We present identities relating the pair-correlation functions and static structure factors of states in the maximal spin multiplet. This allows us to compute these density-density correlation functions of all members of the multiplet using just these correlation functions of the highest-weight state. We apply these relations to obtain energies for many fractional quantum Hall (FQH) states. In particular, we analytically compute the energies of the Halperin-$(1,1,1)$ state as a function of density imbalance and layer separation, and numerically evaluate these energies for many other FQH states.

We study spectral properties of quantum many-body Hamiltonians through a subsystem-based framework. Given a Hamiltonian of the form $H = \sum_{X \subseteq Λ} Φ(X)$ acting on a tensor product Hilbert space, we associate to each subset $S \subseteq Λ$ a subsystem Hamiltonian $H_S$ and its spectrum $\mathcal{S}(S) = σ(H_S)$. This produces a family of spectra indexed by subsystems, allowing spectral data to be organized according to interaction structure. We show that subsystem Hamiltonians admit local approximations: $H_S$ can be approximated by operators supported on finite neighborhoods with an error bounded by $\|H_S - H_{S,r}\| \le |S| e^{-μr} \|Φ\|_μ$. As a consequence, subsystem spectra are stable under truncation in the sense that $d_H(\mathcal{S}(S), σ(H_{S,r})) \le |S| e^{-μr} \|Φ\|_μ.$ We then prove that for disjoint subsets $S_1, S_2 \subseteq Λ$, the subsystem spectrum is approximately additive: $d_H\big(\mathcal{S}(S_1 \cup S_2), \mathcal{S}(S_1) + \mathcal{S}(S_2)\big) \le (|S_1| + |S_2|) e^{-μD} \|Φ\|_μ,$ where $D = d(S_1, S_2)$. In the finite-range case, this relation becomes exact. The results show that spectral properties reflect the locality of interactions not only at the level of operators, but also at the level of spectra. The framework provides a way to study many-body systems in which interaction geometry directly shapes spectral behavior.

We study continuity, Hölder regularity, and $C^{1,1}$-regularity of geodesics between continuous plurisubharmonic functions on bounded domains of $\mathbb{C}^n$. We then derive regularity properties of rooftop envelopes.

In this paper, a new method for investigating Dirichlet's divisor problem is developed. For this purpose, integer points under the graph of a hyperbola are studied. Since many investigations in this direction focus on direct estimates of trigonometric sums and are not suitable for studying means, we shall consider shifts with respect to various parameters to define an optimal one. The method allows for obtaining the best possible estimates in the classical divisor problem.

We introduce a deficiency-based representation and approximation framework for values of the Riemann zeta function. The method is based on comparing two nonlinear accumulation mechanisms: global transformation of a base partial sum and local transformation of each term. Their gap defines a cumulative deficiency functional that yields the exact identity \[ ζ(q)=ζ(p)^{q/p}-D_{\infty}^{(p,q)}, \qquad q>p>1. \] This converts zeta approximation into estimation of a nonlinear deficit. We derive corrected estimators that remove first-order bias and prove the convergence law \[ B_n^{(p,q)}-ζ(q)=O\!\left(n^{-\min(2p-2,q-1)}\right). \] For odd targets, suitable choices of the base exponent recover the natural truncation rate while preserving the structural identity. Numerical experiments for $ζ(3),ζ(5),ζ(7)$ confirm theory, demonstrate strong finite-sample behavior, and illustrate extension to spectral zeta functions. The contribution is structural rather than replacing classical Euler--Maclaurin methods: we provide a unified nonlinear viewpoint on zeta approximation, convexity-induced correction terms, and tunable approximation families.

Commercially-relevant quantum computers will require large numbers of high-performing qubits that can be manufactured, integrated, and controlled at scale. Silicon exchange-only (EO) qubits are a strong candidate modality due to their control-signal simplicity and compatibility with advanced semiconductor manufacturing, but questions remain around the achievability of sufficiently low noise and a scalable control and wiring solution. Here we introduce a quantum processing unit composed of a custom-designed cryogenic CMOS controller, a novel high-density superconducting ribbon cable, and a low-noise EO qubit device. The quantum chip features a three-rail array of 54 exchange-coupled quantum dots, configurable to host up to 18 EO qubits. We integrate and use these components to demonstrate qubit performance for both single-qubit and entangling operations that advances the EO state of the art by an order of magnitude. We further validate this system by implementing a distance-5 repetition code and a quantum error detecting code then make detailed comparisons with simulations. Our approach facilitates a utility-scale quantum computer with manageable operational and capital requirements.

