Papers

Nonverbal communication through head and neck movement is fundamental to human social signalling, yet how robotic neck morphology translates motion into communicative information remains poorly understood. We present an information-theoretic framework characterising robot neck movement as a communication channel, quantifying information transmitted and energy expended across varied configurations. Using a robotic neck platform, we recorded 84 video stimuli spanning three rotational degrees of freedom (DoF), varying amplitude, acceleration, and frequency, measuring Shannon entropy of pixel-change signals alongside energy consumption. A perceptual study validated communicative interpretations of each motion. While humans typically engage one axis per gesture, robots are unconstrained by biological architecture, motivating tests up to 3 DoF. Yet communicative information peaks at two DoF and decreases at three despite rising energy cost, a phenomenon we term the morphological information bottleneck. Motion parameter effects were parameter-dependent, some additive, others non-linear. We introduce the Motor Information Space, a framework mapping entropy against energy to expose communicative efficiency across morphologies, in which the optimal configuration achieves 5.26 bits at competitive energy cost. Perception data further confirm multi-axis movements reduce clarity. These findings challenge the assumption that anatomical completeness improves robotic expressiveness, establishing a quantitative basis for morphological design in robots, especially humanoids.

Solving linear differential equations is a fundamental task in scientific computing and an important primitive for quantum computing. A recent one-ancilla quantum differential equation solver provides a hardware-friendly and locality-preserving approach with provable performance guarantees, making it highly suitable for the early fault-tolerant and near-term regimes. Its simple circuit structure comes with a natural trade-off: the maximum single-run circuit depth scales as $O (1/\epsilon)$ in the target accuracy $\epsilon$. In this work, we reduce this depth by combining the solver with classical step-size postprocessing. By running the one-ancilla solver at a logarithmic number of finite time step sizes and using classical post-processing to cancel leading discretization errors, we reduce the maximum single-run circuit depth to $O(\mathrm{polylog}(1/\epsilon))$ without adding quantum ancillae or sacrificing locality. Technically, extending extrapolation ideas beyond Hamiltonian and Lindbladian dynamics requires regularity estimates for observable maps under nonunitary evolution, which we obtain through a holomorphic extension of the adjoint evolution. Numerical experiments on the Hatano-Nelson model (ODE) and the convection-diffusion equation (PDE) demonstrate the effectiveness of the approach.

We present JAX-FVM, an open-source, fully differentiable finite volume method (FVM) for the two-dimensional compressible Euler and Navier-Stokes equations on unstructured triangular meshes. The solver is written entirely in JAX, so that every operation : mesh connectivity, flux evaluation, slope limiting, and time integration is just-in-time compiled, vectorised, and end-to-end differentiable through automatic differentiation (AD), and runs transparently on CPU or GPU. On the numerical side, JAX-FVM is built around an entropy-conservative Tadmor/Ismail-Roe two-point flux supplemented with entropy-variable Rusanov or Roe dissipation, second-order MUSCL reconstruction of primitive variables with least-squares gradients and Venkatakrishnan limiting, and a family of explicit (RK2-4) and matrix-free implicit (Newton, SDIRK2) time integrators whose Jacobian actions are obtained by AD. The combination of an unstructured-mesh compressible FVM with end-to-end differentiability fills a gap left by existing differentiable CFD frameworks, which are almost exclusively restricted to structured grids or spectral discretisations. We describe the governing equations, the discretisation, the software architecture, and a set of standard verification cases. The code is openly available at https://github.com/guigzair/jax_fvm.

In this paper, we prove the existence of directed strongly regular graphs with parameters (108,11,3,2,1), (108,14,10,0,2), (108,22,12,6,4), (108,23,9,8,4), (108,25,15,8,5), (108,34,18,12,10), (108,38,22,12,14), (108,39,23,14,14), (108,41,35,16,15), (108,42,33,18,15) and (108,46,22,19,20). Further, we obtain directed strongly regular graphs with parameters (54,10,4,1,2), (54,11,4,3,2) and (54,16,7,4,5). The constructions are obtained by considering finite groups acting transitively on 54 and 108 vertices.

In this article, a necessary and sufficient condition is proved for the dynamical irreducibility of a family of polynomials over a finite field. Using this result, an explicit construction of a dynamically irreducible set of polynomials is given over the finite field $\mathbb{F}_{p^p}$ where $p$ is a prime. Moreover, the existence of dynamically irreducible sets of size at least $p^2$ is also established over every finite field $\mathbb{F}_q$ where $q$ is a $p$ power. Finally, a bound on the cardinality of the set is given that needs to be tested for the dynamical irreducibility of a set of polynomials.

