllers mitigate localized disturbances more effectively than those based on classical metrics.

We consider a general class of ``inexactly smooth'' convex functions, providing a universal model capturing as special cases $L$-smooth, $M$-Lipschitz, and Hölder smooth functions, and any combination thereof. Such functions possess a calculus closely following that of smooth functions. Our main results provide inexactly smooth functions with interpolation theorems that are necessary and sufficient up to modest universal constants. These enable analysis of first-order methods for any inexactly smooth convex problem class via solving convex Performance Estimation Problems (PEPs). Further, these enable the extension of Drori and Taylor's constructive approach to algorithm design. From this, we derive an exactly minimax optimal method for $(β,0)$-Hölder smooth problems, methods with the best-known convergence guarantees up to constants for any $(β,p)$-Hölder smooth convex minimization, and a new universal fast backtracking method for any inexactly smooth convex problem.

In this paper, we study a class of double Lambert series and establish several identities and transformation relations for them. These formulae provide useful tools for reducing certain double Lambert series to single Lambert series. As applications, we derive identities related to recent conjectures of Andrews, Dixit, Schultz, and Yee, and of Amdeberhan, Andrews, and Ballantine. We also propose a new proof of a result of Amdeberhan, Andrews, and Ballantine.

We establish the existence and uniqueness of discrete Einstein metrics on trees under Lin-Lu-Yau Ricci curvature using Perron-Frobenius theory. We establish a sharp upper bound for the largest eigenvalue of the associated Ricci matrix in terms of the maximum degree. Turning to structural properties, notably, the existence of a positive-curvature Einstein metric implies the tree must be a caterpillar. Furthermore, these metrics exhibit radial monotonicity, with edge weights decreasing strictly away from the maximal edge.

Let $G$ be a simple connected graph on $n$ vertices, and let $λ_1(G),λ_2(G),\ldots,λ_n(G)$ be the eigenvalues of its adjacency matrix $A(G)$. For $p>0$, define the $p$-energy of $G$ by $\mathcal E_p(G)=\sum_{i=1}^n |λ_i(G)|^p$. We prove that, for every real number $p\ge 2$ and every simple connected graph $G$ on $n$ vertices, $$ \mathcal E_p(G)\ge \mathcal E_p(P_n), $$ where $P_n$ denotes the path on $n$ vertices. Moreover, for each fixed $p>2$, equality holds if and only if $G\cong P_n$. Together with the previously known star-minimality results, this completes the solution of two questions of Nikiforov. The proof combines two different comparison principles. For $2<p<4$, we use a bipartite reduction, a Mellin representation of fractional powers, and a determinant comparison involving matching generating polynomials and tree shifts. For $p\ge4$, we prove a second-order stop-loss comparison for the squared singular values of bipartite graphs. This comparison is established by rank-one spectral-shift estimates, deletion-minimal counterexamples, and a finite certified analysis of the terminal sparse-sun configurations.

This paper shows that the concept of complex frequency, originally introduced to characterize the dynamics of signals with complex values, constitutes a generalization of eigenvalues when applied to the states of linear time-invariant (LTI) systems. Starting from the definition of geometric frequency, which provides a geometrical interpretation of frequency in electric circuits that admits a natural decomposition into symmetric and antisymmetric components associated with amplitude variation and rotational motion, respectively, we show that complex frequency arises as its restriction to the two-dimensional Euclidean plane. For LTI systems, it is shown that the complex frequencies computed from the system's states subject to a non-isometric transformation, coincide with the original system's eigenvalues. This equivalence is demonstrated for diagonalizable systems of any order. The paper provides a unified geometric interpretation of eigenvalues, bridging classical linear system theory with differential geometry of curves. The paper also highlights that this equivalence does not generally hold for nonlinear systems. On the other hand, the geometric frequency of the system can always be defined, providing a geometrical interpretation of the system flow. A variety of examples based on linear and nonlinear circuits illustrate the proposed framework.