Let $\mathbb{F}$ be a sub-modulus field such that $2 \neq 0$. Let $\mathcal{X}$ be a sub-normed linear space over $\mathbb{F}$. Then we show that \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|x-y\|, \|y-x\|\}-(\|x\|+\|y\|) \end{align*} and \begin{align*} \bigg|\|x\|-\|y\|\bigg|\leq \|x\|+\|y\|-\frac{2}{|2|}\|x+y\|+\frac{2}{|2|}\max\{\|y-x\|, \|x-y\|\}. \end{align*} Above inequalities are finite field versions of important Tarski-Maligranda inequalities obained by Maligranda [\textit{Banach J. Math. Anal., 2008}].

The long-standing hypothesis formulated by Goreinov, Tyrtyshnikov and Zamarashkin \cite{GTZ1997} has recently been solved positively by Sengupta and Pautov \cite{SP2026} in the case of two-column matrices. In this paper, we complement their elegant proof with the equality criterion.

This paper combines the decomposition technique ($σ$-stability) in random functional analysis with the deterministic theory of asymptotically pointwise contractions to provide a complete self-contained derivation of a fixed point theorem for random asymptotically pointwise contractions. We assume the contraction function is linear $ψ(t)=λt$ ($λ<1$) and focus on the linear case under the assumption that $G$ is bounded. By choosing $p$ sufficiently large so that $5^{1/p}λ<1$, we apply the deterministic theorem in $L^p(E)$. The paper gives detailed explanations of concepts such as random normed modules, the $(ε,λ)$-topology, and $σ$-stability, and reviews the historical development of fixed point theory in the introduction.

By recent work of \citet{DobrinenICM} and \citet{Balko7} we know that every finite $G$ in the Henson graph $\mathbb{H}_{n+1}$ (the universal ultrahomogeneous $(n+1)$-clique free graph) has exact finite big Ramsey degree $k({G,n})$. That is, there is a positive integer $k({G,n})$ such that for each finite coloring $C$ of the copies of $G$ in $\mathbb{H}_{n+1}$, there is $\tilde{\mathbb{H}}$, a substructure of $\mathbb{H}_{n+1}$ and isomorphic to $\mathbb{H}_{n+1}$, such that in $\tilde{\mathbb{H}}$ at most $k({G,n})$ colors are used on the copies of $G$ in $\tilde{\mathbb{H}}$. Moreover, for exactness, for some coloring and all corresponding $\tilde{\mathbb{H}}$, all $k({G,n})$ colors are needed. The ultimate result here is that if $|G|\geq 2$, then there is a finite computable coloring $C$ such that, for all such $\tilde{\mathbb{H}}$, we have that $\tilde{\mathbb{H}}$ computes $\emptyset^{(|G|-1)}$ (and hence the halting set).

This study examines the domination polynomials of friendship graphs and book graphs, focusing on unanswered questions related to these families [Alikhani, Brown and Jahari, on the domination polynomials of friendship graphs, Filomat \textbf{30}(1) (2016) 169--178]. For the friendship graph $F_n$, with even $n$, we show that the polynomial $D(F_n,x)$ has exactly three real zeros: $0$ and two simple zeros in the intervals $(-2,-1)$ and $(-1,0)$. We further show that these two nonzero zeros have monotonic variation and converge to $-1-\frac{1}{\sqrt2}$ and $-1+\frac{1}{\sqrt2}$, respectively. We obtain the quantitative approximation $(|z|-1)^2\log |z|\le n$ for any complex zeros of $D(F_n,x)$, resulting in the explicit bound $|z|\le 1+\sqrt{\tfrac{n}{\log 2}}$. For book graphs $B_n$, we ascertain the comprehensive limit set of domination roots and establish results about the presence of real roots contingent on parity. We provide a partial answer to the integer-root an issue by establishing that friendship and book graphs have no nonzero integer domination roots, whereas for corona families, the only nonzero integer root is $-2$.