Greenhouse fruit-production control aims to maximize the economic performance (fruit revenue minus operating costs) while operating within system constraints under external weather disturbances. Control methods need to balance the delayed economic benefit of fruit yield with current operating costs. For such problems, model predictive control (MPC) can explicitly handle system constraints under future weather disturbances, but can become computationally demanding when using sufficiently long prediction horizons for (relatively large) nonlinear greenhouse fruit production models. In contrast, reinforcement learning (RL) can learn control policies offline while considering longer-term economic performance, but struggles to enforce system constraints, and performance may degrade under unseen weather trajectories. This work proposes trajectory-selection RL-MPC, a framework that incorporates longer-term economic information of fruit yield into a short-horizon MPC optimization problem. The framework uses an RL rollout trajectory to define a terminal region constraint and terminal cost. Next, a nonlinear MPC solves a short-horizon optimization problem with these terminal ingredients to find a local optimum. Finally, the framework selects and executes the first input from the trajectory with the better objective value, either from the MPC-predicted or the RL rollout trajectory. The method is applied to GreenLight, a large-scale greenhouse tomato production model that exhibits stiff dynamics. The simulation results show that trajectory-selection RL-MPC with a one-hour prediction horizon matches the closed-loop performance of a high-performing guiding policy while significantly improving over standalone MPC with the same horizon.

Although the action of diffeomorphisms on pinor (or spinor) fields is geometrically unambiguous, the definition of a natural Lie derivative is more subtle and requires some understanding of the geometry of the space of metrics at a point. In this work we revisit the classical construction of Kosmann and Bourguignon-Gauduchon and develop the corresponding theory in generalised geometry.

Maximal monotone mappings that are piecewise polyhedral arise from subdifferentials of convex functions and saddle functions that are piecewise linear-quadratic and enter into algorithmic constructions of importance in linear-quadratic optimization and associated splitting methods. The question of whether those constructions preserve maximal monotonicity is then crucial. The usual answers to that invoke constraint qualifications involving the nonemptiness of intersections of certain relative interiors, but it is shown here that the piecewise polyhedral structure allows the relative interiors to be bypassed.

We prove a closed-string remodeling statement for the affine binary dihedral Calabi–Yau orbifold threefold $\mathcal X=[\mathbb C^2/\Gamma]\times\mathbb C$, where $\Gamma$ is a binary dihedral subgroup of $SU(2)$. This target lies outside the toric setting of the Bouchard–Klemm–Mariño–Pasquetti remodeling conjecture: the toric mirror curve is replaced by the type-$D_l$ relativistic Toda curve of Brini–Ma–Strachan, and the Chekhov–Eynard–Orantin topological recursion is replaced by the $\mathbb Z_2$-equivariant topological recursion of Giacchetto–Kramer–Lewański, run in the sign sector of the Toda-curve involution with the Prym kernel as its two-point input. We identify the equivariant orbifold quantum cohomology Frobenius manifold of $\mathcal X$ with the invariant Jacobian Frobenius structure of the Toda curve, and we prove that the B-model $R$-matrix, defined by regularized stationary phase, equals the A-side normalized canonical Givental–Teleman $R$-matrix on a chosen simply connected domain; this equality is anchored at the orbifold point through a semistable degeneration of the Toda curve. Comparing the resulting Givental–Teleman and Dunin-Barkowski–Orantin–Shadrin–Spitz graph sums then identifies, after a parity-twisted leaf substitution, the sign-sector recursion with the descendant Gromov–Witten generating functions of $\mathcal X$ in the stable range ($2g-2+n>0$ with $n>0$), and identifies the recursion free energies with the equivariant Gromov–Witten free energies of $\mathcal X$ for $g\geq2$. We also match the genus-one potential defined by the reduced calibrated tau function with the Gromov–Witten potential up to an additive constant.

We study de Rham functional equations driven by two increasing fractional linear transformations. Our main purpose is to relate the singularity theory of the associated solutions to dyadic potential theory on the binary tree. We first prove an existence and uniqueness theorem for increasing, left-continuous solutions in the full range of linear fractional data, and identify the trapping region in parameter space where the solution is continuous. For a large class of parameters we show that the de Rham solution is the normalized cumulative capacitary function of a multiplicative dyadic capacity. This gives a potential-theoretic model for Möbius de Rham systems. We then sharpen Okamura's Hausdorff-dimensional estimates for the singular measure associated with the solution by replacing Hausdorff dimension with dyadic Riesz capacities at the upper endpoint of Okamura's theorem.