The expected signature uniquely determines the law of a random rough path under a moment-growth condition, yet finite-sample bounds for estimating its truncations from a single long dependent trajectory remain unavailable. We study a strictly stationary stochastic process equipped with a geometric rough-path lift, observed in non-overlapping blocks of equally-spaced samples, and prove a non-asymptotic mean-squared error (MSE) bound for the block-averaging estimator of its truncated expected signature. Under moment and stationarity assumptions together with a direct covariance-decay condition on block signatures -- strictly weaker than $α$-mixing and applicable to long-range-dependent processes -- the error separates into a discretization term and a fluctuation term, with rates determined respectively by path regularity and dependence strength. A levelwise rough-factorial variance analysis keeps finite-truncation constants explicit and yields an optimal allocation rule under a fixed observation budget. We verify the assumptions for independent-coordinate fractional Ornstein--Uhlenbeck processes in three regimes: short-range (Hurst $1/4<H<1/2$), semimartingale ($H=1/2$), and long-range ($H>1/2$); in all three, the block-signature covariance is summable, so the fluctuation term decays at the same rate as in the independent-block case, even under long memory at $H>1/2$. Monte Carlo experiments show empirical slopes steeper than the guaranteed upper-bound rates.

We study attention mechanisms through the lens of a canonical unsupervised problem: principal component analysis (PCA). We show that, when trained on Gaussian data, both softmax and linear attention layers learn parameters that align with the principal eigenvectors of the covariance matrix, thereby establishing a direct and explicit connection with PCA. Our analysis covers both finite and infinite prompt regimes. In the infinite-prompt limit, we prove convergence to globally optimal solutions aligned with the leading spectral direction, while in the finiteprompt setting we show that the same behavior emerges up to sampling effects. We further extend the analysis to an in-context setting with spiked Wishart covariances, where attention successfully recovers the underlying signal direction. These results demonstrate that attention inherently performs PCA-like computations under unsupervised objectives, providing a theoretical foundation for its representation-learning capabilities.

We predict a field-free transverse Josephson diode effect in altermagnets (AMs) with Rashba spin--orbit coupling, achieving diode efficiencies exceeding $3000\%$ and unidirectional transverse supercurrents in four-terminal junctions. In this geometry, a longitudinal phase bias generates transverse supercurrents that exhibit nonreciprocity and a finite anomalous phase shift, while the longitudinal current itself displays a Josephson diode effect. Both responses are tunable via the Néel vector orientation. We further show that the effect remains robust against moderate disorder and imperfect interfaces. These results establish AMs as a promising platform for nonreciprocal superconducting transport, with clear routes toward experimental realization.

We introduce large $p$ asymptotic geometric quantities associated with $p$-capacity, the first $p$-eigenvalue, and the Maz'ya constant on complete noncompact Riemannian manifolds. We prove the hierarchy $$ \mathcal{V}(M)\geq \mathcal C(Ω)\geq Λ(M)=\mathcal M(M)\geq0, $$ and show that, under a centered-ball isoperimetric condition or a rotational symmetry condition, these quantities coincide with the volume entropy or the dimension. In the Hadamard nonpositively curved case it also agrees with the topological entropy of the geodesic flow. As an application, combining with the entropy rigidity theorem, we obtain a characterization of hyperbolic manifolds. We also prove a second-order refinement. For a Hadamard manifold with compact quotient, under certain condition, the first-order large $p$ capacitary limit detects volume entropy, whereas the logarithmic second-order correction detects the rank.

Quantum-inspired tensor networks algorithms have shown to be effective and efficient models for machine learning tasks, including anomaly detection. Here, we propose a highly parallelizable quantum-inspired approach which we call SMT-AD from Superposition of Multiresolution Tensors for Anomaly Detection. It is based upon the superposition of bond-dimension-1 matrix product operators to transform the input data with Fourier-assisted feature embedding, where the number of learnable parameters grows linearly with feature size, embedding resolutions, and the number of additional components in the matrix product operators structure. We demonstrate successful anomaly detection when applied to standard datasets, including credit card transactions, and find that, even with minimal configurations, it achieves competitive performance against established anomaly detection baselines. Furthermore, it provides a straightforward way to reduce the weight of the model and even improve the performance by highlighting the most relevant input features.

Pressure-relief valves, often the critical last line of defence in process engineering, are known to be susceptible to valve chatter. Such behaviour has been shown to arise from a flutter instability, or Hopf bifurcation, associated with the fundamental, quarter-wave acoustic mode of their inlet piping. Here, a novel design concept is proposed and analyzed for eliminating this instability. The concept involves using an oversized valve with reduced lift and adopting a discharge characteristic that enhances the blow-down effect, so that the valve opens immediately to its upper lift limit upon reaching set pressure. The concept is demonstrated numerically using an updated version of a 1D fluid pipe dynamics mathematical model solved using a Lax-Wendroff method. Stability properties are analysed using dynamical systems theory, applied to an improved reduced-order modal model. It is shown how the valve settles to a stable so-called pseudo equilibrium, in contact with the upper stop, provided the coefficient of restitution of is not too large. Such stable operation is reached despite the equivalent regular valve being unstable to the quarter-wave Hopf bifurcation. Parameter studies using the reduced-order model demonstrate the extent of the enhanced stability effect, which is confirmed using the full model for the case of gas service valves.