Model-based multi-agent control requires agents to possess a model of the behavior of others to make strategic decisions. Solution concepts from game theory are often used to model the emergent collective behavior of self-interested agents and have found active use in multi-agent control design. Model predictive games are a class of controllers in which an agent iteratively solves a finite-horizon game to predict the behavior of a multi-agent system and synthesize their own control action. When multiple agents implement these types of controllers, there may exist misspecifications in the respective game models embedded in their controllers, stemming from inaccurate estimates or conjectures of other agents' objectives. This paper analyzes the resulting prediction misalignments and their effects on the system's behavior. We provide criteria for the stability of multi-agent dynamic systems with heterogeneous model predictive game controllers, and quantify the sensitivity of the equilibria to individual agents' game parameters.

In this work, we develop a control-theoretic framework for constrained optimization problems with composite objective functions including non-differentiable terms. Building on the proximal augmented Lagrangian formulation, we construct a plant whose equilibria correspond to the stationary points of the optimization problem. Within this framework, we propose two control strategies - a static controller and a dynamic controller - leading to two novel optimization algorithms. We provide a theoretical analysis, establishing global exponential convergence under strong convexity assumptions. Finally, we demonstrate the effectiveness of the proposed methods through numerical experiments, benchmarking their performance against state-of-the-art approaches.

Let $(M,g,J,ω)$ be an almost Kähler manifold. For any smooth function $f$ on $M$, one can associate an automorphism $ψ\in \mbox{Aut}(TM)$ for which the Kähler form is invariant. Using $ψ$, one can ``twist" the metric $g$ and almost complex structure $J$ to obtain a new almost Kähler structure $(g^ψ,J^ψ,ω)$ on $M$. Let $\widetilde{D}$ denote the Chern connection of $(g^ψ,J^ψ,ω)$ and let $K^{-1}$ denote the anti-canonical bundle of $(TM,J^ψ)$. In the current paper, we give an explicit formula for the local connection 1-form $α$ associated to the pair $(K^{-1},\widetilde{D})$. The Chern-Ricci form of $(g^ψ,J^ψ,ω)$ is then $ρ_{\widetilde{D}}=-dα$. We note that under certain conditions the aforementioned formula assumes a simpler form when applied to the calculation of $α$. We illustrate this with some examples.

As generative AI (GenAI) tools are increasingly used as informal tutors for mathematics learning, future systems may become more proactive and process-aware in deciding when and how to offer support. Yet such support raises an important design tension: help that is timely may also feel interruptive or overly monitoring. To inform the design of process-aware AI tutors, we surveyed 330 secondary school students in China (Grades 7--11) about their preferred tutoring behaviors, attitudes toward proactive intervention, and acceptable use of learning-process data. We found three design-relevant patterns. First, students preferred autonomy-preserving support, such as hints over direct answers. Second, they favored graduated proactive support over constant interruption, preferring small hints first and stronger assistance only as needed. Third, they drew clear privacy boundaries around learning-process data: students were comfortable with problem-solving steps and mistake patterns, but substantially less comfortable with attention- or behavior-related signals. Together, these findings offer early empirical guidance for designing AI tutors that balance timely support with learner agency, and personalization with perceived privacy boundaries in K-12 contexts.

We investigate the arithmetic-harmonic inequality (AHI) index, a bounded and scale-invariant measure of dispersion for positive random variables, defined through the interplay between the mean and its reciprocal. We derive analytical expressions for the AHI index within the generalized inverse Gaussian (GIG) family, encompassing the inverse Gaussian and gamma distributions as important special cases. We study the associated estimator, obtain a tractable expression for its expectation, establish its asymptotic properties, and derive explicit first-order bias approximations. A Monte Carlo study is conducted to evaluate the finite-sample performance of the estimator under various scenarios. An application to GDP per capita data for countries in the Americas illustrates the role of the AHI index within the broader Atkinson family across several values of the inequality-aversion parameter. The results show the good performance of the AHI index as a tractable and interpretable measure of economic dispersion.

As open-source silicon initiatives democratize access to integrated circuit development using multi-project environments, silicon area has become a premium resource. However, minimizing this layout area traditionally forces designers to compromise on core performance specifications. To address this challenge, this paper presents an open-source framework based on a hybrid mixed-variable particle swarm optimization algorithm and the gm/ID methodology to minimize the layout area of complex analog circuits while meeting design requirements. The framework's efficacy is demonstrated by designing a low-power instrumentation amplifier that achieves a 90.33% reduction in gate area over existing implementations.