Online truckload bid acceptance is a closed-loop stochastic decision problem in which a carrier or broker must, in real time, accept or reject a tendered load subject to operational feasibility, fleet repositioning costs, and opportunity cost against future demand. Public, reproducible benchmarks for this problem are scarce: existing routing benchmarks are static, while dynamic-fleet studies typically rely on private operator data. We introduce FreightBidBench, a public-calibrated, dependency-free, closed-loop benchmark in which feasibility (pickup reach, appointment windows, simplified hours-of-service, stochastic yard delays) and economics (service-failure penalty, terminal fleet value, daily price-premium window) are explicit, versioned, and reproducible from public Freight Analysis Framework and U.S. Department of Agriculture truck rate data. We develop two full-horizon hindsight ceilings: a simple LP style relaxation and a tighter Lagrangian-per-truck information relaxation that retains per-truck hours-of-service and sequencing structure and is 20.7% tighter than the LP relaxation on a tight-capacity scenario and 39.3% tighter on a scarce-capacity scenario. We introduce a parametric surrogate-rollout cascade with boundary-band and scarcity-pressure escalation triggers. On ten-seed tight and scarce scenarios, the best simple policy retains 91.0% and 86.5% of rollout profit and the standard-library surrogate 94.2% and 89.3%; a cascade at a single escalation band recovers roughly 98% on both at 40-56% of rollout's mean decision latency, and on the tight scenario is statistically indistinguishable from the rollout teacher (paired-bootstrap 95% CI on the profit delta spans zero).

I formulate black-hole-horizon-induced decoherence of charged branch codes as the leading-soft QED restricted to an exterior algebra, formulated as an open quantum system. The fixed-history Feynman–Vernon identity ${\cal F}[J,J]=1$ remains exact. Decoherence enters through the unequal-history influence factor that survives exterior monitoring and belongs to the complementary horizon output. In the coherent eikonal regime, I derive the completely positive Schur channel $({\cal E}_H^{(0)}\rho)_{ab}=\langle\Phi_b^{H,(0)}|\Phi_a^{H,(0)}\rangle \, \rho_{ab}$. The leading soft input is the eikonal factor, projected onto the horizon radiative algebra. The channel yields Gram-positivity constraints, an exterior quantum-eraser bound, finite-time non-Markovianity tests, soft/hard scaling criteria, and a charged-qutrit interferometer measuring a leading-soft Bargmann holonomy. The holonomy phase is the rephasing-invariant symplectic area of a triangle in horizon soft phase space. I show that its orientation, common-mode, triangulation, and completely positive determinant identities render falsifiable tests beyond pairwise two-path visibility.

We study the output regulator problem for an impedance passive linear plant, using the classical resonant internal model based controller.

Large-scale combinatorial optimization remains demanding for classical heuristics, particularly when dense Quadratic Unconstrained Binary Optimization (QUBO) formulations induce large memory footprints, high CPU utilization, and long execution times. While near-term quantum processors cannot yet deliver unconditional quantum advantage, hybrid architectures can provide practical value by reducing the resource burden. This paper presents a resource-efficiency study of Hybrid Quantum Neighborhood Selection (HQNS), a framework that decomposes large dense QUBO instances into bounded-width quantum subproblems via stochastic frontier selection. We evaluate HQNS on the Maximum Diversity Subset Selection Problem (MDSSP), focusing on the trade-off between solution quality retention and resource consumption. Benchmarks up to N=1000 candidates show that HQNS preserves 99.9908% of the mean diversity score of an 11-restart parallel Simulated Annealing baseline, while reducing wall-clock time by 94.91%, peak CPU utilization by 64.68%, and peak memory usage by 88.61%. The QPU execution time remains bounded within a 6-7 second envelope across scales, indicating that the quantum component is decoupled from the global QUBO dimension when the frontier size is fixed. These results suggest that HQNS provides a resource-aware pathway for deploying hybrid quantum optimization in practical large-scale settings, serving as an efficient architecture for incorporating near-term quantum processors into classical optimization pipelines.