We prove finite-time Type--I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,α}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<α<\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The axial strain and the global vorticity norm blow up at the Type--I rates $-\partial_z u_z(0,0,t)\simeq (T^*-t)^{-1}$ and $\|ω(\cdot,t)\|_{L^\infty}\simeq (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\simeq (T^*-t)^{1/(1-3α)}$. The proof is organized around a Lagrangian clock-and-driver framework. The clock is the meridional Jacobian $J(t)$, and the driver is the compressive axial strain $-\partial_z u_z(0,0,t)$. These variables satisfy, to leading order, a closed Riccati-clock system: the axial strain drives the collapse of $J(t)$, while the collapse of $J(t)$ amplifies the axial strain. We prove that the Euler flow tracks this clock-and-driver model up to the singular time. The main nonlocal obstruction is the pressure Hessian; it is controlled by a non-perturbative strain--pressure Hessian comparison showing that pressure cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $α=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular functions in a weighted Hölder topology.

This study introduces the Bragg Frequency Converter, a spatiotemporal-periodic grating concept that extends conventional Bragg gratings into the dynamic domain for pure parametric frequency conversion. By selectively time-modulating either the high-index or low-index layers of a quarter-wave stack, the structure achieves directional frequency conversion: high-index modulation yields efficient down-conversion, while low-index modulation produces up-conversion. This layer selectivity stems from the asymmetric Bloch mode distribution and associated phase-matching conditions. One practical realization, based on a silicon rib waveguide with periodic sidewall corrugations and selective doping of the high-index segments, is presented and analyzed. A coupled-mode theory is developed to explain the mechanism and validated through full-wave simulations. An experimental setup using optical pumping is also proposed for practical implementation. The theoretical and numerical results establish temporal Bragg gratings as a versatile, reconfigurable platform for spurious-free frequency conversion with applications in optical signal processing and integrated photonics.

We propose a deep photonic neuromorphic network (PNN) architecture based on phase-change material (PCM) synapses and local optical feedback for online, unsupervised Hebbian learning. The proposed architecture combines optical vector-matrix multiplication, non-volatile PCM synaptic weighting, and local coincidence-driven synaptic adaptation within a multilayer photonic crossbar framework compatible with photonic integrated circuits. Unlike conventional PNNs that rely on externally computed gradients, repeated optical-electrical-optical conversions, or global backpropagation, the proposed framework employs local Hebbian learning governed directly by correlated pre- and post-synaptic optical activity. To investigate the feasibility of the proposed learning mechanism, we implemented the PNN design using fiber-optic components, programmable variable optical attenuators, and real-time software control that incorporates PCM thermal dynamics. Supervised and unsupervised learning behaviors were experimentally evaluated under both offline and online learning conditions using representative image-recognition tasks. The experimental results demonstrate adaptive synaptic evolution, successful optical inference, and autonomous pattern encoding through local Hebbian learning under realistic fiber-optic hardware conditions. These results establish a pathway toward future integrated photonic neuromorphic systems capable of scalable and energy-efficient online Hebbian learning.

We give a representation of the extension class associated to a holomorphic fibration by curvature, generalizing the work of Atiyah on holomorphic principal bundles in a natural way. As an application, we obtain a nonlinear analogue of the classical result of Weil on characterizing the existence of flat connections on holomorphic vector bundles over compact Riemann surfaces. We further establish a faithful functor from the category of nonlinear flat bundles reductive of Kähler type to the category of nonlinear Higgs bundles over the same base, which is assumed to be a compact complex manifold of Kähler type. Finally, we establish a notion of nonlinear harmonic bundle and prove that the variation of nonabelian Hodge structure is a nonlinear harmonic bundle in the rank one case and in the semisimple case.