We consider subset recovery in the many-access Gaussian multiple-access channel with a shared spherical codebook, where codewords are drawn independently and uniformly from the hypersphere of radius \( \sqrt{nP} \), the number of active users scales linearly with the blocklength $n$ as \( K_a(n)=βn \) for a constant \( β> 0 \), and the codebook size is \( M_n=n^d \) with \( d>2 \). We identify a geometric property showing that, for \( 0<β<2 \), any transmitted \( K_a(n) \)-subset lies in a single hemisphere with high probability for sufficiently large $n$. We further show that reliable decoding is possible only for \( β< 1/4 \). The overlap between the reliable decoding range of \( β\) and the hemispherical concentration range motivates our approach of two-stage decoding procedure. In the pre-filtering stage, the decoder restricts attention to a sequence of spherical caps \( \{ \hat{\mathcal{H}}_n \} \) that converges in Hausdorff distance to the hemisphere $\hat{\mathcal{H}}$, whose axis is the normalized observation \( \hat{\mathbf{u}}=\mathbf{Y}/\|\mathbf{Y}\| \). In the second stage, maximum-likelihood decoding is performed over the reduced candidate set. We show that the per-user error probability of the pre-filtering stage vanishes as \( n\to\infty \). Moreover, the per-user error probability of the maximum-likelihood stage over the reduced search space decays exponentially with asymptotic exponent \( P/4 \).

We develop a mechanistic dynamical-systems formulation of best response in finite-action games with relational structure on the action set. The proposed neuromorphic decision dynamics realize best response as the stable outcome of an internal state-space process, rather than as an externally imposed choice rule. This provides a deterministic account of commitment formation, symmetry resolution through basins of attraction, and hysteresis and decision persistence under perturbations. For action spaces with circulant coupling, we prove using Lyapunov-Schmidt reduction that the action-coupling operator determines which components of evidence govern decision formation. We further show that the dynamics implicitly compute a geometry-aware utility, converge exponentially to the corresponding best response with rate independent of the number of actions, and switch only when evidence is sufficiently strong. In contrast, supplying the same geometry-aware utility directly to logit dynamics does not recover these properties, showing that relational structure must be embedded in the decision mechanism itself. We illustrate the framework in a repeated coverage game, prove that the induced game is an exact potential game, and show that its Nash equilibria are reached by the neuromorphic dynamics.

Prior research shows that large language models (LLMs) exhibit systematic extrapolation bias when forming predictions from both experimental and real-world data, and that prompt-based approaches appear limited in alleviating this bias. We propose a supervised fine-tuning (SFT) approach that uses Low-Rank Adaptation (LoRA) to train off-the-shelf LLMs on instruction datasets constructed from rational benchmark forecasts. By intervening at the parameter level, SFT changes how LLMs map observed information into forecasts and thereby mitigates extrapolation bias. We evaluate the fine-tuned model in two settings: controlled forecasting experiments and cross-sectional stock return prediction. In both settings, fine-tuning corrects the extrapolative bias out-of-sample, establishing a low-cost and generalizable method for debiasing LLMs.

We explore the connection between black hole shadows, thermodynamic phase structure, and microstructure of charged and rotating black holes within General Relativity and Geometrothermodynamics. Focusing on Reissner-Nordström and Kerr solutions, we establish a criterion to select the most suitable Geometrothermodynamic metric for a system, revealing that the first metric from enthalpy and the second from mass correctly reproduce heat capacity singularities. We show that the shadow radius encodes the same phase information as entropy and introduce Shadow-Microstructure diagrams to extract insights into stability and microscopic interaction types directly from observational bounds. Applying this framework to Sagittarius A*, we constrain the macroscopic parameters and the allowed microscopic thermodynamic phases. Our findings indicate that shadow measurements offer a novel probe of thermodynamic and microscopic aspects of black holes, enabling tests of alternative theories of gravity and thermodynamic frameworks.

We determine the non-reduced components of global nilpotent cones in various cases of interest. In particular, under the appropriate coprimality conditions, we show: (1) the global nilpotent cone for an $L$-twisted $\operatorname{GL}_r$-Hitchin fibration associated to a curve $C$ of genus $g\ge 2$ is nowhere reduced, where $L$ is either the canonical bundle or has degree greater than $2g-2$; (2) the global nilpotent cone for a moduli space of one-dimensional sheaves on a K3, abelian, or del Pezzo surface is nowhere reduced; (3) suppose $\ell$ is a primitive, basepoint-free, big and nef class on a K3 surface, then a general fiber of a Beauville-Mukai system for the class $r\ell$ has primitive homology class if and only if $r=1$. Our methods include group scheme actions on Lagrangian fibrations, a GIT-stratification of global nilpotent cones of Hitchin fibrations, and deformation to the normal cone.