We discuss some useful properties of the solution map (flow) of a nonlinear dynamical system with a finite-dimensional state space. Then, we introduce the Perron-Frobenius semigroup, and we prove that it is a positive strongly continuous semigroup of contractions. We show that, given a nonlinear system and an invariant set, this set is an almost global attractor if and only if certain Perron-Frobenius semigroups associated to the nonlinear system are strongly stable. Unlike other works on the Perron-Frobenius semigroup from the literature, we do not require the existence of a compact and invariant state-space for the dynamical system, we allow trajectories with finite escape time, and we do not require the attractor to be locally (Lyapunov) stable. Two simple examples are used throughout the paper to illustrate the theory.

The capitulations of ideals in $\mathbb{Z}_p^d$-extensions and pseudo-null submodules of the classical Iwasawa modules are closely related as evidenced in the works of Ozaki and Fujii. In this paper, we investigate the analogous situation for the even $K$-groups. As an application, we obtain a new sufficient condition for existence of non-trivial pseudo-null submodules in the classical Iwasawa modules.

We study iterative linearized finite element methods for the numerical approximation of semilinear elliptic boundary value problems with nonlinear reaction terms of asymptotically linear growth. Our approach reaches considerably beyond the classical theory by allowing for nonlinearities that are not necessarily monotone. We investigate a stabilized implicit-explicit (IMEX) iteration combined with first-order finite element discretizations on graded meshes that are able to resolve corner singularities in polygonal domains. To account for the reduced regularity of the analytical solutions, we establish regularity estimates in corner-weighted Sobolev spaces. The principal novelty of the paper is a non-standard analysis of the approximation properties of the discrete Galerkin limits resulting from the iterative process, which yields optimal a priori convergence estimates. Numerical experiments underline the theoretical findings.

We study a bulk-driven nonlinear variant of the Kipnis-Marchioro-Presutti model of stochastic energy diffusion in which local collisions are biased to induce a net energy flow, resembling the effect of an external field. Starting from the microscopic master equation, we derive the hydrodynamic description of the driven system via a local equilibrium approximation, obtaining explicit expressions for the energy current and the associated diffusivity and mobility transport coefficients, which are nonlinear functions of the local energy density. We test our findings in kinetic Monte Carlo simulations of the model and, as a proof of concept, we demonstrate the versatility of this driving mechanism to control nonlinear energy transport by inducing time-crystalline phases. In particular, we show that appropriately designed packing fields induce the spontaneous formation of traveling energy condensates, exhibiting robust long-range temporal order reminiscent of continuous time crystals. Our results provide a simple yet powerful framework to study bulk-driven nonlinear energy diffusion in stochastic many-body systems, offering a bridge between microscopic dynamics, macroscopic transport, and controlled spatiotemporal order.

We prove that a commutative ring $\mathbf R=(R,+,\cdot)$ of characteristic $2$ satisfying the identity $x^{2^n}\approx x$ together with the binary relation $\le$ on $R$ defined by $x\le y$ if $xy=x^2$ forms a meet-semilattice with smallest element $0$. If, moreover, $\mathbf R$ is unitary then we derive two binary term operations $\wedge$ and $\vee$ on $R$ which together with the unary term operation $x':=x+1$ form a Boolean algebra.

We develop a spectral theory of $z$-ideals for commutative semirings. The lattice $\mathsf{ZId}(S)$ of $z$-ideals is a coherent frame for every commutative semiring $S$ – unconditionally, without cancellativity, subtractivity, or Noetherian hypothesis – so the prime spectrum $\mathsf{Spec}_z(S)$ is spectral. Under an explicit finite-type hypothesis on the canonical congruence-generated closure~$g$, the lattice $\mathsf{Id}_{g}(S)$ of $g$-closed ideals is likewise a coherent frame, and $\mathsf{Spec}_g(S)$ is spectral and homeomorphic to the space of prime $g$-congruences. These frame results are accompanied by a regularity criterion: a semiring with all multiplicative idempotents complemented is von Neumann regular if and only if every principal ideal is a $z$-ideal, extending Mason's classical theorem from rings. Separating the maximal-ideal-hull $z$-closure from the maximal-congruence-hull $g$-closure – operations that coincide in rings but diverge in semirings – is a central theme, confirmed by explicit computations in $\mathbb{N}$ and power-set semirings. Both constructions carry a complete functorial formulation.