Human cognition depends on large scale communication constrained by white matter architecture. Although weak connections are abundant in mammalian connectomes, they have long been treated as noise and downweighted because of tractography uncertainty in the human brain, and their relevance to human cognition and large scale functional organization remains unresolved. Across multiple datasets and tractography pipelines, we show that, when tractography derived connectivity weights are interpreted through a nonlinear weighting framework, weak connections make measurable contributions to cognitive prediction, functional connectivity simulation, and structure-function coupling. These effects are selective: nonlinear weighting improves the prediction of general cognitive ability and memory more than that of crystallized intelligence or processing speed, consistent with the notion that weak connections preferentially expand the modal repertoire of brain networks to enhance both large scale integration and fine grained segregation, thereby supporting the functional balance essential for diverse cognitive abilities. Importantly, these effects are replicated in a reliability aware connectome generated by integrating two post tractography filtering methods, in which preserving weak links consistently outperforms conventional thresholding strategies. Finally, we show that weak connections contain functionally informative subsets organized along systems level and transcriptomic gradients. In particular, a specific class of weak connections, predominantly linking visual and motor systems with limbic regions and characterized by negative gene coexpression, exerts a disproportionately large influence on brain function.

This manuscript introduces a novel method to assess tissue oxygen concentration via the quantum entanglement (QE) of photons originating from positronium which is produced within the patient's body during positron emission tomography. We also investigate the possibility of assessing hypoxia by simultaneously detecting positronium lifetime and the positronium decay rate ratio. We introduce two distinct quantum sensing approaches. Method 1 utilizes the correlation between oxygen concentration and ortho-positronium (o-Ps) decay rates, relying on the simultaneous measurement of the mean o-Ps lifetime ($τ_{\mathrm{oPs}}$) and the $3γ$-to-$2γ$ annihilation rate ratio of o-Ps ($R_{\mathrm{oPs-3γ/2γ}}$). Method 2 introduces a novel hypothesis: that the degree of QE is sensitive to the relative contribution of annihilation mechanisms (pick-off vs. conversion), which in turn depends on oxygen concentration. We derive a formula for partial pressure of oxygen ($p\mathrm{O}_2$) as a function of $R_{\mathrm{oPs-3γ/2γ}}$ and $τ_{\mathrm{oPs}}$ and estimate the measurement accuracy required for these parameters - and for the degree of QE - to sense in-vivo oxygen pressure in the range between hypoxic and physoxic conditions. Theoretical models and quantitative estimates for $R_{\mathrm{oPs-3γ/2γ}}$, $τ_{\mathrm{oPs}}$ and for the degree of QE ($C_{\mathrm{QE}}$ ) as a function of $p\mathrm{O}_2$ are provided for water, isopropanol, cyclohexane, isooctane, and adipose tissue. In particular, applying the formulas derived under the working hypothesis that in pick-off process the photons are not entangled, we estimated that for $p\mathrm{O}_2 = 0$, the degree of quantum entanglement $C_{\mathrm{QE}}$ is equal to 0.890 for adipose, 0.886 for isopropanol, 0.867 for water, 0.818 for cyclohexane, and 0.784 for isooctane.

The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different families of functions $f$. The article establishes convergence of the power series of $f(B)$ and $f(T)$ under the operator norm for the Wiener algebra $\mathbb{W}_+$, and convergence under strong operator topology for $f$ in $H^{\infty}$, based on the use of Abel sums. Based on this calculus, it unifies the notion of stationary process invertibility with the operator invertibility of the transfer function $f(T)$.

We give an explicit description of the BPS Lie algebra of any quiver with zero potential, by relating the perverse filtration on the cohomological Hall algebra with certain limit conditions on polynomials. Our results also give a partial description of the perverse filtration for arbitrary potential, which we conjecture is complete in the case of tripled quivers with canonical cubic potential.

For a traveling salesperson problem (TSP) of n cities, we present a compact quantum encoding based on a time-register representation of tours. A candidate route is represented as a sequence of n-1 city labels over discrete time steps, with one fixed start city and the remaining cities encoded in binary registers. We describe three ingredients of the construction: uniform route generation over the route register, a reversible validity oracle, and a phase oracle that encodes the total tour cost. The validity oracle checks both that the non-start city labels form a permutation and, for incomplete graphs, that every directed edge used by the route exists. The cost oracle then accumulates the start-edge, intermediate-transition, and return-edge costs into a tour-dependent phase for valid routes. This yields a coherent superposition of candidate routes with feasibility and tour-length information embedded directly in the quantum state. The complete construction uses O(n log n) qubits, while a naive implementation has worst-case elementary-gate complexity O(n^3 log n). The encoding is compatible with amplitude amplification or spectral filtering techniques such as the quantum singular value transform (QSVT) or Grover's algorithm. However, due to the exponentially small fraction of valid tours, the overall complexity remains exponential even when combined with amplitude amplification.

Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected complex Lie group with $\mathrm{Lie}(G)=\g$ and compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG=A_0\text{-alg}<\RAzp \cup \RAzm>.$ This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all simple complex Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup with a unique bi-invariant (Haar) state.