We give a categorical proof of the projectivity of $N$ in the free topos -- in proof-theoretic terms, the rule of countable choice for intuitionistic higher-order logic -- based on the unpublished proof of Michael Makkai (c.1980). The presentation aims to be self-contained and accessible to any reader acquainted with elementary toposes and their logic.

Energy efficiency has emerged as a critical challenge in modern base stations (BSs), as the power amplifier (PA) consumes a substantial portion of the total power due to its limited efficiency. We investigate waveform and mode adaptation to enhance the energy efficiency of BSs. We propose Switch-DFT, an adaptive switching framework that selects between cyclic prefix orthogonal frequency division multiplexing (CP-OFDM) and discrete Fourier transform-spread-OFDM (DFT-s-OFDM) waveforms, as well as between single-input multiple-output (SIMO) and multiple-input multiple-output (MIMO) modes. Switch-DFT improves efficiency by reducing PA backoff with DFT-s-OFDM and achieves the target rate at lower power by leveraging higher MIMO throughput. This results in superior energy efficiency over a wide range of the spectral efficiencies compared with static configurations.

We derive $(1+2)$D subsystems~$(E1,E2)$ from the (2D inviscid Boussinesq, 3D axisymmetric Euler) equations in the (meridian) plane. The integer $m=1,2$ only appears in two numerical coefficients of subsystem~$(Em)$. Thus we discover a unification. We then study two unified $(1+1)$-dimensional systems, denoted $(R0)$ and $(Z0)$, that are rigorously derived from the $(Em)$. The main point of view in this revision is that these $(1+1)$D systems are not ad hoc model equations and not merely ``symmetry-axis reductions.'' Rather, they arise as exact symmetry-axis/apex restrictions of the full $(1+2)$D system~$(Em)$ obtained from 2D inviscid Boussinesq and 3D axisymmetric Euler, and they already contain the core finite-time singularity mechanism of the full problem. The paper has three main outputs. First, it derives the polar $(1+2)$D subsystem~$(Em)$ from the 2D inviscid Boussinesq equations and from the 3D axisymmetric Euler equations and identifies the exact unified $(1+1)$D systems $(R0)$ and $(Z0)$ carried by the symmetry axes. Second, it proves finite-time blow-up for the resulting apex dynamics and analyzes the associated convective axis reduction. Third, it derives the exact background--remainder equations and formulates a conditional nonlinear stability mechanism: if a compatible full background exists on $[0,T)$ with the adapted coefficient bounds required by the weighted energy method, if the weighted elliptic estimate holds, and if a gap exponent $σ\in(C_{\rm lin},1)$ is available so that the remainder remains below the background scale, then the same finite-time apex blow-up transfers to the full solution.

The surface code is a promising platform for a quantum memory, but its threshold under coherent errors remains incompletely understood. We study maximum-likelihood decoding of the square-lattice surface code in the presence of single-qubit unitary rotations that create electric anyon excitations. We microscopically derive a non-linear sigma model with target space $\mathrm{SO}(2n)/\mathrm{U}(n)$ as the effective long-distance theory of this decoding problem, with distinct replica limits: $n\to1$ for optimal decoding, which assumes knowledge of the coherent rotation angle, and $n\to0$ for suboptimal decoding with imperfect angle information. This exposes a sharp distinction between the two decoders. The suboptimal decoder supports a "thermal-metal" phase, a non-decodable regime that is qualitatively distinct from the conventional non-decodable phase of the surface code under incoherent Pauli errors. By contrast, the metal phase cannot arise in optimal decoding, since the metallic fixed-point becomes unstable in the $n\to 1$ replica limit. We argue that optimal decoding may be possible up to the maximally-coherent rotation angle. Within the sigma model description, we show that the decoding fidelity is related to twist defects of the order-parameter field, yielding quantitative predictions for its system-size dependence near the metallic fixed point for both decoders. We examine our analytic predictions for the decoding fidelity as well as other physical observables with extensive numerical simulations. We discuss how the symmetries and the target space for the sigma model rely on the lattice of the surface code, and how a stable thermal metal phase can arise in optimal decoding when the syndromes reside on a non-bipartite lattice.