Pairwise comparison (PC) via pairwise reciprocal matrices (PRMs) is central to the Analytic Hierarchy Process (AHP). Although the traditional eigenvector method is widely applied to derive priorities, its theoretical robustness in reflecting true priority vectors remains debated. Building upon a previous iteration of this study, this research develops the revised Least Penalty-Squared Prioritization (LPSP) optimization models, including the revised Least Product of Penalty and Direct Squares (LPPDS) and revised Weighted Squares (LPPWS), to minimize the revised Root Mean Penalty-Squared Variance (RMPSV) and the revised Root Mean Penalty-Weighted Square Variance (RMPSWV). However, solving these non-linear formulations is computationally complex for decision-makers. To overcome these limitations, this study proposes the Parallel Osprey Optimized Least Penalty-Squared Prioritization (POO-LPSP) method. By integrating an improved bio-inspired metaheuristic Parallel Osprey Optimization Algorithm (POOA), this framework efficiently solves complex LPSP models to minimize RMPSV and RMPSWV, thereby enhancing prioritization reliability. The practical utility and computational efficiency of the POO-LPSP method are validated through a numerical application focusing on a Generative AI (GAI) vendor selection problem. To extend, POO-LPSP can serve as a robust alternative to Saaty's Eigen system method for AHP applications.

We establish novel quantitative limit theorems for the asymptotic distribution of colours in a generalized Pólya urn. Concretely, we construct explicit rates of convergence for the proportion of balls of each colour in the urn, both in square-mean and almost surely, under a general condition on the replacement matrix. As an application, we revisit three models of random recursive trees studied by Janson (Random Structures & Algorithms 26 (2005), 69–83): random recursive trees, random plane recursive trees, and random recursive $d$-ary trees. For each model, we show that the corresponding outdegree statistics can be cast as generalized Pólya urns, and thereby obtain explicit rates of convergence, both in $L^2$ and almost surely, for the proportion of nodes of each outdegree. In all three cases, the rates we obtain are of order $O(1/n)$ in $L^2$ and almost surely, and are uniform in the outdegree under consideration.

Friedman and Wigderson (1995) introduced a notion of second eigenvalue for hypergraphs that generalizes the second eigenvalue of the adjacency matrix of a graph. We show that $r$-uniform Erdős-Rényi hypergraphs on $n$ vertices exhibit a spectral gap as soon as their expected number of hyperedges $m$ satisfies $m \gg n^{r/2}$. Prior work identified this scale only up to logarithmic factors; removing these factors is the main technical challenge. Our proof overcomes this obstacle through an explicit decomposition of an associated selector process, inspired by a generic decomposition theorem of Talagrand (2021). As a consequence of our techniques, we obtain improved norm bounds for sparse random tensors with independent entries. Finally, under a mild moment equivalence assumption, we extend to tensors a seminal result of Seginer (2000) for random matrices with i.i.d. entries.

In this paper, a residual-safeguarded local Matrix Fisher–Gaussian (MFG) inference method is developed for wrench-based pose estimation on $\mathrm{SO}(3)\times\mathbb{R}^3$. The force/torque measurements are modeled by a quasi-static contact system in which the predicted wrench depends on the unknown object pose through an implicit equilibrium state. Since the resulting nonlinear likelihood is not globally conjugate to the coupled MFG family, a local Bayesian update is constructed by linearizing the reduced wrench residual and matching the induced Gauss–Newton posterior model to a coupled MFG distribution. It is shown that the reduced residual Jacobian has a Schur-complement form, and that the local quadratic posterior admits a closed-form MFG approximation matching the prescribed local first- and second-order posterior coefficients. The same sensitivity model yields a compensated rotational information score, which characterizes the weakest locally informative attitude direction after translational compensation. A residual-safeguarded recentering algorithm is further introduced to update the linearization point only through candidates that decrease the recomputed whitened wrench residual. In the tested sparse prior-mismatch regimes, the resulting estimator reduces residual merit and pose error relative to single-pass and local baseline variants, and controlled robot experiments provide a proof of concept under calibrated quasi-static conditions.

We consider Schrödinger operators with purely imaginary potential $P = - h^{2} \Delta + i V ( x )$ on a bounded domain. Assuming that near its critical points the potential $V$ can be approximated by an homogeneous polynomial, we show that in the limit $h \to 0$ the leftmost eigenvalues of $P$ are asymptotically given by the local model associated to the most degenerated critical points of $V$. We give applications of this result to the associated evolution problem including shear flows in fluid mechanics.