Absolute binding free energy (ABFE) calculations offer a theoretically rigorous approach for predicting protein--ligand binding affinities without the scaffold constraints of relative binding free energy (RBFE) perturbations. However, broad adoption of ABFE in high-throughput hit discovery campaigns has been hindered by high computational costs and a lack of large-scale validation. Here, we present Felis, an open-source, automated, and scalable toolkit designed for high-throughput ABFE calculations. Paired with ByteFF, a previously developed data-driven molecular mechanics force field for drug-like molecules, Felis achieves ranking performance comparable to state-of-the-art RBFE methods on a diverse dataset comprising 43 protein targets and 859 ligands. Furthermore, we demonstrate robust convergence and ranking performance of Felis on a more challenging KRAS(G12D) dataset, where some ligands and the cofactor are highly charged. Crucially, all Felis predictions in this study were generated in a strict zero-shot manner, eschewing custom force-field modifications and alchemical schedule fine-tuning. This demonstrates the viability of Felis as an effective, ready-to-use tool for computational structure-based drug design.

Recent analyses of large-scale structure and redshift surveys have reported significant dipolar anisotropies in the local Universe that are not straightforwardly attributable to a global kinematic boost. When interpreted within standard frameworks, these signals may correspond to coherent bulk flows that have been reported to exhibit tension with $Λ$CDM expectations. On the other hand, signals inferred from different astrophysical probes are not always consistent with the Cosmic Microwave Background (CMB) dipole, challenging the assumption of dipoles that are pure kinematical in origin. In an inhomogeneous universe, the identification of the Hubble frame with a geodesic matter flow is not guaranteed beyond the idealized FLRW limit, particularly once structure formation leads to a non-trivial distribution of velocities and gravitational fields. Within a fully covariant framework, we show that a non-geodesic observer congruence introduces an additional contribution to the propagation of redshift along the past light cone, proportional to the line-of-sight projection of the observer four-acceleration. This generates a dipolar modulation in the redshift itself, which propagates to any observable defined in redshift space. Unlike the standard kinematic dipole associated with a global Lorentz boost, this contribution arises from the kinematics of the observer congruence and depends on its evolution along the past light cone. As a result, it induces a dipolar modulation with a non-trivial redshift dependence. This behaviour provides a concrete observational test of whether the observed dipole is fully accounted for by large-scale structure kinematics or requires additional non-geodesic contributions.

In this work, we study the properties of the pseudoscalar hidden-charm tetraquark states by analyzing their two-body strong decays via the QCD sum rules based on rigorous quark-hadron duality. We take into account the vacuum condensates up to dimension 5 on the QCD side, and obtain the hadronic coupling constants. At last, we obtain the total decay widths $Γ(Z_{c}^{-}) = 326.20^{+4.26}_{-3.11}$ MeV and $Γ(Z_{c}^{+}) = 91.84^{+0.96}_{-0.76}$ MeV, respectively, where the $Z_{c}^{+}$($J^{PC}=0^{-+}$) and $Z_{c}^{-}$($J^{PC}=0^{--}$) denote the pseudoscalar hidden-charm tetraquarks with the diquark-antidiquark structures $[uc]_{A}[\bar{d}\bar{c}]_{V}-[uc]_{V}[\bar{d}\bar{c}]_{A}$ and $[uc]_{A}[\bar{d}\bar{c}]_{V}+[uc]_{V}[\bar{d}\bar{c}]_{A}$, respectively.

We study low-rank tensor methods for the numerical solution of Schrödinger's equation with time-independent and explicitly time-dependent Hamiltonians, motivated by large-scale simulations of many-body quantum systems and quantum computing devices subject to time-dependent control pulses. We outline the recent application of the "basis update and Galerkin" (BUG) method for tensor trains, and describe the established TDVP and TDVP-2 algorithms based on the time-dependent variational principle. For comparison, we also consider the BUG method in the Tucker format. All these approaches enable memory efficient representations of partially entangled quantum states and thereby mitigate the exponential cost of conventional state-vector formulations. The rank-adaptivity relies on the truncated singular value decomposition, in which the rank of a matrix is reduced by setting its smallest singular values to zero, based on a threshold parameter that controls the truncation error. Numerical experiments on representative time-independent and time-dependent Hamiltonian models quantify the tradeoff between accuracy and compression across methods, with particular attention to the interplay between the time-step and the truncation threshold, and how the computational effort scales with the number of sub-systems in the quantum system.