This study presents a multi-fidelity framework for the systematic optimization of genetic algorithm (GA) hyperparameters. The framework integrates three fidelity levels: high-fidelity Fast Fourier Transform (FFT) homogenization for validation, a medium-fidelity 3D convolutional neural network surrogate for rapid property evaluation, and a low-fidelity Gaussian process (GP) surrogate within a Bayesian optimization (BO) framework to guide the hyperparameter search. Various acquisition functions are evaluated, with logNEI achieving the best performance by effectively accounting for the noise inherent in GA evaluations. The proposed framework identifies hyperparameter configurations that enable a 25-generation GA run to achieve elastic modulus values comparable to those obtained in a full 75-generation optimization. Furthermore, introducing a penalized BO objective significantly reduces the number of required lattices with only minor decreases in absolute achieved elastic modulus, revealing a practical trade-off between performance and the number of structures that must be evaluated. High-fidelity FFT validation verifies the effectiveness of the surrogate-driven optimization strategy. The optimized hyperparameters allow for rapid convergence, eliminate the need for lattice mutation, and reduce the overall computational cost by 24% (from 225 to 171 hours) while preserving mechanical performance. These results demonstrate the potential of multi-fidelity optimization as an efficient and practical approach for GA hyperparameter tuning and future experimental lattice design studies.

We establish the noncompact Iwasawa factorization for a Delaunay-type potential associated with affine spheres on the complex plane away from countably many lines. Using the DPW method, we give an explicit description of the factorization in terms of Weierstrass elliptic functions via a reduction to a linear system related to the Tzitzéica equation. As an application, we construct explicit translationally equivariant hyperbolic affine spheres and classify them according to their slice curves. In particular, we show that every such affine sphere is equiaffinely equivalent to one whose slice curve is a circle, hyperbola, or parabola, consistent with the Calabi correspondence between hyperbolic affine spheres and proper convex cones.

We develop stratified renewal theorems for multi-time observations of Markov-additive paths on standard Borel state spaces with nonnegative integer vector increments. Horizon directions with the same active constraints form strata. A compact-uniform inverse expansion yields a global limit in a topology combining hypi-convergence with exact restrictions to each stratum. At Brownian scale, the central theorem determines the joint terminal-edge law at fixed finite families of critical horizons about a common direction, under nondegenerate pairwise Gaussian contrasts. One Gaussian vector selects the exiting coordinates. Selections of the same coordinate at the same threshold share an edge; different selected pairs carry conditionally independent equilibrium marks. These marks retain both endpoint states, the full increment, age and overshoot. Under total-variation convergence of the state marginal to stationarity, pointwise scalar renewal convergence in total variation suffices for the marked argument. Regenerative conditions provide explicit criteria. Stationary scalar level processes retain dependence at fixed integer offsets. Under additional moment conditions, a second-order allocation theorem localizes optimal budgets and, with the marked hypotheses and nonsingular Gaussian contrasts, yields a cost-weighted equilibrium-edge mixture. An exact covariance criterion links uniform tightness of the unperturbed inverse field to the absence of a square-root allocation loss. Finite, denumerable and continuous modulating environments are included.

In recent years, several mathematicians have looked at a class of partitions called the overcubic partition $k$-tuples, for small values of $k$. They have proved several divisibility results satisfied by these partitions. We continue this study and prove a few infinite family of congruences. Our proofs use elementary techniques from $q$-series and number theory.

Chimeric Antigen Receptor (CAR) T cell therapy has emerged as a successful treatment for relapsed or refractory hematological malignancies, particularly for B cell Acute Lymphoblastic Leukemia (B ALL), where CD19 targeted therapies have achieved high initial remission rates. However, relapse after treatment remains a major clinical challenge, frequently associated with antigen escape mechanisms and the emergence of CD19$^-$ leukemic cells. Understanding the interaction between CAR T cells and antigen expression dynamics is, therefore, essential for improving therapeutic efficacy and long-term patient outcomes. In this work, we develop and comparatively analyze novel mathematical models describing CAR T immunotherapy and CD19 dynamics in leukemia. Our proposed framework combines compartmental ordinary differential equation (ODE) formulations as well as partial differential equation (PDE) systems. Both methods are able to capture the evolution of leukemic populations under immune pressure and, in particular, the models explicitly distinguish between CD19$^+$ and CD19$^-$ leukemic cells and incorporate bidirectional phenotypic transitions regulated by CAR T activity. The developed models provide biologically interpretable and computationally efficient tools for studying treatment response, resistance, and relapse mechanisms in CAR T cell therapy. We compare the ability of the different modeling approaches to reproduce CD19 antigen modulation and CAR T efficacy dynamics. We also propose sensitivity analyses to study the parameters' influence on the models' dynamics. Our work contributes to the mathematical understanding of antigen-driven resistance and offers a basis for future optimization and personalization of CAR T therapeutic strategies in leukemia.